The Aperiodical: Recent Episodes
Christian Perfect, Peter Rowlett and Katie Steckles
Double Maths First Thing is excited about the moon!
Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight in doing and thinking about maths.
Annoyingly, immediately after last week’s issue went out, several interesting things popped onto my radar. One of them was Katie and Peter asking why nobody can draw a noughts-and-crosses board properly.
The other was Matt Parker going to the moon — or at least sending code to the moon to estimate pi. The kickstarter is already way beyond its goal, and schools are encouraged to get involved. I keep telling non-mathematicians about it, and they look blank and say “… why?!”. I shake my head at them. IT’S MATHS ON THE MOON! (I seem to be involved in the project. It is great fun working with bizarre restrictions.)
LinksMy favourite link of the week is this, from Abigail Pain, in which she hacks into her cybersecurity homework assignment to avoid having to do it. This falls squarely in the mathematician’s remit of going to extreme lengths to avoid doing any proper work. Of course, there’s also a relevant XKCD.
While my lunar coding hasn’t yet involved any bit-twiddling, it may reach a level where that’s required. This means I’m more than usually interested in a radix ( 2^{51} ) trick for adding things up and a leap year check using bitmasks. Smashing stuff.
Walking legend Robin Houston has pointed me at an online version of Stewart T Coffin’s The Puzzling World of Polyhedral Dissections, which I’ve barely had a chance to glance at, because I would vanish into it for a couple of weeks and DMFT wouldn’t go out on time, my work would go unfinished and neither the kids nor Pete would be fed. If you can read it and summarise for me, that would be great, hmmkay?
What else is good here? Oh yeah square theory. I keep flipping between “this is really obvious” and “that’s actually a really nice model for several things I enjoy” — crosswords, jokes, and proofs.
And my most fun fact of the week: Raisa Smetanina won an Olympic gold medal for cross-country skiing a couple of weeks before her sixth birthday.
CurrentlyThere’s a Finite Group livestream taking place a week today (on Wednesday 25th June) at 2pm UK time. It’s Games Time with James Grime, which is a good title because the names rhyme.
The Carnival of Mathematics is coming home this month, being hosted by Katie at the Aperiodical. If you’ve got a blog that could provide a future stopping-point for the Carnival, let Katie know!
Also, Talking Maths in Public is a couple of months away — whether a Pseudorandom Ensemble show is enough to persuade you or not, bursaries for those who wouldn’t otherwise be able to go are available, but the deadline is this Friday, June 20th at noon UK time.
That’s all I’ve got for this week. If you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive courtesy of my dear friends at the Aperiodical.
Meanwhile, if there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something you want to tell me.
Until next time,
C
In this series of posts, we’ll be featuring mathematical video and streaming channels from all over the internet, by speaking to the creators of the channel and asking them about what they do.
We spoke to Ravi Boppana about his channel, Boppana Math.
Channel title: Boppana Math
Link: youtube.com/@BoppanaMath
Topics covered: Pure mathematics, with a focus on discrete mathematics
Average video length: 15 to 25 minutes
Recommended videos:How Paul Erdős Cracked This Geometry Problem
What is your channel about, and when did it start?My channel, Boppana Math, explores a variety of challenging problems in pure mathematics. I posted my first video in January 2024. I realised that I could reach way more people by creating YouTube videos than by writing research papers.
Who are you? Tell us about yourself. My name is Ravi Boppana. I received my PhD in Computer Science from MIT at age 22, and I was a professor for a dozen years at Rutgers University and New York University, where I received a couple of teaching awards. I co-authored a prealgebra textbook for the Art of Problem Solving.
Who is the intended audience for the channel?As my channel banner says, I try to help mathematics lovers explore beautiful results in mathematics. The topics are typically around undergraduate level, but bright high-school students can enjoy the videos, and so can those who have graduated from college.
What is a typical video like?A typical video introduces an intriguing problem, and then gently guides the viewer to discover its solution. I narrate a scripted voice-over while stepping through a slide presentation in LaTeX. My videos are usually between 15 and 25 minutes long. I publish one video every three months or so.
Why should people watch your videos?I try to make difficult topics feel accessible, and I have been told that my presentation style is calm and clear. I try to discuss topics that haven’t been covered much in other videos.
What are some highlights of the channel so far?
One of my videos, on Hypergraphs and Acute Triangles, received an Honorable Mention at the Summer of Math Exposition in 2024! The most recent video was my first to be viewed more than 100,000 times: How Paul Erdős Cracked This Geometry Problem.
What exciting plans do you have for the future? I am currently working on a video about the card game SET, and I have been talking with a video editor to hopefully increase the video quality and video frequency. Mathematics is inexhaustible, so I will never run out of fun topics.
The UK Government have announced the new set of King’s Birthday Honours. Here’s our selection of particularly mathematical entries for this year. If you spot any more, let us know in the comments and we’ll add to the list.
- Ursula Martin, Emeritus Professor in the Mathematical Institute, University of Oxford, becomes a Dame for services to Science and Education.
- Peter Cox, Professor of Climate System Dynamics in Mathematics, University of Exeter. Appointed CBE for services to Science and to Climate Modelling.
- Tim Cole, Professor Emeritus of Population, Health and Practice, Great Ormond Street Institute of Child Health, University College London. Appointed CBE for services to Medical Statistics.
- Philip Colligan, Chief Executive Officer, Raspberry Pi Foundation. Appointed CBE for services to Engineering and Technology for Societal Impact.
- Rowland Raymond Kao, Professor of Veterinary Epidemiology and Data Science, University of Edinburgh. Appointed CBE for services to Mathematics and Infectious Disease Dynamics.
- Emily Fleur Shuckburgh, Professor of Environmental Data Science and Director, Cambridge Zero, University of Cambridge. Appointed CBE for services to Climate Science and to the Public Communication of Climate Science.
- Davin Crowley-Sweet, Chief Data Officer, National Highways. Appointed OBE for services to the Safety of the Road Network.
- Alexander James Samuel Stewart, Senior Statistician, Office of the Chief Economic Adviser, Scottish Government. Appointed OBE for services to Scottish Economic Statistics.
- Richard Cracknell, Lately Head of Social and General Statistics, House of Commons. Appointed MBE for services to Parliament.
- Sarah Lowe, Senior Research Officer, Welsh Government. Appointed MBE for services to Data Research.
- Dame Jocelyn Bell Burnell, Visiting Professor, University of Oxford. Appointed Member of the Order of the Companions of Honour for services to Astronomy and Physics and to Diversity.
- James Brown, Discipline Leader (Maths), University of Technology, Sydney. Appointed OBE for services to Statistics in the United Kingdom and internationally.
Get the full list from gov.uk. Spot anyone we’ve missed? Let us know in the comments.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of May 2025, is now online at Beauty of Mathematics.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Double Maths First Thing is bringing the thunder
Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight in playing with patterns, puzzles, figures and logic.
First up, an apology: in last week’s issue, I mistyped the number of solvable nonograms: rather than 25,000, there are 25,000,000. As of Tuesday evening, humanity has solved about 27% of them. Good work, fellow humans!
LinksOne fellow human doing good nonogram work is Lucas Cimon, who has developed code to generate nonogram QR codes. He says it will likely be “difficult to solve manually”.
Since we’re looking at puzzles and logic, all-round good egg Tony Mann has pointed me at a remarkable 4-by-4 sudoku and Cracking the Cryptic’s solution video. Meanwhile, there’s a ferocious argument going on in the group chat about Zp-ordle, the daily puzzle game for people who think Wordle needs more p-adic integers. (Which is everyone, right?)
There’s one question on everyone’s lips: “Will Jesus Christ return in an election year?” Eric Neyman explains why the likelihood given on Polymarket is 3%, considerably higher than most analysts would predict.
Meanwhile, here’s Hannah Fry peeling an orange.
And in mental arithmetic news, Niklas Oberhuber estimates some logarithms. (I particularly liked that 5^10 is about 9.8 million, which I’ll store in the Useful Approximations drawer of my brain.)
CurrentlyIt’s mid-month, which means that your local MathsJam is coming up soon (Tuesday 17th in many locations, including Weymouth). Find yours here, or email Katie to find out how to start your own.
The 240th Carnival of Mathematics now has a link at Beauty of Mathematics. Similarly, the TMiP animation playlist for May is live.
That’s all I’ve got for this week. If you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive courtesy of my dear friends at the Aperiodical.
Meanwhile, if there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something you want to tell me.
Until next time,
C
In this series of posts, we’ll be featuring mathematical podcasts from all over the internet, by speaking to the creators of the podcast and asking them about what they do.
We spoke to John Bailer and Rosemary Pennington, the hosts of the podcast Stats + Stories.
Podcast title: Stats + Stories
Website: statsandstories.net
Links: Spotify, Apple Podcasts, Player.fm
Average episode length: 28 minutes
Recommended episode: All of them! Episodes are categorised by topic in the ‘+Topic’ section of the website
What is your podcast about, and when did it start?The Stats+Stories podcast tries to “tell the statistics behind the stories and the stories behind the statistics”. The first episode of the podcast was released in 2013. It was an extension of a collaboration between a statistician (John Bailer) and a journalist (Richard Campbell) who were involved in efforts to promote quantitative literacy at Miami University, a public university located in Oxford, Ohio, USA. Bailer and Campbell taught a News & Numbers class in 2009 to undergraduates who were majoring in the humanities, and thought a podcast would provide a reach well beyond a single class at a particular university.
Who publishes your podcast? Tell us about yourself. Our podcast was established as a collaboration between the Department of Statistics and the Department of Media, Journalism and Film at Miami University in Oxford, Ohio. After 4 years, the American Statistical Association (ASA) became a sponsor, and in 2025, we formally became an ASA podcast. We also collaborate closely with Significance magazine, a joint publication of the Royal Statistical Society (RSS), ASA and the Statistical Society of Australia and with Chance magazine, a publication of ASA.
Currently, our podcast includes two panelists: John Bailer (statistician) and Rosemary Pennington (journalist). John is emeritus professor and founding chair of the Department of Statistics at Miami University. He currently chairs the ASA Excellence in Statistical Reporting Committee and serves on the Accreditation Committee, and previously served on the ASA Board of Directors and as International Statistical Institute (ISI) president. Rosemary is the chair of the Department of Media, Journalism & Film at Miami University. She’s an expert on issues of media representation. Rosemary also worked as a medical/science reporter during her last journalism job.
John BailerRosemary PenningtonWho is the intended audience for the podcast? Our audience includes anyone with an interest in data and how the results of analysing data impacts much of modern life. Our listeners span the range from secondary students who have teachers incorporating the podcast in their classes to professional statisticians and journalists to John’s dad (who is not a secondary student nor statistician nor journalist).
What is a typical episode like?A typical episode focuses on an interesting article, a general statistics/science/journalism news story, the communication of quantitative concepts or some other topic that we think would be of general interest. An episode is a conversation between the panelists and a guest (or two, rarely three). Recently, we recorded episodes related to outcomes in golf tournaments, excellence in statistical reporting, statistical songs, and the winner of the international prize in statistics.
We’re always looking for ways to help make sometimes complicated subjects more understandable for a wide audience. Our guests range from graduate students to knighted individuals to data journalists. We want the show to feel like a conversation among friends, rather than an academic discussion among experts.
A typical episode is approximately 28-30 minutes in length that is broken into segments: intro of topic and guest (~2 min), conversation block 1 (~14 min), program break (~1 min), conversation block 2 (~14 min), closing (~1 min). We originally selected a 28 min target with the thought that this would allow a commuter to listen to a complete episode on the way to or from work or school.
In recent years we released about 48 new episodes each year. In 2025, we are releasing 2 new episodes each month along with an episode from the archives (with 360+ episodes we have a lot of options!).
Why should people listen to your podcast?We have a format with panelists reflecting two disciplines, statistics and journalism, who have conversations with a range of interesting people. We’ve been doing this for a long time now and there’s a familiarity between John and Rosemary that helps make the podcast more conversational. We’re also always thinking of our audience as we are interviewing guests – when John says wants his dad to be able to understand what we’re talking about, he means it.
For Rosemary, an inspiration has been The Infinite Monkey Cage podcast on the BBC. While Stats + Stories might not feature a comedian every episode, it does help take the work of researchers out in the public in a similar way. And, similarly to The Infinite Monkey Cage, while we take our work seriously, we don’t take themselves too seriously. For John, an inspiration has been Freakonomics, Science Friday and the National Public Radio Car Guys .
What are some highlights of the podcast so far?One highlight of the podcast was that it inspired us to write a book about what an educated reader needs to know to be an informed consumer of the news. After writing Statistics Behind the Headlines, we decided to record an audiobook version that is now freely available as a Spotify audiobook.
Another highlight has been the response to it. Lots of podcasts are born only to die premature deaths. Stats + Stories has been able to continue for so long because people are interested in issues of data and communication; they’re also unafraid to connect to John and Rosemary. Episode and contest ideas often come from our listeners. It makes it feel like we’re building a community; or, maybe, just creating a new place for the communities who care about the issues we highlight to come together.
What exciting plans do you have for the future? That puts a lot of pressure on the future :-) Our 375th episode is on the horizon. We have enjoyed celebrating such milestones with contests for our listeners. It wouldn’t surprise us if we did this again. We are also planning on annotating our episode database and providing it as a data set that students, teachers and other analysts could explore, possibly as an R package.
Ultimately, we’re going to continue doing this podcast for as long as people listen and for as long as it’s fun for us.
Double Maths First Thing always takes the weather with it
Hello! My name is Colin and I am a mathematician on a mission to disseminate mathematical joy and the pleasure of figuring things out.
I’m just back from a week in the Peak District, where we discovered that Pete the dog likes neither stepping stones, nor the river that he jumps into to avoid them. Apparently it’s my job to carry wet dogs across rivers, although I don’t remember THAT being in the information pack.
We visited George Green’s windmill in Nottingham, which is unusual in that its science centre doesn’t hide the maths away — Green’s theorem is prominently displayed on banners around the place (although there’s only a limited attempt to explain it.) I’ve just added it to Nerdy Day Trips, which is BACK! (It was very easy to submit, so I recommend adding nerdy day trips you’ve enjoyed.)
I spent the car journey back doing my best to help humanity defeat the evil nonogram: there are about 25,000,000 solvable five-by-five puzzles, and we need to solve them all. After a while, you get into a flow state, it’s quite an experience. [Edited 2025-06-04 to correct number from 25,000]
LinksOn to the links! First up, a couple of questions from dear friend Colin Wright: does this theorem have a name in English?, and is there a good ‘why?’ for Marden’s theorem?
In games news, Boggle has been (effectively) solved, and students at Purdue have broken the robot Rubik’s cube world record, with a time of 0.103 seconds. (Incidentally, the 4×4 robot record fell recently — but that’s still significantly slower than the human world record.)
From the Fields medallists doing interesting things beat, Terry Tao has launched a Lean companion to Analysis I — Lean is a “formal verification system” that checks your proofs hold water. Meanwhile, if you’d like a Tim Gowers lecture on why LLMs aren’t better at finding proofs, you should watch one.
Another piece on my to-read-more-carefully list is Aeva’s article on spline fields, for storing and rendering realistic terrains.
CurrentlyIt’s a new month, so there’s a new Carnival of Mathematics: this month’s host hadn’t been posted at press time, but might be before you read this; you’ll be able to find it at Suzza’s Beauty of Mathematics blog.
There’s also a new TMiP Animation Challenge prompt: if you’ve got something to visualise about curves of pursuit, feel free to give it a go. It’s a good excuse to learn a new skill; the Finite Group generators seem to delight in using unconventional approaches like tikZ, but Manim or possibly Lottie appear to me like more reasonable starting points. Last month’s efforts will soon be linked from the same page at TMiP.
That’s all I’ve got for this week. If you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive courtesy of my dear friends at the Aperiodical.
Meanwhile, if there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something you want to tell me.
Until next time,
C
In this series of posts, we’ll be featuring mathematical video and streaming channels from all over the internet, by speaking to the creators of the channel and asking them about what they do.
We spoke to retired mathematician and A-level maths tutor Jim Simons (not that Jim Simons) about his YouTube channel, in which he covers A-level maths topics.
Channel title: Jim Simons
Link: youtube.com/channel/UCnYszOhEIIdIMYyNx2yjwfg
Topics covered: Mathematics, mainly about A level
Average video length: 20 minutes
Recommended videos: Three pretty geometric theorems, proved by complex numbers; The Binomial Theorem with a Real Exponent; How We Used Log Tables
What is your channel about, and when/why did it start?It is about mathematics, mainly around A level. It started in the pandemic when the Mathematical Association’s annual conference was cancelled, and I had to decide what to do with the talk I had prepared. As every teacher knows, there is no better way to really understand something than to try to teach it, and I’ve thought much more deeply about the A level material now than I ever did when I was being taught it, and I want to share some of that insight.
Who are you? Tell us about yourself.In 2009, I retired from a career in the civil service as a professional mathematician, and took up tutoring, mainly at A level. I have always loved mathematics, and I quickly discovered that I love teaching it too. I joined the Mathematical Association, and have been an active member, writing for its journals, serving on its Teaching Committee, and going to its conference to share my ideas, and to listen to other maths teachers.
Who is the intended audience for the channel?As with my very first video, I mostly imagine myself as talking to an audience of maths teachers who teach A level Maths and Further Maths, or similar levels around the world. So this is not a mass market channel, but of course anyone else interested in that sort of mathematics might enjoy it.
What is a typical video like? I talk over powerpoint and also use geogebra, because I love moving diagrams. I usually try to produce rigorous proofs of the results I discuss. My videos are mostly about 20 minutes long, and it looks as though I do about 5 a year.
Why should people watch your videos?I love hearing other maths teachers talk about their approach to the subject matter, and so I hope they might enjoy mine. I often feature enrichment ideas, or approaches that I think are novel, or sometimes just my favourite proofs of standard results. I often go a bit beyond A level in a way that I hope will be interesting and stimulating. I have become very interested in the history of mathematics, so I often include some history, which I think enriches and humanises the subject.
What are some highlights of the channel so far?Three pretty geometric theorems, proved by complex numbers: This is my most popular video – the results are just so beautiful. Useful ideas for a Further Maths classroom! The history of the results is interesting too.
The Binomial Theorem with a Real Exponent: The result is usually proved using Maclaurin’s theorem, but that’s not taught in English schools until later, which means that it isn’t really proved at all. So I have worked up a proof that doesn’t use calculus at all! This is an approach to proving the result that I have never seen elsewhere, and to judge by the comments, neither have a lot of people. It is quite advanced mathematics, so probably not really for use in the classroom, but I hope rewarding.
How We Used Log Tables: When I was at school, we carried around a book of tables, as today’s kids carry a calculator, and indeed as most of their teachers did. I think teachers often talk about tables when they teach logs, but don’t necessarily have first hand experience of using them in earnest. So I made a video showing how they worked. I tackle some old O level questions that required the use of tables.
What exciting plans do you have for the future?I am too old for long-term plans! But I am currently working on a video concerning a pretty little result about Pythagorean triangles. When I raised it at the recent joint mathematical associations conference, nobody knew it, so I decided to make the video. It is not especially deep, but gives me the opportunity to talk about and prove Heron’s formula for the area of a triangle, and Euclid’s formula for generating Pythagorean triples, both of which I love.
Here’s a round-up of all the mathematical news from the last couple of months we didn’t otherwise cover here.
Prizes and AwardsThe Shaw Prize in Mathematical Sciences 2025 has been awarded to Kenji Fukaya, “for his pioneering work on symplectic geometry, […] and for his subsequent ground-breaking and impactful contributions to symplectic topology, mirror symmetry, and gauge theory.” The prizes, awarded since 2004 and carrying a prize of $1.2m, honour “individuals […] who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence.” (via Paysages Mathématiques)
A new prize for an outstanding PhD thesis in model theory is being established in honor of French mathematician Zoé Chatzidakis, by Fondation Sciences Mathématiques de Paris. They’re looking for donations to form the prize fund. (via Artem Chernikov)
Journal NewsThe editorial board of Mathematical Logic Quarterly, published by Wiley, have resigned. They’ve set up a new diamond open access journal called Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, in homage to an earlier journal of the same name. (via Emily Riehl)
Interlace is a new journal for work at the intersection of mathematics and fiber arts. The editors-in-chief are sarah-marie belcastro (Mathematical Staircase, Inc. & Bryn Mawr) and Carolyn Yackel (Mercer) and they intend to publish annually. (via Colin Wright)
Inventions and DiscoveriesLuciole Math is a new font for maths, designed to be easy to read. It’s the result of a collaboration between the Centre Technique Régional pour la Déficience Visuelle which supports visually impaired young people, the type-design studio typographies.fr and the mathematician Daniel Flipo. (via Le Libre Éducatif)
There’s inevitably been a new π calculation record of 300 trillion digits, from the team behind the Linus Tech Tips YouTube channel. Here’s their video about it.
An optimal solution has been found for Boggle. According to the blog post, “Many people have searched for high-scoring boards before, but no one has ever constructed a computational proof that they’ve found the best one.”
Other NewsMathematicians in the news! ICYMI, there’s a new pope, who excitingly was a math major in college; also Romania’s new president Nicusor Dan is a mathematician with two gold medals from the IMO (his IMO stats). He’s one of only 11 students to solve the famous “Vieta jumping problem” at the 1988 IMO.
The Campaign for Mathematical Sciences has launched a “provision tracker” monitoring the health of university mathematics departments in the UK. It’s gloomy reading.
The International Day of Mathematics (14th March) is looking for a theme for 2026. (via Martin Skrodzki)
Geoff Wain, founder of MathsWorldUK, has died. (via Kit Yates)
Double Maths First Thing shaves newsletters if and only if they don’t shave themselves
Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight in puzzles, problem-solving and practicing maths.
I’m currently wading through the combinatorics of meteorology, which is another way of saying “this walk is a bit muddier than I expected”. Coincidentally, I’m also thinking about the maths of rainstorms, which is much more exciting than the practicalities of navigating the dog through them.
LinksYou don’t have to be in Aoteroa to appreciate Maths Craft New Zealand’s resources — I love things like step-by-step instructions on meanders — and (for example) @welshpixie’s guide to drawing Celtic knots, which inspired Andrew Taylor’s Celtix.
If you feel like maths games should be less knotty and more numerical, Andrew has you covered there, too: his newest offering is Ophex; I am annoyed that I recently lost my perfect streak, so it’s dead to me now.
For a change of pace, you might want to read Toby Lam’s piece on differentiating inverses graphically. I’ve only skimmed it, because it has diagrams with arrows in that remind me of my Erasmus year in France, where I understood the French perfectly, but the maths was gibberish to me.
I have mixed feelings about this piece in defence of Venn Diagrams by Jack Murtagh. It’s lovely as far as it goes (I was surprised to realise that Venn diagrams are less than 150 years old), but it doesn’t go very far — I suspect this is a function of SciAm’s word count limits.
Back in Issue 21, I mentioned Fractran; Tzerjen Wei has a working interpreter in case you want to try it for yourself.
CurrentlyThis afternoon (Wednesday May 28th, 5pm UK time), Paul Lockhart, author of Lockhart’s Lament, is doing an AMA.
You’ve got a few days left to submit anything interesting to Carnival of Mathematics 240, which will be hosted by Suzza at Beauty of Mathematics.
That’s all I’ve got for this week. If you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive courtesy of my dear friends at the Aperiodical.
Meanwhile, if there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something you want to tell me.
Until next time,
C
A conversation about mathematics inspired by Lewis Carroll’s Game of Logic. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by … an object. Presented by Katie Steckles and Peter Rowlett, with special guest Tai-Danae Bradley.
Katie mentions Peter’s The unplanned impact of mathematics, free to read at Nature.
A conversation about infinity inspired by The Library of Babel by Jorge Luis Borges. Presented by Katie Steckles and Peter Rowlett.
I’d like to cut a rectangle into a 3×4 grid of squares. To minimise the number of cuts, should I cut three long strips first, or four short strips? Does it matter?
Here’s the talk I gave at this year’s Big MathsJam Gathering. I called it Please don’t overthink this (I already have)
That’s the short version of the story. I really did spend a lot of time thinking about this!
Recently I presented a maths workshop for some year 3/4 kids in a local school. For those outside England and Wales, that means kids aged 7 to 9.
My plan was to talk about fractions by making up patterns from paper squares of different colours.
In order to do this, I needed lots of paper squares. The school aren’t providing any material themselves (or paying for my time…) so I want to do this on the cheap. So I’ve ruled out just buying lots of origami paper.
I ordered a ream of red A4 paper and pinched a ream of white A4 from my department’s resource room.
So I have to decide how to cut A4 sheets, which are silver rectangles, into squares.
A sheet of A4 paper measures 210×297mm.
The most obvious way that occurred to me was to cut a strip off the long side to end up with a 210mm square, and then halve that in each direction until I get squares of a suitable size. I thought that halving once would give me 105mm squares, which feel about the right size.
But the 87×210mm strip left over felt too big to me.
Then I realised that 210 is 3×70, and 297 is only a little bit more than 280 = 4×70, so I could get 12 70mm squares out of each sheet, only leaving a little strip.
This feels like the most efficient use of the material.
While I was in my office pondering this, my colleague Shweta popped her head in. As I tried different options out, I talked about the practical aspects of cutting up paper.
I know how to fold a rectangle to get the biggest single square, and I think that doing a clean rip along a folded line is one of life’s most sublime experiences.
To subdivide this square, I could fold and rip again. I’d been having a fairly gloomy day, but this cheered me up.
While this method is satisfying, it’s not quick. It was clear that I’d need to use the guillotine in the resource room.
I wondered about how to accurately cut to the right size using the guillotine. Folding to mark a line, as well as being slow, would make it harder to put the paper through the guillotine.
While still in my office, I concluded I’d need to make a template square of the right size by folding, and use that to determine how far to push the paper through the guillotine.
When I got down to the resource room, of course, I saw that the guillotine has a ruler marked on it.
Because I’d decided on 70mm squares, a nice round number, I could line my paper up exactly with one of the marked lines.
But when I started with my first sheet, I spent longer than I like to admit staring at the ruler and trying to work out how to trim off the 17mm from the end. The 17mm mark is under the plastic guard that stops you cutting your fingers off, and the lines are only 5mm apart so it’d be tricky to reliably hit it.
The realisation that hit me was the first time commutativity of operations came into play that day. The second came right at the end.
It’s obvious now that trimming at the 17mm mark, leaving you with pieces 17mm and 280mm long, is equivalent to trimming 280mm, leaving you with pieces 280mm and 17mm long.
So this is my algorithm for cutting the squares I want:
- Push the paper through, portrait, until the bottom edge is on the 280mm mark, and slice.
- Then push through and slice at the 210, 140, and 70mm marks. Even though this is only three cuts, it leaves you with four 70×210mm strips. This was the first appearance of a fencepost error.
- Then turn the strips round and push them through, cutting at the 140 and 70mm marks. This leaves you with three 70mm squares for each strip.
After doing that for a while, I started to wonder: does it matter if I do the horizontal cuts before the vertical ones? Could I save myself a cut or two?
I tried both ways but wasn’t sure I’d counted the cuts properly. I needed to prove how many cuts it takes with each method. And is there an even quicker method that will reveal itself?
I started seeing an algebraic structure: think about the multiset of pieces of paper that I have. I begin with a single $(4,3)$ piece. The two moves I can make are to cut a piece $(a+b, c)$ into two smaller pieces $(a,c)$ and $(b,c)$, or to cut a piece $(a,b+c)$ into two pieces $(a,b)$ and $(a,c)$. For example, $(4,3) \to (3,3) + (1,3)$.
All I have to do is either prove or find some theorems about this system and I can work out the quickest route from $1 \times (4,3)$ to $12 \times (1,1)$.
I can make vertical or horizontal cuts, so it should be fairly simple to draw out the Cayley graph of this thing and find the shortest path in it…
The problem is that at each stage, you haven’t just got two possible moves – horizontal or vertical – you’ve got at least as many moves as different shapes, and some shapes can be cut in a few different ways.
Bonus question: what’s the most different shapes you can have at one point in time? Is it in that picture?
What you need to know about me is that I started a PhD in group theory and then gave up on it. I like abstract algebra in the abstract, but I don’t like actually doing it. I can’t tell from that half-a-diagram where the symmetry is, and I don’t think I’d be able to see it even if I drew the whole thing out.
I was thinking through all this as I was guillotining paper. I tried doing all the vertical cuts first, and then I tried doing all the horizontal cuts first. I wasn’t really keeping count of the number of cuts, because I was also trying to keep track of how much paper I was using. (I ended up making about two thousand squares too many)
Eventually, I realised something that makes the solution completely obvious:
- Each time I make a cut, I have one more piece than I used to.
- I start with one piece.
- I want to end up with twelve pieces.
So it doesn’t matter what I do or in what order: it’ll always take 11 cuts.
That’s really simple! A child could understand it! You could understand it! I could understand it!
But imagine if I hadn’t told you about all the thinking I’d done to get there?
Double Maths First Thing is where there area 10 kinds of people: those who understand hexadecimal, and F others.
Hello! My name is Colin and I am a mathematician on a mission to share joy and delight in maths, beyond and instead of the test. Around here, we count in hexadecimal.
My big news!Slap-Dash Pete, a chocolate-brown Patterdale terrier, who has clearly won the battle over whether he’s allowed on the furniture.This is not remotely maths-related, but we’ve just been adopted by a Patterdale terrier called Slap-Dash Pete. He’s slipped straight into the family as if he’s always been here. You definitely need a picture.
Also not-maths-related, I finished third in my Toastmasters area humorous speaking contest. I spoke about the process of figuring out I’m autistic and how that relates to being a mathematician. I thought it was a good talk, but it wasn’t the best on the day.
Maths events and news!It is currently Maths Week England, an event aiming to help people realise that maths is for everyone and not only for genius. See what’s going on near you!
Much less joyfully, it seems that Ada Lovelace Day Live will no longer be happening. It’s galling that tech giants can’t (or rather, won’t) find a few quid down the back of a beanbag to help make sure there’s space at the table for everyone. The organisation lives on, and the day celebrating women in science lives on, but it’s sad that the flagship event has to stop.
Dates of local MathsJams are all over the place this month. Some have moved to yesterday in support of MWE, but others — including Weymouth — are sticking to the traditional penultimate Tuesday date. Find your local ‘Jam here — if there isn’t one near you, there are instructions on how to start one; alternatively, you may prefer to join in the Online MathsJam that’s usually on the antepenultimate Tuesday. Because of course it is. You just missed it. Sorry.
Links!There’s an interesting discussion on Reddit decrying the state of maths games in general.
Some maths games that very much don’t suck are those created by the legendary Simon Tatham, which have stolen about as much of my work time as Tetris and Slay The Spire. Simon posted on Mathstodon about the ZX Spectrum BASIC manual. For geeks my age (and probably a little older), this is one of the 1980s’ most significant works of literature, the book that taught me how to program (and to develop bad habits that would take years to unlearn.)
(Incidentally, Mathstodon is an excellent community of maths people, and far less shouty than the Other Place. I’ve heard good things about Bluesky, also; I gather it’s possible to bridge between the two, but don’t ask me what that means. I’m already in too deep.)
Another legend, Rob Eastaway, is making a rare screen appearance in a Numberphile video about Philip Henslowe’s diary and the shift from Roman numerals to Arabic.
A third and final legend for this week: Tanya Khovanova is making foams out of felt. A foam is a mathematical object rather than something to make safety equipment out of, it transpires.
That’s all for this week! In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive courtesy of my dear friends at the Aperiodical.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something I should be aware of.
Until next time,
C
A conversation about mathematics inspired by a very special parallelepiped. Presented by Katie Steckles and Peter Rowlett, with special guest Ayliean.
Double Maths First Thing is like a diet MathsJam, some of the flavour but only a hint of the joy.
Hello! My name is Colin and I am a mathematician on a mission to spread delight in my beloved subject.
I spent the weekend at Big MathsJam in Staffordshire and gorged myself full of puzzles, surprises and fascination — my personal highlights were Mats Vermeeren’s talk about why the start lines on athletics tracks are curved the way they are, Vincent van Pelt’s MathsJam Jam song (Mnemonic to the tune of Blondie’s Atomic) and the colouring-in in the quiet room. Now I’m sad that I don’t get to do it for another year.
From around the internetIt’s always cool to see ancient technology in action. Slide rules may not be ancient ancient, but they were no longer in common use by the time I was at school. (I remember we had some Napier’s bones, but nobody knew how to use them. I wish they had, I’d have lapped it up.)
As everyone knows, the Mathematical Villain goes through your spreadsheets turning data into dates. Here are some more times that Excel users failed to, well, excel. (Aside: it’s all so avoidable! A halfway-competent programmer can set up a script that checks for this sort of thing. And if you need one of those, you should let me know.)
In “everything is interesting if you look at it closely enough” news, the horrors of implementing daylight saving rules around the world are fascinating.
I also loved this bit of code golf for finding Fibonacci numbers. The explanation is much more interesting than the code.
I couldn’t explain why, but I’m averse to calculator notebooks. Not my cup of tea. Don’t like Jupyter. I may have had a bad experience with Maple as a student. That doesn’t mean you can’t experiment and enjoy, though!
Lastly, to file under “ridiculous but brilliant projects”, several generators of the Finite Group are challenging humankind to say the new record Mersenne Prime before the next one is discovered. Join the race!
Books!Apparently some people celebrate Actual Christmas rather than (or as well as!) MathsJam. That’s OK, all are welcome. If you’re looking for maths-related books to buy someone so they can add them to the unread pile on the floor, here is a selection of books I’ve either read and enjoyed or have had recommended to me:
- Alex Bellos’s Think Twice
- Rob Eastaway’s Much Ado About Numbers
- Pippa Goldschmidt’s Schrödinger’s Wife
- Matt Parker’s Love Triangle
- Ernő Rubik’s Cubed
- Olivia Waite’s The Lady’s Guide To Celestial Mechanics
(Note: these links don’t necessarily go to the cheapest place to order from. I recommend asking your local independent bookshop — if you don’t have one of those, Gulliver’s in Wimborne deliver across the UK and are lovely people too.)
In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive courtesy of my dear friends at the Aperiodical.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something I should be aware of.
Until next time,
C
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of October 2024, is now online at Math Intersect Programming.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Here’s a roundup of some of the maths-related news from this month we didn’t otherwise cover here!
Maths NewsThe obvious big maths news this month is that a new Mersenne prime has been found: it’s now official that 2¹³⁶²⁷⁹⁸⁴¹−1 is a prime, an indivisible whopper with almost twice as many digits as the previous largest known prime. The last new one was discovered in 2018, so it’s big news for the community, and was discovered by Luke Durant (one of many participants in the mass distributed computing project GIMPS, whose computer was the one that found it).
According to the official GIMPS press release, some wonder if this prime discovery “ends the 28-year reign of ordinary personal computers finding these huge prime numbers”, since Durant used processing power in the cloud, and “developed infrastructure to run and maintain a suite of GIMPS software across many GPU servers”. You can hear an interview with Durant in this Numberphile video, and other maths peeps were quick off the mark with a video from Ayliean and another one from Matt Parker both going out very shortly after the discovery was confirmed.
While most mathematicians agree that this kind of stamp-collecting is a gloriously pointless endeavour, they haven’t seen Ayliean’s new Say The Prime project, or to give it its full name, Can Humans Say The Largest Prime Number Before We Find the Next One?. Only time will tell.
Awards and grantsMeanwhile, the Nobel Prize in Physics has been awarded to John J. Hopfield and Geoffrey E. Hinton “for foundational discoveries and inventions that enable machine learning with artificial neural networks”. Is it physics? Who can say.
The National Academies of Sciences, Engineering, and Medicine in the USA have announced the 2024 Recipients of the Eric and Wendy Schmidt Awards for Excellence in Science Communications, including maths communciator Kyne Santos who creates a mathematical drag TikTok channel, as well as the book Math in Drag and Think Queen podcast, scooped the Independent Science Communicator award. “Her use of vibrant visuals and storytelling make mathematical concepts accessible and engaging, particularly for underrepresented groups in STEM, and her multimedia content challenges stereotypes and promotes inclusivity in STEM fields.”
And finally, the Campaign for the Mathematical Sciences call for proposals for Maths Degrees for the Future Grants is now open, encouraging innovative new designs for maths degrees. (via @campaignmathsci on Twitter).
Double Maths First Thing is Colin’s refuge from the kids’ obsession with Odd Squad.
Hello, and welcome to Double Maths First Thing! My name is Colin and I am a mathematician on a mission to spread joy and delight in my subject.
It’s half-term week, so this is necessarily rushed, brief, and poorly formatted — but it’s Big MathsJam at the weekend, so I expect to more than make up for the brevity next week.
On my list of things to contemplate on the long journey northwards:
- Miles Gould has pointed me at the Capstan Equation, not to be confused with Captain Caveman. Wrapping a line around a cylinder makes it possible to hold much heavier loads than one would naively expect.
- Devine Lu Linvega has a zine about rewriting rules, which appeals to me.
- The train wifi is unable to allow me to watch Sophie Maclean talk about voting paradoxes on Numberphile, much as I would love to.
- The Finite Group’s birthday livestream included Peter Rowlett talking about Carnelli, a game which involves running film titles together — the canonical example is The Empire Strikes Back To The Future. I’m enjoying subverting it (Run Lola Run Lola Run, or On the Waterfronthe Waterfront, or somehow making Zero Dark Thirty and 300 into a loop). What interesting things can you do with it?
- There is a lot of discussion currently about the number of holes in a straw (which is obviously and unambiguously one). If you join the ends together to make a loop, how many holes are there now? If you sew up the top of a sock, how many holes does that contain?
If you’re going to be at Big MathsJam, I hope I’ll see you there! I’ll be talking about HyperRogue, but you risk accidental spoilers if you click through.
In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive courtesy of my dear friends at the Aperiodical.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something I should be aware of.
Until next time,
C
A conversation about mathematics inspired by an area the size of Wales. Presented by Katie Steckles and Peter Rowlett.
Double Maths First Thing is part of Colin’s fight against the forces of tedium.
Hello, and welcome to Double Maths First Thing! My name is Colin and I am a mathematician, on a mission to spread joy and delight in maths.
More from meI promise not to make this whole thing about me, but if I’ve got a blog post about something I find delightful, it would be rude not to share it. Here’s a link that took me a long time to make about the relationship between the binomial expansion and the binomial distribution. The clue’s in the name, right?
New Largest Known Prime!(?)I am decidedly ambivalent about finding larger and larger Mersenne primes. I feel like some of those involved in the hunt are in it for the money, the mersennaries. Even if it’s been six years since the last one, the announcement that there’s a new one is not one that thrills me. I think throwing more compute at the same problem is of limited use. However, it has reminded me about the Lucas-Lehmer test, which is a very nice piece of maths that happens to coincide with the structure of computers, making it efficient (although still lengthy) to calculate.
Some people who are less cynical than me:
- Ayliean
- Matt
A load of ballsSomewhere deep in the list of tabs that seemed like a good idea to open, I found instructions for making a giant windball. It uses some sort of construction kit called makedo, but I’d be surprised if you couldn’t find some butterfly pins and spare cardboard.
I was surprised by a result, which is always a nice feeling: if you’re thinking about balls (settle down back there), you’d expect to see ( \pi ) show up. Finding ( e ) was not on my bingo card.
I was also surprised to find that the word dodecicosacron in FractalKitty’s Mathober challenge was not a typo, but the sort of spiky shape you would avoid in a video game.
Stretching the theme still further, I hadn’t heard of Pappus’s centroid theorem(s), which you could use to work out the volume of a sphere (see! There is a link!) — they’re reasonably obvious once you think about them a little, but it’s still a nice way to approach surfaces and volumes of revolution.
Other nice things!From Reddit, probably to be filed under “absurd but also very impressive”: a computer cuber broke a world record. Not just any world record, but the record for a 121-by-121-by-121 cube. By 69 hours. My understanding is that a 121-cube is just like a 5-cube, only more so — but still, the concentration and dedication you’d need to do that… chapeau! Oh, and they say this is the fifth-largest cube ever solved by a human.
Over on the platform-still-referred-to-as-Twitter-by-everyone-sensible, David K Butler has an interesting way to look at addition and multiplication using parallel and intersecting lines (respectively). I’m always up for a new thing to add to my mental models!
In podcast news, I am given to believe that Sam Hansen is at it again. I’m not sure they ever stopped, honestly; Sam and Sadie Witkowski now co-host Carry The Two, recently with a theme of elections and representation. It’s almost enough to get me to the gym so I can listen to it in peace. Almost.
And — if you’re quick about it — you might be able to subscribe to the Finite Group in time for their first anniversary livestream.
In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive right here at the Aperiodical.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something I should be aware of.
Until next time,
C
Double Maths First Thing is to maths news what the noticeboard outside the coffee shop is to theorems
Hello! My name is Colin and I am a mathematician on a mission to spread delight and joy while making people think.
More from me!I almost forgot (so strongly do I dislike the academic publishing process) that I had a paper published recently about Heron’s formula. One of the reasons I dislike it is that T&F want to charge you £45 of those sterling pounds to read a four-page paper, which is patently ridiculous. I can send you a copy if you want one. You definitely shouldn’t paste the DOI reference, 10.1080/0025570X.2024.2376510, into SciHub, or else the whole publishing system might collapse! (In fact, you can read the proof and the story behind it here).
I’ve also done my stint volunteering for Dorset Coding Day at a couple of local schools. The best questions I was asked were “how many pages were in your books?” and “what’s your favourite Netflix movie?”. Not a single more-of-a-comment, ten-year-olds are brilliant.
And in an email that made me grin from ear, a Tudor living historian emailed me to say he’d read my piece about Henslowe’s trick and is now performing it at events. How amazing is that?
Links from everywhere!Sometimes, people use the word “intuitive” for something that doesn’t line up with my intuition at all — but that’s ok, it’s good to see how other people’s minds work. For example, Gregory Gundersen’s piece on the Black-Scholes equation doesn’t match with how I’d explain it, but it’s still a lovely piece!
I haven’t yet got around to reading this article on the Kelly Criterion, but it’s a topic that always makes me prick up my ears. From my recollection, it turns out that “maximising expected returns” means roughly “small chance of an enormous jackpot, otherwise ruin”, but it’s a fascinating thing to play with.
Another item on my to-read list is this guide to transforming colours with matrices. This feels all sorts of wrong, but at first glance, it seems to work nicely!
Lastly, from memory lane, one of my favourite pieces of mathematical writing: Tim Gowers on deducing the cubic formula. A Fields Medallist explaining how to think about something? Clearly and lucidly? Sign me right up.
I’m also midway through Grant Sanderson explaining Manim to Ben Sparks and now I want to make videos just so I have an excuse to play with it.
Community!In a move closely aligned with my key themes, the Finite Group have opened up their Discord to free-tier members. Among other things, it’s a great source of memes and somewhere you can suffer an endless stream of bad jokes, not all of which are from me. (The amazing live-streams — the next of which is on Wednesday 23rd October at 2pm UK time — are paid content, and worth every penny.)
Gathering4Gardner, best-known for their biennial gatherings that inspired Big MathsJam, but who do all sorts of amazing work, have a fundraising auction starting next week. I refuse to look at it because I have to dispose my income on fixing my laptop, but there might be something there that tickles your fancy!
Speaking of Big MathsJam, Tuesday coming is Little MathsJam Day — find your local Jam here or, failing that, start your own! It’s simple enough that I can do it. Instructions are on that page.
In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive courtesy of my dear friends at the Aperiodical.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something I should be aware of.
Until next time,
C
A conversation about election mathematics inspired by a ballot. Presented by Katie Steckles and Peter Rowlett, with special guest Sam Hansen.
For more from Sam and the Carry the Two podcast check out this episode about Mathematics and Voting.
Double Maths First Thing is Colin’s weekly news bulletin. Although it’s more like a nerfpelletin, honestly.
Hello! My name is Colin and I am a mathematician on a mission to help everyone find the joy and delight in figuring things out.
Art!Up in that London they have these days, the Piccadilly Circus ad boards are being taken over by Olafur Eliasson. While this doesn’t look especially mathematical, a lot of Eliasson’s work is gorgeously so.
I took the kids to a science fair recently, and they tried their hand at marbling with actual paint… and wet paper, which ripped before we’d left the venue. Fortunately, I was reminded that it’s possible to do marbling mathematically. And it’s invertible, so you can recover your original image!
Speaking of inverses, that’s today’s Mathober prompt! FractalKitty is running it again; it’s a prompt-a-day, make-what-you-like challenge. (Personally, I’m trying to write a song verse every day; I know Katie is trying to write a daily crossword clue. The possibilities are endless.)
Computing!Following on from the “computers are magic” thing last week, I’ve stumbled on, but not checked out, Arithmazium, which seems to be an explanation of how computers deal with numbers. Or, from a brief glance, doughnuts.
Shapes!Once upon a time, I wrote about Ailles’ Rectangle — if you inexplicably prefer Wikipedia to my blog, here’s your link. It’s a really neat way to figure out the trig values for 15-75-90 degree triangles, and — if you play about with it a bit, to prove all sorts of identities.
Dave Richeson spotted a naughty cartoon in the New Yorker — not seaside-postcard naughty, more British road-sign naughty.
Obsessions!The Mathematical Objects podcast is off to a flier in Season 8, chatting with Adam Townsend about possibly the greatest MathsJam talk ever.
There is one that rivals it for commitment to the bit: Ben Ashforth’s calendar odyssey.
I’ll be speaking at this year’s Big MathsJam, but I promise I will not be visiting every cell on the border of Camelot. It’s barely a month away. Eek!
In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive right here at the Aperiodical.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something I should be aware of.
Until next time,
C
A conversation about mathematics inspired by a space-filling curve. Presented by Katie Steckles and Peter Rowlett.
Double Maths First Thing is Colin’s weekly assortment of mathematical news. Or what time’s manacle.
Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight through the medium of mathematics. It’s Wednesday morning and it’s time for Double Maths First Thing.
On magicIt’s probably a bit gauche to start with an article I wrote, but it’s certainly something that caught my eye: Rob Eastaway (all-round good egg and author of Much Ado About Numbers, available wherever good books are made available) sent me a page from the diary of an Elizabethan impresario describing a card trick. Here’s my description of how it works — with a bit of help from young Bill.
A different kind of magic goes on under the bonnet of your average computer. (Computers should definitely have bonnets. “Oooths, looks like your fan belt’s gone, that’s going to be expensive.”) When you input a number, the computer takes it in as a string of characters. How does that get turned into an Actual Number? It’s surprisingly complicated.
On beauty
You say you have found beauty
In Euler’s identity
It’s basic trigonometry
It’s Pi Day, I’m in a huff
Even when it’s not Pi Day, I get in a huff about the framing of Euler’s identity as “the most beautiful equation”. Andrew Stacey articulates it a lot more clearly than I would, and with less swearing.
I’ll accept that dance can be beautiful (I have a cousin who’s a professional choreographer, and who has a very stern Disapproving Look, so I have to say that). Here’s a nice piece about different styles of dance notation; my only criticism is that they don’t make a joke about Scottish country dancing needing a Ceilidh table.
On stupidity and getting things wrongAnother article that’s had me nodding along and saying “YES!” is this from Math For Love: it makes the powerful point that feeling stupid is an important part of becoming smarter — and it’s an entirely different thing from being stupid.
One thing that always makes me feel stupid is how our experience of the world is pretty much limited to an incredibly narrow shell — a plane at 30,000 feet is 0.14% of an Earth radius up in the air. It turns out that GPS and route-trackers generally are just… not very good at elevation.
This also prompted me to look up: the Earth is neither smoother nor rounder than a billiard ball, but it’s pretty close.
On my way outA couple of final things: I recently stumbled on Cyrille Rossant’s Awesome Maths List — I’m sure some of you know of resources that belong on there, and he seems receptive to pull requests.
I’ll end with something else about me: this year, I’m doing something I’ve never done before. I’m going into my kids’ school to run a lunchtime code-breaking club around the National Cipher Challenge for years 5 and 6. (The headteacher almost bit my hand off, it sounds like they’re studying Bletchley Park this term). I’ve done the challenge before, it’s just the wrangling young’uns that’s new.
In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
If you’ve missed the previous issues of DMFT or — somehow — this one, you can find the archive courtesy of my dear friends at the Aperiodical.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something I should be aware of.
Until next time,
C
In a dimly-lit tavern on the South Bank of the Thames, Philip Henslowe — builder and owner of the Rose Theatre — is celebrating the success of Shakespeare’s latest blockbuster, Henry VI Part I, among the cutthroats, actors and other lowlife of London. He spreads thirteen playing cards on a table in a circle. “Pick a card,” he grins. “Any card.”
Henslowe, one of Elizabethan theatre’s most important figures, kept a diary. It’s mainly the accounts of the theatre and records of loans, but among the administrivia are some gems — including the following card trick:
Dulwich College Archive MS VII f18v — with kind permission of the Governors of Dulwich CollegeMy dogged team of researchers is looking into it, but there are very few documented card tricks from this era — and most of them are sleight-of-hand or forces. Tiago says this might be related to something written by Pacioli in 1478, and I understand there are Italian deck-stacking tricks from the first half of the 16th century. While it’s relatively unremarkable now, it seems quite sophisticated for its time.
Luckily, the trick was transcribed by W. W. Greg barely 300 years after having been scrawled out by Henslowe:
Now, I don’t know about you, but I’m not fluent in shorthand-infused streams-of-consciousness written in Early Modern English. Maths communication has evidently come a long way in the last 400 years. Here’s the best I can do as a more-or-less faithful translation:
Take 12 cards and the jack of clubs and lay them in a circle like a clock, all face down except for the jack. Put the jack at the bottom like on your watch [ed: I have never owned a watch with the jack of clubs at the bottom of it, but let’s roll with it], laid out like [the picture]. Then ask the volunteer what time they will get up and to keep it to themself.
Tell them to pick a card to count from [ed: It’s unclear to me whether the trickster or the volunteer picks the card — it doesn’t make a difference, so I’d let the volunteer do it]. Starting from this card and moving clockwise, they should count from their card up to 15 — so if they picked 7, they should count on eight cards.
Going around the circle, you count aloud clockwise while pointing at the cards, saying “15” on the first card clockwise from the jack, “16” on the second and so on up to 26. Tell the volunteer that whichever number you said when you pointed at their current card, they should count anticlockwise from their secret number up to that number.
When they flip the card they land on, it will be the number they first thought of.
I presume “a proved” is Early Middle English for “and everyone said WOW! That’s amazing.”
But it doesn’t work.If you follow the instructions — which, like a game of Telephone that started centuries before the telephone was invented, have been written down from Henslowe’s memory, transcribed by an expert from unclear manuscript, and then translated into modern-day English by someone unqualified to do so. Hi! — you’ll find your “tada!” falls flat, because it’s not their secret number.
Let’s try it: I get up when I want, except on Wednesdays when I’m rudely awakened by the dustmen at 6am. And, rolling a 13-sided die¹ to decide where to start, I get card #3. I need to count on 9 clockwise from there (to make it up to 15), so I end up on card #12. That’s been given the number 26, so I need to count counterclockwise from my number (6) up to 26 — that is, 20 cards backwards. That takes me to 5.
¹ Yes, I do own a 13-sided die. Why do you ask?
Close, but no not-yet-introduced-to-England cigar.
It turns out that, whatever card you start from, and whichever number you pick, you’ll end up on the card immediately before your secret number. This suggests an easy fix: start your counting-aloud from 14 at card #1.
In case you want to do the trick yourself correctly, here are instructions for my version:
- Tell your volunteer to think of a secret number from 1 to 12 (don’t let me control your mind!).
- Tell them to pick a card (any card! don’t tell me what it is) and count clockwise from their chosen card, starting from their secret number and ending at 15. Have them tell you the card they land on.
- Now tell them the numbers attached to the card: whatever “time” their card is on the clock face, give them a number 13 higher.
- They must now move anticlockwise from their chosen card, starting from their secret number, ending on the number attached to the card.
- Turn over the card they ended up on and say “abracadabra!”.
My glamorous assistant Bill goes through the trick with me. Better magicians than me — which is pretty much everyone — will have ideas about improving the patter and performance.NOW it works. But why?You know what else, apart from the later Shakespeare plays, telephones, and cigars, hadn’t arrived in Elizabethan London? I’ll tell you: modular arithmetic. At least, modular arithmetic as we know it — working with remainders goes back to at least Sun Zi in the third century CE, but Euler and Gauss’s formalisations of it were still 150 years away.
I don’t know what Henslowe’s mathematical background was — he was certainly competent at regular arithmetic — so I don’t know whether he understood why the trick worked, whether he came up with it himself, or anything about the history of it. All the same, I’m certain he wouldn’t have used the modulo function.
(In case you’re one of today’s lucky 10,000: modular arithmetic uses the remainder left over when you divide by a given number, like on a clock: 16:00 is the same as 4pm, and we’d say we were working “modulo 12” or “mod 12”, because we are lazy and modulo is far too long a word. The numbers 4 and 16 have the same remainder when you divide them by 12. In this problem, we’ll be working modulo 13.)
Let’s say you’ve picked secret number (s) and you decide to start from card #(c). You’re going to count on ( 15-s ) cards from there, so you end up at card #( (c + 15 – s ) ). (We can think of card 14 as the same as card 1 and so on.)
The number I assign to it is 13 more than the card number. Modulo 13, that’s just the card number — but doing it this way ensures we don’t have to deal with negative numbers. (Negative numbers had probably reached England by this point, but I don’t imagine they were the kind of thing you’d want to have in a card trick.)
In any case, the volunteer is currently at card #( (c +15 – s ) ) and has been given the target # ( (c +28 – s )) to count to, starting at their secret number ( s ). That means they’re going to move ( (c +28 – 2s) ) cards back the way they came, starting at card #( (c + 15 -s ) ). Moving backwards makes it a subtraction, so we work out ( (c + 15 – s ) – (c +28 – 2s ) ) to see that we end up on card #( (s – 13)).
And, because there are 13 cards, that’s the same as card #(s), which has the volunteer’s secret number written on it.
Boom.
One more twist, thoughWhen I talked to young Bill about it, he asked a tremendous mathematical question: “would it work with a number other than 15?” The kid is ten years old, and already making me mutter “good GRIEF, where did that come from?” about three times a month.
The answer is… you don’t need it to be 15. In fact, the first half of the trick is mathematically irrelevant². You could ask them to spell out their secret number in a language of their choice, you could ask them to add their age to their best Parkrun time in minutes, you could ask them to spin a coin and pick the card it lands closest to. It doesn’t matter in the slightest, as long as they pick a card.
² That doesn’t mean it’s not an important part of the trick! I think it’s helpful to demonstrate how you want the final bit counted, and it misdirects the volunteer/audience into thinking there must be something clever going on.
If that’s card #( C ), then they subtract ( (C + 13) – s ) from it — which again leaves you on card #( (s – 13) ), which is card #( s ).
Even knowing the maths behind it, I think this is still a pretty impressive trick. To someone frequenting a smoky Elizabethan tavern, it must have looked like, well, magic.
Thanks to Rob Eastaway for sending me the trick. His book on the maths of Shakespeare, Much Ado About Numbers, is available wherever good books etc. Thanks also to Paul O’Malley and Tiago Hirth for historical help, and to Calista Lucy and the Governors of Dulwich College for permission to reproduce the manuscript page.
Double Maths First Thing is Colin’s weekly newsletter. Usually several letters, arranged into words.
Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight through the medium of mathematics. It’s Wednesday morning and it’s time for Double Maths First Thing.
Number City!There’s only one place to start this week: on an Orcadian all-weather hockey pitch, where Katie Steckles is at it again. A crack team of maths communicators built towers of boxes representing the prime factors of the numbers up to 64. I love this sort of large-scale outreach project — low barrier to entry (anyone can doodle a number on a box), high curiosity factor (“what are all those nerds doing on the hockey pitch? Is that… Matt Parker? With SEVEN FROM NUMBERBLOCKS?!“), and plenty of depth available for those who seek it.
Meanwhile, Andrew Taylor created a game off of it, because when you’re Andrew Taylor, there are more important things than sleep. You’re given a picture of a small section of Number City and need to deduce where you are.
The only thing I don’t like is that it’s clearly not Number City — it’s a Factory Town.
A Number-Picking PuzzleI secretly write down a number between 1 and 100 and you have to guess it. You pay me £1 every time you make a guess (I’ll tell you “higher”, “lower” or correct) and I’ll pay you £6 once you get the right answer.
How should you play if you know I’ve picked at random? How should you play if we’re both playing to win? Who wins in the long run?
The reason I ask is, Steve Ballmer (one-time CEO of Microsoft) used to ask this as a coding interview question; his answer was controversial.
There’s some analysis from John Graham-Cumming and from Possibly Wrong — I haven’t gone through it myself, because I’m still trying to figure out the answer when it’s a number from 1 to 3.
Classical mathsI’ve included the next three links as a challenge. Don’t get me wrong, I think they’re good, solid mathematical blog posts, made available for free, so I’m not going to complain — and yet I have a nagging feeling they could be done better. Could you explain these things more clearly?
Eli Bendersky has some notes on the Euler formula (I shall redact my rant about how the very idea of “the most beautiful equation” is offensive and wrong and replace it with one grumbling about Eli not properly LaTeXing up his functions).
John D Cook, meanwhile, has some advice on mentally approximating logs and trig functions.
And finally, Adrian Biagioli has an explanation of Perlin noise.
Not enough maths in your life?I mean, who has? In case you’re one of today’s lucky 10,000 who don’t know about Chris Smith, he’s been running a school maths department newsletter for… well, I subscribed a decade ago and it’s gone from issue 293 to 690 in the meantime, so I imagine you can work it out. If you want to be part of the fun — a seemingly endless supply of jokes, puzzles and news, not to mention a milk rota that I’m too far in to ask about now — you can sign up by sending him an email.
In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website.
Until next time,
C
A conversation about mathematics inspired by a low bridge sign. Presented by Katie Steckles and Peter Rowlett, with special guest Adam Townsend.
The plot discussed around 11 minutes and various other photos are available on Adam’s Height Hunt website (spoilers for the episode’s twists and turns!).
Double Maths First Thing is Colin’s weekly news summary. Or autumnal, if you’re reading this after the equinox. You can sign up to receive it in your inbox on a Wednesday morning here.
Hello! My name is Colin and I am a mathematician. It’s Wednesday morning, and it’s Double Maths First Thing.
Shape-ologyOver on the Talking Maths In Public WhatsApp group, we’ve been looking at collapsible polyhedra, which Barney Maunder-Taylor calls Flatonic Solids. He’s not the only one, though: here’s a satisfying Instagram reel and an article by Liz Meenan in case you want to make your own.
It also reminded me that you can do cool things with pop-ups, whether or not you have the book.
Speaking of booksTom Briggs has been compiling suggestions of maths books that aren’t about teaching. I’m given to believe he might be making his own addition to the list in due course.
Peter Rowlett and his son have been reading Gulliver’s Travels, and found an interesting early description of something computery. A biased generator of randomness that produces plausible English? I bet the venture capitalists would be all over that.
SudokuI recently had cause to revisit the Miracle Sudoku video — memorably described at the time by Ben Orlin:
You’re about to spend the next 25 minutes watching a guy solve a sudoku.
Not only that, but it’s going to be the highlight of your day.
The highlight of my day recently was coming across Phistomephel’s ring, which is a neat consequence of standard sudoku rules.
Tony Mann pointed me at another Cracking the Cryptic video with the same energy — the frustrations and feelings of stupidity that come with not having the answer yet, followed by the sheer joy of having worked out something clever.
Another (and significantly shorter) video plausibly worth your time is Alyssa Williams and Christian Scott at G4G discussing how to set variant sudoku.
Joy in mathsBack to taking pleasure in maths, here’s a short interview with Talithia Williams, PhD: I loved the bit about maths appreciation, and trying to change the mindset that maths is about doing calculations to pass a test.
Another article that caught my eye this week was about climbing. Or rather, spotting an error on the climbing wall and getting it fixed. It’s interesting for several reasons, but what grabbed my attention was what I think of as x-ray vision: the power to see that something looks off, and the insistence that it be put right. That strikes me as a very mathematical thing. (And, speaking for myself, possibly an autistic thing. Drives me MAD when people don’t care about breaking the rules, I tell you.)
For your listening pleasureThis week, I have mostly been listening to:
- Tim Harford’s Cautionary Tales
- Bec Hill and Matt Parker’s A Problem Squared
- Evelyn J Lamb and Kevin Knudson’s My Favorite Theorem
- Katie Steckles and Peter Rowlett’s Mathematical Objects — I understand that series 8 is imminent.
I’ve not yet picked up the TMiP podcast, but we all should. And Sam Hansen would give me endless, deserved grief if I didn’t mention Relatively Prime.
The week aheadThanks to September ending on a Monday, the monthly MathsJam meet-up is coming around distressingly quickly — those that meet on the traditional penultimate Tuesday will do so on September 17th. You can find your local MathsJam here — I’ll be at the Weymouth one.
Also, if you’re planning to go to Big MathsJam in November, early-bird pricing ends on Sunday.
There’s a Finite Group livestream on Friday, September 13th at 9pm BST — Katie and Ayliean are putting the ‘fun’ into ‘fundamental theorems’, it says here.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website.
Until next time,
C
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of August 2024, is now online at Maths for Life.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Double Maths First Thing is Colin’s weekly news round-up. Or round-down, if the fractional part is smaller than a half. You can sign up to receive it in your inbox on a Wednesday morning here.
Hello! My name is Colin and I am a mathematician. Welcome to issue 0 of Double Maths First Thing, in which I highlight some of the mathematical things that have caught my eye this week.
Let’s talk about ( \pi ) and powersFirst up, a nod to physicists Arnab Priya Saha and Aninda Sinha for doing something with no real application: they “accidentally discovered a new formula for pi”. There’s a bit about it in Scientific American, a Numberphile video, and a paper in Physical Review Letters (open access). I’ve not worked through it in detail, but it’s got a Pochhammer symbol in it, so it must be good.
I promise this isn’t always going to be about pi, but I also stumbled on a proof that pi is irrational — again, I’ve not worked through the details, but it looks like it would be accessible to a good A-level class with a bit of hand-holding.
Via reddit, a surprisingly tricky problem with a lovely twist in the tail: show that ( 3^k + 5^k = n^3 ) has no solutions for ( k > 1 ). (There’s a hint and a spoiler over on mathstodon.)
Somewhere to visit: W5, BelfastI’ve recently been on holiday in Northern Ireland. We visited W5 in Belfast, which is a pretty cool science museum — lots of hands-on stuff, including a build-your-own Scalextric-style car, bottle rockets and a green-screen bit where you can present the news about the alien invasion. On the minus side… there are lots of missed opportunities for highlighting the maths that underpins it all. Still, it’s a fun half-day if you’re all Titanic-ed out.
Maths in the newsIn the proper news, the Guardian had a long read about Field’s Medallist Alexander Grothendieck; although it too is a bit maths-light, it’s understandable given quite how heavy Grothendieck’s maths is. Katie Steckles also pointed me at the devastating news that UK railcard discounts are dropping from 34% to 33.4%, which strikes me as the sort of thing that probably costs more to implement than it could possibly save the train operators.
Upcoming mathsIf you’re in the market for more maths, I can heartily recommend both the Finite Group, whose next livestream is on Friday September 13th, and Big MathsJam, which is a gathering of amazing geeks the first weekend of November. Early-bird tickets are (just about) still available; I have mine already. There’s also a day of recreational maths lectures in memory of David Singmaster on Saturday September 21st in London or online, and a New Scientist event about how maths explains the world the following Saturday, also in London.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website.
Until next time,
C
Here’s a quick round-up of some news stories from this month.
AwardsThe Royal Society has announced its award winners for 2024, which include mathematicians Ingrid Daubechies (Bakerian Medal/lecture for the physical sciences), Hannah Fry (David Attenborough Award/lecture for public engagement) and Philip Maini (Sylvester medal for mathematical research).
Ingrid Daubechies, speaking at ICM 2018 in Rio de Janeiro (Photo: Rodrigo Leao/R2)Hannah Fry at the The European Data of Tomorrow Conference in 207 (photo: Sebastiaan ter Burg)Philip Maini in 2015 (photo: Royal Society)And the joint IMA/LMS Christopher Zeeman Medal for 2024 has been awarded to Brady Haran for his work in communicating mathematics via the Numberphile channel on YouTube.
Other newsThe Protect Pure Maths campaign group has relaunched as the Campaign for Mathematical Sciences, encompassing a broader remit to promote and support mathematical activity in the UK.
In research news, an elliptic curve with rank at least 29 has been found by researchers Noam Elkies and Zev Klagsbrun. The previous record was rank ≥28, found by Elkies in 2006. (via Robin Houston)
[y2 + xy = x3 – 27006183241630922218434652145297453784768054621836357954737385x + 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497]
Pierre Cartier, mathematician and Bourbaki member, has died aged 92.
In this series of posts, we’ll be featuring mathematical video and streaming channels from all over the internet, by speaking to the creators of the channel and asking them about what they do.
We spoke to Jon Chase, aka Oort Kuiper the Science Rapper, about his TikTok channel and how he’s been using it to share mathematical raps.
Channel title: Science Rapper (Sci Comm)
Link: tiktok.com/@science.rapper.sc
Topics covered: KS4 maths raps, plus more general STEM
Average video length: 45-75 seconds
Recommended videos: Factorising to solve a quadratic, 8 Circle theorems
What is your channel about, and why did it start?It mostly started as a way to share some of the stuff I have done in STEM, to get an idea of what worked on TikTok. After trying out some other avenues on TikTok, I soon discovered that it was maths raps that gained the most interest, and so I decided to focus on that.
Who are you? Tell us about yourself. I’m a science communicator that makes raps to communicate STEM ideas and information. The focus on maths is predominantly targeted towards education-based TikTok audiences, as this platform seems to be where those audiences are engaging with the content most. Other than maths on TikTok, I do plenty of other science communication ranging from stage shows, workshops and presenting on screen, to writing articles – as well as a few books.
What is a typical video like, and why should people watch?The videos are just short raps about a particular topic in maths – either overviewing it, or providing rapped worked examples. It’s quick and easy and catchy (depending on your preferences, of course). It’s a very unique way to teach/share maths.
@science.rapper.sc #learn #lawsofindices #index #base #multiplying #dividing #mathsonmonday #mathematics #indices #maths #stem #mathsrap #oortkuiper #mathsrapgcse
♬ original sound – Science Rapper (SciComm)
What are some highlights of the channel so far?It was exciting to see my factorising video going viral! It’s now got over 400,000 likes and over 3,000 comments – and it’s always fun watching the comments unfold: essentially, the audience educates each other through their comments, as well as arguing about methods.
What exciting plans do you have for the future? I’m hoping to cover more content from the GCSE curriculum, to provide greater access to engaging and quick maths revision!
Earlier this week I posted a matrix multiplication worksheet on Mastodon.
If you do some of these, you might spot what’s funny about them. For example.
[ \Large \begin{bmatrix}
\color{navy}{4} & \color{navy}{8}\
\color{navy}{2} & \color{navy}{3}
\end{bmatrix} \begin{bmatrix}
\color{cyan}{8} & \color{cyan}{8}\
\color{cyan}{2} & \color{cyan}{7}
\end{bmatrix} = \begin{bmatrix}
\color{navy}{4}\color{cyan}{8} & \color{navy}{8}\color{cyan}{8}\
\color{navy}{2}\color{cyan}{2} & \color{navy}{3}\color{cyan}{7}
\end{bmatrix} ]
That is, the answer to each question can be made by treating the element in the first matrix as the first digit and the corresponding element in the second matrix as the second digit in the answer element. This is not how matrix multiplication works, and ought to be funny if I hadn’t totally over-explained the joke!
I saw one of these in a meme that Katie posted in the Finite Group chat and it got me thinking about how these work.
If we set up the matrices like this
[ \begin{bmatrix}
a & b\
c & d
\end{bmatrix} \begin{bmatrix}
e & f\
g & h
\end{bmatrix} = \begin{bmatrix}
10a+e & 10b+f\
10c+g & 10d+h
\end{bmatrix} ]
Then we establish four equations with eight unknowns.
[ \begin{align}
ae + bg &= 10a+e\
af+bh &= 10b+f\
ce+dg &= 10c+g\
cf+dh &= 10d+h
\end{align}]
Since there are more unknowns than equations, these don’t have a single solution. What I wanted was to find integer solutions with all values single-digits. I wrote some Python code to find these. I removed some that look overly symmetrical – either the rows of the matrix are identical, or the same matrix is repeated. This left 73 items.
From these 73 items, I wrote a second Python script that picks 20 of them at random and builds these into a LaTeX worksheet. For the Mastodon post I reformatted this into the shape and size that I thought would display better on social media, and added in one of the squared matrices for an extra hint something weird is up, hoping people might notice this isn’t just a boring post about matrix multiplication practice!
You can view these scripts and associated files on GitHub.
Here’s a round-up of some news we didn’t cover on the Aperiodical in the last couple of months.
Research NewsIn computation news, the fifth Busy Beaver number has been found. This Quanta article gives a good writeup. (via TheHigherGeometer)
In “are we nearly there yet” news, 202 trillion digits of pi have been calculated, breaking the previous record. The computation used the Chudnovsky algorithm (pictured below) and took around 100 days to crunch. (via Robin Houston)
A new James Maynard paper potentially rules out some exceptions to Riemann Hypothesis, making a big step forward in understanding the structure of the prime numbers.
And finally, the Antikythera mechanism has been theorised to be connected to the lunar rather than the solar calendar, based on (possibly sketchy) gravitational wave research. Dubious!
Awards and announcementsSarah Hart’s excellent maths/literature book Once Upon a Prime has been awarded the 2024 Euler Book Prize, which is awarded annually to authors of “exceptional mathematics books that significantly impact public perception and understanding of mathematics”.
In case there weren’t enough books about maths and literature (we’re looking at you, Rob Eastaway) the incredible Ben Orlin of Math With Bad Drawings has announced his next book will also touch on the overlap of words and numbers, and will be out on 3rd September.
And earlier this month, Tim Harford gave the inaugural Vicky Neale Public Lecture at Oxford University. The lecture recognises the invaluable contribution to mathematical education of the late Vicky Neale, and Tim spoke about “how data built the modern world – and how we can use it to build a better one”. The lecture can be re-watched on the Oxford Mathematics YouTube Channel.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of July 2024, is now online at Theorem of the Day.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Hi! My name is Colin, and I am a PROPER mathematician now. I’ve made a contribution to the Online Encyclopaedia of Integer Sequences.
The OEISIf you don’t know about the OEIS, then congratulations! You’re one of today’s lucky 10,000. Except for possibly Wikipedia, the OEIS is probably the most important and useful mathematical community resource on the internet. The main use case is, you’re doing some maths and you find a sequence. You wonder whether it’s something new or interesting. So you type it in to the OEIS search bar, and if it exists, you’re told what it is, whether there are formulas for it, different suggestions of where it crops up, lists of terms… it’s like a Who’s Who of number sequences.
There’s seemingly no limit to what it contains. For example, it contains the decimal form of the n-th color mentioned in the song “I Can Sing a Rainbow”, which I would call borderline “not maths at all”. At the other end of the spectrum (ahahaha), it also contains the all 1s sequence, which is borderline “not a sequence at all”. The idea seems to be “if it’s a reasonable sequence someone has ever had cause to think about, it belongs here”.
Now, I’m very fond of saying I’m a mathematician. I have two degrees, several academic papers, a number of published books, an equation named after me and no proper job. But at the same time, my impostor syndrome insists that a proper mathematician would have at least something in the OEIS.
I am now doubting whether I really qualify as an impostor.
The questionThis all started — as most of my maths seems to these days — from a question posed by my 10-year-old, Bill.
“Does Pascal’s triangle go up the way, too?”
If you’re not familiar with Pascal’s triangle, then congratulations! Etc. You might want to visit Matt Enlow’s piece on it in the Math-off final.
Meanwhile, here are the first few rows:
[ \begin{array}{ccc ccc ccc ccc ccc}
&&&&&&& 1\
&&&&&& 1 && 1 \
&&&&& 1 && 2 && 1 \
&&&& 1 && 3 && 3 && 1 \
&&& 1 && 4 && 6 && 4 && 1 \
&& 1 && 5 && 10 && 10 && 5 && 1 \
&1 && 6 && 15 && 20 && 15 && 6 && 1 \
1 && 7 && 21 && 35 && 35 && 21 && 7 && 1 \
\end{array} ]
Translating that into how boring grown-up mathematicians speak, he was asking whether you can add rows above the 0th row — what would the -1st row look like? The -5th?
We had a good chat about this — how we could change the rules slightly to make it work, but remain consistent, what patterns he could find, and eventually he wandered off with a pad of paper to see what he could fill in for himself.
I was going to ask him about what would happen with fractions, before I nerdsniped myself asking “what would happen with complex powers? What’s the ( i)th row of Pascal’s Triangle?”
There’s a trick for generating the ( n )th row of Pascal’s triangle:
- Let your current entry be 1 and your fraction be ( \frac{n}{1} ).
- Write down your current entry.
- Multiply it by your fraction to get a new current entry.
- Subtract one from the top of your fraction and add one to its bottom.
- Go back to step 2 until either your entry becomes 0 or you get bored (or both).
So, for example, the fourth row would be:
- Write down 1, and multiply by ( \frac{4}{1} ).
- Write down 4, and multiply by ( \frac{3}{2} ).
- Write down 6, and multiply by ( \frac{2}{3} ).
- Then 4, 1, and 0 — so we stop.
We get the numbers 1, 4, 6, 4, 1, as we would have hoped.
This also works for non-natural numbers. We could find the -3rd row to answer Bill’s question:
- Write down 1, and multiply by ( \frac{-3}{1} ).
- Write down -3, and multiply by ( \frac{-4}{2} ).
- Write down 6, and multiply by ( \frac{-5}{3} ).
- Then -10, 15, -21, and we get bored so we stop.
We get the numbers 1, -3, 6, -10, 15, -21 and can hypothesise or prove that these are the triangular numbers with alternating signs.
So what about the ( i )th row?Well, it works just the same way, only with slightly trickier calculations.
- Write down 1, and multiply by ( \frac{i}{1} )
- Write down ( i ), and multiply by ( \frac{i-1}{2} )
- Write down ( \frac{-1 – i}{2} ), and multiply by ( \frac{i-2}{3} )
- Honestly, I’m bored already. We have machines for this sort of thing.
It’s also a lot easier if we just ignore the bottom of the fraction — it’s trivial to say “we need to divide the ( k )th term by ( k!),” and it doesn’t affect the working out at all.
And if we do that, we get a Gaussian integer sequence:
[ 1, i, (-1-i), (3+i), -10, (40-10i), (-190+90i), \dots ]
Now, the OEIS, like any good hip-hop artist, prefers to keep it real. You are as likely to find Gaussian integers in the OEIS as in the work of Grandmaster Flash (who is surprisingly mediocre at chess). However, I reasoned, the real and imaginary parts might be there separately.
And boom, there they were:
- Expansion of e.g.f. ( \cos ( \ln(1+x) ) )
- Expansion of e.g.f. ( \sin ( \ln(1+x) ) )
Wait. What?The OEIS is useful not just as a research tool, but as an educational tool: it’s forever throwing up unexpected links and ideas and threads to pull at. Here, for example, it raises the question “what on earth is an e.g.f.?”
Luckily, Wikipedia is also a useful research and educational tool: the e.g.f. is the exponential generating function. Let’s start with a sidetrack into (common or garden) generating functions, which are the sort of dark magic it’s worth messing about with.
A generating function is a way of turning a sequence of numbers into a function, the better to understand its properties. For exampe, if you take the Fibonacci sequence, with terms 0, 1, 1, 2, 3, 5, …, the associated generating function is
[ g(x) = 0 + 1x + 1x^2 + 2x^3 + 3x^4 + 5x^5 + \dots ]
Generating functions make certain manipulations very simple: for example, you can “move the sequence along” if you multiply by a power of ( x). As a result, you can use the definition of the Fibonacci sequence to find that ( x^2 g(x) = x g(x) + g(x) ) — each term is the sum of the previous two — and rearrange to find that ( g(x) = \frac{1}{x^2 – x – 1} ). As a result, you can say “Let’s put ( x = 100 ) into that” and find that ( \frac{1}{9,899} = 0.00 01 01 02 03 05 08 13 \dots ) . (After a while, the carries get in the way and break the sequence.) There are all sorts of cool applications in number theory and probability.
In particular, the binomial expansion of ( (1 + x)^n ) is the generating function for the ( n )th row of Pascal’s triangle — that is, the coefficient of ( x^k ) in the generating function of the ( n )th row is ( nCr(n, k) ).
So what about the exponential generating function?
Rather than taking the coefficients from ( a_0 + a_1 x + a_2 x^2 + \dots ), the e.g.f. takes its coefficients from ( b_0 + b_1 x + b_2 \frac{x^2}{2!} + b_3 \frac{x^3}{3!} + \dots ). It’s the same idea, but the manipulations are slightly different (you shift by differentiating or integrating, for example.)
When we’re looking at the ( k )th element of a row in Pascal’s triangle, it’s the same as multiplying the coefficient by ( k! ) — or rather, you get ( nPr(n, k) ) .
So: what I worked out above, the Gaussian integer sequence, is the e.g.f. of ( (1 + x)^i ).
And where does trigonometry come into it?It wasn’t immediately obvious to me how that was linked to the OEIS sequence, until I took a few minutes to work it through.
If we combined the sequences to make the Gaussian integers, we’d get ( \cos( \ln(1+x)) + i \sin(\ln(1+x))). And that’s a familiar form: ( \cos(z) + i \sin(z) \equiv e^{iz} ).
So our sequences combine to make ( e^{i \ln(1+x)}) ), which works out to be ( (1+x)^i ). Lovely!
And lastly, the permutations observation from before means that each term can be written as ( nPr(i, k) ).
There’s a slight wrinkle, though: it’s not exactly reasonable to pick ( k ) items from a set containing ( i ) of them. Even the formula of ( nPr(n, k) = \frac{n!}{(n-k)!}) ) isn’t well-defined: you can’t really take the factorial of something that isn’t an integer.
However, there’s a handy generalisation of the factorial function to all complex numbers, the gamma function, defined such that ( \Gamma(n+1) \equiv n! ) on the natural numbers.
As a consequence, we can extend the permutation function in a similar way: on the natural numbers, ( nPr(n, k ) \equiv \frac{\Gamma(n+1)}{\Gamma(n+1-k)}) ), and (having checked with the Mathematical Authorities), we’re allowed to use that definition for all complex numbers (apart from the unnatural integers, where ( \Gamma(x) ) is undefined. But that’s for another day.)
Where were we? Oh, yes. Each term of our Gaussian integer sequence is ( nPr(i, k) ), which we can write as ( \frac{\Gamma(i+1)}{\Gamma(i+1-k)}) ); the two sequences correspond to the real and imaginary parts of this.
And that formula, or anything like it, was missing from the OEIS.
Now it isn’t, and I can finally cut up my impostor card and say with confidence: my name is Colin, and I am a mathematician.
Here’s the final match of The Big Internet Math-Off.
Over the past month, we’ve heard from 16 interesting mathematicians and whittled them down to just 2. Today, we’re pitting Matt Enlow against Angela Tabiri to determine The World’s Most Interesting Mathematician (2024, of the people who I asked to take part and were available).
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Matt Enlow – Hidden in Pascale’s TriangleDedicationI would like to dedicate today’s excursion to my 11th- and 12th-grade math teacher, John Barrow, who passed away on June 20th of this year. It’s safe to say that I wouldn’t be here, doing this, if it weren’t for him. He encouraged me to double up in math my junior year, and then in my senior year, he taught me AP Calculus BC one-on-one. I don’t think I appreciated at the time what a privilege that was—particularly to be doing it with such a kind-hearted, caring, and inspirational teacher. Rest in peace, Mr. Barrow.
ExcursionToday we will be looking at a mathematical “object” that has been a favorite of mathematicians for millenia: Pascal’s Triangle.
A lot of people already know that the sums of the entries in the rows of Pascal’s Triangle are the powers of two:
But what you may not know is that if, instead, you multiply the entries in each row together, you get…
… uh… numbers that get really big, really fast!
Okay, so maybe that’s not so much of a surprise. Particularly since the entries in the triangle increase in size rather quickly themselves.
But suppose we wanted to try to get a better handle on just how quickly those products are increasing. One way we could do that is by calculating the ratios of successive products. In other words: How many times greater is each row’s product than the previous row’s product? If, for example, those ratios were to approach some constant, we could say that the products were increasing approximately exponentially.
First, let’s get a little more data. (By convention, we call the “row” with a single 1 in it the (0^{\text{th}}) row, and the next row, with two 1’s in it, the (1^{\text{st}}) row, etc.)
Whoa. Those are indeed some big numbers. In the interest of conserving space, I’ll write them using scientific notation.
But before I do that… I’m noticing something. Look at the curve formed by the lead digits of the products. I can tell just by looking at that curve that these numbers are increasing more quickly than merely exponentially. If the numbers were increasing exponentially, then that curve would not be curved at all; it would look more like a straight line. For example, check out the list of the first 50 powers of 162 below.
Technically, that is what exponential growth looks like: a nice, straight line!
Okay. So. Anyway. Scientific notation.
Okay. Now let’s calculate the ratios of successive products, as mentioned earlier:
As expected, these ratios themselves are increasing rather quickly. Ah, but… Notice the steep slope of the lead digits. That looks much straighter than the curve in the earlier list of products! So maybe… the sequence of ratios is growing exponentially? Let’s look at the ratios… of the ratios! (We’ll call these the “second ratios.”)
Aha! (… Again, this is where you yell “Aha!” really loudly.) These second ratios are much better behaved. They are increasing, but it looks like the rate at which they are increasing… is decreasing. So a natural thing to wonder at this point is, are these second ratios approaching a specific value, or will they increase without bound?
(… I think I just felt a few shivers of ecstasy from some of you out there who see where this might be going…)
LimitsLet’s introduce some notation to help us try to get an answer to this question. Define (p(n)) as the product of the entries in the (n)th row of Pascal’s Triangle. In other words, [p(n):=\prod_{k=0}^{n}\binom{n}{k}=\prod_{k=0}^{n}\frac{n!}{(n-k)!k!}.]
Now. How can we create an expression for those second ratios using (p(n))? Well, the sequence of first ratios begins [\frac{p(1)}{p(0)}, \frac{p(2)}{p(1)}, \frac{p(3)}{p(2)}, \frac{p(4)}{p(3)},\ldots,\frac{p(n+1)}{p(n)},\ldots] which means that the sequence of second ratios begins [\frac{p(0)p(2)}{p(1)^2}, \frac{p(1)p(3)}{p(2)^2}, \frac{p(2)p(4)}{p(3)^2},\ldots,\frac{p(n-1)p(n+1)}{p(n)^2},\ldots]
So it looks like what we’re hoping to determine is the value (if it exists) of [\lim_{n\to\infty}\frac{p(n-1)\,p(n+1)}{p(n)^2}.]
The prospect of using our definition of (p(n)) to evaluate this limit seems daunting; the notation-wrangling alone is enough to make us want to leave it to someone else.
Fortunately, this (p) function of ours has a nifty property (proven in the Postscript): For all values of (n), [\frac{p(n+1)}{p(n)}=\frac{(n+1)^n}{n!}.]
You may recognize the expression on the left as representing our first ratios. So if you plug values of 0, 1, 2, 3, etc. in for (n) in the right-hand expression, you will see that you get the values of those first ratios calculated earlier: [\frac{1^0}{0!}=1,\qquad \frac{2^1}{1!}=2,\qquad \frac{3^2}{2!}=4.5,\qquad \frac{4^3}{3!} \approx 10.7,\qquad \text{etc.}]
Now let’s take another look at that expression for which we’re trying to find a limit, and make use of that nifty property: [\frac{p(n-1)p(n+1)}{p(n)^2} = \frac{\ \frac{p(n+1)}{p(n)}\ }{\ \frac{p(n)}{p(n-1)}\ } = \frac{\frac{(n+1)^n}{n!}}{\frac{n^{n-1}}{(n-1)!}}] And this expression actually simplifies quite nicely: [\frac{\frac{(n+1)^n}{n!}}{\frac{n^{n-1}}{(n-1)!}} = \frac{(n-1)!\,(n+1)^n}{n!\;n^{n-1}} \ = \frac{n+1}{n}\left(\frac{n+1}{n}\right)^{n-1} = \left(1+\frac{1}{n}\right)^n.] And, as most calculus students know, [\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n \approx 2.71828,] better known as Euler’s number, or simply (e).
I was scandalized when I first encountered this. What on earth is (e) doing in Pascal’s Triangle?!? Even though we just proved it, it’s still hard to make sense of.
Showing that something is true does not necessarily illuminate why it’s true. I think I’ve done the former here, but I think it would take some more contemplation on my part before I’m able to do the latter.
I would love to hear your attempts at such illumination!
Oh, and by the way: Those of you who read the title of my pitch and thought, “Wait a minute—there’s no ‘e’ in ‘Pascal’!”… Well, now you know the truth!
Postscript: Proof that (\frac{p(n+1)}{p(n)}=\frac{(n+1)^n}{n!})[\begin{aligned} \frac{p(n+1)}{p(n)} &= \frac{\prod_{k=0}^{n+1}\frac{(n+1)!}{(n+1-k)!k!}}{\prod_{k=0}^{n}\frac{n!}{(n-k)!k!}} & & \text{by definition}\[0.5em] &= \frac{\prod_{k=0}^{n}\frac{(n+1)!}{(n+1-k)!k!}}{\prod_{k=0}^{n}\frac{n!}{(n-k)!k!}} & & \text{remove a “1” from the top product} \[0.5em] &= \prod_{k=0}^{n}\frac{\ \frac{(n+1)!}{(n+1-k)!k!}\ }{\ \frac{n!}{(n-k)!k!}\ } & & \text{combine products} \[0.5em] &= \prod_{k=0}^{n}\frac{(n+1)!(n-k)!k!}{(n+1-k)!k!n!} & & \text{invert and multiply} \[0.5em] &= \prod_{k=0}^{n}\frac{n+1}{n+1-k} & & \text{simplify} \[0.5em] &= \frac{\prod_{k=0}^{n}(n+1)}{\prod_{k=0}^{n}(n+1-k)} & & \text{separate products} \[0.5em] &= \frac{(n+1)^{n+1}}{(n+1)!} & & \text{simplify} \[0.5em] &= \frac{(n+1)^n}{n!} & & \text{cancel an }n+1 \end{aligned}]
Matt Enlow teaches mathematics at the Dana Hall School in Wellesley, MA. You can follow him on X, BlueSky and Mathstodon.
Angela Tabiri – #MathsMotivationWe start off today’s pitch with a hands-on activity. For this, you need three circular objects, a thread, a rule, a paper and a pen. Using the thread, find the length of the circular object by wrapping the thread around the object and measuring it on the rule. Record this in one column on the paper. Next, locate the centre of the circular object and measure the diameter (from one end of the circle through the centre to the opposite end). Record this diameter in another column on the paper. In a third column, divide the length of the circular object by the diameter. Repeat this process for the two remaining circular objects. What do you observe? Irrespective of the size of the circular object, the ratio of the length (circumference) to the diameter is a constant, called Pi (π).
In a quest to find the area of a circle, we unwrap the circle to get a right angled triangle with base (2\pi r) and height (r). Then the area of this right angled triangle is (\frac{1}{2} \times 2 \pi r \times r ) which is equal to (\pi r ^2). Hence, the area of the circle is (\pi r^2).
Let us continue to the world of prime numbers to learn some interesting facts. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … They seem to be random with no pattern. However, if we take any prime number greater than or equal to 5 and square it, the result is divisible by 24 with a remainder of 1. For instance (5^2 = 24 + 1), (7^2 = 2 \times 24 + 1), (11^2 = 5 \times 24 + 1), …
Recall that the first few digits of the number (\pi) are
[ \pi = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 8628 03482 53421 17067 \ldots ]
Recall the prime chunks from Katie’s pitch. Then, the first prime chunk after 3 is 14159. Squaring this results in ( 14159^2 = 8,353,220 \times 24 + 1). As an assignment, you can verify this for the other prime chunks in (\pi).
For our third interesting fact, we consider the sequence 1, 1, 2, 3, 5, 8, … called the Fibonacci sequence. If we draw golden rectangles as pitched by Matt, we can find the ratio of the sides of the rectangles. This ratio gives the golden ratio (\Phi). The golden ratio is common in nature, in particular on the body. The proportion of the length of the body from the head to toe, to the length from the head to the navel exhibits the golden ratio. This is evident in Leonardo Da Vinci’s The Vitruvian Man. Also, the ratio of the length from the top of the nose of an adult to the centre of the lip to the length from the centre of the lip to the chin exhibits the golden ratio. Cosmetic surgeons use the golden ratio in reconstructing the face.
Why should we care about all these interesting facts? They make maths fun and rewarding as diverse careers can be pursued after a strong foundation in mathematics. Example: Data scientist, software engineering, sound engineer, accounting, … Thus, your love for mathematics can be translated into a rewarding career.
If you found this pitch interesting and would like me to be crowned the World’s Most Interesting Mathematician, vote for me 😇 and subscribe to the Femafricmaths YouTube channel.
Medaase! Asante Sana! A dupe! Na gode! Merci! Mweebale! Murakoze Urakoze! Thank you!
Angela Tabiri is a mathematician and youth mentoring in STEM expert from Ghana. She is the founder of Femafricmaths, a non profit organisation that promotes female African mathematicians to highlight the diversity in careers after a degree in mathematics. You can follow Femafricmaths on YouTube, Instagram, Facebook and X.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.Voting is now closed: Angela Tabiri is the World’s Most Interesting Mathematician (2024, of the people who I asked to take part and were available)!
Congratulations Angela, and thanks to Matt and everyone else for playing this year!
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of June 2024, is now online at Cavmaths.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Here’s the second semi-final match of The Big Internet Math-Off. Today, we’re pitting Angela Tabiri against Ayliean.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Angela Tabiri – Six everyday examples of mathematics applicationsMathematics is difficult. Mathematics is abstract. What will I use maths for after high school? In this contest, we demonstrate six applications of mathematics in everyday life.
Starting your morning in Ghana usually involves a breakfast of “koko” and “koose”. Koko is a porridge made from millet while koose is made from beans. The koko seller stores the porridge in an aluminium cylindrical object. Using the mathematics of computing the volume, the koko seller can compute the quantity of porridge and how much the portions need to be sold in order to make profit. The volume of the cylinder can be explained as each layer of the cylinder consisting of circular surfaces so the sum of all these circular surfaces gives the volume of (\pi r^2 h).
When joining a road from a side street, you estimate the speed of the driver in the lane you are joining to see if the vehicle will be a safe distance from you by the time you join the road. If the estimation shows that the speed is high and you are not safe to join the road, you need to wait for the road to clear up in order to safely join. This involves speed, calculated as the ratio of distance to time.
In buildings which are tiled, mathematics is used to compute the area of the floor to estimate the quantity of tiles needed to tile the floor. If there are special designs to be incorporated, maths is used to compute the precise location where the designs need to be in order for the artwork of the tiles to come out beautifully.
When window frames are not fixed properly, the smallest space will allow water to seep in when it rains. If the right angle is accurately fixed at the corners of the window frame, this will reduce the chances of rain seeping through the window.
Malaria is a challenge especially in developing countries. Anytime you test positive for malaria, immediate treatment is recommended. The treatment prescribed is the stage of malaria. Mathematics is used in determining the dosages to ensure that they are taken in the right quantities and at the right time.
The wall clock shows a twelve hour time when we have twenty four hours. With modular arithmetic, we can convert times from 13:00 to 24:00 to times between 0.00 t0 12:00. We do this by computing the remainder when divided by 12. For instance: 13:00 becomes 1:00pm since 13 leaves a remainder of 1 when divided by 12.
In conclusion, mathematics is the foundation for most professions due to its universality. A deep understanding of mathematical concepts and availability of interesting learning materials will make it accessible to all.
mathsiseverywhere #mathsisfun
Angela Tabiri is a mathematician and youth mentoring in STEM expert from Ghana. She is the founder of Femafricmaths, a non profit organisation that promotes female African mathematicians to highlight the diversity in careers after a degree in mathematics. You can follow Femafricmaths on YouTube, Instagram, Facebook and X.
Ayliean – Counting Stars (it’s surprisingly difficult to count to 5)Ayliean (noun): Mathsy, arty, crochet crafty, origami, activist, zine author known for making badges and trouble. You can follow them on YouTube, as @Ayliean on all social media, or look at their homepage.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the final!
We’re giving the competitors a break before the final round. Come back on the 23rd for the grand final, or check out the announcement post for your follow-along wall chart!
Here’s the first semi-final match of The Big Internet Math-Off. Today, we’re pitting Fran Herr against Matt Enlow.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Fran Herr – (Con)figuring out spacesA frustrating scenarioImagine driving down a narrow road with cars parked on both sides. Suddenly you meet another car driving in the opposite direction. First, you look around for a space to pull over, but you have no luck. So you are forced to back up until you reach the next intersection so that you and the other car can pass each other. Thankfully cars can drive in reverse!
But now we construct a nightmarish hypothetical by placing two cars on a one-lane road with dead ends on both sides. The poor cars can never switch places by driving. We can see this by building another space to model the situation.
The configuration space is the set of all possible positions of the two cars on the road—including those reachable if we, acting like a child with toy cars, pick them up and place them down again. We can designate each car’s position by a number between 0 and 1: we give ‘0’ to the West end of the road and ‘1’ to the East end. Then each configuration is given by a point in (\mathbb R^2) of the form [(\text{position of car A}, \text{ position of car B}).] In total, the configuration space consists of all points in the unit square ([0,1] \times [0,1]) except for the diagonal (points of the form ((x,x))) since the two cars must be in different positions. The fact that the two cars cannot switch their positions (by themselves) is shown by how the configuration space is not connected.
Two cars trapped on a road. The green subset on the right is the configuration space.Now we mercifully add an alley partway down the road. Then one car can pull aside to let the other pass and thus, all configurations of the two cars on this road can be reached by driving. This means that its configuration space is connected. We can build a model of this configuration space by taking a few of the positions and connecting them by paths if they are definitely reachable by driving.
A model of the configuration space of two cars on a road with an alley. The red and blue dots symbolize cars.Whether a space is connected or not is a very coarse description; there are some more refined questions we can ask. Topologists are interested in getting information about a space by extracting some algebraic structure. One such object we will look at today is the fundamental group.
A brief description of the fundamental groupSuppose you are walking a dog on a leash in a field with a tree in the middle. Suddenly Fido bolts off to chase a squirrel, but you stand still in astonishment. Ten seconds later, you call him and he comes running back. Whether or not he made a lap around the tree will make a big difference for you! If not, you can easily pull the leash tight. But if he has, then now you have to go around the tree to untangle the leash. If the squirrel ran around the tree several times and Fido followed him, you will have to make several circles to untangle the leash. For you, the exact path that Fido made does not matter. You only care about the number of times (and in which direction) you now have to walk around the tree.
The fundamental group consists of loops up to wiggling and contracting, which mathematicians call homotopy. This wiggling and contracting is like pulling the leash tight after Fido’s joy-run. And from our little thought experiment, we see that each loop is characterized by the number of times and the direction it wraps around the tree (we can choose positive numbers for counterclockwise wraps and negative numbers for clockwise wraps). Also, loops add in the same way integers do. If Fido runs once around the tree, twice, it is the same as if he ran two times around the tree. (1 + 1 = 2) If he runs around once around clockwise and once around counterclockwise, he will untangle the leash himself. (1 + -1 = 0)
From this, we can see that the fundamental group of the field with the tree is the integers (\mathbb Z). And likewise, the fundamental group of the configuration space of the second road— the one with the alley— is also (\mathbb Z).
Dancers on stageLet’s look at a more complicated configuration space. Imagine a group of (k) dancers on a stage. We will consider the configuration space consisting of all positions of those dancers. We will not be concerned with placement of each dancer’s limbs, but will instead only keep track of their spot on the 2-dimensional stage. Certainly the configuration space is connected since the dancers can get from any one configuration to any other. So let’s try to find the fundamental group to get more information.
To simplify this situation, we will model each of the (k) dancers as a point in (\mathbb R^2). Then the configuration space consists of all lists of (k) distinct points in the plane. It is written as [\mathop{\mathrm{Conf}}_k(\mathbb R^2) := {(a_1, \dots, a_k) \: | \: a_i \in \mathbb R^2, \: a_i \neq a_j \text{ if } i \neq j}.] A loop in (\mathop{\mathrm{Conf}}_k(\mathbb R^2)) corresponds to a movement sequence where every dancer returns to their starting position. How can we record such a dance? We could record a video of the dancers, but when we watch it, we will only see one frame at a time. Instead, we can introduce another dimension.
Suppose we have a video of the points moving in the plane which corresponds to the dance. It starts at time 0 and ends at time 1. Then we slice it into many frames and lay them on top of each other with the first frame on the top of the stack and the last frame at the bottom.
Here is an example if a dance of 3 points. When we stack the “frames” of the video on top of each other, the moving points trace out braiding strands.If we track one point moving from (t = 0) to (t = 1), it will trace out a curve in space. The curves will weave around each other like braiding strings. In this way, each loop in the configuration space can be realized as a braid on (k) strands.
The braid groupA braid is a weaving of (k) strands in three-dimensional space with the top ends pinned in a starting position. There are some braids where the strands return to their starting order at the end of the braid; we call these perfect braids.
Two kinds of braids. The left one is a perfect braid, the right one is not.For a point ((a_1, \dots a_k)) in our configuration space, either we care about the order of these (k) entries, or we don’t. In our dance metaphor, this is the difference between dancers who are each in a unique costume and distinguishable vs. dancers who are in uniform costume and indistinguishable.
Mathematically speaking, what we have already defined is the ordered configuration space. The unordered configuration space (\mathop{\mathrm{UConf}}_k(\mathbb R^2)) is given by (\mathop{\mathrm{Conf}}_k(\mathbb R^2) / S_n) where (S_n) is the symmetric group and it acts by permuting the points. Loops in (\mathop{\mathrm{Conf}}_k(\mathbb R^2)) correspond with perfect braids, where all strands come back to their own spot. And loops in (\mathop{\mathrm{UConf}}_k(\mathbb R^2)) correspond with braids where strands return to any spot.
Then we note that braids do form a group. The identity is given by keeping all strands straight. The inverse of a braid is given by reflecting the braid across the vertical axis. Two braids are added by stacking one on top of another. I will leave you to think about why the collection of pure braids forms a subgroup of all braids.
Group properties for braids.So we have explained (but not showed) that the fundamental group of (\mathop{\mathrm{UConf}}_k(\mathbb R^2)) is the braid group on (k) strands, denoted (B_k). And the fundamental group of (\mathop{\mathrm{Conf}}_k(\mathbb R^2)) is the pure braid group on (k) strands, denoted (P_k).
I love this piece of math because (\mathop{\mathrm{Conf}}_k(\mathbb R^2)) seems like a highly inaccessible space. It contains (2k) dimensions since there are 2 coordinates specified for the location of each of the (k) points. However, by changing perspective, we find that its fundamental group is a very understandable and beautiful object. This is one example of how topology can help us access the inaccessible.
Fran Herr is a PhD student in mathematics at the University of Chicago. She studies low dimensional topology and geometric group theory. You can follow her on YouTube and X.
Matt Enlow – A Sine SurpriseNote: I drafted all four of my pitches before the Big Math-Off started, because I knew that I would not have time to write them “on demand.” What I did not count on was the possibility that another participant would choose one of my four “cool math things” to talk about in one of their pitches. Yet this is what has happened! What are the odds?!? (No, seriously.) I decided to just forge ahead as planned, present my pitch, and let the voters decide how much the déjà vu matters to them. Thank you for understanding, and enjoy!
I remember the moment that my antipathy towards calculators began. I had only been teaching for a year or two (we won’t talk about which century this was), and I was teaching a summer school class. When I saw a student—a sophomore in high school—reach for his calculator to determine the value of (6 \times 7), I may have involuntarily let out a small shriek.
In retrospect, that may not have been the most empathetic response. But ever since then I have had to work hard to overcome my fuddy-duddy-Luddy bitterness over what the ubiquity of calculators has done to the learning—and, yes, the enjoyment—of mathematics.
But in today’s excursion, I admit an exception to that bitterness, and present to you a phenomenon that quite possibly would have never occurred to the Eulers or Archimedeses of history, due to the severe lack of digital calculators at the time.
Let’s Begin!Please take a moment to try to locate for yourself a scientific calculator. It must satisfy the following requirements:
- It must have buttons for trigonometric functions (“sin,” “cos,” and “tan”).
- It must be in “Degree Mode,” meaning that it will interpret any argument fed to a trig function as being an angle measured in degrees (rather than radians). If entering
sin(90)gives you1, rather than0.893997, then you’re all set! - It must automatically write sufficiently large or small numerical results using scientific notation (such as
6.022x10²³or6.022e23).
(If you don’t have such a calculator handy, don’t worry. I got you.)
Now I would like you to evaluate, in turn, each of the following expressions, noting the outputs:
Take your time. I’ll be right here.
For those without calculators, the animation below is a fairly good simulation of what you might have experienced:
Ahem Allow me:
WHAT.
ON.
EARTH…
… is (\pi) doing here?!? WHY?!? WHAT?!? HOW?!? What the heck do numbers that are just strings of 5’s have to do with (\pi)?!?
Before we proceed, let’s just take a moment. This shock, this bafflement, this magic, this wonder, this mystery—this is what keeps mathematicians doing mathematics. Yes, we’ll eventually look into it a little more closely, and get some of our “Why” questions answered, but this moment, this not-knowing, this state of being mystified, this reminder that the universe still holds so much magic in it, is worth savoring.
And no matter how many times you have this experience, mathematics will always have more in store.
Stating Our ClaimOkay, so what is going on here? Let’s start by trying to state what it is that we’re seeing:
As the number of 5’s in the denominator increases,
(\sin!\left(\frac{1}{555\ldots 5}^{\circ}\right)) approaches (\pi) times 10 to some negative power.
This is okay for a first pass, but we can do better. For example: Is there a relationship between the number of fives we use and the exponent on 10 in the result? An examination of our data seems to suggest that if we use (n) 5’s, then the exponent on 10 will be (-(n+2)).
Now… Can we write an expression, in terms of (n), for a number written as a string of (n) 5’s? We can, by making use of the fact that the number (\underbrace{999 \ldots 9}_{n\text{ 9’s}}) can be written as (10^n-1). This means that
[\underbrace{555 \ldots 5}_{n\text{ 5’s}} = \frac{5}{9}(10^n-1),]
and its reciprocal, (\frac{1}{555 \ldots 5}), can be written as (\frac{9}{5(10^n-1)}.)
So now we can rewrite our claim more precisely:
As (n \rightarrow \infty), (\sin!\left(\frac{9}{5(10^n-1)}^{\circ}\right) \rightarrow \pi \times 10^{-(n+2)}).
Next comes the piece that really makes the trick “work.”
Radians vs. DegreesOne fact about the sine function is that when you give it an input value close to zero, the output is very close to the input. Stated more precisely:
As (x \rightarrow 0), (\sin(x) \rightarrow x).
However, this statement is only true if (x) is measured in radians.
In each of the graphs below, the graphs of (y=x) and (y=\sin x) are plotted.
Note that when (x) is in radians, the two graphs almost coincide near (x=0), but when (x) is in degrees, the two graphs are very different.
It’s clear that the successive inputs that we fed our calculator were approaching zero. But we need to look at what those inputs of (\frac{1}{555…5}) degrees look like when they are converted to radians. We can do this using the conversion factor (\frac{\pi\text{ radians}}{180^{\circ}}):
[\left(\frac{9}{5(10^n-1)}\right)^{\circ}=\left(\frac{9}{5(10^n-1)}\right)^{\circ}\times\frac{\pi\text{ radians}}{180^{\circ}}=\frac{\pi}{100(10^n-1)}\text{ radians}]
Aha! (… This is the part where you yell “AHA!” really loudly.) Suddenly all of our fives and nines have disappeared, and it’s powers of 10 as far as the eye can see.
As (n) approaches infinity, we can see that this latter expression approaches (\frac{\pi}{100(10^n)}), or (\pi \times 10^{-(n+2)}.) And with this, we have proven our claim that
As (n \rightarrow \infty), (\sin!\left(\frac{1}{555…5}^{\circ}\right) \rightarrow \pi \times 10^{-(n+2)}).
So, looking back… What made this “trick” so surprising? It really just seems to come down to the human-introduced division of a circle into 360 pieces. So one thing this might lead one to wonder is whether similar tricks would exist if we had decided to use a unit of angle measurement that divided a circle into a different number of parts. For example, a less-frequently-used unit of angle measurement (in mathematics, anyway) is the gradian. There are 400 gradians in a full circle. Could you create a similar trick with a calculator that’s in “gradian mode?” If not, what subdivisions of circles would easily lend themselves to such a trick?
As always, there are plenty of other directions our wondering brains might go, and I hope you chase those rabbits to your hearts’ content and delight.
Matt Enlow teaches mathematics at the Dana Hall School in Wellesley, MA. You can follow him on X, BlueSky and Mathstodon.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the final!
Come back tomorrow for the other semi-final match, pitting Angela Tabiri against Ayliean, or check out the announcement post for your follow-along wall chart!
Here’s the last quarter-final match of The Big Internet Math-Off. Today, we’re pitting Ayliean against Dave Richeson.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Ayliean – Geometry and Oat MilkOne of the things I love about learning Maths is the extra layer of appreciation it adds to everything in the world around me.
Sometimes it will be the leaves of a plant, the movement of a crowd, the tiles on a pavement… this week, it was a weird shaped oat milk sachet.
Here are some links of things for you to cut out and make yourself
A single printable page containing the net of a Flexahedron, and just enough space in the margin for a Tri-Hexa-Flexagon!
The stunning Möbius Flexahedron from OIST (it even comes in rainbow colour!)
The printable Truchet Tiles Zine, I wanna see these hiding in cafes for people to find and play with – get them printed, get them out there!
I hope you have as much fun cutting and folding these as I did – and please show me anything you make!
Stay nerdy,
Ayliean :)
Ayliean (noun): Mathsy, arty, crochet crafty, origami, activist, zine author known for making badges and trouble. You can follow them on YouTube, as @Ayliean on all social media, or look at their homepage.
Dave Richeson – A Doubly Surprising Calculator TrickWho doesn’t love a calculator trick? It could be a simple one, like typing 0.7734, 376006, 57738461375, or some other carefully chosen number (yes, pre-teen boys, I know 5318008 is your favorite) into a calculator with a seven-segment display and turning it upside down to read HELLO, GOOGLE, or SLEIGHBELLS. It could also be more mathematical, such as having your friend perform a seemingly random sequence of steps that produces a pre-determined number. (Here’s one I just made up: type in any five-digit number, type it a second time [getting 2718227182, for instance], divide this by 11, divide this by your original five-digit number, subtract 1, divide by 9, and you will always get 1010.)
Being a mathematician, I was doubtful I could be surprised by a newcalculator trick, but when someone showed me this one, my mouth droppedopen.
To perform this simple trick, you will need a calculator that can do trigonometry.
Calculator trick
- Set your calculator to degree mode.
- Type several 5’s, such as 555555.
- Press the (1/x) button.
- Press the (\sin) button.
What do you notice? Cool, right?
Starting with 555555 and following these steps, the calculator on myphone spits out the value 0.000000031415958. After some 0’s are digitsthat look suspiciously like—but not exactly—the first digits of (\pi) ((3.141592653589793\ldots))!
Expressed another way, the calculator is telling me that [10^8\cdot \sin(1/555555)=3.1415958\ldots,] which agrees with (\pi) for the first six digits. Typing inmore 5’s is even better: the first 13 digits of [10^{16}\cdot \sin(1/55555555555555)=3.141592653589825\ldots] are correct. Surely, this isn’t acoincidence. What’s going on?
The interested and mathematically inclined reader may want to stopreading and see if they can give a justification. (But if you figure itout, come back since there’s a second trick that builds on this firstone.)
As a hint, this trick relies on the following three facts.
Fact 1. (\frac{1}{180}=0.00\bar{5}).
Fact 2. If (\theta) is measured in radians and (\theta) is approximately 0, then (\sin\theta\approx \theta).
Fact 3. Degrees and radians are related via theequation [\theta_{\text{rad}}=\frac{\pi}{180}\cdot\theta_{\text{deg}}.]
Fact 2 is illustrated in the figure below. In the circled region, thegraphs of (y=x) and (y=\sin x) are almostindistinguishable.
Let’s see why the trick works. Suppose we begin with a (k)-digit number of all 5’s: (\overbrace{555\ldots 5}^k). By Fact 1, [\begin{aligned}
\frac{1}{180}&=0.00\bar{5}\ &\approx 0.00\overbrace{555\ldots 5}^k\ &=(555\ldots 5)\cdot 10^{-k-2}. \end{aligned}] Rearranging terms, this is equivalent to stating that [\frac{1}{555\ldots 5}\approx 180\cdot 10^{-k-2}.]
Fact 2 says that when (\theta) is given in radians and is approximately 0, (\sin(\theta)\approx \theta.) What if (\theta) is given in degrees? Let (\sin_\text{rad}) and (\sin_\text{deg}) be the radian-mode and the degree-mode sine functions, respectively. Then, using the degrees-to-radians conversion in Fact 3, [\sin_\text{deg}(\theta)=\sin_\text{rad}\left(\frac{\pi}{180}\theta\right)\approx \frac{\pi}{180}\cdot \theta.]
Putting this all together, we see that when our calculator is in degree mode, [\begin{aligned} \sin\left(\frac{1}{555\ldots 5}\right)&\approx\frac{\pi}{180}\cdot\frac{1}{555\ldots 5}\[0.5em]&\approx\frac{\pi} {180}(180\cdot 10^{-k-2})\[0.5em]&=\pi\cdot 10^{-k-2}.\end{aligned}]
So, (10^{k+2}\cdot\sin\left(\frac{1}{555\ldots 5}\right)\approx\pi.) That explains the trick!
[Performance note: This trick works for any number of 5’s in step 2.However, if the final value has many leading 0’s but not enough totrigger scientific notation, the “punchline” may be difficult torecognize. For instance, the input 5555555555555 yields0.000000000000003. For an iPhone calculator, I’d avoid using 10 to 13fives. If your friend does type that many, casually encourage them totype in a few more.]
The trick has one more surprise up its sleeve. We know that the digits following the string of 0’s is approximately (\pi.) How far off is it? What is [10^{k+2}\cdot\sin\left(\frac{1}{555\ldots 5}\right)-\pi?]
Here’s how to compute the error on your calculator.
Calculator trick (revisited)
Calculate (\sin\left(\frac{1}{555\ldots 5}\right)) as in steps 1-4.
- Press the (\times) button.
- Type in “10”.
- Press the (x^y) button.
- If you had typed in (k) 5’s in step (2), type in the number (k+2).
- Press “(-)”.
- Press “(\pi)”.
- Press “=”.
When I computed the error for my first example (with (k=6)), I got (0.000003141595795.) The number (\pi) appears again!
Wait. What? Why?
To understand why the error is related to (\pi), we must go back to the explanation of the first trick. We saw that [\frac{1}{555\ldots 5}\approx 180\cdot 10^{-k-2},] but in our analysis of the error, we need to be more exact and obtain an equality at this stage, not an approximation.
So we don’t have to wade through too much notation, let’s see the calculations for our six-digit number, 555555: [\begin{aligned} \frac{1}{180}&=0.00\bar{5}\ &=0.00555555\bar{5}\ &=0.00555555+0.00000000\bar{5}\ &=555555\cdot10^{-8}+10^{-6}\frac{1}{180}. \end{aligned}] Moving these quantities around, we can rewrite that equality as [\frac{1}{555555}=180\cdot
\frac{10^{-8}}{1-10^{-6}}.]
The exact same method applies if our initial number had (k) 5’s. We’d obtain the equality [\frac{1}{555\ldots 5}=180\cdot\frac{10^{-k-2}}{1-10^{-k}}.]
Because the calculator is in degree mode,
[\begin{aligned} \sin\left(\frac{1}{555\ldots 5}\right)&\approx\frac{\pi}{180}\cdot \frac{1}{555\ldots 5}\ &=\frac{\pi}{180}\left(180\cdot \frac{10^{-k-2}}{1-10^{-k}}\right)\ &=\frac{\pi\cdot 10^{-k-2}}{1-10^{-k}}.
\end{aligned}] This implies that [10^{k+2}\cdot \sin\left(\frac{1}{555\ldots 5}\right)\approx\frac{\pi}{1-10^{-k}}\approx \pi,] as we concluded in the first trick. Here we are using the fact that when (k) is large (or even not too large) (10^{-k}\approx 0,) so (\frac{1}{1-10^{-k}}\approx 1).
But now, we can examine the error in this approximation: [\begin{aligned} 10^{k+2}\sin\left(\frac{1}{555\ldots 5}\right)-\pi&\approx\frac{\pi}{1-10^{-k}}-\pi\ &=\frac{\pi 10^k}{10^{k}-1}- \frac{\pi(10^{k}-1)}{10^{k}-1}\ &=\pi\cdot \frac{1}{10^k-1}\ &\approx \pi\cdot 10^{-k}, \end{aligned}] as claimed!
Acknowledgments: I don’t know who discovered this trick. If you know, leave it in the comments! But the details of how I learned about it are found in my blog and in the comments (from almost 15 years ago!).
Dave Richeson is a professor of mathematics and the John J. & Ann Curley Faculty Chair in the Liberal Arts at Dickinson College in Carlisle, Pennsylvania, USA, and is the author of Euler’s Gem (Princeton University Press, 2008) and Tales of Impossibility (Princeton University Press, 2019). You can follow him on Mathstodon and X, or look at his homepage.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the semi-final.
We’ve got a little break before the next round of the competition. Come back on the 17th for the first semi-final match, pitting Matt Enlow against Fran Herr, or check out the announcement post for your follow-along wall chart!
Here’s the third quarter-final match of The Big Internet Math-Off. Today, we’re pitting Fran Watson against Fran Herr. It’s an extravafranza!
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Fran Watson – Learn about rotationFran invites you to hum along, join in with the actions or just listen and enjoy!
Fran Watson is a teacher and communicator of mathematics originally hailing from Cornwall but now living in Cambridgeshire (by way of Cardiff in between – locations today brought to you by the letter C!) She loves puzzles, origami, games and musical theatre and will endeavour to weave these passions into her pitches.
Fran Herr – All knotted upTake a long paper strip, twist one end (180^{\circ}), and attach the short endswith some tape. This produces a Möbius loop.
If have spent some time in the math communication world, you mightknow the result of cutting this shape along the center line. But, ifthis is new for you, take a moment and try the experiment for yourself!(Or just watch the video below.)
Why do we get only one loop? My favorite explanation is to view theresulting longer loop as a thickened version of the boundary of theMöbius loop. This experiment tells us that the boundary of the Möbiusloop is a single unknotted circle.
Consider repeating this process with more twists. For a Möbius loopwith (k) half-twists, what is theresult of cutting it in half? If you want to know the answer in fullgenerality, I made a video on thisexact topic a few years ago. I will demonstrate one more example here: athrice-twisted Möbius loop.
This little activity tells us that the boundary of the thrice-twistedMöbius loop is the trefoil knot. Have I surprised you yet?
A mathematical knot is an continuous embedding of the circle into (\mathbb R^3). Think of a piece of string with the ends attached. Two knots are the same (ambiently isotopic) if you can “wiggle them around” in (\mathbb R^3) to look the same. We can also have links which are embeddings of a finite disjoint collection of circles into (\mathbb R^3). I will often abuse terminology and call them all “knots”. Some of my favorite examples are below.
Knots are a bit squirrely in character. For example, mathematiciansdon’t yet know an algorithm to tell if two knots are the same! One wayto study knots is to instead study a Seifert surface of a knot.This is an (orientable)surface with the given knot (or link) as the boundary. Although the“orientable” condition does have some uses, it doesn’t matter for ourpurposes so we will consider both orientable and non-orientable Seifertsurfaces. Our investigation above reveals that the thrice-twisted Möbiusloop is a (non-orientable) Seifert surface for the trefoil knot.
Given a knot, there is a simple algorithm to generate a correspondingSeifert surface (two surfaces, in fact). First draw a planar knotdiagram which sections the plane into different faces. We can colorthese faces red and blue in an alternating way so that no two facessharing a bounding edge are the same color.
Place a vertex in each red face and connect each pair of verticesonce for each crossing between the corresponding faces. Mark these edges(L) or (R) for a left-hand or right-hand twist.Repeat with the blue faces. Then we have generated two planar graphswhich are dual toeach other by construction. Each of these graphs encode a Seifertsurface. Place a disk at each vertex of the graph and a twisted bandwith a right or left twist for each edge. We are left with a surfacethat has the given knot as a boundary.
Examples of left and right hand crossings for the knotdiagram. They correspond with twisting a band is the indicateddirection.Consider the example above for the trefoil knot. The red graphrepresents the thrice-twisted Möbius loop; the blue graph provides analternative Seifert surface. The trefoil is an alternatingknot, meaning that all its crossings have the same orientation.Thus, we have omitted the left/right labeling for this example.
Last year I became acquainted with Shiying Dong’s fantastic work intopological crochet. Shiying constructs Seifert surfaces with naturalcrochet principles by performing a sort of inverse of the processdescribed above. She starts with a graph— some vertices and edges— andimagines each edge as a “ribbon” with thickness. Then she applies aright-hand twist to each edge and reattaches it. If she starts with theblue graph in the figure above (consisting of two vertices with threeedges between them), she obtains the orientable Seifert surface for thetrefoil.
A crocheted orientable Seifert surface of thetrefoil.This is the first crochet project that I learned from Shiying and the first that she teaches in her workshops. You can learn to make one for yourself with my tutorial video or the videos on Shiying’s channel! But without yarn and a crochet hook on hand, we can perform the same process on a graph using pen and paper.
The graph algorithm performed to obtain the orientabletrefoil Seifert surface.For a general planar graph, the steps are as follows.
- Draw a planar graph (G). It can have double edges and loops, but no edges should cross. The cyclic order of the edges at each vertex is fixed by the embedding into the plane.
- We will construct a new graph (H). Put a vertex of (H) on each edge of (G) in a new color.
- A given edge of (G) has four “neighboring edges”. If the endpoints of edge (e) are vertices (v) and (w), then the neighboring edges of (e) are on its right and left sides in the cyclic order at each vertex (v) and (w). Some edges may be double counted.
- Add an edge in (H) between two vertices if the corresponding edges in (G) are neighbors. Each vertex in (H) should have degree 4.
- At each vertex of (H), draw a right-handed crossing.
- You are left with a knot or a link!
The graph algorithm performed on a randomgraph.In this process, we have made a universal choice for the handednessof each crossing. This means that all knots and links generated in thisway will be alternating. A modification would be to choose thehandedness of each crossing individually.
I have really enjoyed trying this on my favorite graphs. Sometimesthe resulting knot is surprising and reveals an unexpected connection.In general, it seems difficult to determine which knot or link weobtain—or even the number of link components.
What are the next graphs that you would try? Here, I’ll choose theedge graphs of the Platonic solids. Let’s start with the tetrahedron.Draw four vertices and connect every pair of vertices with an edge. Whatlink do we get when we apply with the process above?
A crocheted version of the Seifert surface resulting fromthe tetrahedron.Try for yourself and you will see that we meet another old friend:the Borromean rings. If we continue with the other platonic solids, weobtain two more woven links, one for the cube/octahedron, and one forthe icosahedron/dodecahedron. (Recall that dual graphs will give thesame link.) These links can be nicely woven by pipe cleaners; thanks tomy friend Elliot Kienzle for pointing that out!
Some other great examples to try are the wheel graphs andthe edge graphs of prisms.What corresponding graphs and knots can you find? Whether you engagewith this question by crocheting surfaces or by doodling graphs, thereare many surprises and delights to be found.
The Seifert surface generated using the edge graph of thecube (left) and the surface from the edge graph of the tetrahedron(right).Fran Herr is a PhD student in mathematics at the University of Chicago. She studies low dimensional topology and geometric group theory. You can follow her on YouTube and X.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the semi-final.
Come back tomorrow for the last quarter-final match, pitting Ayliean against Dave Richeson, or check out the announcement post for your follow-along wall chart!
Here’s the second quarter-final match of The Big Internet Math-Off. Today, we’re pitting Angela Tabiri against Howie Hua.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Angela Tabiri – A Quantum WorldAnytime we add two numbers, say 2 and 3, the order in which we add the numbers does not matter. That is 2+3 = 3+2. However, when we consider subtraction and compute 3-2 and 2-3, the results are unequal. The order in which we subtract numbers matters. This property is called the non-commutative property. This video explains the non-commutative property in the English language. Words with the same alphabets but different meanings are called anagrams. For instance ‘ear’ and ‘are’ illustrate an anagram. Thus, the order in which we arrange alphabets matter since a different arrangement might have a different meaning.
Prime numbers play a critical role in facilitating a safe way to send and receive messages. The safety in the transfer of messages is guaranteed by the hardness of the problem of decomposing a large number into the prime numbers that multiply to it. Classical computers are unable to do such complex computations in a short time. However, with the latest advancements in quantum computers, such computations will be done with no complexity in the computation. The year 2025 has been declared by the UN General Assembly as the International Year of Quantum Science and Technology. Anytime you think about the word quantum, remember it means the non commutative property. This video gave examples of quantum things we see in our world.
Can you think about a quantum property in everyday life? Comment with your examples.
Angela Tabiri is a mathematician and youth mentoring in STEM expert from Ghana. She is the founder of Femafricmaths, a non profit organisation that promotes female African mathematicians to highlight the diversity in careers after a degree in mathematics. You can follow Femafricmaths on YouTube, Instagram, Facebook and X.
Howie Hua – My favorite mental math trick and why it worksHowie shares his favorite mental math trick that will surely impress your friends:
Howie Hua teaches math to future elementary school teachers at Fresno State. He also likes to make math explainer videos and math memes. You can find all his socials on his linktree.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the semi-final.
Come back tomorrow for the third quarter-final match, an all-Frans affair pitting Fran Watson against Fran Herr, or check out the announcement post for your follow-along wall chart!
Here’s the first quarter-final match of The Big Internet Math-Off. Today, we’re pitting Benjamin Dickman against Matt Enlow.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Benjamin Dickman – Thinking Outside the Magic BoxPrelude: If you’ve arrived at Round 2 of the Big Internet Math-Off 2024 less as a mathemagician and more as an equation solver, rest assured: We will find a solution to (157x + 225y = 1) over the integers by the end of this post. (And hopefully encounter deeper and more fascinating mathematics along the way!)
In the previous round, I wrote about what I learned in the previous summer at PCMI. In the current round, I am writing about what I’m learning in the current summer at PROMYS for Teachers. This involves continued fractions, which I will call cons, and something called the Magic Box for an algorithm described below.
Cons pop up in all sorts of con-texts; often, one is seeking (increasingly accurate) rational estimates for irrational numbers. Here, though, we start off in (\mathbb{Q}) (but see the Notes for less rational matters).
Con-sider, for reasons not yet clear, the con ( 1 + 1/(2 + 1/(3 + 1/(4 + 1/5)))) which can be written:
[ \huge{1 + \frac{1}{2 + \frac{1}{3 + \frac{1}{4 + \frac{1}{5}}}}} ]
Pros with cons would express this as [1; 2, 3, 4, 5].
WolframAlphacomputes that this con is equivalent to 225/157. Let’s use themagic box to compute the same!
| 🪄 | ✨ | 1 | 2 | 3 | 4 | 5 | | --- | --- | --- | --- | --- | --- | --- | | 0 | 1 | | 1 | 0 |
Figure 1. Does drawing the Magic Box make you a con artist?The setup has three rows: the top one has a bit ofmagic followed by the numbers that appear in the con; the middle rowbegins with 0, 1, then blanks underneath each con number; the bottom rowbegins with 1, 0, then blanks underneath blanks.
Let’s carry out the algorithm row by row (you might skip ahead to Figure 2 for a con-crete example worked out).
For the middle row, multiply its column header by the entry one earlier in the row, and add this product to the entry two earlier in the row.
This begins with: (1 \times 1 + 0 = \color{#cf2e2e}{1}), then (2 \times 1 + 1 = \color{#ff6900}{3}), then ( 3 \times 3 + 1 = \color{#00d084}{10} ).
For the bottom row, do the same [I’ll copy and paste]: multiply itscolumn header by the entry one earlier in the row, and add this productto the entry two earlier in the row.
This begins with: (1 \times 0 + 1 = \color{#cf2e2e}{1}), then (2 \times 1 + 0 = \color{#ff6900}{2}), then (3 \times 2 + 1 = \color{#00d084}{7} ).
The result (e.g., for step three: multiply the underlined numbers then add the italic number for 10):
| 🪄 | ✨ | 1 | 2 | 3 | 4 | 5 | | --- | --- | --- | --- | --- | --- | --- | | 0 | 1 | 1 | 3 | 10 | 43 | 225 | | 1 | 0 | 1 | 2 | 7 | 30 | 157 |
Figure 2. The Magic Box with a con-crete example worked out.Success! The final column, in purple, reveals the con (if not the trick): 225/157. But what did we calculate along the way?
The rainbow colored columns indicate the convergents that provide estimates for 225/157, and which can be found by computing the initial segments of our con:
[ \begin{align}
1 &= {\color{#cf2e2e} \frac{1}{1}} \[1em]
1 + \frac{1}{2} &= {\color{#ff6900} \frac{3}{2}} \[1em]
1 + \frac{1}{2 + \frac{1}{3}} &= {\color{#00d084} \frac{10}{7}} \[1em]
1 + \frac{1}{2 + \frac{1}{3 + \frac{1}{4}}} &= {\color{#0693e3} \frac{43}{30}}
\end{align} ]
Our fourth convergent is ( 43/30 = 1.4\bar{3}\ldots ) which is impressively close to (225/157 = 1.433121\ldots)
But wait – there’s more! Look at the 2×2 tables formed as we moved along the middle and bottom rows, and take the difference of their cross products. In other words, consider (ad-bc) for each sub-table of the form:
| a | b | | c | d |
Figure 3. That meeting could have been an email and this table could have been an emoji: 🔡What do we get as we move across the 2×2 tables in Figure 4’s Magic Box?
[\begin{align}
-1 &= 0 \times 0 – 1 \times 1 \[0.5em]
1 &= 1 \times {\color{#cf2e2e} 1} – {\color{#cf2e2e} 1} \times 0 \[0.5em]
-1 &= {\color{#cf2e2e} 1} \times {\color{#ff6900} 2} – {\color{#ff6900} 3} \times 1 \[0.5em]
1 &= {\color{#ff6900} 3} \times {\color{#00d084} 7} – {\color{#00d084} 10} \times {\color{#ff6900} 2} \[0.5em]
-1 &= {\color{#00d084} 10} \times {\color{#0693e3} 30} – {\color{#0d93e3} 43} \times {\color{#00d084} 7} \[0.5em]
1 &= {\color{#0693e3} 43} \times {\color{#9b51e0} 157} – {\color{#9b51e0} 225} \times {\color{#0693e3} 30}
\end{align} ]
Observe that this last line provides an integral solution to (157x + 225y = 1), as promised in the prelude! In particular, we can take ((x, y) = (43, -30)).
A real mathemagician never reveals their tricks, but in the spirit ofthis being a con-test, I include exploratory problems asNotes for the reader looking to understand a bitmore.
Notes0. I ended my previous entry with a tombstone, and I started this onewith a title-acronym of TOMB.
- You may have seen (otherwise, you will see) that the golden ratio– which, incidentally, is Mr. Enlow’s topic for today! – is representedas the infinite con: [1; 1, 1, 1, …]
If you fill out the Magic Box below, what do you notice? What do you wonder?
| 🪄 | ✨ | 1 | 1 | 1 | 1 | 1 | | --- | --- | --- | --- | --- | --- | --- | | 0 | 1 | 1 | 2 | 3 | | 1 | 0 | 1 |
- Suppose our con is [1; 2, 2, 2, …], i.e., an initial 1 and then infinitely many 2s. Can you fill out the Magic Box below and use its first few convergents to guess the value of this con?
| 🪄 | ✨ | 1 | 2 | 2 | 2 | 2 | | --- | --- | --- | --- | --- | --- | --- | | 0 | 1 | 1 | 3 | 7 | | 1 | 0 | 1 |
-
At some point in this writeup, there was a con-tention that thecross product differences are always ±1. In particular, they alternatebetween -1 and 1. Is this true? (Why or why not?)
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(This lengthier note can be skipped without loss of con-tinuity.)If you’re wondering about how to find the con for a given rational number, then read on; otherwise, skip to note 5.
If you’ve seen the Euclidean Algorithm (aka Euclid’s Algorithm) then you already know how to find the gcd (greatest common divisor) of two integers. For example, ( \gcd(157, 225) ) can be found with repeated division:
[ \begin{align}
225 &= {\color{#cf2e2e} 1} \times 157 + 68 \
157 &= {\color{#ff6900} 2} \times 68 + 21 \
68 &= {\color{#00d084} 3} \times 21 + 5 \
21 &= {\color{#0d93e3} 4} \times 5 + 1 \
5 &= {\color{#9b51e0} 5} \times 1 + 0
\end{align} ]
The last line’s 1 is the greatest common divisor, i.e., (\gcd(157, 225) = 1).
Notice that the rainbow colored quotients are precisely the ones found in our continued fraction of [1; 2, 3, 4, 5]. (If this is new to you, consider it further food for thought!) Euclid’s Algorithm can be unwound through a rather involved process to find integer solutions for, in this case, the equation (157x + 225y = 1). With our Magic Box method above, though, we used *[1; 2, 3, 4, 5]* to find such an ((x, y)) solution rather more efficiently!
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We saw the con for ( \frac{225}{157} ). How might you find the con for a number like (\sqrt{7})?
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If you extend the con explored above as [1; 2, 3, 4, 5, 6, 7, 8, …] then you get into some very deep mathematics. In particular, you end up with (I_0(2)/I_1(2) ), where (I_n(z)) is the modified Bessel function of the first kind. WolframAlpha calculates this expression as ≈ 1.43313 using a ratio of summed unit fractions whose denominators are factorial products. Indeed, this is con-vincingly close to our computed convergent of ( \frac{225}{157} = 1.433121 \ldots )
| Input interpretation | Result | | --- | --- | | [ \frac{ \sum_{n=0}^\infty \frac{1}{(n!)^2} }{\sum_{n=0}^\infty \frac{1}{n!(n+1)!}} ] | [ \frac{I_0(2)}{I_1(2)} \approx 1.43313 ] |
Con-gratulations for reading this far and please vote to your heart’scon-tent!
Matt Enlow – Golden Squares & A CircleFilmmaker David Lynch began his creative career as a painter, but he has said that he got into making movies because he found himself wanting his paintings to move. I sometimes find myself in a similar position. For example, consider the following image:
For those unfamiliar, this is indeed a golden rectangle—meaning that it has just the right proprtions so that when it is dissected into a square and a smaller rectangle, the smaller rectangle has the same proportions as the original. This means that a golden rectangle can be “dissected” into the infinite spiral of squares you see above.
It can be calculated that the ratio of a golden rectangle’s length to its width is (\frac{1+\sqrt{5}}{2} \approx 1.618). This number is referred to as the golden ratio, and it is often represented by the Greek letter (\varphi ) (phi).
When I see a static image like the one above, I find myself wanting it to move.
Like… What if each square touched the next smaller square only at a corner, but the squares were free to move around otherwise? What if we could… unfurl that spiral of squares?
I have been an avid Mathematica user for decades now, and I frequently use it to help me answer questions like these by creating animations. Here is an unfurling of the squares:
As satisfying as this is, it does only lead me to more wondering. What would it look like if we went past that fully-extended “straight line” of squares?
In every frame of this animation, the sequence of squares shrinks down to a particular limit point:
And now I’m wondering… Watch that limit point as it moves. What path does it appear to trace?
That sure appears to be a circle! But is it? And if it is, where is its center located? What is its radius?
This is where complex numbers come in handy.
The Complex PlaneLet’s think of these squares as living in the complex plane, with the largest one having its lower-left corner at the origin (0), and its upper-right corner at (1+i). Each succesive, smaller square is rotated counter-clockwise by an angle of (\theta ), and (\theta ) is what varies (from (0) to (2\pi )) over the course of the animations.
Let’s call the limit point we’re looking for (z). We need to find an expression for (z) in terms of (\theta ), and then see how (z) changes as (\theta ) completes a full rotation (from (0) to (2\pi )).
One way to find this expression for (z) is to notice the self-similarity of the diagram above. Through a sequence of linear transformations, this diagram can be copied onto itself, leaving only the point (z) fixed.
So the (z) that we are seeking is the complex number that is unchanged by this sequence of transformations. In other words, we want to solve this equation for (z): [ \left(\frac{1}{\varphi }\; z\right)e^{\theta i}+(1+i)=z ]
When we do, we get [ z=\frac{1+i}{1-\frac{1}{\varphi }e^{\theta i}}. ]
Now, this doesn’t exactly look like the equation of a circle. But we can show that it is, using the following logical steps (which, in the interest of not making this pitch even longer, I invite you to verify for yourself):
- The expression (e^{\theta i}), as a function of (\theta ), represents the unit circle in the complex plane.
- The formula we derived for (z) can be thought of as a Möbius transformation of (e^{\theta i}).
- In the complex plane, Möbius transformations map circles to circles.
- Therefore, our formula for (z), as a function of (\theta), represents a circle in the complex plane.
So we’ve now established that this path is indeed a circle. But we still haven’t found its center or radius.
Evaluating (z) when (\theta = 0) and when (\theta = \pi ) will give us the endpoints of a diameter of the circle. When (\theta = 0), (z=\frac{1+i}{1-\frac{1}{\varphi }}). Using some very helpful properties of (\varphi ), we can show that this is equal to ((\varphi +1)(1+i)). Similar logic can show that when (\theta = \pi ), (z=(\varphi -1)(1+i)).
The center will be located at the average of these two endpoints, which is (\varphi (1+i)=\varphi + \varphi \, i), or the coordinates ((\varphi , \varphi )) in the coordinate plane. Which just so happens to be where the second and third squares connect when (\theta =0):
The radius of the circle would simply be half of the distance between the two diameter endpoints we found earlier: [ \frac{1}{2}\left|(\varphi +1)(1+i) – (\varphi -1)(1+i)\right| = \frac{1}{2}\left|1+i\right| \left|2\right| = \sqrt{2} ]
So the circular path traced by the limit point has its center at ((\varphi ,\varphi )), and a radius of (\sqrt{2}).
I hope this excursion has caused even more questions to come up for you! In case you could use some prodding, here are a couple more images that might set off some ideas and wonderings…
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the semi-final.
Come back tomorrow for the second quarter-final match, pitting Angela Tabiri against Howie Hua, or check out the announcement post for your follow-along wall chart!
Here’s the eighth and final match in Round 1 of The Big Internet Math-Off. Today, we’re pitting Dave Richeson against Kit Yates.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Dave Richeson – A surprising application of graph theory to structural designIt is probably no surprise that an architect or structural engineer would need to know mathematics—geometry, trigonometry, and differential equations come to mind immediately, but graph theory? Not your first thought!
In preparation for this post, I designed and 3D-printed the objects shown below.
The gray rods slide over the purple pins to form a square grid of any desired size. The photo below shows them assembled into a grid of five squares by four squares. Even though the rods cannot stretch, compress, or bend, they can swivel on the pin joints at every corner, so the structure is not rigid. A light touch distorts the configuration, turning the squares into other rhombuses.
This is where the white beams come in. They are exactly the length of the diagonals of the squares. Placing one of these bracing bars across a square in either direction keeps it from distorting. We could put one crossbeam in every square to make the structure rigid. But can we get by with fewer? If so, where should we place the crossbeams? What is the fewest number needed to stabilize the structure?
Below, we see two 5 × 4 structures with 10 crossbeams each. Is either one rigid? If it is rigid, does it need all 10 beams?
As we see below, the first structure is not a rigid bracing. Although some rhombuses remain square, others are not sufficiently braced to keep their original shape. However, the second bracing is rigid, and in fact, it is overbraced. We can remove two beams and still have a rigid bracing, also shown below.
As it turns out, there is a straightforward way to determine if a bracing is rigid, but it requires some basic ideas from graph theory.
The first thing to observe is that in any braced structure—rigid or deformed—all the vertical bars in a row are parallel, and all the horizontal bars in a column are parallel. We can see that in the deformed structure above.
A crossbeam in a rhombus forces it to be square. If the square is in the ith row and jth column, then all the vertical bars in the ith row are perpendicular to all the horizontal bars in the jth column. The photo below shows how one white crossbeam forces the yellow bars to be perpendicular to the red bars.
To achieve a rigid bracing, we need the vertical bars in every row to be perpendicular to the horizontal bars in every column. Here is where the graph theory comes in.
First, we construct what is called a bipartite graph from any bracing structure. Create two collections of vertices, one representing the rows of squares and one representing the columns of squares. Draw an edge from the vertex for the ith row to the vertex for the jth column if there is a crossbeam in the ith row and jth column. The graphs for our two structures are shown below.
Because a bracing bar makes the horizontal bars in that column perpendicular to the vertical bars in that row, it follows that if there is a path in the graph from the ith row to the jth column, then, following the sequence of edges, we conclude that the vertical bars in the ith row are perpendicular to the horizontal bars in the jth column. So, a structure is rigid provided we can find a path in the graph from any vertex to any other vertex! In the language of graph theory, we obtain the following remarkable conclusion.
A braced structure is rigid if and only if the associated bipartite graph is connected.
Moreover, a graph theorist can tell you that the smallest connected graph for a given set of vertices is a tree—a connected graph with no cycles. A tree that’s also a bipartite graph with m + n vertices has m + n − 1 edges.
Returning again to our two initial examples, there are five rows and four columns, so we should be able to brace the structure with 5 + 4 − 1 = 8 crossbeams. The structure that is not rigid has 10 crossbeams, but, as we see below, the bipartite graph is not connected. On the other hand, the bipartite graph associated with the rigid structure is connected. Since it also has 10 crossbeams, we can remove two crossbeams (the two dashed edges) and still have a rigid structure.
Now, let’s consider a different question. What if the crossbeams are replaced by wires? The wires are not rigid (they can collapse) but cannot stretch. Now, how do we cross-brace the structure so it is rigid? Below, we see two structures, each with 10 tension bracing wires. Are either of them rigid?
We leave the full analysis to the interested reader, but we will give the punchline. It was irrelevant whether we oriented a rigid bracing bar northeast-southwest or northwest-southeast inside a square, but the orientation makes a crucial difference for a tension cable.
Again, draw a bipartite graph, but this time, the edges are directed–that is, they have arrowheads indicating a direction. If the tension bracing runs northeast-southwest, draw the arrow from the row vertex to the column vertex. If the bracing runs northwest-southeast, draw an arrow from the column vertex to the row vertex.
We can show that a structure with tension wires is rigid provided we can get from any vertex to any vertex by following the arrows. In graph theory terminology, we say the following.
A structure with tension bracings is rigid if and only if the associated directed graph is strongly connected.
Moreover, if the structure has m rows and n columns, we need m + n + 1 cables.
Below are the graphs for our two structures. They both have the necessary 5 + 4 + 1 = 10 cables. However, the first graph is strongly connected, but the second one is not.
In the second graph, getting from columns 1 or 4 to rows 2 or 3 is impossible. Indeed, as we see below, the structure is not rigid.
To 3D-print your own rods and pins, download the STL files from Thingiverse. To read more about this topic, see, for instance, Sections 2.6 and 4.4 of Counting on Frameworks by Jack E. Graver.
Dave Richeson is a professor of mathematics and the John J. & Ann Curley Faculty Chair in the Liberal Arts at Dickinson College in Carlisle, Pennsylvania, USA, and is the author of Euler’s Gem (Princeton University Press, 2008) and Tales of Impossibility (Princeton University Press, 2019). You can follow him on Mathstodon and X, or look at his homepage.
Kit Yates – When will we ever need to use maths?“When will we ever need to use this?” It’s the question asked perennially in maths classrooms up and down the country and indeed around the world. But is it a fair one, and how should we answer it?
The pandemic has provided some very upfront answers. For the last few years we have been hearing regularly about the potential for cases to rise exponentially. News programmes carried regular features on the reproduction number, R. Others reported, in vain, that we might nearly have reached elusive mathematically defined herd immunity thresholds.
We relied on mathematical models, not only to understand the current situation, but to predict what might happen in the future, from the impact of mitigations to the effectiveness of vaccines. We used maths to determine the most efficient order to deliver jabs during the vaccine roll out and to plan the roadmap out of lockdown in early 2021. Maths was front and centre much of the time.
Trying to get a picture of the current situation during the acute phase of the pandemic was like putting together a jigsaw that you had to assemble using maths.Even outside times of crisis we see maths in the newspaper headlines every day. We use it to establish whether our politicians are telling the truth about unemployment. Maths allows us to monitor exchange rates during currency crashes. It is invaluable to opinion pollsters determining the popularity of our political parties and to fact checkers holding politicians to account.
Away from the front-page headlines, maths is the language of science. It appears everywhere from physics to engineering and chemistry – aiding us in understanding the origins of the universe and building bridges that won’t collapse in the wind. Perhaps a little more surprisingly, maths is also increasingly integral to biology. Scientists in my own specialist area of mathematical biology, are helping to develop treatments for diseases and to answer the question of how the leopard got its spots.
Mathematicians are trying to answer a range of different questions in biology, including “How the leopard got its spots”. Photo by Pexels User.Beyond the academy, we are increasingly employing maths in sport to enhance the performance of our top athletes and in the movies to create computer-generated images of scenes that couldn’t exist in reality. More mundanely we are frequently using maths in our everyday lives when we go shopping or when we are following a recipe, when we tell the time or when we budget for the future. Much of the time we do it without even realising it.
Certainly, much of the maths we learn early on in school we use directly in our everyday lives. Other topics that we might have learned later, or perhaps we never got round to, are essential for the functioning of modern society even if we don’t often see their use directly.
There are certainly bits of maths (particularly pure maths) for which is it harder to imagine a direct use. But isn’t this true of every subject? Should we hold geography, for example, to the same exacting standards of utility we expect of maths. I don’t remember the last time I put my hard-won knowledge of ox-bow lakes to use. Similarly, in chemistry, when was the last time you needed to write down the chemical reaction diagram depicting esterification? Probably not recently.
This is not to denigrate these subjects, but to point out that this is not a maths specific issue. Perhaps maths suffers more because it is harder to visualise the direct application of an algebraic equation than it is to picture the flow of water in a river, for example. We can all remember sitting by a river watching the water flow past, but fewer of us, I would suggest, can imagine laying down our picnic blanket on the complex plane of an Argand diagram.
By necessity maths tends to deal in generalities and therefore abstractions from reality. But, at least in part, it is the generality – the abstractness – which makes mathematics so pervasive.
At university I teach students that a single abstract equation can describe the spread of heat through your radiator, the diffusion of a drop of food colouring in a glass of water and the random dispersion of cells on a petri dish. With such a diverse range of applications you can start to see how powerful it is to study a seemingly abstract and lifeless equation for the deep insights it can provide about these ostensibly unrelated systems.
It was not for nothing that Philosopher Eugene Wigner wrote of “The unreasonable effectiveness of mathematics” for describing the natural world. Many simple mathematical ideas come up over and over again in different areas. The normal distribution – or bell curve – for example can be used to describe people’s IQs as well as their heights and, by a strange quirk of fate, the density of that diffusing drop of food colouring in the glass I described above amongst hundreds of other applications.
Maths can be “unreasonably effective” at describing natural phenomena like these fractal ice crystal patterns. Photo by Алена.The equation 2+2=4 describes what happens when you have two apples and your friend gives you two more. But this isn’t an equation about apples, it’s about bananas or oranges or grapes or absolutely anything, in fact, that comes in discrete packages. So general is the equation that it would be odd to narrow its applicability by forcing it into a specific real-world context.
It may be that the problem mathematics faces is that there are too many applications. Perhaps we should be responding to the question “When will we ever need to use this?” with the counter “When will you not?”
Kit Yates is an author, communicator and academic mathematical biologist who is interested in sharing stories about the places where maths can impact our lives without us even realising it. You can follow him on Mastodon and X, or look at his homepage.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the quarter-final.
We’ve got a day off before the next round of the competition. Come back on the 10th for the first quarter-final match, pitting Competitor 3 against Competitor 4, or check out the announcement post for your follow-along wall chart!
Here’s the seventh match in Round 1 of The Big Internet Math-Off. Today, we’re pitting Fran Herr against Tom Edgar.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Fran Herr – Modular Multiplication and Dancing PlanetsThis is a story about fantastical patterns emerging from a pair of integers, and it has more to do with geometry than number theory (sighs with relief). Start with a positive integer (m) and place (m) evenly spaced points around a circle, labeling them (0, 1, \dots, m -1). Then choose another integer (a) to be the multiplier. For each point labeled (p), draw a chord of the circle connecting (p) and (ap \bmod m). The resulting arrangement of chords is our main object of study; we will call it a modular multiplication table and denote it (\mathop{\mathrm{MMT}}(m,a)).
A picture of (\mathop{\mathrm{MMT}}(12, 2))You may know these pictures by another name: “string art,” “light caustics,” “spirographs,” or “curve stitching,” for example. They are popular across the math communication world. The following question has been partially answered, but today we dive deeper than you have likely been before!
Given the values for (m) and (a), what pattern will (\mathop{\mathrm{MMT}}(m,a)) create?
From the small selection of modular multiplication tables below, one can see that they are an eclectic bunch.
I encourage you to draw some of these tables for yourself as you read! There are many tools to do so online (made by people who know how to make webapps); one that I like is this one. You can also download my python code or write a quick script yourself.
We see our first pattern by fixing (a) at a “small” value and increasing (m); as we do this, a well-known curveemerges. For example, the picture below shows this process for (a = 2) and we see everyone’s favoriteplane curve: the LOVEly cardioid <3.
(a=2, m=25)(a=2, m=50)(a=2, m=100)This curve, to which all the chords are tangent, is called theenvelope of the chords. As we repeat this experiment with (a = 3, 4, 5,\dots) we notice apattern.
These curves are called epicycloids. The epicycloid with(a – 1) humps is made by rolling acircle with radius (\frac{1}{a+1})around a fixed circle with radius (\frac{a-1}{a+1}) and tracking a point onthe boundary of the outer circle. These curves are not the main focusfor today, but they are neat and you can learn more about them from thewikipedia page.As we experiment with small values of (a) (between (a=2) and (a =12) approximately), we might be led to believe that the envelopeof (\mathop{\mathrm{MMT}}(m,a)), forsufficiently large (m), is always theepicycloid with (a – 1) humps. But infact, this is a hasty conclusion.
When we look at higher multipliers, we see the large variety ofpatterns showcased earlier. We notice that the numerical relationshipbetween (m) and (a) plays a key role in determining thepattern. For example, tables of the form (\mathop{\mathrm{MMT}}(2a, a)) have acommon shape independent of the choice for (a). A similar phenomenon happens fortables of the form (\mathop{\mathrm{MMT}}(2a– 2, a)), and (\mathop{\mathrm{MMT}}(2a + 2, a)).
Of course, you should not take me at my word! I encourage you tocheck by drawing more tables from the three families above.Bonus: Can you explain why we get these threepatterns?
Another way to write these three families is to assume (m) is even and express them by
[\mathop{\mathrm{MMT}}\left(m,\tfrac{m}{2}\right), \hspace{20pt} \mathop{\mathrm{MMT}}\left(m,\tfrac{m}{2} + 1\right), \hspace{10pt} \text{and} \hspace{10pt} \mathop{\mathrm{MMT}}\left(m, \tfrac{m}{2} – 1\right).]
A natural generalization is to replace ‘2’ by an arbitrary ‘(b)’: assume (m) is divisible by positive integer (0 < b < m) and consider tables of the form
[\mathop{\mathrm{MMT}}\left(m, \tfrac{m}{b}\right), \hspace{20pt} \mathop{\mathrm{MMT}}\left(m,
\tfrac{m}{b} + 1\right), \hspace{10pt} \text{and} \hspace{10pt} \mathop{\mathrm{MMT}}\left(m, \tfrac{m}{b} – 1\right).]
The resulting designs for (b = 4) are shown below.
There are some beautiful patterns to understand here, and I encourage you to run some experiments yourself. But here we press forward. A few years ago, I became obsessed with modular multiplication tables and I went down this same line of exploration. The next question I asked was, “what if I dropped the assumption that (b) must divide (m)?” In this case we would still like to keep (a) as an integer, and so (for the first family) there are two natural choices for the multiplier:
[\mathop{\mathrm{MMT}}\left(m, \left\lceil \frac{m}{b} \right\rceil\right) \hspace{10pt} \text{and} \hspace{10pt} \mathop{\mathrm{MMT}}\left(m, \left\lfloor \frac{m}{b} \right\rfloor\right).]
Also, for each (b), there are (b-1) categories for (m) given by its equivalence class modulo (b). These families of tables display some amazing designs.
Figure 1 is a “table of tables”. Each table is of the form (\mathop{\mathrm{MMT}}(m, \lceil \frac{m}{b}
\rceil)) for some positive integers with (b < m). The rows are indexed by the value for (b) and the columns are indexed by (r) where (m \equiv r \mod b). We have taken (m) to be relatively large so as to see the pattern clearly. Most values for (m) are approximately (200).
I encourage you to pause and ponder this array. What patterns do younotice? Do the designs remind you of other curves we have looked at? Whydo you think these patterns might be emerging?
A “table of tables”! Each table is of the form (\mathop{MMT}(m, \lceil \frac{m}{b} \rceil)) for some positive integers with (b < m). The rows are indexed by the value for (b) and the columns are indexed by (r) where (m \equiv r \mod b).Let’s look at one particular example from this collection. The table(\mathop{\mathrm{MMT}}(100,34))satisfies (m \equiv 1 \mod 3) and(a = \lceil \frac{m}{3} \rceil). Lastyear, I decided to draw this on a Valentine’s day card (seemedappropriate). I started by connecting 1 to 34, 2 to 68, and so on… butI soon noticed a pattern in the way the chords developed.
As we plot the chords one at a time, we notice that we can form thesame pattern by moving the initial endpoint of the chord 3 spaces andthe terminal endpoint 2 spaces. This observation unlocks a newperspective on these modular multiplication tables.
Imagine two planets, named A and B, on the same circular orbit arounda central sun. Suppose an infinitely stretchy tether connects the twoplanets. At any one moment this forms a chord of their circular orbit.This setup is very similar to an animation by Matt Henderson, posted onTwitter,except in our case both planets are on the same orbit. We will call thissystem a planet dance.
Formally, a planet dance, denoted by (\mathcal{P}(\alpha, \beta)) is defined bytwo integers (\alpha) and (\beta). It consists of the set of directedchords of the unit circle with initial point (e^{2\pi i \alpha t}) and terminal point(e^{2\pi i \beta t}), for all (t \in [0, 1]).
We can realize a modular multiplication table (\mathop{\mathrm{MMT}}(m,a)) as a discretesampling of a planet dance. Set planet B orbiting (a) times faster than planet A, take apicture of the system at (m) regularintervals along planet A’s orbit, and overlay all these images. This iscalled an (m)-sampling of aplanet dance (\mathcal{P}(\alpha,\beta)) and it is defined as a finite subset of (\mathcal{P}(\alpha, \beta)); those chordsfor which (t = \frac{k}{m}) forinteger (k) with (0 \leq k \leq m-1). We denote thissampling by (\mathcal{S}(\alpha, \beta,m)).
Planet dances open up a whole new world; we can choose (\alpha) and (\beta) so that (\frac{\beta}{\alpha}) is not necessarilyan integer. I again encourage you to graph these and look for patterns.You can do so again with my code or you cansee the envelope of the curve by graphing the epicycloid in desmos.
Our observations above indicate that (\mathcal{S}(1, 34, 100)) and (\mathcal{S}(2, 3, 100)) actually producethe same set of chords. But why is that?
Now, finally, we can enter into topology land! A directed chord onthe circle is uniquely determined by the position of the two endpoints,planet A and planet B. Hence, the space of all such possible chords is acircle times a circle: (\mathbb S^1 \times\mathbb S^1 = T^2)… a torus! So each point on the toruscorresponds to a directed chord of the circle, and a planet dance is acontinuous set of directed chords forming a linear path on thetorus.
We model our torus as the unit square in (\mathbb R^2) with the opposite edgesidentified. We can think of this as the quotient space (\mathbb R^2 / \mathbb Z^2) where weidentify points ((x,y) \sim (x + 1, y) \sim(x, y + 1)). In this model, (\mathcal{P}(\alpha, \beta)) is given bythe line (\alpha y = \beta x) in(\mathbb R^2) under this quotientmap. This looks like a set of parallel diagonal line segments on theunit square.
An example of linear loops on the torus. The blue linesdepict (y = \frac{3}{2}x) whichcorresponds with (\mathcal{P}(2, 3))and the orange lines depicts (y = 9x)which corresponds to (\mathcal{P}(1,9)). The black dots indicate that (\mathcal{P}(1, 9)) is sampled at a rate of(25), corresponding with (\mathop{\mathrm{MMT}}(25,9)).When we graph our two favorite planet dances ((\mathcal{P}(1, 34)) and (\mathcal{P}(3, 2))), we see that theyintersect exactly 100 times on the torus, and that these intersectionpoints are at regular intervals because everything is linear. So when wesample either dance at a rate of 100, we get the same set of chords. Andnow, the mysteries begin to unravel.
(\mathcal{S}(1,34,100))next to the set of chords it represents.For a modular multiplication table (\mathop{\mathrm{MMT}}(m,a)), we can graphthe sampling points of (\mathcal{S}(1, a,m)) on the unit square, and this reveals which planet dance isreally the envelope of the curve: just connect the dots in theway that feel more “natural”. In our example above, we see that theblack sampling points are positioned to trace out the line (y = \frac{2}{3}x) more naturally than(y = 34x).
This phenomenon is a visual example of aliasing, awell-studied aspect of signal processing (hello appliedmathematicians!). The (\mathcal{P}(1,a)) planet dance is undersampled as an MMT, and it endsup looking like a different planet dance. There is a lot more to uncoverhere! If you would like to read more about these ideas, I’ve written afullpaper on it. But to end, I leave you with some more pictures ofMMT’s and the corresponding sampled linear loops.
A table of the form (\mathop{\mathrm{MMT}}(2a + 2, a)) alongwith (\mathcal{P}(a, 1)) sampled atrate (2a + 2).A table of the form (\mathop{\mathrm{MMT}}(3a, a)) along with(\mathcal{P}(a, 1)) sampled at rate(3a).A table of the form (\mathop{\mathrm{MMT}}(4a – 4, a)) alongwith (\mathcal{P}(1, a)) sampled atrate (4a – 4).Fran Herr is a PhD student in mathematics at the University of Chicago. She studies low dimensional topology and geometric group theory. You can follow her on YouTube and X.
Tom Edgar – A Rational Day at the Hilbert HotelI want to tell you about a curious conundrum I encountered during my time working at the Hilbert Hotel. Maybe you’ve heard about the standard problem from the mystical lodge before, but in case not, here’s a quick recap of the classical story. The Hilbert Hotel contains infinitely many rooms, one for each natural number (0,1,2,3,\ldots ). As legend goes, one night the hotel was full when a new traveler arrived and requested a room. The savvy hotel manager simply asked everyone currently staying in the hotel to move to the next room, thus opening room 0 for the new traveler. A similar strategy works if any finite number, say (n), of new travelers need rooms: simply repeat the process (n) times. Even more striking, if an infinite number of people (one for each natural number) need a room in the filled hotel, the hotel can accommodate them by simply sending the guest in room (m) to room (2m), thus opening all the odd rooms (see the video for an animation of the two different scenarios).
The classic Hilbert Hotel dilemma helps us learn to make sense of one-to-one correspondences between infinite sets. The last scenario succeeds because the set of natural numbers ({ 0,1,2,\ldots } ) is in one-to-one correspondence with the set of even natural numbers ({ 0,2,4,6,\ldots } ) via the map (m\mapsto 2m) and the set of odd natural numbers ({ 1,3,5,\ldots } ) via the map (m\mapsto 2m+1.) The existence of these two correspondences allows the infinitely full hotel to make room for the infinitely many new guests.
I expected working at the hotel to be an easy gig because I thought I knew all the necessary tricks. But the job tested me immediately. On the first day after the hotel had been closed for renovations, an infinite collection of people, one for each positive rational number, requested a room. The positive rational numbers are numbers that can be written in the form (p/q) where (p) and (q) are both positive natural numbers. I wasn’t worried because I knew there is a one-to-one correspondence between the natural numbers and the positive rationals. Figure 1 distills the short form of the proof.
Figure 1: There exists a one-to-one correspondence between the natural numbers and the positive rationals (or more precisely pairs of positive natural numbers) since we can traverse anti-diagonals in order and eventually count to every rational.
The idea is that you can write down pairs of positive integers in a two dimensional grid, and then you can line them up in an ordered list (corresponding to the natural numbers) by traversing the anti-diagonals starting at the top left. The resulting ordered list is [ 1/1,2/1,1/2, 3/1,2/2,1/3,4/1,3/2,2/3,1/4,\ldots ]
I used this ordered list to assign rooms to the rational guests and each headed to their room. Easy-peasy.
Later that day, the hotel owner, David, came by visibly frustrated. He commended me for accommodating the infinite collection of guests, but he expressed disappointment that I left so many rooms vacant. After all, the slogan for the hotel was “The Hilbert Hotel: Where every room is full but we always have space for you.” I had broken the cardinal rule. Each rational number has many representations; for instance the number 1 can be written as 1/1, 2/2, 3/3, etc, So by my list, the rooms I assigned for 2/2, 3/3, etc. remained empty since the guest 1 occupied only room 1. In fact, each rational number has infinitely may representations so I had left infinitely many rooms vacant!
I realized my mistake: the listing of rational numbers from the two-dimensional grid does not create a one-to-one correspondence. It only guarantees the existence of such a one-to-one correspondence! Instead the grid argument provides a one-to-one correspondence between the natural numbers and the set of ordered pairs of positive natural numbers. One way to get the correspondence between natural numbers and rationals is to skip the rationals you have already written in the list earlier. But I wanted to fix my mistake in a more prescriptive way, finding a formulaic way to write down the list of positive rationals so that each appears once and only once in the list.
I turned to an unexpected friend, one of my favorite mathematical objects: Pascal’s triangle (which, perhaps should be attributed to Omar Khayyam or even Halayudha). To construct this triangular array, begin and end each row with a 1, and, for each other entry, add the two entries in the previous row directly above the entry. I reduced the arithmetic triangle modulo 2, that is, I replaced odd numbers with (1) and even numbers with (0), and then summed along the shallow diagonals as shown in figure 2 (or see the video for an animation of the construction of the triangle, the reduction modulo 2 by shading entries blue if they are odd, and the summing of shaded entries along shallow diagonals).
Figure 2: Pascal’s triangle with odd entries shaded (left) and the triangle reduced modulo 2 along with sums of the entries on shallow diagonals (right) with the first few terms of Stern’s Diatomic sequence.
This process results in a famous integer sequence called Stern’s diatomic sequence: [ 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5,\ldots , ]
which can also be defined recursively by (a_{2n}=a_n) and (a_{2n+1}=a_n+a_{n+1}) along with the initial conditions (a_1=1) and (a_2=1.) Essentially, to find the next term in the sequence, if you are at an even index, copy down the term that is halfway to the desired term; and if you are at an odd index, add the two terms that are halfway to the term. (Notice the similarity with the Fibonacci sequence, which arises as the diagonal sums in Pascal’s triangle!)
Stern’s diatomic sequence has a truly amazing property that solved my resort riddle. By taking successive ratios of terms in the sequence, we get the following list of rational numbers: [ 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4/1,\ldots ]
The magic? Each positive rational number appears once and only once in this list.
While I don’t have time to discuss the proof of this fact, suffice it to say that I used this fascinating list to completely fill the Hilbert Hotel with the positive rationals. My solution pleased David, and I kept my job for a little while longer. But eventually I quit that job to investigate more properties of this intriguing sequence. If you find yourself wanting to know more, I recommend “On Stern’s Diatomic Sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4,…” by Sam Northshield or “Recounting the Rationals” by Neil Calkin and Herbert Wilf.
Tom Edgar is a math professor and the outgoing editor of Math Horizons. He enjoys thinking about and animating so-called “proofs without words.” You can follow him on YouTube and Instagram, or look at his homepage.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the quarter-final.
Come back tomorrow for our eighth match, the last in round 1, pitting Dave Richeson against Kit Yates, or check out the announcement post for your follow-along wall chart!
Here’s the sixth match in Round 1 of The Big Internet Math-Off. Today, we’re pitting Ayliean against K.P. Hart.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Ayliean – Maths PathsHello! Here is my pitch on the maths of paths (kinda), its a bit of a windy road!
But the real treasure was the maths we found on the way.
Here is a bonus fact for actually coming to The Aperiodical and checking out both pitches: All odd anti-prisms have eulerian paths so in theory can be made by balloon modelling a single balloon!
Bonus unlisted video:
Here is a link to the Demaine and Hart paper mentioned.
And I’ll take this opportunity to big up the maths community for always doing cool things like this, creating opportunities for each other, and taking an interest in each other’s work.
Ayliean (noun): Mathsy, arty, crochet crafty, origami, activist, zine author known for making badges and trouble. You can follow them on YouTube, as @Ayliean on all social media, or look at their homepage.
K.P. Hart – A rope around the EarthOne of the things I like about mathematics is that you can do mental experiments, things that would be impossible to do in real life.
Why would you do that? Well, it’s fun to think up ridiculous scenarios, “What if …”.
But occasionally such an experiment teaches you something unexpected, and you end up with some new insights.
Today’s experiment starts with a rope that fits snugly all around the Earth, say along the equator, or across the North and South Poles over the (0)- and (180)-degree meridians. So the length of the rope will be equal to the circumference of the Earth.
(Here we see the power of mathematics: we temporarily turn the earth into a perfect solid ball of rock, so that we have no problem doing the snug fitting. And we will keep our feet dry.)
Well, actually, our rope is a bit longer than the circumference and when we tie the ends together we end up with a closed rope around the earth that is exactly one metre longer than the circumference.
So, the rope is a bit of a loose fit now. The question is: How much looser?
Looseness 1We could enlist a number of people to help us pull up the rope from the surface until it is a nice circle, concentric with the circle where we intended to fit the rope.
Our first way of measuring the looseness is the height we, and all the other people, have to pull up the rope in this way.
Ponder this for a while. If you already know the answer think of how you would explain this to a friend and what they may learn from this.
Looseness 2We enlist nobody but we, at the North Pole, put the rope on a hook and pull it up until it is completely taut. (Imagine the Earth hanging suspended from that hook.)
Our second way of measuring the looseness is the height to which we have to raise the hook.
Figure 1: The two questions in one picture.Figure 1, from issue 1 of volume 24 of Pythagoras shows the what we are thinking about: in A the rope fits snugly around the Earth; in B the rope is lifted up everywhere to the same height; and in C we have a powerful being holding up the rope over the North Pole.
The answer to Looseness 1Many people know the answer, but when I mention this question there often is someone who gives a wrong answer. The reason usually is that they are a bit too fast in their conclusion: “Earth is quite big, a metre is almost nothing, so, a millimetre maybe, or even less?”.
They would probably answer “No” to the question in the picture
Figure 2: Will a mouse be able to pass under it?
From issue 2 of volume 44 of Pythagoras.But if you pick a piece of paper and a pencil and write down what we are asking, the answer is easily found. We have the famous formula (C=2\pi R) for the circumference of a circle of radius (R). And we want to know how much (R) should increase so that (C) increases by (1). Well [C+1 = 2\pi R +1 =2\pi R+\frac{2\pi}{2\pi}=2\pi\left(R+\frac1{2\pi}\right)] We must increase (R) by (1/(2\pi)), which is about (0.159).
So our army of rope-raisers will have to lift the rope almost (16\,\mathrm{cm}). And the mouse will pass easily under the rope.
InsightWhat did we learn? The important thing that we learned is that the initial size of (C), or (R), does not matter in this problem: to increase (C) by one metre, increase (R) by (1/(2\pi)) metre.
And how do we know this? Well, we never used the values of (C) and (R) in the calculation, only their relation.
The answer to Looseness 2Figure 3 shows another picture from issue 2 of volume 44.
Figure 3: Will a polar bear be able to pass under it?It asks whether the second version of looseness will allow a polar bear to walk under the rope. Before you say “Yes” I should warn you that the picture is not to scale at all.
What I like about this particular problem is that you can answer it on a few different levels. Here I will give an answer that everybody with a decent calculator can verify.
In the video I will discuss further ramifications of this question and show how you might attack the problem with some first-year university Calculus under your belt.
First a pictureOn of the first rules of mathematics is: draw a picture, if at all possible.
Figure 4: A sketch of the problem.In Figure 4 we see the important part of the rope: the part above the surface of the Earth. Here is what the letters mean.
- (h) is the height of the hook, what we want to know.
- (R) is the radius of the Earth.
- (A) is where the rope detaches from the surface.
- (d) is the distance from (A) to the North Pole.
- (\varepsilon) is one half of the extra metre of rope.
- (\alpha) is the latitude, measured from the North Pole and in radians, of (A).
Now we stare at the picture for a while until we see some relations between the known quantities — (R) and (\varepsilon) — and the unknown quantities.
Since we work in radians we have (d=\alpha\cdot R). So (d) and (\alpha) are closely related.
Next: Pythagoras’ theorem (we have a right angle at (A)) tells us that [(R+h)^2=R^2+(d+\varepsilon)^2] which helps us to express (h) as a function of (R), (\varepsilon), and (d) (or (\alpha)) [h=-R+\sqrt{R^2+(d+\varepsilon)^2}=-R+\sqrt{R^2+(\alpha R+\varepsilon)^2}] So, all we need to do is calculate (\alpha) and we’re done.
We can find an equation for (\alpha) alone via another relation between (\alpha) and (d): [\tan\alpha=\frac{d+\varepsilon}{R}] if we put in (\alpha=d/R), then we get an equation just for (\alpha): [\tan\alpha=\alpha+\frac\varepsilon{R}] Well then, solve the equation for (\alpha), plug that value in the formula for (h) and Bob’s our uncle.
This is the sticky bit: the equation has no nice solution in closed form. And here is where a decent calculator comes in handy.
But first we need to put in the known values (R) and (\varepsilon). The latter is easy: (\varepsilon=\frac12) (in metres). For the former we use the original definition of the metre. It was chosen so that the meridian through Paris would measure exactly (10\,000\,\mathrm{km}) from the North Pole to the equator. This means that the circumference (C) is equal to (40\,000\,000\,\mathrm{m}), and hence that (R=\frac{40000000}{2\pi}\,\mathrm{m}). Putting that all together we see that we must solve [\tan\alpha – \alpha = \frac\pi{40000000}] Already in 2004 many calculators came with a solve button that would give you approximate solutions to equations; they would have no problem with this one.
You can also get a solution from Wolfram Alpha, you’ll get many and you need the one closest to (0).
I used the Maple program, it gave me (\alpha=\mathtt{0.006176391781}). If we plug that into [h=-R+\sqrt{R^2+(\alpha R+\varepsilon)^2}] we get (h=\mathtt{121.430}\,\mathrm{m}).
So there we have it: our polar bear will probably not even notice the rope as it is held quite high over the pole by our hook.
What’s in the videoIn the video I will show how you can attack the problem using a bit of first-year university Calculus, most notably Taylor polynomials.
In the end we find an approximate formula for (h) in terms of (R) and (\varepsilon); that was the reason why I used (\varepsilon) and not just (\frac12) in the calculations: I was thinking ahead to this moment: [h\approx \frac{\sqrt[3]9}2\cdot\varepsilon^{\frac23}\cdot R^{\frac 13}] This formula explains why different values of (R) give noticeably different results.
Different values? Yes: the metre has changed length since its first definition, and in the time before calculators constants like (R) were found in tables, quite often rounded to the nearest kilometre, or nearest ten kilometres, or …. Better ways of measuring lead to changes too.
All that affects the outcome of our calculations, but not too much: all answers are roughly between (121.4\,\mathrm{m}) and (121.5\,\mathrm{m}).
K.P. Hart is a mathematician, general topologist, and occasional set theorist; he also writes for the Dutch math journal Pythagoras. You can follow him on Mathstodon, Bluesky, YouTube, or look at his homepage.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the quarter-final.
Come back tomorrow for our seventh match in round 1, pitting Fran Herr against Tom Edgar, or check out the announcement post for your follow-along wall chart!
Here’s the fifth match in Round 1 of The Big Internet Math-Off. Today, we’re pitting Matt Peperell against Fran Watson.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Matt Peperell – An unexpected use for the Fibonacci sequenceA reminderThe Fibonacci sequence is a sequence in which each term is the sum of the previous two, starting with 0 and 1 (less commonly 1 and 1, or 1 and 2). The sequence in its most common form begins as follows: (0, 1, 1, 2, 3, 5, 8, 13, 21, 34,55, 89, \ldots)
Adjacent pairs of this sequence taken as a quotient converge alternately higher and lower on a value known as the golden ratio, written as (\phi). Despite the approximations being rational, the converged-on value is, in fact, irrational and is approximately equal to 1.618033988749.
A slightly different valueAn exact quotient which has a similar value to this is (\frac{25146}{15625} = 1.609344). This ratio precisely links SI kilometres to miles, with the mile being the longer unit.
Approximating the conversion between miles and kilometresMore common applications and occurrences of the Fibonacci sequence involve pentagons, snail shells and sunflower seeds. But outside of nature, using two adjacent terms of the Fibonacci sequence allows us to convert between miles and kilometres with a theoretical error of (\frac{\phi-1.609344}{1.609344} \approx 0.53997\%), though the actual error value will vary and depend on the integer rounding. Some examples:
| Miles | Approximation in km | Actual value in km | Error | | --- | --- | --- | --- | | 5 | 8 | 8.04672 | -0.581% | | 13 | 21 | 20.921472 | +0.375% |
What about values not found in the Fibonacci sequence?We can perform simple calculations on these values. E.g. multiplication by 10 is trivial:
| Miles | Approximation in km | Actual value in km | Error | | --- | --- | --- | --- | | 50 | 80 | 80.4672 | -0.581% | | 80 | 130 | 128.74752 | +0.973% |
We can also decompose values into partitions, convert those individually, and then perform addition:
| Miles | Approximation in km | Actual value in km | Error | | --- | --- | --- | --- | | 58 = 50 + 8 | 80 + 13 = 93 | 88.51392 | -0.366% | | 173 = 80 + 80 + 13 | 130 + 130 + 21 = 281 | 278.416512 | +0.928% |
Note that the errors can compound or counteract, depending on whether the terms used are in odd or even positions in the series.
Converting other unitsWe do not need a conversion factor directly similar to the golden ratio; we can operate at powers of 10. This gives us access to some other possible conversions:
- Cubic inches to cubic centimetres
- 1 cubic inch = 16.387064 cm3
- Magnitude: (\times 10)
- Approximate error 1.262…%
- Worked example:
- 10 cubic inches
- 10 = 5 + 5 → 8 + 8 = 16
- (16 \times 10 = 160) cm3 (real value is 163.87064)
- Milligrams to grains
- 1 mg = 0.01543236 grains
- Magnitude: (\div 100)
- Approximate error 4.847…%
- Worked example:
- 8 mg
- 8 → 13
- (13 \div 100 = 0.13) grains (real value is 0.1234589)
- Minutes to hours
- 1 minute = 0.0166…hrs
- Magnitude: (\div 100)
- Approximate error 2.918…%
- Worked example: 130 mins,
- Worked example:
- 130 mins
- (130 = 13 \times 10 \to 21 \times 10 = 210),
- (210 \div 100 = 2.10) hrs (real value is 2.166…)
As before, conversion with this technique gives an approximation: further mistakes creep in due to integer rounding.
Can you come up with any other unit conversions? Remember to take care with the magnitude.
Matt Peperell is a London-based recreational mathematician living a double-life as a software developer. Outside of work and when not doing maths he likes bellringing and playing board games. You can follow him on Mathstodon.
Fran Watson – Winning at playing rather than playing to winAs an enthusiastic player of games, I particularly love those that have an obvious puzzle or mathsy element to them, but also tend to look for the maths wherever possible and “Kingdomino” has been subjected to this a lot since lockdown when it was a staunch favourite.
Even without any more information about the game, think about the questions you already want to ask about this picture: Is it significant that some of the tiles are upside down? Why is the tile in the middle different? What do the colours mean? Where do you start? How do you win? The list goes on! Learning games involves lots of question posing (in my experience at least) and even if you try to remember the rules from the start, sometimes you need to be immersed, before being ready to ask the questions needed to make sense of it for yourself – a lot like learning mathematics.
If you don’t already know this game I’ll give a brief outline to start with, but if you do, you might like to look at the picture above, think about what order the pieces could have been played in and then skip ahead to paragraph 4. The picture above is my board at the end of a game. Everyone playing (usually 2-4 people) is aiming to build their own 5×5 board in front of them, comprising a starting tile (the square tile with the dashed outline) and 12 other domino-type pieces which are laid, one per round, in any orientation according to the following rules. Tiles must either be placed next to one of the four sides of the starting square or have at least one end matched in colour/land type to a tile that’s already been laid, and once they are placed tiles can’t be moved (unless you’re feeling kind to your opponents!) Players are aiming to score as highly as possible at the end of the game by multiplying the number of sections of any contiguous area of the same colour/land type by the total number of crowns in that area. (Alongside the land types of forest, pasture, swamp and sea, I’ve always thought of the yellow squares as desert, but realise now on closer scrutiny of the pictures that they’re wheat fields!)
So my board would score 24 points, as the starting tile isn’t worth any points and neither are the three other areas that don’t have any crowns in them. And there the game usually ends…
… only now is when it can get interesting if you dismantle your board and using the same tiles each, allow rebuilding, abiding by the same rules, but with the difference that as you know all your tiles in advance, you might make new choices about how to assemble them. The starting tile doesn’t have to end up in the middle, as shown in a few examples below (ignore the numbers on the back of the dominoes, they aren’t to do with scoring, only to do with the order of play when selecting them as your next choice).
In fact there are some positions of the starting square that might mean you can’t complete the 5×5 array, leaving some of your tiles unplayed so that there are “holes” in your board (this is a whole other investigative track to happily womble down, particularly for people who’ve never met tiling puzzles before). You might find that you’re doing a whole load of visualising, experimenting, comparing and figuring out, without even realising you’re working mathematical muscles!
So now you’ve a second total score, which could be the maximum possible for the tiles you have (I think mine would be 30 in the picture above) and might mean that your position in the leaderboard is different to the end of the original game. Finding the difference between your two totals is another nice metric for “most improved” player (6 extra points between my first and second goes here) and sometimes helping your opponents make even higher scores with their tiles (upon invitation of course) is oddly satisfying, and I say this as someone who is usually fiercely competitive!
Other questions have arisen whilst playing about what tiles could come up/are available in the game as a whole (a little like wanting to know the letter frequencies of Scrabble tiles to see if you’ve already had all the Js) but the only information provided in the rules booklet is this:
Although you’re given some information, you’re not told what the make-up of the 48 individual dominoes are, so unless your memory helps you out from previous games, each tile turned over is a surprise at the beginning of a turn. The exception to this in our house, is tile number 25. This was excitedly chewed by a friend’s dog in his eagerness to play! So if all this has whetted your appetite for more maths in the form of “Kingdomino” (in which I had also over looked the Kingdom reference, until recently – I should clearly pay more attention to the instructions and less to the winning it would seem) I will finish by bequeathing you the dubious insider advantage of knowing what tile number 25 looks like after being mangled by Ozzy!
Fran Watson is a teacher and communicator of mathematics originally hailing from Cornwall but now living in Cambridgeshire (by way of Cardiff in between – locations today brought to you by the letter C!) She loves puzzles, origami, games and musical theatre and will endeavour to weave these passions into her pitches.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the quarter-final.
Come back tomorrow for our sixth match in round 1, pitting Ayliean against K.P. Hart, or check out the announcement post for your follow-along wall chart!
Here’s the fourth match in Round 1 of The Big Internet Math-Off. Today, we’re pitting Mats Vermeeren against Howie Hua.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Mats Vermeeren – Waves of predatorsIn mathematical biology, one of the simplest models of population dynamics is the Lotka-Volterra model. It is a system of two differential equations, modelling the interaction of the populations of two species: a predator and a prey. The mathematics behind it has a surprising connection to the dynamics of waves in a shallow canal.
The Lotka-Volterra predator-prey modelImagine a population of cute rabbits living in a large grassland. There’s plenty of food to be found, so, rabbits being rabbits, you may expect the population size to grow exponentially. The population size at time ( t ) may be approximated by a function ( x(t) ) satisfying the differential equation
[ \frac{\mathrm d x}{\mathrm d t} = \alpha x ]
for some constant ( \alpha > 0 ).
Unfortunately for our long-eared friends, a group of foxes has moved into the grassland. Let’s denote the size of the fox population by ( y(t) ). This introduces a new term to the differential equation:
[ \frac{\mathrm d x}{\mathrm d t} = \alpha x – \beta x y \,, ]
where ( \beta > 0 ) is another constant. The new term slows down the growth of the rabbit population. It is proportional to the product ( x y ) of both population sizes. This reflects the observations that (i) if there are more foxes, they will eat more rabbits and (ii) if there are more rabbits, the same number of foxes will catch more rabbits, relying less on other food sources.
If we factorise the right hand side, the differential equation reads
[ \frac{\mathrm d x}{\mathrm d t} = (\alpha – \beta y) x \,. \tag{1} ]
There is a similar differential equation for the population size of the foxes:
[ \frac{\mathrm d y}{\mathrm d t} = (\delta x – \gamma) y \,, \tag{2} ]
where ( \gamma ) and ( \delta ) are positive constants. If the expression in brackets were constant, this would again give exponential growth (or decay). The dependence on ( x ) of this factor means the growth of the fox population is faster the more rabbits there are. But if there are too few rabbits (not enough food), then ( \delta x – \gamma ) becomes negative and the fox population decays.
The system of equations (1)-(2) is known as the Lotka-Volterra predator-prey model. For most biological applications it is too simple to capture the real population dynamics, but it’s mathematical structure is quite pleasing. A typical solution looks like this:
Population over time of rabbits (white) and foxes (orange) for a typical solution to the Lotka-Volterra equations.The populations fluctuate in a periodic way: after a certain amount of time, the population sizes are exactly what they were in the beginning. Then they repeat the same rise and fall over and over again.
The periodicity can be explained by the fact that the system has a conserved quantity
[ F(x,y) = \alpha \log(y) – \beta y + \gamma \log(x) – \delta x \,. ]
“Conserved quantity” means exactly what it says on the tin: ( F(x,y) ) does not change over time. You can check that
[ \frac{d F}{d t} = \frac{\partial F}{\partial x} \frac{d x}{d t} + \frac{\partial F}{\partial y} \frac{d y}{d t} ]
is zero by virtue of the Lotka-Volterra equations (1)-(2). We can also confirm this graphically by drawing a contour plot of ( F(x,y) ) and comparing it to the curve that a solution traces in the ( (x,y) )-plane. We see that the solution stays on one of the level sets of ( F ):
Contour lines of the conserved quantify (F) (yellow) and the path traced by a solution to the Lotka-Volterra equations (white).The Volterra ChainTo keep our equations simple, let’s set all the constants equal to 1 and consider the system
[ \begin{cases}
\displaystyle \frac{\mathrm d x}{\mathrm d t} = (1 – y) x \,, \
\displaystyle \frac{\mathrm d y}{\mathrm d t} = (x – 1) y \,.
\end{cases} ]
What if we add another species to the ecosystem? One that eats foxes. If we denote its population size by ( z ), then the system of equations becomes
[ \begin{cases}
\displaystyle \frac{\mathrm d x}{\mathrm d t} = (1 – y) x \,, \
\displaystyle \frac{\mathrm d y}{\mathrm d t} = (x – z) y \,, \
\displaystyle \frac{\mathrm d z}{\mathrm d t} = (y – 1) z \,.
\end{cases} ]
Now why stop at three? Consider a whole sequence of species ( q_1, \ldots, q_n ), such that each species eats the species whose index is one less. This leads to the system of equations
[ \begin{cases}
\displaystyle \frac{\mathrm d q_1}{\mathrm d t} = (1 – q_2) q_1 \,, \
\displaystyle \frac{\mathrm d q_i}{\mathrm d t} = (q_{i-1} – q_{i+1}) q_i \qquad \text{ for } i = 2, \ldots, n-1 \,, \
\displaystyle \frac{\mathrm d q_n}{\mathrm d t} = (q_{n-1} – 1) q_n \,.
\end{cases} ]
This system of equations is known as the Volterra chain. As a biological model, it probably isn’t very realistic. It assumes that there is a long food chain where each species exclusively eats the one just below it. But this isn’t a biology-off, it’s a math-off. And mathematically, the Volterra chain is an interesting object.
The Volterra Chain is a discrete analogue of certain wave equations, where the continuous space variable of a wave equation is replaced by the discrete sequence of species forming our food chain. If we take a large number of species (n), it is not hard to see the wavy nature of this system:
A disturbance in the population sizes of many species (numbered 1,2,3,…) propagates like a wave.In this picture we started with a spike in population of species number 1, the lowest on the food chain. This perturbation ripples through the other species like a wave. After the wave has passed, population numbers settle back into the equilibrium.
Such a solitary wave, one that travels along without changing its shape, is called a soliton.
Solitons and water wavesThere are several ways to approach the theory of soliton equations, but on some level they all come back to our observation that the two-species Lotka-Volterra model has a conserved quantity. Soliton equations have a lot of conserved quantities (infinitely many in fact). And it turns out that if you have an equation that conserves many different abstract things, you can deduce that it will also conserve some very tangible features, like the shape of a traveling wave.
Another equation that famously exhibits solitons is the Korteweg-de Vries (KdV) equation, modelling waves in a shallow canal. You can see an example of a KdV soliton in this video from the Scripps Institution of Oceanography:
The mathematical description of the solitons and other features of the KdV equation mimics that of the Volterra chain. In fact, you can consider the KdV equation as a limit of the Volterra chain with infinitely many species!
Mats Vermeeren is a Research Fellow and Lecturer at Loughborough University, UK. He is a Dutch-speaking Belgian. Dutch is one of the few languages in which the word for “maths” does not derive from the Greek “máthèma”, so his parents had no idea what they predestined him for. You can follow him on YouTube and Mathstodon, or look at his homepage.
Impress your friends by visualizing 1/3+1/9+1/27+1/81+… using any random bookHowie shows how you can physically show the sum of this infinite series with any book!
Howie Hua teaches math to future elementary school teachers at Fresno State. He also likes to make math explainer videos and math memes. You can find all his socials on his linktree.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the quarter-final.
Come back tomorrow for our fifth match in round 1, pitting Matt Peperell against Fran Watson, or check out the announcement post for your follow-along wall chart!
Here’s the third match in Round 1 of The Big Internet Math-Off. Today, we’re pitting Matt Enlow against Sam Kay.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Matt Enlow – Triangle of MediansThis excursion begins with a simple wondering I had several years ago: If you take the three medians of a triangle (the segments joining each vertex to the midpoint of the opposite side), can you always make a new triangle having those medians as its three sides?
(Those who are so inclined might pause here and form an opinion as to the answer before proceeding.)
The answer, it turns out, is “Yes!”
Now, I hope that you don’t take the above animation as anything resembling proof of my claim. Who knows what kind of trickery I might have employed? Maybe the specific triangle I chose to use is special in some way, and it doesn’t work for all triangles. Maybe I altered the lengths or angles of the medians as they slid across the screen. Don’t let me off the hook!
So how could we show that this is true in all cases?
One way would be to use the formula for the lengths of the medians (m_a), (m_b), and (m_c) in terms of the original side lengths (a), (b), and (c):
[m_a=\frac{\sqrt{2b^2+2c^2-a^2}}{2}\text{, etc.,}]
then use those, along with the fact that (a), (b), and (c) satisfy the Triangle Inequality (the sum of any two side lengths is greater than the third), to show that (m_a), (m_b), and (m_c) also satisfy the Triangle Inequality.
… Like I said… that’s one way. But it’s not a particularly appealing one.
A much simpler way would be to think about the sides and medians as vectors.
Let’s let (A), (B), and (C) be the vertices of our triangle. The median from vertex (A) can be thought of as the vector (\frac{1}{2}(\overrightarrow{AB}+\overrightarrow{AC})):
Likewise, the median from (B) is the vector (\frac{1}{2}(\overrightarrow{BC}+\overrightarrow{BA})), and the median from (C) is the vector (\frac{1}{2}(\overrightarrow{CA}+\overrightarrow{CB})).
If we can show that the sum of these three median-vectors is zero, then that will suffice to show that they can be thought of as the three sides of a triangle. Fortunately, this is straightforward to do:
[\begin{align} &\;\frac{1}{2}\left(\overrightarrow{AB}+\overrightarrow{AC}\right)+ \frac{1}{2}\left(\overrightarrow{BC}+\overrightarrow{BA}\right)+ \frac{1}{2}\left(\overrightarrow{CA}+\overrightarrow{CB}\right) \[0.5em] =&\; \frac{1}{2}\left(\overrightarrow{AB}+\overrightarrow{AC}+ \overrightarrow{BC}+\overrightarrow{BA}+ \overrightarrow{CA}+\overrightarrow{CB}\right) \[0.5em] =&\; \frac{1}{2}\left(\left(\overrightarrow{AB}+\overrightarrow{BA}\right)+ \left(\overrightarrow{BC}+\overrightarrow{CB}\right)+ \left(\overrightarrow{AC}+\overrightarrow{CA}\right)\right) \[0.5em] =&\;\frac{1}{2}(0+0+0) \[0.5em] =&\;0 \end{align}]
A second, more visual proof requires taking six copies of your triangle (along with its medians), and making them into a hexagon. You can then clearly see a triangle whose sides are twice the lengths of the medians:
Scaling this triangle down by a factor of (\frac{1}{2}) gives us the desired “triangle of medians.”
I like to tell my students that the best questions lead not only to answers, but to more questions. So where can we go from here? Every triangle has a “median triangle.” What else can be said about how these two triangles are related?
This question is intentionally extremely open-ended; I’d love for you to let your own wondering take you down your own path.
The path I traveled led me to discover that the area of the median triangle is (\frac{3}{4}) that of the original triangle.
Again, if you are so inclined, take some time to think about why that might be true before proceeding.
As always, there are many ways one could go about showing that this is true. Below is an animation of my favorite, which makes use of the following facts:
- Joining the midpoints of the three sides divides the triangle into four congruent triangles, and
- drawing the three medians of the triangle divides the triangle into six regions of equal area.
I hope you enjoyed this excursion as much as I did! Here is a little preview of coming attractions, should I be blessed with the opportunity to share them with you in this competition:
Quarter-finals: How to use squares and the Golden Ratio to make a circle
Semi-finals: Using a calculator (and some trigonometry) to (metaphorically) pull a rabbit out of a hat
Finals: Just when you thought you’d seen all of the gems hidden in Pascal’s Triangle…
Matt Enlow teaches mathematics at the Dana Hall School in Wellesley, MA. You can follow him on X, BlueSky and Mathstodon.
Sam Kay – Counting Down to DoomsdayI Know What Day Of The Week You Were Born On.
No, I’m not performing one of those scams where I state ‘Sunday’ and hope to impress one-seventh of the people reading this piece. My neat little party trick involves asking someone when they were born, and roughly 10 seconds later giving them which day of the week that was.
How on Earth is this possible?
There are two routes a keen mathematician would come up with; either I know that the Gregorian calendar has a 28-year cycle, and so remembering 28 years’ worth of days and dates is a doable task, or I take a modular arithmetic approach. The latter is clearly easier to do.
The Doomsday algorithm is a method, an algorithm, popularised by Group Theory legend John Conway that pinpoints exactly what day of the week any given day in recent history was or soon-to-be future would be. It begins with the day of the year that Conway deems Doomsday which will always be the last day of February, no matter if it’s a leap year or not. At the time of writing (2024) it is a Thursday.
The first part of the algorithm uses Doomsday to figure out the day of the week for any day in the same year. An astounding result emerges when considering the even months that follow: the 4th of April, 6th of June, 8th of August, 10th of October, and 12th of December all fall on Doomsday. That is, this year 4/4, 6/6, 8/8, 10/10, and 12/12 are all Thursdays. From here, for any desired date you can simply count up or down from Doomsday.
Suppose I am hosting a birthday party on the 10th of August and I need to let my friends know which night of the week to keep available. I know that the 8th is a Thursday, and so the 10th being two days later makes it a Saturday. They probably have the day off anyway.
I will give a quick note on modular arithmetic as it is the heart of this process. Since we are working with days of the week, there are only seven cycling objects we need to keep track of. That is, after counting seven days we arrive back at the day we started on, which essentially counts zero days. This process works under arithmetic modulo 7. It is convention to relabel the days in terms of numbers as such:
0 – Sunday, 1 – Monday, 2 – Tuesday, 3 – Wednesday,
4 – Thursday, 5 – Friday, 6 – Saturday.
For the eager Aperiodical enjoyers reading this on the day of release, today is Wednesday, now relabelled as 3. 10 days from now will be 3+10 ≡ 3+3 = 6 modulo 7, which is a Saturday. All this is saying is that 10 days from now will be the same day as 3 days from now, because 10 ≡ 3 modulo 7.
There are other key Doomsday dates to remember in the year; a common mnemonic is “working a 9-5 in a 7-11 store” because 9/5, 5/9, 7/11, and 11/7 all match with Doomsday too. If my bar steward tells me I need to work late for a fresher’s week event on the 27th of September, I want to know which day of the week I should catch up on sleep for. The mnemonic tells us the 5th of September this year is a Thursday, 4, and the 27th is 18 days away. 18 ≡ 4 modulo 7, and 4+4 ≡ 1 modulo 7. This leaves us with a Monday.
March’s neat trick is that all multiples of 7 are Doomsdays*. For January and February, it us up to each mathemagician to decide which reference dates to use. This is because they are different dependent on the leap year-ness. I like to note that in the years that are not leap years, February has all multiples of 7 being Doomsday and January follows the same as October in that the 10th is Doomsday. In the case of a leap year, February now has all multiples + 1 being Doomsday and January’s Doomsday now follows the same as April and July (4th/11th).
*yes, that means Pi Day is a Doomsday! No, Tau Day is not a Doomsday.
It might already be an impressive feat knowing what day of the week any day in the year is. But the whole trick of knowing a person’s birth-day requires knowing Doomsday for any year at hand. This requires a bit more finesse.
In order to do this, one notes that Doomsday 2000 was a Tuesday, and Doomsday 1900 was a Wednesday. The importance of this is best described with an example. Let’s say we wanted to work out Doomsday for fifteen years’ time, 2039. It is common knowledge that moving up one year in the calendar shifts the days of the week by one. We first take Doomsday 2000 and add the number of years. 39 more days than Tuesday is 2 + 39 ≡ 2 + 4 = 6 modulo 7, a Saturday.
But we mustn’t forget leap years! Leap years shift the week by two days each. One asks how many leap years are between 2000 and 2039; it’s a 39-year difference, and since leap years occur every four years that leaves us with 9 leap years slotted in. Saturday plus 9 days is 6 + 9 ≡ 6 + 2 ≡ 1 modulo 7, a Monday. Doomsday 2039 is a Monday.
Having to count how many years and leap years there are between 1900 and 2000 is tedious work though, and not efficient. There is one trick I have come across that involves the multiples of 12. Another astounding fact is that the (n)th multiple of 12 years after a century has its Doomsday (n) days after the century’s Doomsday. For example, Doomsday 1948 is Wednesday + 4 days = Sunday. Doomsday 2072 is Tuesday + 6 days = Monday.
The final part is to put these two ideas together and test this out on your willing family and friends. Heck, try it out on your own birthday and ask your mum if it was right – she’s bound to remember. Take, for example, Alice’s birthdate on 10th June 1996: 10th June is 4 days more than Doomsday, and Doomsday 1996 is Wednesday + 8 days using the multiple of 12s trick. 3 + 8 ≡ 4, so 10th June 1996 is 4 + 4 ≡ 1, a Monday.
My closing thought will be a quick disclaimer. I did say this was a neat little party trick, so by all means perform this at parties. Just make sure that the party in question contains people that know the day of the week they were born on. Otherwise you end up looking like a bit of a creep.
Sam Kay is a maths student at Durham University where he hosts the Chalkboard Ultra podcast and spends too much time thinking about spinors. Outside of maths, Sam runs one of the university’s jazz bands. You can follow him on X, and Chalkboard Ultra on X and Instagram.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the quarter-final.
Come back tomorrow for our fourth match in round 1, pitting Mats Vermeeren against Howie Hua, or check out the announcement post for your follow-along wall chart!
Here’s the second match in Round 1 of The Big Internet Math-Off. Today, we’re pitting Angela Tabiri against Max Hughes.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Angela Tabiri – Understanding the maths behind machine learningThis video gives a brief introduction to the mathematics for machine learning in non technical language.
The abstract mathematics we do, have real life applications.
Which other applications of vectors in real life can you think about?
Angela Tabiri is a mathematician and youth mentoring in STEM expert from Ghana. She is the founder of Femafricmaths, a non profit organisation that promotes female African mathematicians to highlight the diversity in careers after a degree in mathematics. You can follow Femafricmaths on YouTube, Instagram, Facebook and X.
Max Hughes – Constructing a mathematical pride dressThe interesting piece of mathematics I would like to put forward for Round 1 of this year’s “Big Internet Math-Off” is the Leonardo Dome, a mathematical structure built following geometric rules that was first designed by Leondaro da Vinci over 500 years ago. To truly represent the majesty of the dome, I decided to document myself turning it into the base for a giant pride themed dress to celebrate pride in mathematics. See the video below:
Max Hughes is the coordinator of MathsCity Leeds, who spends their free time playing table-top roleplaying games and reading comic books, whilst being engaged with fun mathsy projects on the side. You can follow them on Instagram.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the quarter-final.
Come back tomorrow for our third match in round 1, pitting Matt Enlow against Sam Kay, or check out the announcement post for your follow-along wall chart!
It’s math-off time!
Here’s the first match in this year’s Big Internet Math-Off. Today, we’re pitting Katie Steckles against Benjamin Dickman.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Katie Steckles – Prime Chunks of πI’m always on the lookout for interesting maths, and one place I often find some of the most interesting stuff is in the crossovers between different mathematical ideas. What happens when you look at the intersection of topological shapes and colouring problems, or the parts where functions meet fractals, or the crossover between flexagons and the fold and cut theorem? (I’m not saying any of these are potential ideas for my entries in future rounds of the Math Off, but I’m also not not not not saying that.)
One of my favourite mathematical crossovers comes between two numerical classics of the genre: prime numbers, and π. Everyone’s (\frac{\tau}{\pi})th favourite circle constant, meeting the indivisible building blocks of the integers. What could possibly go wrong?
We all know that the digits of π go on forever, and never repeat. This makes them a useful source if you ever want a string of seemingly random digits, since they keep going and there’s not a discernible pattern in them. But what if we look for a pattern in them? And in particular, if we look for prime numbers?
For example, the first digit (3) is a prime. Success! But for the sake of argument, let’s keep going. The next digit (1) is, assuming we’re doing it right, not a prime number.
But if we look a little further, and include the following digit as well, we get the number 14, which is… also not a prime. But 141, which we make by extending a little further, is… also not a prime number.
With a little patience, we’ll find (having been disappointed once more by 1415, which is shockingly divisible by 5), that we can go as far as 14159, which is gloriously prime. It’s also immediately followed by 2 – straight in the bucket of primes.
This can continue – the disappointment of 6 and 65 are outweighed by the beauty of the prime that is 653. It’s followed by 5, one of the classic small primes, then (not 8 but) 89, then 7.
9 isn’t prime (as much as my subconscious brain wants to include it whenever I give a list of primes), nor is 93, or 932 – but 9323 is, giving us another wonderful entry on our list.
And you might imagine we can carry on this way, theoretically forever – chunking off prime pieces of π and serving them up with ice cream. But while the digits of π do theoretically provide an infinite string of primes using this method, not all of them are quite so easy to swallow.
At this point in the decimal expansion of π, we find an unhelpful succession of frustratingly even digits: 8, 4, 6, 2, 6, 4 – none of which are of any use to us at all in making another prime. In fact, the next chunk of digits we can section off into a prime number is actually
846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837297804995105973173281609631859502445945534690830264252230825334468503526193118817101000313783875288658753320838142061717766914730359825349042875546873115956286388235378759375195778185778053217122680661300192787661119590921642019893809525720106548586327886593615338182796823030195203530185296899577362259941389124972177528347913151557485724245415069595082953311686172785588907509838175463746493931925506040092770167113900984882401285836160356370766010471018194295559619894676783744944825537977472684710404753464620804668425906949129331367702898915210475216205696602405803815019351125338243003558764024749647326391419927260426992279678235478163600934172164121992458631503028618297455570674983850549458858692699569092721079750930295532116534498720275596023648066549911988183479775356636980742654252786255181841757467289097777279380008164706001614524919217321721477235014144197356854816136115735255213347574184946843852332390739414333454776241686251898356948556209921922218427255025425688767179049460165346680498862723279178608578438382796797668145410095388378636095068006422512520511739298489608412848862694560424196528502221066118630674427862203919494504712371378696095636437191728746776465757396241389086583264599581339047802759009946576407895126946839835259570982582262052248940772671947826848260147699090264013639443745530506820349625245174939965143142980919065925093722169646151570985838741059788595977297549893016175392846813826868386894277415599185592524595395943104997252468084598727364469584865383673622262609912460805124388439045124413654976278079771569143599770012961608944169486855584840635342207222582848864815845602850601684273945226746767889525213852254995466672782398645659611635488623057745649803559363456817432411251507606947945109659609402522887971089314566913686722874894056010150330861792868092087476091782493858900971490967598526136554978189312978482168299894872265880485756401427047755513237964145152374623436454285844479526586782105114135473573952311342716610213596953623144295248493718711014576540359027993440374200731057853906219838744780847848968332144571386875194350643021845319104848100537061468067491927819119793995206141966342875444064374512371819217999839101591956181467514269123974894090718649423196156794520809514655022523160388193014209376213785595663893778708303906979207
Yes, that’s a prime number 3057 digits long. After teasing us with manageable 2-, 3- and 4-digit beauties, and even some delicate single-digit primes, the decimals of π have dumped in our lap a heap of digits so vast it’s almost boring.
This seems like an incredible coincidence. What would possibly possess whichever number devil wrote this number to make it have such a strange distribution of prime lengths? But when I thought about this a little longer, I gradually realised what was going on here.
While the prime numbers, much like the digits of π, are mysterious and unpredictable, there are some patterns they’re known to follow. The Prime Number Theorem tells us that for a given value N, the number of primes less than N can be relatively well predicted.
The prime counting function (\pi(N) = \frac{N}{\ln(N)}) tells me that there are roughly 21 primes less than 100 (in fact there are 25), 144 primes below 1000 (actually there’s 168) and 1086 primes below 10,000. But these ranges are getting bigger much quicker than the number of primes is: the prime numbers get further apart as the numbers get larger, and the density is decreasing.
This also means that given a particular number of digits, the probability that any given random string of digits of that length is a prime will get smaller very quickly as the numbers of digits gets larger; roughly 8.6% of 5-digit numbers are prime, and only 7.2% of 6-digit numbers.
So if we’re looking for primes in a specific string of digits, we’re relying heavily on these probabilities to line up if we want to keep finding primes – and once we get into 6 or 7 digits, that probability gets pretty low. All we’d need to trip us up is a successive string of, say, digits divisible by 2 or 5, none of which can occur at the end of a prime number written in base 10, and we’d… oh.
So that chunk of even digits is much more or a problem than it initially seems – by the time we get to numbers that start 846264, the primes are so spaced out that none of the possible digits placed in that next spot would give a prime. And of the numbers starting 8462643, only one of them – 84626431 – is prime (but sadly π just gives us another 3). Once we reach this length of string, the probability of hitting another prime is so small, it’s easy to see how it takes a very long time before we get to the next one.
It doesn’t get much better after that either – the next digits of π are 73, then 467 (two perfectly manageable primes), but then we dive off into a 14650-digit monster. (for reference, A047777 in OEIS is the primes, and A121267 gives the corresponding lengths).
Another nice consequence of this is that it’s not a phenomenon unique to π – other infinite non-repeating decimal expansions also suffer from gigantic prime chunks. e starts with a 2, then a 7, then the next prime is 649 digits long. The square root of 2, with its unhelpful ‘14142’ opener, takes 55 digits before we even get to the end of our first prime. But even though the maths tells us this kind of thing is likely, it still somehow feels a little unexpected.
So the next time you’re looking for an interesting fact about π to share, try chopping it up into prime numbers and watch people’s faces when you tell them what happens.
Katie Steckles is a mathematician based in Manchester, who gives talks and workshops and writes about mathematics. She finished her PhD in 2011, and since then has talked about maths at universities, schools events, festivals, on BBC radio and TV, in books and on the internet. You can find her on Mathstodon, Instagram, and as part of The Finite Group (and here on The Aperiodical).
Benjamin Dickman – Sharing is CaringI’m hopeful that The Big Internet Math-Off will bring mathematical joys – and perhaps even surprises! – to its readers. Thinking back, here are a couple of mathematical tidbits that I enjoyed and/or surprised me back when I was young and still had light behind my eyes:
- I always enjoyed sequences and series, and I recall learning that the sum (\sum_{n\geq 1} 1/n^p) converges when (p \gt 1). Seeing that it diverged for (p = 1) was enjoyable, but not knowing what it converged to for (p \gt 1) was an early frustration of mine. Pleasantly, I learned that at least one value is well known: (\sum_{n\geq 1} 1/n^2 = \pi^2/6).
- Mathematicians finish their proofs in different ways: after (n) too many jokes about “Q.E.D.” standing for “Quite Easily Done,” I have come to enjoy the various boxes that are often referred to as tombstones. A reminder not only that mathematics is alive, but also that every time you use a result or its proof, you are essentially invoking a zombie.
For today’s mathematical vivacity let us consider the following problem:
Randomly pick two positive integers; what is the probability that their only shared factor is 1?
Now, you cannot really pick two elements at random from the positive integers; so, let us be a bit more precise and say… pick two positive integers in [1, really big]. The mathematical term for their largest shared factor is the greatest common divisor, which is often abbreviated as gcd, and we say that positive integers whose only shared factor is 1 are relatively prime. This means that the sharing that we are caring about can be rephrased as:
What is the probability that two “randomly” chosen positive integers are relatively prime?
We will solve this problem in mere moments, but first: What does your intuition tell you? If you want to test your intuition, you could try typing gcd(a,b) into WolframAlpha where each of a and b is formed by mashing a number pad. I tried this just now and, indeed, they are relatively prime. Give it a shot!
My intuition unfolds as follows: With a probability problem, the answer is probably 0 or 1; it could be ½ to be tricky; and, in cases where the problem is unfamiliar, it is probably 1/e or something related. If you want to gather more data from WolframAlpha, go for it; I ended up using GPT to simulate this 100,000 times for positive integers chosen in the range [1, 1050]. The simulation found the proportion with a gcd of 1 to be 0.60813.
My intuition now suggests that this is converging to (1-1/e = 0.6321\ldots) I’ve seen enough problems to make that guess with confidence of, idk, about (1/e); the next step could be to verify it, but that is difficult to do for a conjecture that is flat out wrong.
Okay: Let’s investigate the problem and find the correct answer! Imagine the first quadrant in the xy-plane and make a dot at ((a,b)) when (\gcd(a,b) = 1). We will focus only on points with coordinates that are both positive integers, which transforms our problem into: Zooming further and further out, what proportion of the total coordinates have dots?
Here is a graph of the dots as (a) and (b) range from 1 to 1000:
Table 1. This is what my nightmares look like.Visualization is a great tool for making sense of mathematics problems. And. I do not find this particular graph to be especially helpful – at least for finding the exact answer. The graph does suggest that the ratio probably (?) is not going to zero. But who knows!
Let’s zoom in a bit and consider integers chosen in [1, 100]:
Table 2. This is what my daymares look like.(A sensemaking checkpoint: Why are there no black dots along the main diagonal? And why are there so few along the horizontal and vertical lines emanating from 30?)
Rather than staring at the black dots in search of an epiphany, let us consider what is happening for the white dots. We know that they indicate a gcd that is greater than 1; so, let’s try putting dots at ((a,b)) when (\gcd(a,b) = 2); then try it for (\gcd(a,b) = 3). Maybe we’ll try (\gcd(a,b) = 4) later.
Table 3. This is what my dreams look like.Now, the astute reader (either the one who is already astute, or the one who will momentarily become astute) will notice that the three graphs above have different ranges from which the integers were chosen. The idea is as follows: Consider points ((a,b)) such that (\gcd(a,b) = 1). We can now consider ((2a,2b)) to match them up with points that have a gcd of 2. Similarly, we could triple them and consider ((3a,3b)) to match them up with points that have a gcd of 3.
When we look at Table 3, we see that the picture for gcd=1 among [1,8] looks rather like gcd=2 among [1,2×8], and both look rather like the image for gcd=3 among [1,3×8]. (Look at the dots and blank spaces!) If we pursued this further, our next graph would investigate gcd=4 among [1,4×8]. Perhaps you are now prepared to predict what it will look like:
Table 4. This is what my daydreams look like.At this point, we are nearing the finish line. The remainder of the argument, modulo a bit of hand waving, goes as follows:
Color the points with gcd=1 in a first color; color those with gcd=2 in a second color, and observe that the latter match with the former using a transformation of ((a,b) \to (2a,2b)), which means that, in some sense, they appear only 1/4 as often. Similarly, we color those with gcd=3 in a third color and, due to the matching of ((a,b) \to (3a,3b)), they appear only 1/9 as often as the first batch of points. Continuing in this way for all positive integers, let us denote by p the proportion that we are interested in, namely, those with gcd=1. Then, the proportion with gcd=2 can be expressed as (p/4), the proportion with gcd=3 as (p/9), and so forth, from which we can sum them all up to find:
[ p + \frac{p}{4} + \frac{p}{9} + \frac{p}{16} + \ldots + \frac{p}{n^2} + \ldots ]
How can we interpret this series? Well, the first term is the probability we are interested in, i.e., that two randomly selected positive integers have gcd=1; the second term is the probability that they have gcd=2; etc. Continuing on (and on and on) we can view this infinite series as accounting for all possible gcd’s; since any pair of numbers must have some gcd, the series at hand covers all possible scenarios without overlap. In particular, it sums to 1:
[ p + \frac{p}{4} + \frac{p}{9} + \frac{p}{16} + \ldots + \frac{p}{n^2} + \ldots = 1 ]
Understanding this step (or, at the least, making peace with it) prepares us to solve for p: We factor it out from the left hand side to find:
[ p\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \ldots\right) = p\left(\sum_{n \geq 1} \frac{1}{n^2} \right) = p\left(\frac{\pi^2}{6}\right) = 1 ]
where we used the first item at the beginning of this post to rewrite that parenthetical infinite series as (\pi^2/6). This leads us to an answer of (p=6/\pi^2 = 0.6079\ldots), which fits rather nicely with our earlier simulation! And that is the answer: (6/\pi^2) (just like my intuition suggested… 👀)
The reader is encouraged to press forward with other explorations; you might consider the probability that among three “randomly” chosen positive integers the probability that their gcd (the greatest common divisor that goes evenly into all three integers) is 1. A similar argument, perhaps even graphed with different colored points in a cube, leads to an analogous answer with our initial series changed to have an exponent of three: (1/\left(\sum_{n \geq 1} \frac{1}{n^3}\right)).
You may wonder whether this last number’s denominator has a snazzy formula like (\pi^2/6). The answer is… unknown! In fact, not much is known about this number. It’s called Apéry’s Constant and named after Roger Apéry – a mathematician who proved it’s irrational. In fact, our friend Roger was so pleased with his proof that it’s an irrational number that he memorialized it as a Q.E.D. on his own life: a rather fitting way to close out our write-up:
Notes
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This write-up is based on the wonderfully assembled PCMI (Park City Mathematics Institute) Summer 2023 materials.
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Thank you to Jay Cummings for bringing this gravestone to my attention in a twxxt.
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You can find a bit more about Apéry’s Constant in the Aperiodical in an appropriately named post called Apéryodical. (The assertion proved here is stated in that post without proof!)
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If you want to try your hand at a further problem, we closed with a “proof” concerning the case of three positive integers that share no common divisor other than 1. A variation on this question would be to ask for the probability that, among three randomly selected positive integers, any two of the three have gcd=1; in other words, what is the probability that they are pairwise relatively prime? What does your intuition tell you?
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Generalize the problem in Note 4 to the case of k randomly selected positive integers being pairwise relatively prime. What does my intuition tell me?
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A different proof approach uses something called Möbius inversion. Write out your own explanation using this alternative approach. I don’t really know what Möbius inversion is, so I will appreciate it if you can make it understandable to me!
Benjamin Dickman is a mathematics teacher at a girls’ day school in New York City and a two-time Fulbrighter. His doctoral dissertation was on creativity and problem posing with the times table. He created the word game FiddleBrix! You can find him on X and Bluesky.
So, which bit of maths has tickled your fancy the most? Vote now!
Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the quarter-final.
Come back tomorrow for our second match in round 1, pitting Angela Tabiri against Max Hughes, or check out the announcement post for your follow-along wall chart!
I’m slowly working to (sort of) recreate Martin Gardner’s cover images from Scientific American, the so-called Gardner’s Dozen.
This time it’s the turn of the March 1964 issue. In the article ‘The remarkable lore of the prime numbers’, later included as chapter 9 in Martin Gardner’s Sixth Book of Mathematical Games from Scientific American, Gardner describes how Stanislaw Ulam in a boring meeting doodled a grid of numbers, spiralling out, then circled the primes. “To his surprise the primes seemed to have an uncanny tendency to crowd into straight lines.” These Ulam sprials, discovered the year before, contain lines related to prime-generating functions, which I have written about recently.
I tried to write code for this that worked as the original doodle – starting in the middle with 1 and working from cell to cell outwards in an anticlockwise spiral.
To start, I set up a document like this. I’m going to use TikZ for the drawing, as usual, and etoolbox for some conditional logic.
\documentclass{standalone}\usepackage{tikz}\usepackage{etoolbox}\begin{document}% \end{document}
To move from cell to cell, I set some global counters to keep track of the current number (n), location, and the direction of the move to the next cell. The location is a pair of coordinates (x and y, initially (0,0)) and the direction of the move to the next cell is also a pair of values (xdir and ydir, initially (1,0)).
\newcounter{n}%\setcounter{n}{1}%\newcounter{x}%\setcounter{x}{0}%\newcounter{y}%\setcounter{y}{0}%\newcounter{xdir}%\setcounter{xdir}{1}%\newcounter{ydir}%\setcounter{ydir}{0}%
The value of a counter called A is displayed in the document using a command \theA, but if the value is needed for a calculation then it is accessed using \value{A}. The % at the end of each line mean we ignore the whitespace, meaning the tikzpicture that we’re about to draw doesn’t have a load of blank spaces before it.
Next I drew a grid. I used scope to offset this by (0.5,0.5). This is so that I could place the numbers at integer coordinates even though the command grid puts its lines through integer points. Basically this means I don’t have to faff with lots of .5 values in the rest of the code.
\begin{tikzpicture} \begin{scope}[shift={(0.5,0.5)}] \draw (-5,-5) grid (5,5); \end{scope}\end{tikzpicture}
Next I drew a node at the centre of the grid to hold the number 1. This goes within the tikzpicture above, as do all the drawing commands.
\node at (0,0) {\then};
I’m now looking at a grid which contains the number 1. Not the most exciting, but it’s a start!
Drawing the other numbers is a matter of setting up some loops. After the initial 1, the doodle draws:
- one cell then turn
- one cell then turn
- two cells then turn
- two cells then turn
- …
And so on until nine cells. First I set a loop on \i from 1 to 9 to represent that the length of the run of cells grows from 1 to 9. Within this, I put a loop on \k from 1 to 2 because, as noted above, each length of cells is repeated twice. Within this, I draw the actual cells using a loop on j which ranges from 1 to the current value of \i. These loops look like this:
\foreach \i in {1,...,9}{ \foreach \k in {1,2}{ \foreach \j in {1,...,\i}{ } }}
Within the inner loop, I first increase n.
\addtocounter{n}{1}
Then I move x and y to the next cell position. This means increasing x by xdir and y by ydir. I first use \pgfmathsetmacro to do the calculation and store the result in \x and \y, then use these values to update my x and y counters and to draw the new number. I use global counters in this way rather than the commands \x and \y directly because the values of \x and \y don’t persist between loops.
\pgfmathsetmacro{\x}{\value{x}+\value{xdir}};\pgfmathsetmacro{\y}{\value{y}+\value{ydir}};\setcounter{x}{\x};\setcounter{y}{\y};\node at (\x,\y) {\then};
At the moment, this will just keep drawing cells in a straight line. This is because I haven’t implemented the turns. This is a bit fiddly, but I was thinking of (xdir,ydir) as a vector pointing the way to the next cell. As such, the procession is:
- move to the right: (1,0);
- move to the cell above: (0,1);
- move to the left: (-1,0);
- move to the cell below: (0,-1).
I need to update these to the next direction outside the \j loop but inside the \k loop. From the procession above, I notice that if xdir is 0 in the current iteration, then ydir is going to be 0 in the next, and if xdir is currently non-zero then it is going to be 0 in the next iteration. In the cases where xdir is zero, if ydir is 1 then xdir becomes -1 and otherwise it becomes 1. Similarly, if xdir is non-zero then either xdir is 1 and ydir should change to 1 or else ydir should change to -1.
I implement this as a series of if statements using \ifnumequal from etoolbox. These are used to set the values of the counters xdir and ydir.
\ifnumequal{\value{xdir}}{0}{ \ifnumequal{\value{ydir}}{1}{ \setcounter{xdir}{-1}; % left }{ \setcounter{xdir}{1}; % right } \setcounter{ydir}{0};}{ \ifnumequal{\value{xdir}}{1}{ \setcounter{ydir}{1}; % up }{ \setcounter{ydir}{-1}; % down } \setcounter{xdir}{0};}
Now I’m looking at a grid that spirals numbers, but only up to 91.
The reason for this is that each length of cells drawn occurs twice until the last. Instead of going into two runs of 10 cells, the cover image stops at 100. To fix this I need to alter the \k loop so that it runs a third time when \i is 9. This is a bit fiddly, but you can see how I did it in the full code below.
The other important aspect of this diagram is that the primes are highlighted with red text and green diagonals. Let’s focus on the red text. Without wanting to write a prime number checker into my LaTeX, I figured I would do what I’d do if I were doodling this by hand – cross-reference with a list of primes.
I did this using commands from etoolbox. First, before my tikzpicture, I set up a command \col to hold the current colour (initially black), and a list of \primes.
\newcommand{\col}{black}%\newcommand{\primes}{}%\forcsvlist{\listadd\primes}{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97}%
Now inside my \j loop, after incrementing the counter, I tested whether the current value of n (\then) is in the list of primes. If it is, then I set \col to red, and if not black.
\xifinlist{\then}{\primes}{\renewcommand{\col}{red}}{\renewcommand{\col}{black}}
Finally, I changed the command that draws each number to set the text to the colour stored in \col.
\node[\col] at (\x,\y) {\then};
Putting all this together, I’m now looking at a spiral up to 91 with primes marked.
From here, the rest is fairly cosmetic or similar to what we have already done. A bit of colouring, a twist to the \k loop adds the numbers 92-100, some conditional statements on the current direction draw the thick border to indicate the direction of the spiral, and a pair of lists identify which primes are crossed with one diagonal or the other.
View the full code.
The final result is this, an Ulam sprial to 100.
In 2018 I ran a just-for-fun competition to find “The World’s Most Interesting Mathematician”. It was so much fun that I ran it again in 2019 and 2020. And then big things happened in my life and the wider world and I haven’t had the energy to do it again.
Until now!
Each match will pit two interesting maths things against each other. The mathematician who gives the most interesting things in each group, as decided by you, goes on to share another fun maths thing in the next round. In order to make the whole thing hang together, we’re going to call the person who wins The World’s Most Interesting Mathematician (2024)*.
* of the 16 people I contacted, who were available in July, and wanted to take part.
I’ve asked the competitors to come up with maths topics they find interesting. I don’t need new things, or things that they came up with – just the kind of thing that you’d tell a fun maths friend about when you bump into them.
The tournament will start on the 1st of July. Each match will be a post here on The Aperiodical, where the two competitors will each make a pitch for something they find interesting. At the end there’ll be a poll where you can vote for the thing you found most interesting. Each poll will be open for 24 hours, and then the person with the most votes will be victorious in that match and continue to the next round.
Without further ado, here are the charming people who will be trying to out-maths each other to victory this year, in random order:
Angela Tabiri is a mathematician and youth mentoring in STEM expert from Ghana. She is the founder of Femafricmaths, a non profit organisation that promotes female African mathematicians to highlight the diversity in careers after a degree in mathematics. You can follow Femafricmaths on YouTube, Instagram, Facebook and X.
Sam Kay is a maths student at Durham University where he hosts the Chalkboard Ultra podcast and spends too much time thinking about spinors. Outside of maths, Sam runs one of the university’s jazz bands. You can follow him on X, and Chalkboard Ultra on X and Instagram.
K.P. Hart is a mathematician, general topologist, and occasional set theorist; he also writes for the Dutch math journal Pythagoras. You can follow him on Mathstodon, Bluesky, YouTube, or look at his homepage.
Tom Edgar is a math professor and the outgoing editor of Math Horizons. He enjoys thinking about and animating so-called “proofs without words.” You can follow him on YouTube and Instagram, or look at his homepage.
Katie Steckles is a mathematician based in Manchester, who gives talks and workshops and writes about mathematics. She finished her PhD in 2011, and since then has talked about maths at universities, schools events, festivals, on BBC radio and TV, in books and on the internet. You can find her on Mathstodon, Instagram, and as part of The Finite Group (and here on The Aperiodical).
Howie Hua teaches math to future elementary school teachers at Fresno State. He also likes to make math explainer videos and math memes. You can find all his socials on his linktree.
Mats Vermeeren is a Research Fellow and Lecturer at Loughborough University, UK. He is a Dutch-speaking Belgian. Dutch is one of the few languages in which the word for “maths” does not derive from the Greek “máthèma”, so his parents had no idea what they predestined him for. You can follow him on YouTube and Mathstodon, or look at his homepage.
Dave Richeson is a professor of mathematics and the John J. & Ann Curley Faculty Chair in the Liberal Arts at Dickinson College in Carlisle, Pennsylvania, USA, and is the author of Euler’s Gem (Princeton University Press, 2008) and Tales of Impossibility (Princeton University Press, 2019). You can follow him on Mathstodon and X, or look at his homepage.
Max Hughes is the coordinator of MathsCity Leeds, who spends their free time playing table-top roleplaying games and reading comic books, whilst being engaged with fun mathsy projects on the side. You can follow them on Instagram.
Fran Herr is a PhD student in mathematics at the University of Chicago. She studies low dimensional topology and geometric group theory. You can follow her on YouTube and X.
Ayliean (noun): Mathsy, arty, crochet crafty, origami, activist, zine author known for making badges and trouble. You can follow them on YouTube, as @Ayliean on all social media, or look at their homepage.
Kit Yates is an author, communicator and academic mathematical biologist who is interested in sharing stories about the places where maths can impact our lives without us even realising it. You can follow him on Mastodon and X, or look at his homepage.
Benjamin Dickman is a mathematics teacher at a girls’ day school in New York City and a two-time Fulbrighter. His doctoral dissertation was on creativity and problem posing with the times table. He created the word game FiddleBrix! You can find him on X and Bluesky.
Matt Peperell is a London-based recreational mathematician living a double-life as a software developer. Outside of work and when not doing maths he likes bellringing and playing board games. You can follow him on Mathstodon.
Matt Enlow teaches mathematics at the Dana Hall School in Wellesley, MA. You can follow him on X, BlueSky and Mathstodon.
Fran Watson is a teacher and communicator of mathematics originally hailing from Cornwall but now living in Cambridgeshire (by way of Cardiff in between – locations today brought to you by the letter C!) She loves puzzles, origami, games and musical theatre and will endeavour to weave these passions into her pitches.
You’ve got $2 \times (2^4-1) = 30$ bits of fun maths to look forward to over the next month. I’m sure there’ll be some old favourites, and plenty of stuff you’ve never heard of – looking at the list of things that are going to come up, there were a fair few things I’d never seen before.
Of course, no Summer knock-out tournament would be complete without a wall-chart to print out and follow along at home, so I’ve made one:
Here’s the full tournament schedule in text form:
| Date | Match | Mathematician 1 | Mathematician 2 | | --- | --- | --- | --- | | 2024-07-01 | Match 1 | Katie Steckles | Benjamin Dickman | | 2024-07-02 | Match 2 | Angela Tabiri | Max Hughes | | 2024-07-03 | Match 3 | Matt Enlow | Sam Kay | | 2024-07-04 | Match 4 | Mats Vermeeren | Howie Hua | | 2024-07-05 | Match 5 | Matt Peperell | Fran Watson | | 2024-07-06 | Match 6 | Ayliean MacDonald | Klaas Pieter Hart | | 2024-07-07 | Match 7 | Fran Herr | Tom Edgar | | 2024-07-08 | Match 8 | Dave Richeson | Kit Yates | | 2024-07-10 | Quarter-final 1 | | 2024-07-11 | Quarter-final 2 | | 2024-07-12 | Quarter-final 3 | | 2024-07-13 | Quarter-final 4 | | 2024-07-17 | Semi-final 1 | | 2024-07-18 | Semi-final 2 | | 2024-07-23 | Final |
The mathematicians are relying on your support to carry them all the way to the title of World’s Most Interesting Mathematican (2024, of the 16 people I contacted who were available in July and wanted to take part).
Follow along on social media – we’ll be tooting at @aperiodical@mathstodon.xyz and the competitors will be posting on their own channels. And please post your ideas for interesting bits of maths with the hashtag #BigMathOff.
It used to live, unloved, in the A-level formula book: a mysterious result relating the area of a triangle to its sides. The most interesting thing about it was its name: Heron’s formula. (As far as I can make out, the chap’s name was Hero of Alexandria, and if you do a possessive in Greek it goes into the genitive case, which makes it Heron’s Formula. You might want to debate this; I regretfully decline.)
The Call To AdventureLove Triangle is available wherever good books etc. are, from June 20th – and signed preorders are available from Maths Gear
So there I was, peacefully proofreading Matt Parker’s forthcoming book Love Triangle when I was shocked by a vicious — and frankly unprovoked — assault on the very idea of Heron’s formula.Yeah yeah yeah. Let ABC be a triangle with area ( \Delta ), side lengths ( a), (b) and (c), and semiperimeter (s = \frac{a+b+c}{2}).
It was never my favourite formula, sure, but it occasionally saved a step. And there’s a certain sleekness to it, at least if you square it to get its nicest form: ( \Delta^2 = s(s-a)(s-b)(s-c) ). Of Matt’s many complaints about our poor first-century-CE friend’s contribution to the literature, one did strike me as valid: there isn’t a nice proof of it. You can make like Hero himself and do some jiggerypokery with a degenerate cyclic quadrilateral, you can follow Sir Isaac Newton’s lead and wrangle the algebra, or you can trigonometry yourself to death.
This surprised me. Surely a nice symmetric formula should have a nice symmetric proof? According to the Wikipedia citations, I’m not the only person to think that. It was also something that interested John Horton Conway — a clue that led me to a lovely proof.
The First ThresholdOnce upon a time, back in the heady days when Christian had an occasional few minutes of free time, this esteemed organ ran a Big Math[s]-off. One of my few successful submissions was a proof without words of Conway’s Circle Theorem. It was a very nice proof without words; however, I now have a better and clearer one.
Conway’s Circle Theorem states:
Given triangle ABC, extend sides AB and CB through B by a length equal to side AC. Similarly, extend sides BA and CA and through A and sides AC and BC through C to define the six Conway points. Then the ends of these sides lie on a circle (the Conway Circle) which is concentric with the incircle of ABC.
I mean, what?! I’ve proved this several ways and it still seems absurdly neat.
The simplest proof I have looks like this:
The radii of the incircle that reach the edges, and the segments connecting the incentre to the vertices, naturally divide triangle ABC into six triangles in three congruent pairs. These can be rearranged to make a rectangle as shown; the most distant point on the rectangle is a Conway point.
You can do the same thing either way from any of the three inradii, giving you the six Conway points; all of the rectangles are congruent, so their diagonals have the same length. In other words, all six Conway points are the same distance from the incentre, so they lie on a circle that’s concentric with it ( \blacksquare )
That’s tidy and all, but what’s really interesting is that the rectangle shown has the same area as triangle ABC does — it’s made up of the same six triangles. One of the rectangle’s edges is equal to ABC‘s inradius and the other is equal to its semiperimeter. And the only other place I’ve ever seen a semiperimeter mentioned is in Heron’s formula.
Approach to the inmost caveWe’re currently simultaneously close but some way off: we can see that ( \Delta = rs) — but there’s no immediate sign of ( (s-a) ) and its friends. They’re there if you look closely, though: each of the six triangles has a leg of the right form. For example, the red triangle has one leg of length ( r ) and the other of length ( s-c ).
Unfortunately, multiplying them together is non-trivial and is going to take a little work.
You’ll also see a lot of dry, technical alphabet soup that I’ve tried to avoid here — think of this as a somewhat coherent argument rather than a proof.
Let’s sidetrack for a moment and talk about the difference between finding and presenting a proof. If you look at the “final version” of this proof, currently going through peer review at Mathematics Magazine, you’ll just see a clever sequence of triangle manipulations. You won’t see the dead-ends and mis-steps, of which there were many. For example, the first time I noticed the semiperimeter thing, I thought “I just need to prove that ( r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}} ).” I then realised how much heavy lifting that “just” was doing and set it aside until spurred back into action by a Parker diatribe.
I spotted there was another way: if I could show that the base of the rectangle was equal to ( \frac{(s-a)(s-b)(s-c)}{r} ), then I could multiply the two areas together to get ( \Delta^2 ). So that’s what I did.
The battle beginsWe’re going to need some letters, I’m afraid.
This is the same triangle ABC as before, with the same triangles, but it’s helpful to label things. Call the inradius ( r ). The half-angles at A, B and C are ( T ), ( U ) and ( V ) (respectively), and the segments linking each to the incircle have lengths ( rt ), ( ru ) and ( rv ) — it makes the algebra simpler later to keep a factor of ( r ) in there. Lastly, the segment linking A to the incentre has length ( rt’ ).
The main idea of this proof is that you can multiply lengths together by scaling triangles.
Triangle 1Start by building a triangle similar to a blue triangle (the ones containing point A) and match its “( r )” leg to the “( rv )” leg of a pink triangle (containing C).
This scales the blue triangle by a factor of ( v ), so the new blue triangle’s other leg has length ( rtv ). Its hypotenuse has length ( rt’v ), but we won’t need that until later on.
Triangle 2Now I’m going to scale a yellow triangle (containing B) so its “( r )” leg matches the “(rtv)” leg of the new blue triangle. Because it’s scaled by a factor of ( tv ), its other leg has length ( rtuv ).
A couple of claims here: firstly and easily, the angle between the blue and yellow hypotenuses is ( V ) — this must be the case because ( 2T + 2U + 2V = \pi ) and the non-right-angles of our yellow monster add up to ( \frac{\pi}{2} ). Secondly, I claim that the apex of the yellow triangle (marked P) is a Conway point. That takes a bit more justification, and two more triangles.
Triangles 3 and 4If I drop a perpendicular to the blue hypotenuse at C, I get a triangle that’s similar to the pink triangle. Its scale factor is ( \frac{t’}{v} ), and the new leg has length ( rt’ ).
What about the other triangle, with CP as an edge? That has the same angles as the triangle connecting A and B with the incentre — and it has a pair of corresponding sides the same length, so the two are congruent. In particular, it means that CP has length ( rt + ru = c ). A point on BC extended, a distance of ( c ) away? That’s a Conway point.
Slaying the beastThe rectangle from Conway’s Circle Theorem earlier had area ( rs ), so we can write ( \Delta = r^2(t+u+v) )
But we also just worked out that the yellow triangle’s leg — which corresponds to the “( s )” edge of the rectangle — has length ( rtuv ), so we can write the rectangle area as ( \Delta = r^2 tuv ).
Multiplying the two ( \Delta ) equations together gives something I’m going to arrange as ( \Delta^2 = r(t+u+v)(rt)(ru)(rv) ).
Meanwhile, you can remember from earlier that ( rt = s-a ), ( ru = s-b ) and ( rv = s-c ) and out jumps ( \Delta^2 = s(s-a)(s-b)(s-c) ), which is Heron’s formula ( \blacksquare )
Triumphant returnJust because I wondered and looked it up: Euclid flourished about as long before Hero did as Newton did before me. Assuming that writing for the Aperiodical is what counts as flourishing.
Doris Schattschneider — who didn’t exactly coerce me into writing this up as a proper paper, but from certain people the phrase “I do hope…” has as much weight as a command — tells me she isn’t aware of a previous proof of Heron’s formula that only uses synthetic geometry. I don’t really know what that means, but I gather it’s something to do with what Euclid could have done. I find it a little hard to believe that nobody in the last couple of millennia has stumbled on something akin to this, but in any case, my approach to figuring things out is similar to the fabulous Moose Allain‘s approach to jokes (I paraphrase, and I speak only for me): it doesn’t matter if someone else thought of it first, you’re still allowed to take joy from thinking it up yourself, and you’re still allowed to share it.
This proof brought me great joy. And maybe it’ll help Matt see another side of Heron’s formula.
Here’s a round-up of some of the mathematical news we saw last month.
Maths NewsThomas Hales and Koundinya Vajjha have claimed a proof of Mahler’s first conjecture, that the most unpackable centrally symmetric convex disk in the plane is a smoothed polygon. (via Greg Egan)
There’s also a been a proof of the geometric Langlands conjecture published, as outlined in this New Scientist article.
Zhouli Xu has claimed a proof of the Kervaire invariant one problem in dimension 126. (via Kyle Ormsby)
And finally, Hidetoshi Mino has counted all the magic squares of order 6. Up to rotations and reflections, there are 17,753,889,197,660,635,632. (via Walter Trump)
Awards and AppointmentsThe inaugural Jean-Pierre Demailly Prize for Open Science in Mathematics has been awarded to zbMath Open, “for its broad scope, recent policy changes, and commitment to accessibility and sustainability”. (via the European Mathematical Society)
It’s been announced that the first President of the newly-formed Academy for the Mathematical Sciences (AcadMathSci) will be Professor Alison Etheridge OBE FRS, a professor in Probability at the University of Oxford, and a world expert on stochastic processes and their applications. She will take up the role on 17 June 2024.
The Shaw Prize in Mathematical Sciences 2024 has been awarded to Peter Sarnak, “for his development of the arithmetic theory of thin groups and the affine sieve, by bringing together number theory, analysis, combinatorics, dynamics, geometry and spectral theory.” (via Paysages Mathématiques)
Other News“Des chiffres et des lettres”, the French gameshow on which Countdown is based, has been cancelled after more than 50 years. (via Sarah Dal)
The UK Government has issued a call for £6m funding to set up a National Academy focused on Mathematical Sciences (NAM). Confusingly, this isn’t the same thing as the fledgling Academy for the Mathematical Sciences (AcadMathSci), though AcadMathSci may well bid to become the NAM. Clear?
And sadly, award-winning mathematician and co-founder of the Simons Foundation Jim Simons has died. (via Alberto Ramos)
The UK Government have announced the new set of King’s Birthday Honours. Here’s our selection of particularly mathematical entries for this year. If you spot any more, let us know in the comments and we’ll add to the list.
- Philippa Bonay, Director, Operations, Office for National Statistics. Appointed OBE for Public and Charitable Services.
- Anne Davis, Professor of Mathematical Physics, University of Cambridge. Appointed OBE for services to Higher Education and to Scientific Research.
- Paul Fannon, Fellow, Christ’s College, Cambridge, and Volunteer, United Kingdom Mathematics Trust. Appointed OBE for services to Education.
- Ian Hall, Professor of Mathematical Epidemiology and Statistics, University of Manchester and Senior Principal Modeller, UK Health Security Agency. Appointed OBE for services to Public Health, to Epidemiology and to Adult Social Care, particularly during Covid-19.
- David Marshall, Lately Director of Census, Northern Ireland Statistics and Research Agency (now Northern Ireland chief electoral officer). Appointed OBE for services to Official Statistics and Census-taking in Northern Ireland.
- Bruno Reddy, Founder and Chief Executive Officer, Maths Circle, Ampthill, Bedfordshire, and creator of Times Tables Rock Stars. Appointed OBE for services to Education.
- Sam Rose, Deputy Director, Data and Analysis Division, Department for Transport. Appointed OBE for services to Advanced Analytics
- Matthew Woollard, Professor of Data Policy and Governance, UK Data Archive, University of Essex. Appointed OBE for services to Data Science
- George McMath, Lately Deputy Principal, Northern Ireland Statistics and Research Agency, Northern Ireland Civil Service. Appointed MBE for services to the Northern Ireland Census.
Get the full list from gov.uk. Spot anyone we’ve missed? Let us know in the comments.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of May 2024, is now online at Girls’ Angle.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
A few weeks ago I heard someone casually refer to ‘that formula of Euler’s that generates primes’. I hadn’t heard of this, but it turns out that in 1772 Euler produced this formula:
[ f(x) = x^2 + x + 41\text{.} ]
Using this, (f(0)=41), which is prime. (f(1)=43), which is also prime. (f(2)=47) is another prime. In fact this sequence of primes continues for an incredible forty integer inputs until (f(40)=41^2). It might generate more primes for higher inputs, but what’s interesting here is the uninterrupted sequence of forty primes.
This got me wondering. Clearly (f(0)) is prime because 41 is prime, so that much will work for any function
[ f(x) = x^2 + x + p ]
for prime (p), since (f(0)=0^2+0+p=p). Are there other values of (p) that generate a sequence of primes? Are there any values of (p) that generate longer sequences of primes?
I wrote some code to investigate this. Lately, I’ve taken to writing C++ when I need a bit of code, for practice, so I wrote this in C++.
I figured the cases where (f(0)) is prime but (f(1)) isn’t weren’t that interesting, since (f(0)) is trivially prime. In fact, (f(x)=x g(x)+p=p) when (x=0) for any prime (p), but saying so doesn’t seem worth the effort.
So I kept track of the primes (p) whose functions (f(x)=x^2+x+p) generate more than one prime, and the lengths of the sequences of primes generated by each of these. This produced a pair of integer sequences.
I put the primes that work into the OEIS and saw that I had generated a list of the smaller twin in each pair of twin primes. I was momentarily spooked by this, until I realised it was obvious. Since (f(0)=p) and (f(1)=1^2+1+p=p+2), any prime this works for will generate at least a twin prime pair (p,p+2).
What about the lengths of the sequences of consecutive primes generated? The table below shows the sequences of consecutive primes generated for small values of (p). Most primes that generate a sequence produce just two, and (p=41) definitely stands out by generating forty.
| (p) | (f(x)) | Primes generated | Number of consecutive primes generated | | 3 | (x^2+x+3) | 3, 5 | 2 | | 5 | (x^2+x+5) | 5, 7, 11, 17 | 4 | | 11 | (x^2+x+11) | 11, 13, 17, 23, 31, 41, 53, 67, 83, 101 | 10 | | 17 | (x^2+x+17) | 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257 | 16 | | 29 | (x^2+x+29) | 29, 31 | 2 |
I was pleased to see this sequence of lengths of primes generated was not in the OEIS. So I submitted it, and it is now, along with the code I wrote. (I discovered along the way that the version where sequences of length one are included was already in the database.)
Anyway, I amused myself by having some C++ code published, and by citing Euler in a mathematical work. Enjoy: A371896.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of April 2024, is now online at Ioanna Georgiou’s blog. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
This is a guest post by Elliott Baxby, a maths undergraduate student who wants to share an appreciation of geometrical proofs.
I remember the days well when I first learnt about loci and constructions – what a wonderful thing. Granted, I love doing them now; to be able to appreciate how Euclid developed his incredible proofs on geometry.
In school, it was a slightly different story. Whilst I was meant to be constructing triangles and drawing a locus of a point, the school-supplied compasses had other ideas – slipping around unhelpfully, making them useless for the task. Understandably, I’d often put down the compass, chat with my friends, and sneakily eat crisps when the teacher wasn’t looking.
A lot has changed since then; I now have a working compass. But more than that, I have become a mathematician, mathematics teacher and all round mathematics nerd! I don’t think a day goes by where I have not been involved in some mathematical activity. My favourite on the weekend is working out how long it will be till my takeaway arrives! But I digress. Writing is a passion of mine as it allows me to share the facts and curiosities of this truly wonderful subject.
In this article I aim to share some key ideas that allowed me to develop my interest in geometry, and to appreciate the wonders hidden in plain, or rather, plane sight. Enjoy some of my favourite facts about shapes and lines, while I finish my packet of salt and vinegar crisps.
Angles in a triangleOne of the first things I learnt at school was that angles in a triangle sum to 180 degrees. But I never knew why, or saw a proof. Proof is vital in mathematics as proof allows us to confirm theories and conjectures that can help us progress our mathematical knowledge. If something is proved, then we can always assume it to be true!
I first came across this proof when I was researching different ways to teach angles in parallel lines. We will therefore be using proven facts on angles in parallel lines in this proof.
The first proof I saw of the angle sum was one using parallel lines. We start off with a triangle whose interior angles are all different (called a scalene triangle). In order to write an equation including all three angles, we first draw two lines, both parallel to the base of the triangle:
The lines (a) and (b), which are two sides of the triangle, touch both parallel lines – we say they are transversal lines. We can use some existing results about angles and parallel lines to make some deductions:
(\angle A = \angle \alpha) (‘alternate angles are equal’).
(\angle B = \angle \beta ) (‘alternate angles are equal’).
( \angle \alpha + \angle C + \angle \beta = 180^{\circ} ) (‘angles on straight lines add up to 180 degrees’)
( \Rightarrow \angle A + \angle B + \angle C = 180^{\circ} ) QED
Here, we were able to use the fact that alternate angles are equal, as this is a proven fact. This can save time when we’re proving something that can build on existing mathematical theories.
Proof of equal sides (isosceles)Here we have a triangle with two sides of the same length. We want to show that if this is the case, then the two base angles are equal. This proof is relatively simple, and relies on a powerful tool in geometry!
We bisect the angle at (C), which, in this case, will intersect at the midpoint of (AB) (which we denote (c)), as the sides (a) and (b) are the same length. It follows then, using one other existing result, that:
( \Delta BCc \simeq \Delta ACc) (by ‘side-angle-side’)
( \Rightarrow \angle B = \angle A) QED.
Congruent triangles make an appearance in a lot of geometrical proofs as they allow us to confirm certain angles or sides are equal, allowing us to draw conclusions. A related idea is that of similar triangles, which Thales used to measure the height of the Great Pyramid! But how did he do that?
Thales and the Great PyramidAlthough the Great Pyramid, and Thales, are both three-dimensional, we can model this problem by focusing on the 2D plane cutting through the pyramid, as shown in the diagram.
Thales wanted to know the height of the Great Pyramid, (d) and to do this is placed a vertical pole (BC), of height (a), in front of the Great Pyramid. He then measured the length of the shadow cast by the pole, (c) and the Great Pyramid, (f).
Assuming the sun’s rays are parallel, Thales drew the conclusion that the triangles formed by the tall objects and their shadows must therefore be similar, and so the height of the pole and the Great Pyramid must be in the same proportion. That is to say:
[ \frac{d}{a}=\frac{f}{c}]
Thales will know the lengths of (a), (c) and (f) so, with some rearranging, he can find the height of the Great Pyramid:
[ d = \frac{af}{c}]
PythagorasThere are over 350 proofs of the Pythagorean Theorem! So many ways to prove such a simple yet powerful result. I have not seen or worked out all of them (I took a break when I got to Euclid’s proof), but the one pictured aboveis so far, my favourite, because it uses a lot of nice algebra.
The proof goes as follows: we start by enclosing a square with four congruent triangles, as seen above. We then want to work out the total area of the congruent triangles:
Each blue triangle has area ( \frac{1}{2}ab ), so the total area of the four blue triangles is (2ab).
We can also find the area of the triangles in another way. Because the four triangles are congruent, the sides form a larger square of length ( (a+b) ), and the area of the pink square will be (c^2). From this, we can work out the total area of the triangles a different way:
Total area of blue triangles (= (a+b)^2 – c^2)
We now have two ways to write the area of the blue triangles, so we can equate these two expressions:
[ (a+b)^2 – c^2 = 2ab]
[ a^2 + 2ab + b^2 – c^2 = 2ab ]
[ a^2 + b^2 – c^2 = 0 \qquad \textrm{(Subtracted }2ab\textrm{ from both sides)}]
[ a^2 + b^2 = c^2 \qquad \textrm{(Added }c^2\textrm{ to both sides)}]
QED.
The proof is complete!
Geometry and AlgebraMathematics started to become a passion for me when I first learned about expanding quadratics – it was the first topic I revised when preparing for my GCSEs. I remember spending ages on this topic, because I kept making mistakes when multiplying negatives and positives, but I kept persevering. I even rushed my tea to go and continue to expand brackets! But don’t worry, I took some crisps with me.
The obsession stemmed from the fact that it was extremely fun! I knew I had an end goal and I had to work towards it – double checking to make sure each step I took was correct. Algebra became one of my favourite pastimes, and increased my love for mathematics. So, when I heard there was a link between expanding brackets and geometry, I was excited to learn more.
If we expanded ((a+b)^2), we would get (a^2 +2ab + b^2). But why? It may not initially seem that obvious. We can prove this using the distributive property – but that’s not what this post is about… so let’s use geometry!
We start by drawing a square of side length ((a+b)). We then divide the square up into different sections: we can make a square of length (a)(blue square), then cut out 2 congruent rectangles with dimensions (a) by (b) (green rectangles). We are then left with a pink square that has side length (b).
If we work out the areas of each of these 4 shapes, the sum of these areas will equal the total area of the initial square.
The sum of the areas:
[ \textcolor{blue}{a^2 \textrm{ (blue) } } + \textcolor{green}{ab \textrm{ (green)}} + \textcolor{green}{ab \textrm{ (green)}} + \textcolor{magenta}{ b^2 \textrm{ (pink) }} ]
Therefore, this will be equal to the area of the initial square, which measures ((a+b)) on each side:
[ (a+b)^2 = a^2 + 2ab + b^2 ]
I do quite like this result as it links to my first ever real enjoyment of learning mathematics!
PuzzleI would like to end with a little puzzle for you. Can you work out the area of the green section, in this triangle with circle arcs centred at each corner?
The solution is below.
Geometry RulesI am now reaching the end of my crisps and so, like at school, it is time to call it a day. Geometry is fascinating. There is no denying that. The theorems, proofs, applications are truly something to behold, and we can see the connections between solving equations and drawing shapes! But this article barely scratches the surface of the wonder that is geometry so, when I get a new packet of crisps, I will be sure to share more of the fascinations geometry has to offer.
Area Puzzle solutionThe triangle is equilateral, since all the sides are the same length. Using Pythagoras to work out the height, (h), of the triangle:
[h^2 + 4^2 = 8^2]
[h^2 + 16 = 64]
[h^2 = 48 ]
[ h = 6.928\ (4 s.f.)]
Area of triangle:
[ \frac{1}{2}\cdot 8 \cdot 6.928 = 27.71\ (4 s.f.)]
Area of sectors:
[ \frac{60^{\circ}}{360^{\circ}}\cdot \pi(4^2)]
[ \frac{16\pi}{6} = \frac{8\pi}{3}]
As there are 3 congruent sectors, total area of sectors:
[ 3 \cdot \big( \frac{8\pi}{3} \big) = 8\pi]
Therefore, area of the green section:
[ A = 27.71 – 8\pi]
[A = 2.587\ (4 s.f.)]
In this series of posts, we’ll be featuring mathematical podcasts from all over the internet, by speaking to the creators of the podcast and asking them about what they do.
We spoke to Marcello Seri and Marit van Straaten from the Bernoulli Institute at the University of Groningen, about their podcast, It’s Not Just Numbers.
Podcast title: It’s Not Just Numbers
Website: podcasters.spotify.com/pod/show/not-just-numbers
Links: Apple Podcasts, Spotify, Google Podcasts
Average episode length: 1 hour
Recommended episodes: Intro Episode, Teaching Mathematics (S1E03)
What is your podcast about, and when did it start?The idea for the podcast was born during the pandemic. In a break between online lectures we discussed with some students how hard and time consuming it can be to solve the homework exercises, and this somehow led us to talk about their perceived idea of their lecturers. It was quite mind-blowing how far from the truth this was, and how much the stereotypes from mathematical movies were shaping their impressions. We think that part of this came from the fact that they always see the “lecturer/supervisor” side of us and rarely have a chance to see the rest.
To counter this, It’s Not Just Numbers aims to address some common misconceptions and stereotypes around mathematics by showing the human side of it. In each episode we invite two mathematicians to talk about their drives, their aspirations, and the other hobbies they have outside mathematics. We use the discussion as an opportunity to get to know our guests and also present different sides of what being a professional mathematician entails. Moreover, we discuss a topic that is related to Mathematics, for instance what is involved when teaching Mathematics.
Marcello SeriMarit van StraatenThe podcast is hosted by us: Marcello Seri, an associate professor in mathematics at the Bernoulli Institute of the University of Groningen, and Marit van Straaten, a final year Master student at our institute. We record with the invited guests using facilities provided either by our faculty, or by FSE Radio, a group of student podcasters at our Faculty with their own recording room. For the editing and the planning we take turns among each other depending on how busy we are. The podcast is hosted on Spotify for Podcasters (formerly Anchor) and available on all of the major podcast platforms.
Who is the intended audience for the podcast?Broadly speaking, everybody that is curious about mathematics and mathematicians and what they are like. Looking at our stats, it seems that so far we are reaching young people interested in knowing what being a mathematician is like, be they just curious, potentially interested in enrolling in a mathematics track or already studying some scientific subject and curious to know what mathematics lecturers are
like and what they do.
What is a typical episode like?The episodes are released on a monthly basis, the first Sunday or Monday of the month. They are structured in two parts: in the first half we interview our two guests to get to know them. We look into how they ended up becoming mathematicians, what motivates them and what they do besides mathematics. In the second part we look into some mathematics-related themes. This can vary a lot, from covering different aspects of our jobs as mathematicians to exploring what you could do outside academia as a mathematician.
The episodes are planned less as an interview and more as a discussion among the four of us, so while we have some initial idea and we stick to the two-part structure, the discussion flows freely wherever our collective dialogue brings it. The idea is that this gives a more spontaneous and truthful picture of who and what we are like.
In this first season we focused more on mathematics: teaching and studying it, engaging with people outside our university, and looking briefly at some fields of mathematics and how they relate to each other. In the second season, which will start some time in the fall, we will shift more towards the “aftermath”, what working as a mathematician can be like. Many of our listeners were curious to hear about the experience of doing a PhD, what is the academic career path like for mathematicians and, in general, what people do with mathematics outside academia.
Keeping the format that we have now, we plan to get a bit more into these topics, with the plan to also start looking also outside our department, and perhaps outside our university. We would also like to explore what you can do with mathematics “outside mathematics”: for example, we are trying to plan an episode about philosophy of mathematics for next year.
Why should people listen? Why is it different from other mathematical podcasts?Our podcast is less about specific mathematical topics and more about the mathematicians and the surroundings of being a mathematician. This makes the podcast accessible to a broad audience, as a mathematical background is not required.
We aspire to show the people behind the trade and how varied and different their experiences are, hopefully demystifying some usual misconceptions and stereotypes and providing a glimpse to many aspects of being a mathematician that are rarely discussed. So far I have not seen other podcasts going in this direction in the same way as we do here, although I can see some analogy at least in our objective of bridging the gap between people and mathematicians with the Chalkboard Ultra podcast previously featured here.
What are some highlights of the podcast so far?Recently, we made two episodes about applied mathematics and theoretical mathematics, discussing the boundary between the two fields. It was a good opportunity to show that the difference between mathematics and applied mathematics is not as vast as students sometimes perceive, and that it is quite common that people switch from one field to another.
Besides this, many other insights can be taken from every episode. Each person brings with them a whole different view, making each episode special in its own way. Being able to hear the experiences of your lecturers as a student is extremely valuable. It really helped to put my (Marit) own struggles into perspective, and we have heard from other students that the insights they gained from our podcast have significantly helped them in their academic related challenges. The ability to help fellow students is the biggest highlight of all!
With the emphasis on occasionally, I’m occasionally working to (sort of) recreate Martin Gardner’s cover images from Scientific American, the so-called Gardner’s Dozen.
This time I’m looking at the cover image from the November 1959 issue. The column is ‘How three modern mathematicians disproved a celebrated conjecture of Leonhard Euler’, about the so-called Euler’s Spoilers, the story of three mathematicians – Parker, Bose and Shrikhande – who had disproved a conjecture of Euler’s about Latin squares. The column was reprinted as chapter 14 in his New Mathematical Diversions from Scientific American.
Orthogonal Latin squares, also known as Graeco-Latin squares, are really a pair of overlapping Latin squares – traditionally one in Latin letters and the other in Greek – with the additional property that every possible pair of one Latin and one Greek letter appears exactly once.
Here’s a 3 by 3 orthogonal Latin square in Latin letters A, B, C and Greek letters α, β, γ. Notice how each of the Latin letters appears exactly once on each row and column, and the same is true of the Greek letters, and that every Latin letter is paired with every Greek letter exactly once.
| Aβ | Bα | Cγ | | Cα | Aγ | Bβ | | Bγ | Cβ | Aα |
Euler had conjectured that no orthogonal Latin squares of order (4k+2) exist, on the basis that he couldn’t make one either 2 by 2 or 6 by 6. In fact, squares exist for all (n>1) except (n=2) and (n=6).
The cover image is a 10 by 10 orthogonal Latin square. Since (k=2) gives (4k+2=10), this is a magazine cover counterexample of Euler’s conjecture.
Notice how in the cover image at the bottom left there is a 3 by 3 sub-square that forms a self-contained 3 by 3 orthogonal Latin square in three colours? In this post, I’ll focus on drawing this small sub-square, to keep things simple.
Using the Graeco-Latin square above, we can associate the squares with letters like this:
- A: big white square;
- B: big grey square;
- C: big black square;
- α: small white square;
- β: small grey square;
- γ: small black square.
In TikZ, we can draw a rectangle easily enough. This code will draw a rectangle from (0,0) to (1,1) with no border (draw=none) and filled grey (using the American spelling). Put it inside a \begin{tikzpicture} environment, and don’t forget to \usepackage{tikz} in the document header.
\draw[draw=none,fill=gray] (0,0) rectangle (1,1);
We can layer TikZ graphics on top of each other, so to make a small rectangle inside a larger one we can just draw the big one then draw the small one on top, like this.
\draw[draw=none,fill=gray] (0,0) rectangle (1,1);\draw[draw=none,fill=white] (0.3,0.3) rectangle (0.7,0.7);
This looks like this:
We can position this on the image either by changing the values of the coordinates or by using an environment called scope, which has the advantage that you don’t have to work out the 0.3 and 0.7 versions of every square. Here’s the scope approach, shifting the code above so it draws a square with bottom left corner at coordinates (5,5).
\begin{scope}[shift={(5,5)}] \draw[draw=none,fill=gray] (0,0) rectangle (1,1); \draw[draw=none,fill=white] (0.3,0.3) rectangle (0.7,0.7);\end{scope}
Of course, I could now go through and draw a lot of squares by copying and pasting lots of copies of this code which I edit for position and colour, but I prefer to do this with a for loop. For loops in TikZ have the nice property that you could loop over several variables at the same time. For example, this loop draws three squares in a row – at coordinates (0,0), (1,0) and (2,0).
\foreach \i/\j in {0/0,1/0,2/0}{ \begin{scope}[shift={(\i,\j)}] \draw[draw=none,fill=gray] (0,0) rectangle (1,1); \draw[draw=none,fill=white] (0.3,0.3) rectangle (0.7,0.7); \end{scope}}
Here we don’t want to draw the same colours each time. We can pass the colours as loop variables also. Now we are looping over four variables, \i, \j, \outer and \inner, all drawn from a list of values.
\foreach \i/\j/\outer/\inner in {0/0/white/gray,1/0/gray/white,2/0/black/black}{ \begin{scope}[shift={(\i,\j)}] \draw[draw=none,fill=\outer] (0,0) rectangle (1,1); \draw[draw=none,fill=\inner] (0.3,0.3) rectangle (0.7,0.7); \end{scope}}
We can feed this a full list of coordinates and colours and get a 3 by 3 orthogonal Latin square as desired.
One problem you might notice with this is that in the Scientific American cover image when the inner and outer colours are the same, the inner square is given a little border, and we haven’t done that. We’ll deal with that later.
First I want to deal with a problem that won’t affect the overall look of the image and might just be me: I got pretty bored already of typing all the coordinates, and I’m conscious I’m very likely to get something wrong. I think it would be better to give it a list of pairs of colours and have the program arrange it into a grid for me. This makes the coding more complicated, though, so your preference might be to do the extra typing.
To do it my way, I’m going to define \i and \j outside of the loop and update them myself rather than letting \foreach do it. The ‘pgf’ in these commands refers to the fact that we are programming in PGF, which underpins TikZ. \pgfmathsetmacro is similar to LaTeX’s \newcommand only with extra mathematical functionality.
\pgfmathsetmacro{\i}{0};\pgfmathsetmacro{\j}{0};
We can increment \i using this command. The ‘truncate’ here means we are turning the output into an integer, whereas the result of a calculation naturally wants to be a float (e.g. 1.0).
\pgfmathtruncatemacro{\i}{\i+1};
There is a little niggle here. When you use \pgfmathtruncatemacro inside a loop, it acts on a local variable that only changes for this iteration of the loop. One way to deal with this is by passing a parameter remember to the \foreach command, which tells it to pull the value from the previous iteration.
Putting that together, we now have some code that will draw our squares in a single row.
\pgfmathsetmacro{\i}{0};\pgfmathsetmacro{\j}{0};\foreach[remember=\i as \i,remember=\j as \j] \outer/\inner in {white/gray,gray/white,black/black,black/white,white/black,gray/gray,gray/black,black/gray,white/white}{ \begin{scope}[shift={(\i,\j)}] \draw[draw=none,fill=\outer] (0,0) rectangle (1,1); \draw[draw=none,fill=\inner] (0.3,0.3) rectangle (0.7,0.7); \end{scope} \pgfmathtruncatemacro{\i}{\i+1};}
That isn’t quite what we wanted!
When \i gets big enough that we reach the end of the row, we want to do two things:
- reset
\ito zero, like a carriage return on an old-fashioned typewriter; - decrease
\jby one, like a line feed (reading the square from top to bottom).
To do this I’m going to use a TeX if statement. These are a bit clunky, and there are more sophisticated ways, but here this will do what we need. The format is like this:
\if\a\b % do something\else % something else\fi
This tests whether \a equals \b and takes the first action if it does and the second if not, and the \fi closes the statement.
We are counting from \i=0, so want the line to reset when \i=2. We test this using \if\i2.
\pgfmathsetmacro{\i}{0};\pgfmathsetmacro{\j}{0};\foreach[remember=\i as \i,remember=\j as \j] \outer/\inner in {white/gray,gray/white,black/black,black/white,white/black,gray/gray,gray/black,black/gray,white/white}{ \begin{scope}[shift={(\i,\j)}] \draw[draw=none,fill=\outer] (0,0) rectangle (1,1); \draw[draw=none,fill=\inner] (0.3,0.3) rectangle (0.7,0.7); \end{scope} \if\i2 \pgfmathtruncatemacro{\i}{0}; % CR \pgfmathtruncatemacro{\j}{\j-1}; % LF \else \pgfmathtruncatemacro{\i}{\i+1}; \fi}
Now we are back to the 3 by 3 grid with all the right colours, except the inner squares are missing their borders. In the original image, the inner squares only have borders when the inner and outer colours match, and the border colour isn’t the same for each colour. This means the code to deal with this is going to have to be complicated if statement.
Without getting more complicated, basic if statements in TeX work better on numbers than strings, so first I’m going to convert my colours to numbers. Actually this will help me write out the square since it means an orthogonal Latin square something like this, where the number on the left is the Latin letter/big square and the number on the right is the Greek letter/small square.
| 0/1 | 1/0 | 2/2 | | 2/0 | 0/2 | 1/1 | | 1/2 | 2/1 | 0/0 |
Here I define my colours using the xcolor package, calling them 0, 1 and 2. Do this in the header before \begin{document}.
\usepackage{xcolor}\definecolor{0}{rgb}{0.98, 0.96, 0.96} % white\definecolor{1}{rgb}{0.40, 0.36, 0.38} % grey\definecolor{2}{rgb}{0.06, 0.05, 0.03} % black
For the if statement, I’m going to create a new command in the header of my LaTeX document (outside the TikZ environment) that takes in the current colour and returns a sensible border. In this case, if the value of the input (#1) is grey (1) then we return black (2), and otherwise we return grey (1).
\newcommand{\innerline}[1]{\if#112\else1\fi}
Now we need an if statement inside our loop to detect when the inner and outer colours match. If they do, instead of draw=none to produce no border, the inner square needs to call our new command \innerline. Putting all that together, here’s the full example.
\documentclass{standalone}\usepackage{tikz}\usepackage{xcolor}\newcommand{\innerline}[1]{\if#112\else1\fi}\definecolor{0}{rgb}{0.98, 0.96, 0.96} % white\definecolor{1}{rgb}{0.40, 0.36, 0.38} % grey\definecolor{2}{rgb}{0.06, 0.05, 0.03} % black\begin{document} \begin{tikzpicture} \pgfmathsetmacro{\i}{0}; \pgfmathsetmacro{\j}{0}; \foreach[remember=\i as \i,remember=\j as \j] \outer/\inner in {0/1,1/0,2/2,2/0,0/2,1/1,1/2,2/1,0/0}{ \begin{scope}[shift={(\i,\j)}] \draw[draw=none,fill=\outer] (0,0) rectangle (1,1); \if\outer\inner \draw[draw=\innerline{\inner},fill=\inner] (0.3,0.3) rectangle (0.7,0.7); \else \draw[draw=none,fill=\inner] (0.3,0.3) rectangle (0.7,0.7); \fi \end{scope} \if\i2 \pgfmathtruncatemacro{\i}{0}; % CR \pgfmathtruncatemacro{\j}{\j-1}; % LF \else \pgfmathtruncatemacro{\i}{\i+1}; \fi } \end{tikzpicture}\end{document}
And the output:
The full version is a bit more complicated, but follows the same basic principle. It defines nine colours, so has a more complicated command for choosing the inner border colour, and of course uses a longer list of colour pairs in the for loop. It also does the carriage return and line feed later, when \i=9.
View the full code.
Here’s the result, a Latin square Euler thought wasn’t possible.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of March 2024, is now online at Tom Rocks Maths.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
In this series of posts, we’ll be featuring mathematical podcasts from all over the internet, by speaking to the creators of the podcast and asking them about what they do.
We spoke to Louie and Sam, two students at Durham University, who host a podcast there for their uni’s student radio station Purple Radio.
Podcast title: Chalkboard Ultra
Website: purpleradio.co.uk
Links: Spotify, Apple Podcasts, Purple Radio On Demand
Average episode length: General conversations 25-30 minutes, interviews 40-50 minutes.
Recommended episode: Shakespeare by Chance (our introductory episode), and Maths in Motion with Dr Adam Townsend.
What is your podcast about, and when/why did it start?Something we noticed growing up was a lot of people were turned away from maths at the secondary school or sixth form age and never really came back. They never saw the day-to-day uses, but it doesn’t have to be that. Sometimes it’s the pure curiosity that leads us to where we are. Chalkboard Ultra is a student-run podcast that discusses interesting concepts throughout all of mathematics, with a focus on the unseen areas that often slip under the radar… or behind the chalkboard in our case!
Who’s the team behind Chalkboard Ultra?Louie and Sam are undergraduate mathematicians at Durham University. Louie studies Maths and Stats and gravitates towards the applied statistical concepts, whereas Sam is more enthused by mathematical physics and the geometry of space around us. We publish our podcast through Purple Radio, Durham University’s award-winning radio station that provides students with experience in broadcasting media. We as hosts are grateful for use of their facilities.
LouieSamWho is the intended audience for the podcast? This podcast is aimed at those who are mathematically curious, but don’t know where to begin without picking up a calculator or reliving past traumas of secondary school. Yes, it is a maths podcast, but it is a podcast at the end of the day, and should be a conversation exploring our own interests while guiding the listener through our intuitions. We don’t expect any prior knowledge of maths at any level; everything is explained via a story without any calculations. However, for those that would like more information, we have ‘Chalkboard Ultra Book Reviews’ in which we recommend any literature that could take the listener on their own personal journey of exploration.
What is a typical episode like? What is the format? How long are episodes? How often are they released?The student-led episodes are conversations between the co-hosts, which are structured in a way that tells a story with an overarching theme of something that relates to maths. Some examples include the infinite monkey theorem, time travel, numerology, and the art of gambling. These are our main episodes and last 25-30 minutes. On occasion, we invite special guests – whether that be fellow students or associate professors – to relive their own journey through academia and discuss what they find engaging. These tend to be longer episodes of 40-50 minutes. The episodes are released weekly through our university’s term time. Most episodes also feature an ‘outtakes’ or ‘post-credit’ session, just for fun.
Why is it different from other mathematical podcasts?Too many times, have students refrain from bettering their understanding by not attending office hours or not asking questions. This is down to viewing lecturers and tutorial leaders on a different level of hierarchy. But this shouldn’t be the case! Almost all professors were once in our shoes, curious, and seeking help, and there is always a story to tell behind that. We are students, and we talk to our very own university professors and some PhD students to find out more of their own experiences with maths, where they came from and where their research may take them. This bridges the student-lecturer gap and hopes to bring together the academic community.
What are some highlights of the podcast so far?Most of our favourite moments appear as post-credit sessions. Episode 6 features a series of maths jokes and bad maths puns. In Episode 9, one of our guests reduces Bertrand Russell’s Principia Mathematica to 100 pages of mathematical foreplay. In Episode 10, Sam gets (nearly) cancelled after his attempts to strap a piece of buttered toast to a cat. On the back of this, in Episode 13 Louie summons a demon and banishes Sam to another realm (London).
What exciting plans do you have for the future? Both Louie and Sam are Masters students and therefore have one more year of Chalkboard Ultra. This means one more year of interesting concepts, exciting guests, and perhaps interviewing some of the more popular people in the world of maths communication. However, in the far future we may take it past the university and go rogue. Or maybe a new generation of chalklings will take control of the airwaves.
A few months ago a group of us launched a membership club, The Finite Group, which you can join!
A big update is the lineup — your membership now supports the work of and gives you access to content from mathematician and TikTok star Ayliean MacDonald, as well as Chalkdust’s Matthew Scroggs and The Aperiodical’s Katie Steckles and me. Membership gives you access to a chat community and monthly livestreams. For a taste of the livestreams, check out this π minute video!
The big news is that the next livestream will be free to view live online on 27th March from 5-6pm GMT. All four of us will be working through the recent ‘100 Mathematical Conventions Questions’ quiz that’s been dividing (a small subset of) the internet. The stream will be available live, and a recording will be available to members afterwards.
In this series of posts, we’ll be featuring mathematical video and streaming channels from all over the internet, by speaking to the creators of the channel and asking them about what they do.
We spoke to Wolfram about their CEO Stephen Wolfram and his Twitch streaming channel.
Channel title: Stephen_Wolfram
Link: twitch.tv/stephen_wolfram
Topics covered: Science & Technology, Language Design, Business
Average stream length: 1 hour 15 min
Recommended videos:
- Math Storytelling: Part One
- Live CEOing Ep 748: Language Design Review of Calculus & Algebra Features for 14.0
- Any of the Science & Technology Q&A for Kids and Others. What is the channel about, and when did it start?Stephen’s channel is a broad attempt at transparency in software development (LiveCEOing is a series of actual livestreamed internal design review meetings), democratizing learning about computer science, technology, mathematics, physics and all things computational (his weekly Q&A series), and a behind-the-scenes look at scientific research and what’s possible now with modern computation (his Research working sessions), as well as special events/focus on certain areas he’s actively exploring.
Stephen began livestreaming about 5 years ago with internal software design meetings which he thought would be interesting. The livestreams turned out to be useful internally with the benefits of recorded meetings immediately available, instant feedback from users, and allowing for positive recruiting efforts to viewers who expressed interest. These livestreams help others understand what Wolfram Research does.
Once the pandemic happened, Stephen wanted to find a way to engage students and share his own knowledge as he enjoys interacting with them . He began the Science & Technology Q&A for Kids (and others) series which turned out to be a helpful exercise to clarify Stephen’s own thoughts. Stephen then added a Business, Innovation, and Managing Life Q&A series to explore business ideas and think abstractly about things in his own life. Here’s a collection of links to all of Stephen’s livestream series.
Who is the stream host?Stephen Wolfram is the creator of Mathematica, Wolfram|Alpha and the Wolfram Language; the author of A New Kind of Science; the originator of the Wolfram Physics Project; and the founder and CEO of Wolfram Research. Over the course of more than four decades, he has been a pioneer in the development and application of computational thinking—and has been responsible for many discoveries, inventions and innovations in science, technology and business.
Who is the intended audience for the channel?Anyone interested in computation, science or business – from students to professors, researchers to professionals. Stephen covers a wide range of topics in his weekly series, and many have underlying mathematical concepts he explores live in his responses to audience questions.
What is a typical video like?The streams vary from “Live CEOing” – working corporate meetings – to “Q&As for kids [and others]”, to Live Coding Competitions, to streams, Q&As and working sessions focused on the Wolfram Physics Project. Stephen’s “day job” as the CEO of Wolfram Research keeps him heavily involved in the design and implementation of the Wolfram Language. As new functionality is “born”, you’re invited to see the process, ask questions, influence real-life software design and help name new functions!
You might see some new Wolfram Language function being tried out (often based on code that’s only days or even hours old). You might see a discussion about software engineering, or trends in machine learning, or the philosophy of science, or how to handle some issue of popular culture, or what it’s going to take to fix some conceptual bug. You might see some new area get started, you might see some specific piece of Wolfram Language documentation get finished, or you might see a piece of final visual design get done.
There’s quite a range of people in our meetings, with a whole diversity of accents and backgrounds and specialties. And it’s pretty common for us to need to call in some extra person with specific expertise we hadn’t thought was needed.
Occasionally, Stephen also livestreams special events (Celebrating 35 Years of Mathematica) and research working sessions to offer a behind-the-scenes look at scientific research and what’s possible now with modern computation.
Why should people watch? Why is it different to other mathematical video content?Stephen’s livestreams provide insight and context, not from a book or curriculum, but actual real-life CEO and Physics-meets-Mathematics-and-Computation perspective.
What are some highlights of the channel so far? Some favourites:
- Physics Project launch
- The Long Story of How Neural Nets Got to Where They Are: A Conversation with Terry Sejnowski
- History & Future of Science & Technology Q&A series
You know how loads of things in maths are named for the wrong person? In 1996, a fun quiz appeared in The Mathematical Gazette based on history of maths misconceptions. It contained a series of questions where the obvious answer is not correct, such as “Who discovered Cramer’s rule?”, “Did Pascal discover the Pascal triangle?” and “Who first published Simpson’s rule?”
I was looking for a demo to show my students that generative AI programs are not producing accurate knowledge when I thought of this quiz. I put its questions to ChatGPT to see how it did. The point of the exercise is that these systems just parrot back words from their training data without any concept of truth, so if the training data is full of misconceptions, so too will be the responses. But these are misconceptions from the 1990s, so how much influence will they have on the responses?
You can see how ChatGPT did when I gave it the quiz in a short, free to read, open access paper in the British Journal for the History of Mathematics: Generative AI and accuracy in the history of mathematics.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of Feburary 2024, is now online at Fractal Kitty.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Here’s some mathematical news we didn’t otherwise cover this month.
A collaborative project involving Dennis Gaitsgory and several pals claim they are compiling a proof of the geometric Langlands conjecture, consisting of a series of papers. (via Anton Hilado)
It’s been announced that all European Mathematical Society journals will be diamond open access in 2024. It follows their Subscribe To Open programme, and means that “for the first time the [EMS] Press’s annual journal output will be entirely open access, with a blend of S2O and Diamond publications”.
Maths history news: it’s been making the rounds this month that the invention of the decimal point was actually much earlier than thought, and can be pushed back 150 years to the work of Giovanni Bianchini in the 1440s. The findings have been published by mathematician and maths historian Glen Van Brummelen, who previously spoke about this in 2020 to maths podcaster Sam Hansen for an episode of their Relatively Prime podcast.
And finally: Richard Parker (pictured), one of the authors of the legendary Atlas of Finite Groups, has died, and is remembered fondly by fellow Atlas bod Robin Wilson in this lovely blog post. (via Peter Cameron)
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of August 2023, is now online at Reflections and Tangents.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
A conversation about mathematics inspired by an old textbook, Mathematics in Theory and Practice, edited by Warwick Sawyer. Presented by Katie Steckles and Peter Rowlett.
Here’s a selection of mathematical stories that crossed our desk in August.
Maths Research NewsResearchers have discovered that a shape can be designed to trace almost any infinite periodic trajectory when rolling down a slope, as seen in this Nature.com video (via Jeroen van Dorp)
A new diamond open access journal, Innovations in Graph Theory, has been founded. The first issue of the journal is expected to appear in 2024. (via Peter Cameron)
And in important publication news for silly season: Erik Demaine and Martin Demaine have achieved “the true ideal of an unordered set of equal authors, where every author comes first”. Their paper Every Author as First Author proposes a new standard for writing author names on papers and in bibliographies, which places every author as a first author, with the names all superimposed on top of each other, including details of the \namestack LaTeX command for this purpose. The results are predictably hilarious (see below). (via Nalini Joshi)
Other NewsAlison Kiddle has been posting daily conversation prompts involving LEGO to stimulate mathematical thinking on their blog every day in August, and people have been responding on Twitter and Mastodon.
Mathematician and logician Peter Aczel has died, as has Ian G. Macdonald (who introduced Macdonald polynomials).
In this series of posts, we’ll be featuring mathematical podcasts from all over the internet, by speaking to the creators of the podcast and asking them about what they do.
We spoke to Andy Lumley, Head of Learning Technology for educational charity MEI, about their podcast Digging For The Why.
Podcast title: Digging For The Why
Website:https://mei.org.uk/digging-for-the-why-a-new-mei-podcast-for-key-stage-2-3-teachers/
Links: Apple Podcasts, Spotify, Google Podcasts, Amazon Music, Stitcher, Buzzsprout
Average episode length: 45 minutes
Recommended episode: Episode 7 with Richard Perring has some real nuggets in there!
What is your podcast about, and how did it start?DFTW started out as a classroom mantra whilst I (Andy, right) was still teaching. I always strived to encourage my students to ask why and I always asked them why to really try and gauge understanding. When I joined MEI and met Alison, we shared so many similar thoughts and traits that we decided to start a podcast around the idea, and importance, of asking why. DFTW is essentially that – digging for the why. We talk to people from the maths community (teachers, lecturers, authors) about why we should dig for the why to encourage better conversations, better learning, and better understanding. Season 1 focused on the transition from primary to secondary but season 2 (out soon!) goes much broader but still focuses on the benefits of asking why.
Who publishes your podcast? Tell us about your organisation. MEI (Mathematics in Education and Industry) is a national maths charity with the goal of improving maths education for all. DFTW is our first podcast and could, hopefully, lead to more down the line. I (Andy Lumley) am the Head of Learning Technology for MEI (in charge of Integral alongside other projects) and Alison Hopper (left) is a Maths Education Support Specialist for primary maths.
Who is the intended audience for the podcast? Season 1 is aimed at anybody teaching years 5-8, primarily, but we think anybody involved in teaching maths at any level can find useful things in every episode. Season 2 is for anybody involved in maths education!
What is a typical episode like?DFTW follows a simple format – Alison and I discuss a given topic (e.g. Mastery in the classroom) and try to have a guest on each episode to bring their expertise to the show. Episodes tend to be 30-45 minutes long (some go a bit longer when there is just too much to talk about!) as we tried to make it a commute length listen. Season 1 was released weekly in the spring/summer of 2022, and season 2 will be released weekly in the new academic year 2023/24.
Why should people listen? Why is it different to other mathematical podcasts?As with all maths teachers, ideas and thoughts differ. It is always valuable to listen to fellow educationalists discuss topics and themes even if it is something you feel really good about as you never know what might stick with you.
What are some highlights of the podcast so far?The real highlight for me (Andy) personally has been learning so much about the primary sector of teaching maths. The arrogant secondary teacher in me wants to travel back in time and really ensure that NQT Andy gives real value to the 11-year mathematical journey his students have already been on. There is so much talk about Mastery in the maths classroom at the moment so to hear from Steph Kirk in Episode 2 about how she is able to influence the teaching in her school as it only had year 7s at the time was really insightful.
What exciting plans do you have for the future? Season 2 features a wider range of guests from the ever-amazing Ben Sparks through to world renowned author Alex Bellos so we can’t wait to get them out there later in 2023. Beyond that, we plan to expand further and try to take DFTW international with our guests and see what the rest of the world thinks about the need to dig for the why!
A conversation about mathematics inspired by a scone. Presented by Katie Steckles and Peter Rowlett, with special guest Sophie Maclean.
I was interviewed by Nira Chamberlain, President of the Mathematical Association. I am the twelfth person to whom he has asked his question “what is the point of mathematics?” Hoping to offer something a little different, I spoke about teaching students the role mathematical modelling can play in sustainability.
In this series of posts, we’ll be featuring mathematical podcasts from all over the internet, by speaking to the creators of the podcast and asking them about what they do. We spoke to Beth Goodliff from the National Centre for Excellence in the Teaching of Mathematics, about their in-house podcast aimed at maths teachers. Podcast…
Martin Gardner’s long-running column in Scientific American made it onto the front cover of the magazine twelve times. Gathering 4 Gardner refers to these cover stories as “A Gardner’s Dozen“, while pointing out that these aren’t his ‘greatest hits’ and the magazine artists didn’t necessarily reproduce the graphics as he would have liked them. Nevertheless,…
A conversation about mathematics inspired by the Joukowsky aerofoil. Presented by Katie Steckles and Peter Rowlett.
We asked guest author Elliott Baxby to take a look at Andrew Pontzen’s latest book, The Universe in a Box: A New Cosmic History. Ever since I became interested in mathematics, I have always wanted to learn more about science. I love mathematics, and I can easily spend most of the day reading about it…
Mathematician and ninja mathematical-thinking-prompter Alison Kiddle has been posting an image each day for the whole of August, each prompting some kind of mathematical question or discussion. If you have small mathematicians in your life and enjoy #tmwyk (talking maths with your kids), or are yourself a mathematician of any size continuing to marvel at…
In the 1901 paper that named the game Nim and provided its mathematical analysis, Charles Bouton defined “safe combinations”, positions that if you leave the game in this state, your opponent cannot win. In combinatorial game theory, these are (\mathcal{P}) positions (the previous player has already won), as opposed to (\mathcal{N}) positions (the next player…
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of July 2023, is now online at Tony’s Maths Blog. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Here’s some mathematical news that didn’t make it on to the site otherwise this month. Maths News There’s been more abc conjecture drama: Peter Scholze and Jakob Stix are in line for a ¥140m (around £766k) prize for their paper pointing out the flaw in Mochizuki’s claimed abc proof – if they publish it in…
A conversation about mathematics inspired by a guitar. Presented by Katie Steckles and Peter Rowlett, with special guest Sam Hartburn.
We spoke to friend of the site, award-winning maths communicator and past math-off competitor Kyle Evans about his Edinburgh Fringe show for 2023, which is about maths. Who are you (as if we don’t already know)? I’m Kyle D Evans, I’m a teacher by day and entertainer/performer/presenter of all things mathematical by… well, also by…
A conversation about mathematics inspired by a 1960s game designed to teach set theory. Presented by Katie Steckles and Peter Rowlett. On-Sets: A Vintage Set Theory Game by Peter Rowlett is free to read in Math Horizons.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of June 2023, is now online at Double Root. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Hexaflexagons are great. If you haven’t seen one before, you’re about to have a lovely time. If you have seen them before, the reason I’m writing about them is that I’ve made a webpage that helps you create a template for a hexaflexagon with your choice of picture on each of the three faces. For…
A couple of days ago, a question occurred to me: What’s the furthest I’ve ever been from anyone else? My first guess was on the order of 100km – I’ve travelled a decent amount, to some fairly empty places. But I quickly realised that was way too high, because almost all of my long-distance travel…
A conversation about mathematics and literature inspired by a book. Presented by Katie Steckles and Peter Rowlett with special guest Sarah Hart, author of Once Upon a Prime: The Wondrous Connections Between Mathematics and Literature.
The UK Government have announced the first set of King’s Birthday Honours for King Charles III. Here’s our selection of particularly mathematical entries for this year. If you spot any more, let us know in the comments and we’ll add to the list. Get the full list of honours on gov.uk.
A conversation about mathematics inspired by a Battenberg cake. Presented by Katie Steckles and Peter Rowlett.
This is a guest post by David Benjamin, who’s previously written several other guest posts on various topics. It’s unavoidable that part of doing mathematics will always involve arithmetic: the simple calculations, additions and multiplications that so much else is built on. But the beauty of mathematics is that even these basic operations can be…
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of May 2023, is now online at Eddie’s Math and Calculator Blog. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
A conversation about mathematics inspired by the new aperiodic monotile. Presented by Katie Steckles and Peter Rowlett, with special guest Chaim Goodman-Strauss.
The paper announcing the discover is An aperiodic monotile by David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss.
Chaim was recording from MoMath in New York, which will be running a creative artwork competition based on the monotile with UKMT. Chaim also mentioned a meeting in Oxford: Hatfest: celebrating the discovery of an Aperiodic Monotile.
Note: This podcast was recorded after the discovery of the ‘hat’ and ‘turtle’ monotiles but before the announcement of the ‘spectre’ monotile. Confused? Don’t worry, we explain in the episode!
Surely you didn’t expect news about aperiodic tilings to appear at regular intervals? You know how it is – you wait ages for a new aperiodic monotile discovery to come along, then two come in quick succession.
In March, we covered the discovery of an aperiodic monotile. The team of authors behind that discovery have been continuing their work and this week have an even bigger announcement.
The only slightly dissatisfying aspect of the previous discovery, as you may recall, was that in order to tile the plane, the Hat (and Turtle) tile each needed roughly one in every six tiles to be mirrored. This raised the question of whether a tiling using a single aperiodic tile would be possible without reflections.
It turns out the authors had kind of already found one – a tile they referred to in their previous paper as “Tile(1,1)” was the basis for a chiral aperiodic tile – one which doesn’t need mirroring in order to fully tile.
It’s actually the midpoint of the continuum of possible tile shapes that contains the hat and turtle, but had been previously considered uninteresting, since it can tile periodically if used along with its reflection. However, if you don’t use reflection, it gets more interesting.
Tile(1,1) has now been shown to be weakly chiral – meaning that if you use it without reflections, it must tile non-periodically.
This means we immediately have an infinite family of tiles with the property we’re looking for – by replacing each of the straight edges with anything that’s asymmetrical and oriented consistently, we can force the tiles to be oriented all the same way, producing an aperiodic monotile in the strong sense. (This is similar to what Edmund Harris shared in our math-off as a way to force Penrose tiles to be aperiodic.) The authors have used a simple curved line, creating a vaguely ghost-like shape they’ve christened The Spectre (because it doesn’t have a reflection, lol)
Figure from the paper: “The 14-sided polygon Tile(1, 1), shown on the left, is a weakly chiral aperiodic monotile: if by fiat we forbid tilings that mix unreflected and reflected tiles, then it admits only non-periodic tilings. By modifying its edges, as shown in the centre and right for example, we obtain strictly chiral aperiodic monotiles called “Spectres” that admit only non-periodic tilings even when reflections are permitted.”If you’d like more context and an idea of how the proof has been achieved you can check out Craig Kaplan’s webpage and his excellent Mastodon thread about it.
This is a review of David Acheson’s new book, which we were kindly sent a copy of to read.
In The Spirit of Mathematics: Algebra And All That, David has pulled together a collection of what he refers to as ‘elegant mathematics using only simple materials’ – neat, short algebraic proofs and definitions, models of physical systems and mathematical tricks and curiosities.
He includes all the classics, from proof by induction to Fibonacci numbers to hitting a snooker ball, and each is presented with enthusiasm, alongside stories of mathematicians – and fearlessly including all the equations and derivations (if every equation really did halve your readership, as Stephen Hawking believed, this would be a very brave book to publish). But the maths is well-explained and very approachable, and it’s refreshing to see it featured so prominently outside of a textbook.
The book is also filled with helpful diagrams and illustrations, as well as humorous asides, cartoons and pictures of many mathematicians (sadly, only one female mathematician is featured, and she’s included only for her joke about how hard she’s found it to get a proof…) – but the book is well-produced and clearly laid out, with well-defined, short chapters each with a clearly defined topic.
The result is a compendium of intriguing ideas which would fascinate and compel a keen mathematician wanting to learn more, and provide hours of intrigue and jumping-off points for further investigation. Most topics are only covered briefly, so a deeper understanding would need research elsewhere, but for an enthusiastic reader this would happen naturally. Each discovery is motivated by a real-world example, or an interesting puzzle or curiosity, and all the key topics from algebra are touched on in one way or another.
However, this book wouldn’t suit an inexperienced mathematician – given which steps in the calculations are described as ‘simple’, a reasonable level of maths is assumed, and I’d imagine a strong GCSE or A-level student, particularly one already keen to learn more, would get much more out of it than a younger student. It’d also suit an adult wishing to refresh their mathematical knowledge from school and pick up some new ideas. But despite the blurb on the back claiming ‘for those who dread the subject, this book may be an eye-opener’, I suspect that such a reader might struggle in places.
Overall, this is a well-presented celebration of the best parts of mathematics, and showcases just how powerful maths can be.
It’s been a busy few months! As per our name, here’s an aperiodically-timed round up of things that have happened in the world of maths in the last few months.
Maths Research BreakthroughsAccording to Terry Tao, there’s been a big achievement in Ramsey theory. Tao says:
the long standing upper bound of $(4+o(1))^k$ of the size $R(k,k)$ of a graph required to force either a clique or independent set of size $k$ has finally been reduced to $(4-\varepsilon)^k$ for some positive constant $\varepsilon$. From what I understand, they have developed a new “Book algorithm” to more efficiently locate cliques and independent sets based on recursively finding companion graphs that they call “books”.
A later update adds that the argument gives $\varepsilon = 2^{-7}$. You can read the work directly in the ArXiV paper.
(via Terence Tao on Mastodon)
There’s also a Twitter thread in which Tim Gowers describes the experience of attending one of the Ramsey Theory seminars – the coauthors delivered seminars about it in different places – and calls the problem “perhaps the top open problem in extremal combinatorics”.
According to a New Scientist article (£), there’s been new research (from a paper on PsyArXiV) into different ways of projecting perspective onto a flat image, including how the human brain perceives it. New Scientist connects this to why the moon looks so small in photos versus reality, and how first person video games represent the world.
And of course the big news has been the discovery of the first true aperiodic monotile, described by Henry Segerman as “a shape that forces aperiodicity through geometry alone, with no additional constraints applied via matching conditions” (via Henry Segerman). As well as our own extensive write-up here, Andrew Stacey has announced that his development version of the TikZ library for drawing Penrose (and similar) tiles has been updated to include the new aperiodical hat and has also released an update focused on drawing the new polykite monotiles and clusters (announcements via Andrew Stacey on Twitter and Andrew Stacey on Mastodon).
Other Quick-Fire Maths News The latest Chalkdust Magazine (Issue 17) was released at 9am on Monday 22nd May. The magazine contains articles on teaching maths in prisons, modelling penguin huddles using techniques from fluid mechanics, and Möbius strips. You can read it online or order copies on their website. * There’s a sale 50% off Springer yellow maths books until the end of June (with a separate UK page for the same sale). (via Filip W) * Tim Harford’s done a kids’ book – The Truth Detective equips kids with the mathematical and statistical tools to make good judgements about the world and dig out the truth. It looks good! * They’ve appointed a new director of GCHQ, and as well as being the agency’s first female director, Anne Keast-Butler is a mathematician. * The New York Times have launched a new daily game, Digits, that’s pretty much just the numbers game from Countdown. * The Inuit Kaktovik numeral system, invented by school students who noticed that their native language uses base twenty to name numbers, and wanted symbols to match, is now available in Unicode – as detailed in this great Scientific American writeup, which tells the fascinating story. (via Token Sane Person)* * The Royal Society has put online its collection of letters and manuscripts going back to the society’s founding, including letters from Fermat, Newton and Leibniz.
People seeking thingsThe British Society for the History of Mathematics is asking for nominations for its Neumann Prize, awarded to a general interest history of maths book.
The PolyPlane project plans to create a beautiful art project featuring polyhedra arranged in a room by their numbers of faces, edges and vertices on three axes (which thanks to Euler’s identity, will all lie in a beautiful diagonal plane) and is seeking volunteers to make polyhedra to include in the work (via Henry Segerman)
The Bernoulli Center has issued a call for research program proposals in mathematics, theoretical physics and theoretical computer science. The deadline for submissions is 18th June 2023. (via Terence Tao)
Fluid dynamicists at the University of Leeds are running a photography competition, asking for students aged 7-14, in teams of up to 4, to submit a photo or collection of photos showcasing fluid dynamics phenomena in action. The closing date is 9th June.
The German mathematical union is offering two prizes for representing maths in the media, including a journalism prize and one for ‘for outstanding achievements in presenting mathematics to the public’, which can go to a non-journalist. (via Martin Skrodzki)
And 3Blue1Brown himself, Grant Sanderson, has launched this year’s Summer of Mathematical Exposition competition, awarding prizes for the best online maths explanations. The closing date is 18th August, and more details are available on the SOME website.
Awards & EventsIt’s been awards season! Luis Caffarelli has won the 2023 Abel prize, C.R. Rao has been awarded the 2023 International Prize in Statistics and the IMA Gold Medal 2022 has been awarded to mathematical biologist Philip Maini.
Ukraine was awarded best European team at the European Girls’ Mathematical Olympiad 2023, which took place in April in Slovenia. Слава Україні! (via Rob Corless)
MathsCity in Leeds has announced a half-term board games event from from Saturday 27th May – Sunday 4th June (tickets available on their website), with chances to play some favourite mathematical board games including Laser Maze, Genius Square and Rush Hour.
And if you’re interested in maths communication in any form, registration for the 2023 Talking Maths in Public conference, which is taking place in Newcastle upon Tyne and fully hybrid online, is now open. Tickets cost just £125 (£30 online) for three days of workshops, networking and discussions on all kinds of topics around maths communication, and a chance to meet others who work or participate in sharing maths.
Maths Education NewsThe UK Prime Minister Rishi Sunak has announced he’s setting up a review to tackle the UK’s ‘anti-maths mindset’. This follows his comments in January about making maths compulsory to 18, which were met with mixed reviews.
The Oak National Academy, which was set up in the pandemic and provides teaching resources for schools, is developing a new maths curriculum and teaching resources for secondary and primary maths in partnership with MEI.
Five UK maths education organisations (ATM, AMET, The MA, NAMA, and NANAMIC) have voted to create a new charitable organisation AMiE (Association for Mathematics in Education) and to explore merging into it.
Sad newsThere have been several mathematical death announcements recently, including:
- combinatorist and number theorist Vera Sós (via David Eppstein)
- maths puzzles and games giant Ivan Moscovich
- topologist André Haefliger (via David Roberts)
And with great sadness we share the death of friend-of-the-site Vicky Neale, who was a pillar of the mathematical outreach community and an inspiration to many. She died earlier this month after a long illness, which she spoke about on her podcast Maths + Cancer. The University of Oxford has set up a tribute page which is full of stories, memories and messages thanking her for her great work and influence.
This is a guest post from maths communicator Max Hughes. If you’re thinking about going to the Talking Maths in Public conference this summer, read on to find out what it’s like.
I first attended Talking Maths in Public in Cambridge at the end of August of 2019. At the time I was just about to go into my second year of a maths degree – knowing that I wanted to go into maths communication and outreach after finishing university, and keen to learn more.
Having never attended a conference before, I wasn’t quite sure what to expect when I arrived in Cambridge the evening before it kicked off. Luckily for me, I didn’t need to spend the evening alone in the hotel dwelling on my pre-conference nerves, as there was a planned pub meet up later in the evening for those who had already arrived in Cambridge.
Full of trepidation, I stepped into the lift of the hotel I was staying in, unaware that when the doors opened, I would be transported to a fantastic world of nerdery that I have never since left: the UK maths communication community. This may sound dramatic, but much like Charlie taking his first steps into the chocolate room of Willy Wonka’s factory, or Dorothy opening a door into the technicoloured world of Oz, I knew for certain that this was a seminal moment in my life.
Congregated before me in the hotel lobby was an almost perfect microcosm of the conference attendees. In my first moments at this gathering, I was greeted by a lecturer from my university, multiple popular mathematics YouTubers and speakers, a freelance editor, a couple of teachers and a mathematical knitting enthusiast… to name just a few. The other attendees made me feel almost instantly at ease in this new environment, and I went to the pub confident and excited to see what the rest of the conference would bring.
Presenters Brady Haran, Anand Jagatia and Jen (Primrose Kitten) talking about Maths on YouTubeDay one of the conference started with some useful icebreaker activities followed by an engrossing talk on “Maths on YouTube”. After a short break there was a choice of 5 different workshops to choose from, presented by attendees demonstrating their own mathematical engagement activities. Forever a fan of the fantasy genre, I chose to attend an intriguing workshop that used dragons to engage and excite people about the wonders of mathematics.
The second day started off with my favourite session of the conference; a series of lightning talks by speakers and attendees talking about their work, that covered a diverse range of different types of events and projects, helping to paint a picture of the maths communication landscape not just in the UK, but globally. Following this was an important session on promoting inclusivity within engagement activities, as well as some smaller group discussion sessions and a panel on writing about mathematics. The final keynote session of the day was particularly spellbinding, with magician Neil Kelso informing the crowd on how to astonish audiences, showing that maths and magic have an unexpected amount in common.
By far the best part of the conference was the people, from the attendees brimming with mathematical possibility to the organisers who were passionate and invested in running a conference that was in equal parts informative, accepting and fun. The final morning of the conference was centred around just this, the people, in a range of networking events. This is how I found myself punting along the river Cam with fellow attendees solving maths problems to aid us in a “treasure punt”.
Throughout the entire conference a palpable air of jovial mathematics permeated the space. Whether it was impromptu conversations on mathematical cabaret, tense games of “The Mind”, or mathematical discourse on the toilet blackboards (yes, that’s a thing at the Isaac Newton Institute), the creative expression of mathematical joy was awe inspiring. From exploring the ins and outs of a beautiful city to being engrossed in deep mathematical conversation with a newly made friend, there were new perspectives around every corner.
I can honestly say that Talking Maths in Public 2019 acted as the perfect first step in my journey into the world of maths communication. The skills, contacts, and inspiration I gained empowered me to start properly communicating maths, creating a domino effect that led me to being employed as a maths outreach professional who truly feels part of a community of UK based mathematics communicators.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of April 2023, is now online at Cassandra Lee Yieng’s blog.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of March 2023, is now online at Theorem of the Day.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Maths communicators: assemble! It’s that time again, when everyone’s favourite biannual maths communication conference happens (every two years, in case you weren’t sure). Talking Maths in Public is a conference for people who work in, or otherwise participate in, communicating mathematics to the public.
The event runs from 31st August – 2nd September 2023, and will take place at in the Herschel building at Newcastle University, and online. Sessions will include keynotes on science communication research and communicating maths online, a workshop on audience research, panels on Maths that Moves and Everyday Maths, as well as discussion sessions, skills workshops, networking and lightning talks – and the whole event costs £125 (day rates and bursaries available).
If you’re a maths or maths-adjacent communicator, we can recommend the event highly as a chance to meet other people who talk, write, blog, make videos, draw, sing or otherwise share their love of maths, and to pick up some new ideas and skills too. We’ll all be there! Details and booking are on the programme page at talkingmathsinpublic.uk/programme, and previous events have all sold out so don’t miss your chance!
Actual aperiodicity news on The Aperiodical!
This is probably the biggest aperiodicity news we’ll ever cover here: David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss have produced a single shape which tiles the plane, and can’t be arranged to have translational symmetry.
And it’s so simple!
The “hat” aperiodic monotile.Geometers have been looking for a shape with these properties for over 60 years, and until this example was found it wasn’t clear that one would exist at all.
The tile is made of eight kites – the shape you get by cutting a hexagon up through the midpoints of its edges.
In fact, they show that there’s a whole continuous family of aperiodic monotiles, obtained by changing the lengths of the edges in the shape shown above. Here’s an animation by Craig Kaplan showing a continuous transformation through the whole family:
Note that there are three points in the animation where the shape is degenerate (at the start, middle and end) because two adjacent edges become parallel, and those shapes can tile periodically.
The authors have put together a website to accompany their paper proving the shape is an aperiodic monotile. Have a go at reading the paper: it’s really well written, and starts with a detailed introduction describing the problem and its history.
There’s also an interactive tool for producing patches of the tiling. It wasn’t immediately clear to me how it works: you pick which of the basic clusters H, T, P or F you want to start with, and then click “Build supertiles” to apply the substitution process and end up with a bigger patch of tiles.
It’s worth noting that this is just a preprint, so a mistake in the proof might be found, but this announcement is credible: the authors are well-known geometers who have been working on this and similar problems for a long time, and the outline of the proof looks coherent.
The authors call their shape the “einstein hat”, punning on the German “ein” – one, “stein” – stone (or tile). It’s fairly safe to predict that if the “einstein” part sticks, future generations will be confused about whether Albert Einstein was involved. Opinion differs on whether the shape looks more like a hat or a t-shirt.
Hat, t-shirt, trousers?An explanation of what it’s all about that you can handwave your way through at the dinner tableThink about the 2D plane – an infinite, flat surface. How can you completely cover it up? If you’ve got an infinite supply of tiles, can you arrange them together on the plane so that there are no gaps?
If the tiles can be any shape you like, you can put them down however you like and then fill in any gaps with just the right shape. So it’s more interesting to restrict yourself to a certain, finite, set of different tile shapes.
Tilings of the plane with squares, triangles, and a mix of triangles and hexagons.You can do this with infinitely many squares of the same size, or with a mix of equilateral triangles and regular hexagons. If all you’ve got is regular pentagons, you can’t do it: no matter how you arrange them, eventually you’ll end up with a gap that’s too small to put a pentagon tile in.
Pentagons don’t tile the planeThe next question is: once you’ve put the tiles down, are there any symmetries? If you just used squares, then you can move every tile one space down and it’ll look exactly the same as it did before.
Tiling with squares has translation symmetryIs it possible to arrange the tiles so that there’s no translation symmetry – so that each point in the plane looks completely unique? This is called a non-periodic tiling.
If you split up a square into a few rectangles with the same proportions, you can produce a non-periodic tiling of the plane by by arranging them in a different configuration depending on their position on the plane. But you could also arrange them the same way everywhere, so there would be translation symmetry.
The interesting question is: are there any tiles, or sets of tiles, that can cover the plane, but never with translation symmetry – a truly aperiodic tiling?
The answer is yes: most famously, Roger Penrose found a pair of shapes – a kite and a dart, with specific edge lengths, or alternately a pair of rhombi, marked so that they obey certain edge-matching rules – that together tile the plane, but can never produce translation symmetry. Versions of the shapes which encode the matching rules, with chunks removed and added from the correct edges to force the matching (like the ones shared by Edmund Harriss in our Math-Off) constitute true aperiodic tile sets.
Penrose tiling with rhombi. From wikimedia, by Inductiveload, in the public domain.It’s also possible to tile the plane aperiodically using a single tile, called a monotile – for example, the pinwheel tiling consists entirely of copies of a right-angled triangle with sides of length $1$, $2$ and $\sqrt{5}$ – but this shape could also form a periodic tiling, and in order to force the tiling to be aperiodic, matching rules are needed.
What nobody knew until now was whether there’s a single tile shape that generates only aperiodic tilings, without needing to specify matching rules – an aperiodic monotile.
That’s what Smith, Myers, Kaplan and Goodman-Strauss have found. They’ve proved that it tiles the plane, which is the easy part, and then proved that it must tile aperiodically. They came up with a new technique for proving this – actually, two: they proved it twice, just to be sure.
That’s the short version of the story. If your dinner companions are still interested, here’s some more explanation of how the aperiodicity proof works. Maybe pause for a bit, make sure your dinner isn’t getting too cold, and do some finger exercises to prepare for all the handwaving you’re about to do.
The authors show that no matter how you put the tiles down, it will always be possible to divide it up so that each tile belongs to one of a set of four clusters – specific arrangements of 1, 2 or 4 tiles – and that the edges on adjacent clusters can only match up in certain ways.
The four clusters of tiles. From An Aperiodic Monotile.This part of the proof is done with computer assistance: there are lots of cases, and it’s likely you’d make a mistake while trying to draw them out on paper, so instead the authors rely on verifying that the code for their checking program is correct.
They then show that these clusters can themselves always be separated into larger groups called metatiles, which have the same symmetries as the basic tiles. So if the tiling when looked at as a collection of single tiles has translational symmetry, then looking at it as a collection of metatiles must also have that symmetry.
And then they show that the metatile tiling can’t have translational symmetry! So the monotile tiling doesn’t either!
The four metatile shapes.To show that the metatile tiling is aperiodic, they just do the same trick again, forever: they show that the metatiles form clusters, and after a few steps the clusters from one step look the same as the clusters from the previous step, except bigger. These self-similar shapes are called supertiles. (Good job the proof ends after this step, because they’re running out of words for “bigger than”!)
Once you’ve identified the supertiles, you can perform a substitution to obtain the next step of the clustering process.
A patch of the tiling, with several layers of supertiles superimposed.Remember that we’ve supposed you’re already looking at a complete tiling of the plane, and you’ve found a patch of adjacent supertiles. Replace each supertile with a certain arrangement of copies of the four possible supertiles, and the bigger patch of tiles you end up with must exactly match the tiling you’ve got, covering more of the plane than the patch you started looking at.
Because there’s no translation symmetry inside the supertiles, then there’s no translation symmetry among the metatiles, and hence the original tiles.
ReactionsThe shape is really easy to make. I’ve created a GitHub repository of files representing the shape in various formats, for use in graphic design or 3D printing.
Dan Piker added some Truchet-like markings to the tile to make this nice pattern:
Dave Richeson was quick off the mark to print the tile on his 3D printer:
He’s put his model file on Thingiverse for anyone else who wants to print their own.
Travis Howse skipped a dimension and used his laser cutter to produce a set of tiles:
Adam Goucher has blogged about the paper, noting that the ratio of flipped tiles to unflipped tiles is $\phi^4$.
Dan Anderson has drawn the monotile using Mathigon’s interactive geometry tool, Polypad.
This meme by John May, who admits that it’s terrible, will not help with the einstein/Einstein confusion:
Read the paperReally, read the paper! It’s very well-written, and deserves a lot of credit for going to extra lengths in the introduction to set the scene and provide the gist of the proof. If you just want to skim it, I suggest reading to the end of section 1.2 (“Outline”), and then the introductory paragraphs of each section after that. The subsections largely deal with the fiddly case-by-case checking that should be verified by somebody, but that needn’t be you.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of February 2023 is now online at SamHartburn.co.uk.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Here’s a round-up of the news stories not covered on the site over the past month.
Prizes and AppointmentsBaroness Ingrid Daubechies is the first woman to be awarded the Wolf Prize in Mathematics. Awarded annually to outstanding scientists and artists from around the world since 1978, the award consists of a certificate and a monetary award of $100,000. (via Nalini Joshi)
Maths communicator and TikToker Ayliean MacDonald has been appointed the first Community Mathematician at MathsCity Leeds. Ayliean will run a series of workshops and events at MathsCity, and wants to make maths a multi-sensory experience – sessions will include maths art activities, craft workshops, and maths-inspired food tasting!
The New Government chief scientific adviser Professor Dame Angela McLean is a mathematical biologist. Her PhD thesis was on ‘Mathematical models of the epidemiology of measles in developing countries’ and she has been active in creating models of COVID as a high-profile member of SAGE and SPI-M-O.
Other NewsThe OEIS foundation is looking to raise $3m to fund a full-time managing editor. Founded by Neil Sloane in 1964, the site has so far been run by volunteers, but now a committee of board members has been set up to help raise the necessary funds for an endowment. They have also released the entire source data of the encyclopedia on GitHub, under a Creative Commons Attribute Share-Alike licence. Previously, the data was available in a less-convenient form and only under a licence forbidding commercial use.
Humans can beat AI at Go again! As this article in the FT reports, Amateur Kellin Pelrine has found and exploited weakness in strategy systems that have otherwise dominated strategies used by the game’s grandmasters. (via @moreisdifferent)
The Office for Statistics Regulation has written to HM Treasury to tell it off for tweeting a graph with a non-zero vertical axis. The graph, which showed inflation statistics for January, started from 8% and “gives a misleading impression of the scale of the deceleration in inflation”.
And finally: well-loved mathematician and metagrobologist David Singmaster has died. He passed away earlier this month, and Lucas Garron has been collecting people’s memories of David Singmaster.
In December I organised a series of online public maths talks called What Can Mathematicians Do?
The recordings of the talks are now online, free for anyone to watch. You could go to the official page I put up on Newcastle University’s website, or you could just watch them here!
First, Tanya Gleadow talked about the maths of drawing with lasers, and I stepped in at the last minute to talk about the Herschel enneahedron:
In the second session, Abi Kirk talked about Euler’s formula for polyhedra, and Amy Mason shared a method for deciding what to watch next on Netflix:
Third, Chetna Petal talked about her mathematical career in “I introduce myself as a mathematician… yes, really” and then Lucy Rycroft-Smith talked about the maths of menstruation (yes, really!)
In the penultimate session, Sophie MacLean showed how to get rich by applying maths to stock trading, and Lauren Gilbert talked about her experiences as a disabled physics student at Newcastle:
Finally, Naomi Wray enthused about the dozenal system for writing numbers, and Matt Mack described how to make art using the Travelling Salesman Problem:
So there you go!
I reckon the series was a moderate success: I did gather ten disabled mathematicians to talk to the public about maths, but the format didn’t work too well. We tried to time the sessions so that schools could take part in the last week of school, but didn’t get much take-up. I don’t know if I should have been more persistent with reminding teachers who signed up about the sessions, or if it just wasn’t practical. It was interesting to work with BSL interpreters for the first time, and I’m glad to have made contact with all of the speakers, some of whom I hadn’t worked with before.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of January 2023 is now online at Ioanna Georgiou’s blog.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
We asked guest author Elliott Baxby to take a look at John Allen Paulos’ latest book, Who’s Counting.
Mathematics is an increasingly complex subject, and we are often taught it in an abstract manner. John Allen Paulos delves into the hidden mathematics within everyday life, and illustrates how it permeates everything from politics to pop culture – for example, how game show hosts use mathematics for puzzles like the classic Monty Hall problem.
The book is a collection of essays from Paulos’ ABC News column, on a huge range of topics from card shuffling and the butterfly effect to error correcting codes and COVID, and even the Bible code. As it’s a collection of separate columns, it doesn’t always flow fluently – I did find myself losing focus on some of the topics covered, particularly ones that didn’t interest me as much. This was mainly down to the content though – the writing style is extremely accessible and at times witty.
The book included some interesting puzzles and questions, which were challenging and engaging, and included solutions to each problem – very helpful for a Saturday night maths challenge! I even showed some to my friends, who at times were truly puzzled. I loved the idea of puzzles being a means of sneaking cleverly designed mathematical problems onto TV game shows. It goes to show maths is everywhere!
I enjoyed the sections on probability and logic as these are topics I’m particularly interested in. One chapter also explored the constant $e$, where it came from and where else it pops up – a very interesting read. It does deserve more attention, as π seems to be the main mathematical constant you hear about, and I appreciated seeing $e$ being explored in more depth.
This book would suit anyone who seeks to see a different side of mathematics – which we aren’t often taught in school – and how it manifests itself in politics and the world around us. That said, it would be better for someone with an A-level mathematics background, as some of the topics could be challenging for a less experienced reader.
It’s mostly enjoyable and has a good depth of knowledge, including questions to test your mind. While I didn’t find all of it completely engaging, there are definitely some points made in the book that I’ll refer back to in the future!
This is a guest post by Storm Reinbolt, outlining a historical mathematical incident which almost caused a misdefinition!
π is an irrational number that is equal to 3.1415926535 (to 10 digits). Things could have been different, however, if Dr. Edward J. Goodwin succeeded in passing Indiana Bill No. 246. This bill would have completely changed π and mathematics as a whole.
In 1894, Dr. Goodwin, a physician who dabbled in mathematics, claimed to have solved some of the most complex problems in math. Among these was the problem of squaring the circle, which was proposed to be impossible by the French Academy in 1775. This is impossible due to the fact the area of a circle is $\pi \cdot r^2$, where $r$ is the radius, and the area of a square is $s^2$, where $s$ is the length of each side.
This was proven by Ferdinand von Lindemann in 1882, and is what makes squaring a circle impossible.
In order to square a circle, $\pi \cdot r^2$ must be equal to $s^2$. For example, if $r=1$, we would have $\pi \cdot 1^2 = s^2$, or $\pi = s^2$. This would mean that each side of the square is equal to the square root of π, and since π is transcendental, there’s no algebraic expression that could describe π.
Regardless, Goodwin claimed to have done it, and published his paper to American Mathematical Monthly in 1894. It was gibberish, and no amount of understanding in mathematics would make his work comprehensible. He claimed nine different values of π across his many works, with one claim going as far as $9.2376\ldots$, “the biggest overestimate of π in the history of mathematics” (A History of Pi). When his theories weren’t becoming popular, he decided to take them to the Indiana State Legislature on January 18, 1897.
The Indiana Pi Legislature took place here, in the Indiana Statehouse. Photo CC BY-SA 3.0 Massimo Catarinella, from WikipediaGoodwin had convinced his state representative, Taylor I. Record, to introduce House Bill 246 (Indiana Bill No. 246). House Bill 246 would make Goodwin’s method of squaring the circle a part of Indiana law. However, those in the legislature either didn’t understand or didn’t even glance at the bill – and the House Committee on Canals decided to pass it. Dr. Goodwin’s ridiculous bill was now headed to the senate.
At the statehouse where the senate took up the bill was Professor Clarence Abiathar Waldo, a mathematics professor from New York. When Waldo heard what the bill was about, he was shocked to discover he was in the middle of a debate on a fundamental principle of mathematics. He decided to intervene and talk to the senators about the repercussions the bill would have on everything mathematics, and was able to stop the bill from passing the second chamber.
After Waldo’s intervention, it was clear to everyone that the people involved in the attempted passing of the bill, including Dr. Goodwin, were all wrong, and it was ridiculous to define mathematical truth by law.
The next issue of the Carnival of Mathematics, rounding up blog posts from the months of November and December 2022, is now online at Ganit Charcha.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
This is a guest post by David Benjamin.
Rational numbers, when written in decimal, either have a terminating string of digits, like $\frac{3}{8}=0.375$, or produce an infinite repeating string: one well-known example is $\frac{1}{7}=0.142857142857142857…$, and for a full list of reciprocals and their decimal strings, the Aperiodical’s own Christian Lawson-Perfect has built a website which generates a full list.
I’ve collected some interesting observations about the patterns generated by the cycles of recurring decimals, and in particular several relating to $\frac{1}{7}$.
Maximum recurring cycles and the number patterns they generateFor anyone interested, there’s a surprising connection between reptend primes and Fermat primes.
The recurring cycle for $\frac{1}{7}$, which is $142857$, has a length of $6$ digits – and when the length of the cycle produced from a fraction is one less than the denominator of the fraction, the cycle is a maximum recurring cycle. When the denominator is a prime number $>5$, the rational number will always produce a recurring cycle, and many primes produce a maximum cycle. Such primes are called reptend primes and the first fourteen are $7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167$ (OEIS A001913).
The decimal expansion of general fractions of the form $\frac{n}{7}$ also have the interesting property that it contains the same digits in the same cyclic pattern for any value of $n$ up to $6$, but with the start digit shifted along each time: $\frac{2}{7} = 0.285714285714285714…$, $\frac{3}{7} = 0.428571428571428571…$ and so on.
This is not unrelated to the fact that $142857$ is the first cyclic number, and the only one not beginning with zero. Cyclic numbers (OEIS A180340) are so named because when a cyclic number $n$ is multiplied by the numbers from $2$ to $n-1$, it contains the same digits in a different order.
$142857 × 2 = 285714$
$142857 × 4 = 571428$
$142857 × 5 = 714285$
$142857 × 6 = 857142$
If we multiply by multiples of $7$, an interesting pattern appears, which when an addition is applied, shows a further connection:
$142857 × 7 = 999999$
$142857 × 14 = 1999998 \rightarrow 1 + 999998 = 999999$
$142857 × 21 = 2999997 \rightarrow 2 + 999997 = 999999$
$142857 × 392 = 55999944 \rightarrow 55 + 999944 = 999999$, where $392$ is $56 × 7$
Multiplication by any number greater than $7$ but not a multiple of $7$ can also be seen to produce, via an addition, more cycles of $142857$:
$142857 × 9 = 1285713 \rightarrow 1 + 285713 = 285714$
$142857 × 37 = 5285709 \rightarrow 5 + 285709 = 285714$
$142857 × 127 = 18142839 \rightarrow 18 + 142839 = 142857$
$142857 × 3123 = 446142411 \rightarrow 446 + 142411 = 142857$
$142857$ is also a Kaprekar number (OEIS A006886) – when squared, the two halves of the number sum to the original number. Examples include $45^{2}=2025$ for which $20+25=45$, and $297^{2}=88209$ for which $88+209=297$. For our cyclic number, $142857^{2}=20408122449$ and $20408+122449 = 142857$.
Dattatreya Ramachandra Kaprekar (1905-1986) was an Indian mathematician who said of himself, “a drunkard wants to go on drinking wine to remain in that pleasurable state. The same is the case with me in so far as numbers are concerned”.
Adding subsets of digits of cyclic numbers can also be seen to create interesting patterns:
[14+28+57=99 \qquad 142+857=999 \qquad 1428+5714+2857=9999 ]
We can play the same trick with the second cyclic number $0588235294117647$: $94117647+05882352=99999999$; and using the third cyclic number $052631578947368421$ we have $947368421+052631578=999999999$.
We can also play with squares of subsets of the digits of cyclic numbers:
[857^{2}-142^{2}=714285 \qquad 94117647^{2} – 5882352^{2}=8823529411764705 ]
For other prime denominators, we don’t always get cyclic patterns like this. The denominator $3$ gives two different cycles, $33333…$ and $66666…$ as does the denominator $13$ ($0769230769230769…$ and $1538461538461538…$). The denominator $11$ yields five different cycles. (For more, read Conway and Guy’s The Book of Numbers).
There is an extraordinary outcome for the set of proper fractions with prime denominator $19$. The maximum recurring cycle for $\frac{1}{19},\frac{2}{19},\frac{3}{19},…..\frac{18}{19}$, when laid out in tabular form, produces a magic square: its rows, columns and leading diagonals have the total $81$.
| 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 | | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 | 0 | 5 | 2 | 6 | 3 | | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 | 0 | 5 | | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 | 0 | 5 | 2 | 6 | | 3 | 6 | 8 | 4 | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | | 4 | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 | 0 | | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 | 0 | 5 | 2 | | 6 | 8 | 4 | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | | 7 | 3 | 6 | 8 | 4 | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | | 8 | 4 | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 |
The sum of each row, column and leading diagonal is 81The first row of the table is the recurring cycle for the rational $\frac{1}{19}$, and the second row is the cycle for the rational $\frac{2}{19}$. The digits for each subsequent row continue in a similar way leading to the final row, the cycle for the rational $\frac{18}{19}$.
Cycles by additionAnother nice pattern here is the remarkable fact that the cycle $052631578947368421$, the recurring section from $\frac{1}{19}$, can be generated from the the powers of $2$. Writing the powers of two in subsequent rows, starting on the right, and then adding the columns gives:
| 1 | | 2 | | 4 | | 8 | | 1 | 6 | | 3 | 2 | | 6 | 4 | | 1 | 2 | 8 | | 2 | 5 | 6 | | 5 | 1 | 2 | | 1 | 0 | 2 | 4 | | 2 | 0 | 4 | 8 | | 4 | 0 | 9 | 6 | | 8 | 1 | 9 | 2 | | 1 | 6 | 3 | 8 | 4 | | 3 | 2 | 7 | 6 | 8 | | 6 | 5 | 5 | 3 | 6 | | 1 | 3 | 1 | 0 | 7 | 2 | | 2 | 6 | 2 | 1 | 4 | 4 | | 5 | 2 | 4 | 2 | 8 | 8 | | . | . | . | . | . | 6 | | . | . | . | . | . | . | 2 | 1 | 0 | 5 | 2 | 6 | 3 | 1 | 5 | 7 | 8 | 9 | 4 | 7 | 3 | 6 | 8 | 4 | 2 | 1 |
$142857$, the recurring cycle for $\frac{1}{7}$, similarly occurs in this sequence of infinite additions. It begins with $7$ at the top, and each subsequent row is obtained by multiplying by $5$ and moving the end of the number to the left one each time. The cycle $142857$ is produced by adding the columns generated.
| 7 | | 3 | 5 | | 1 | 7 | 5 | | 8 | 7 | 5 | | 4 | 3 | 7 | 5 | | 2 | 1 | 8 | 7 | 5 | | 1 | 0 | 9 | 3 | 7 | 5 | | 5 | 4 | 6 | 8 | 7 | 5 | | 2 | 7 | 3 | 4 | 3 | 7 | 5 | | 1 | 3 | 6 | 7 | 1 | 8 | 7 | 5 | | 6 | 8 | 3 | 5 | 9 | 3 | 7 | 5 | | 3 | 4 | 1 | 7 | 9 | 6 | 8 | 7 | 5 | | . | 0 | 8 | 9 | 8 | 4 | 3 | 7 | 5 | | . | 4 | 9 | 2 | 1 | 8 | 7 | 5 | | . | . | . | . | . | . | . | 7 | 1 | 4 | 2 | 8 | 5 | 7 | 1 | 4 | 2 | 8 | 5 | 7 |
The $\frac{1}{7}$ Cycle and Conic SectionsI was surprised to discover in David Wells’ excellent Penguin Dictionary of Curious and Interesting Numbers that the points (1,4) (4,2) (2,8) (8,5) (5,7) and (7,1), formed by overlapping pairs of the digits in the $\frac{1}{7}$ cycle, lie on an ellipse, called the one-seventh ellipse:
The one-seventh ellipse:
$19x^{2}+36xy+41y^{2}-333x-531y+1638=0$Better still, we can concatenate pairs of digits from the cycle. The points created, (14, 28), (42, 85), (28, 57), (85, 71), (57, 14) and (71, 42), also lie on an ellipse, shown below:
$-165104x^{2}+160804xy-41651y^{2}+8385498x-3836349y-7999600=0$Wells also stated that the points determined by the period of $\frac{1}{13}=0.076923076923….$ lie on a hyperbola. To find the equation of this conic, I contacted Professor Marc Chamberland – a co-author of the paper ‘A Generalization of the One-Seventh Ellipse‘. Marc sent this explanation in his reply:
Using the notation from our paper, we let $a=0, b=7, c=6$ and $S=9$. Theorem 1 implies that the points $(0,7), (7,6), (6,9), (9,2), (2,3)$ and $(3,0)$ all lie on a conic. Putting these points on the bivariate quadratic curve $Ax^{2} + Bxy + Cy^{2} +Dx + Ey + F = 0$, we can solve for the coefficients: $A = 141, B = 134, C=9, D = -1872, E = -684, F = 4347$.
Marc Chamberland
$141x^{2} + 131xy + 9y^{2} -1872x -684y + 4374 = 0$ Hyperbola created in DesmosThe conics were a huge and fascinating surprise but I don’t know if there is any conclusion or use for their connection to the cycles. However, there is a connection to the final part of the Pascal pentalogy as the paper A Generalization of the One-Seventh Ellipse makes use of Pascal’s Hexagrammum Mysticum Theorem.
Further readingIf you’ve found this interesting, the following links are intended for any reader wishing to look further into the content:
- In particular, I recommend Tony Padilla’s Numberphile video explaining how the cyclic permutations are formed.
- MacTutor biography of Dattatreya Ramachandra Kaprekar, which includes some more number patterns
- You can read more about Cyclic numbers, Midy’s Theorem, Artin’s Constant and the general bivariate quadratic curve at Wolfram MathWorld.
As part of the 24 Hour Maths Game Show which took place at the end of October 2022, our own Christian Lawson-Perfect designed a maths/games crossover gameshow format to end them all – a mashup of hexagon-fighting TV quiz Blockbusters, and his own personal obsession: interesting mathematical factoids. Welcome to Blockbusters of Interesting Maths!
The premise of Blockbusters of Interesting Maths is simple. Start by collecting three maths communicators – in this case, me (Katie Steckles), Sheffield Hallam Uni maths lecturer and recreational maths fan Alex Corner, and mathematician and juggler Colin Wright. Colin is a self-described “torturer of adults and confuser of children”, but to clarify, he mostly does that using interesting maths. Alex teaches on the SHU Game Theory and Recreational Maths module with our own Peter Rowlett, and was prepared to have a good go at coming up with some interesting facts. I, on the other hand, have come across far too many interesting maths facts in my time, and can definitely half-remember most of them.
Between us, we’re pitted against Christian’s board of randomly chosen words – from Ogden’s Basic English, a collection of 850 common English words, from which he’s deliberately removed a chunk of the mathematical and scientific terminology. To make our way across the board, we pick a letter and find the word hiding behind, and are then charged with coming up with some kind of interesting maths fact relating to that word.
Christian’s judgement on whether our maths fact was interesting enough is final, and we’ve got to make an unbroken line from one edge of the board to another. If we fail to come up with a sufficiently interesting fact, or our fact is deemed too tangential to the word in question, that tile is blocked off.
Since we could never do anything the easy/conventional way, instead of a tessellation of hexagons, CLP’s gone for the Cairo pentagonal tiling, so each cell is only adjacent to five others instead of six. His web gadget, a version of which can still be found online for anyone to use (BYO interesting mathematicians), was deployed live on the Game Show to challenge the three of us, and the below is a blow-by-blow of what went down, with links to some of the things we talked about.
We’ve also included some additional facts from Christian, who is also a font of interesting maths facts and is making up for the fact that he didn’t get to play himself. Next time!
Word 1: MoonMy initial instinct was to pass over to Colin, as he’s got a whole bit about calculating the distance to the moon using a pendulum, but instead he gave some interesting facts: the distance to the moon is pretty much exactly about 10 earth circumferences (~40 megametres), and it creates tides on opposite sides of the earth at the same time.
Christian: I can’t remember if anyone talked about all the different ways of counting a lunar month… I like the word sidereal and have no idea how many syllables it has.
Word 2: UnderminedWe failed to come up with sufficient interesting maths for this – a bit of discussion about publishing results before someone else who’s working on them was deemed to be too depressing.
Word 3: ZebraAlex talked about the work of Alan Turing on abiogenesis – mathematical models that can be used to describe patterns found on animal fur, including leopard spots and zebra stripes. Christian confirmed this was to do with reaction-diffusion models.
Christian: Back in 2008, a Simon Scarle published a paper connecting Turing’s work on reaction-diffusion to his other work on computability, through simulations of cardiac arrhythmia on the Xbox 360. I’ve never known what to do with this information. If you want to play with reaction-diffusion models yourself, there’s a good simulator called Ready, which we wrote about it in 2012.
Word 4: QuantityI waffled briefly about the history of counting and the Ishango bone, which is an interesting historical artefact linked to early mathematical activity, and which it turns out I’d got mixed up with the Lebombo bone, which is an even older one.
Word 5: StalkColin took this as a verb, and talked about predator-prey dynamics, particularly related to pursuit predation, including ambush and persistence behaviour in hunting. For each type of hunting, the animal has to weigh the probability of a successful catch against the amount of energy expended on the chase.
Word 6: EngineerAfter a brief digression about which direction the real line points in (since we’d missed the opportunity to connect the board top-to-bottom, which most of us hadn’t realised was a thing), Alex couldn’t think of anything to say, so we lost this one.
Word 7: InterceptAfter mentioning the mathematical use of the word, I managed to just about describe a particular maths problem this reminded me of that involved chasing something that’s swimming in a river (Christian mentioned this was covered in Dara Ó Briain’s School of Hard Sums, and it turns out there’s a writeup on Marcus Du Sautoy’s blog), and we then went on to another puzzle about a cat in a pond, which Ben Sparks has done a great video about.
Christian: Talking of interception reminded me of this fun paper describing a strategy for avoiding being intercepted while mapping an unfriendly subway system.
Word 8: ControlColin covered a couple of topics – starting with control theory, which Colin compared to riding a unicycle. The trick is to keep the wheel under you, by (e.g) pedalling faster if you’re falling forwards, which can be understood by solving fairly straightforward differential equations – as unicycling robots often do.
He also talked about controlling a dog’s behaviour, and how rewarding good behaviour every time means the effect of training wears off more quickly, whereas rewarding it randomly some of the time means the effect lasts longer – this is related to spaced repetition as a learning technique.
Word 9: ChargeBack over to Alex, who took electrical inspiration and used it as a chance to talk about capacitor laws. There were lots of nice relationships between different physical laws and it all got a bit physics, and as a result was rejected by Christian, so we lost this one.
Word 10: SaltI took the opportunity to talk about mathematical crystal structures, bond angles and 3D lattices (and got in a Kathleen Ollerenshaw mention). Christian also connected it to the structures of viruses, and mentioned Hamish Todd’s lovely videos.
Word 11: MixtureColin riffed on ratios in mixtures, from concrete to cake recipes, and then moved on to mixed techniques. Combinatorics, for example, uses a variety of different techniques you have to try in different combinations in order to solve a problem, and Colin explained how maths research, particularly in applied contexts, a mixture of techniques can be most powerful. Von Neumann showed that mixed strategies are always more effective in game theory!
Word 12: ClientAfter a brief digression about profit-loss models in economics, I jumped in with a mention of the version of internet protocols used in communication with objects in space, which Colin then ran with – talking about comms in trading (which relies on the speed of light to make sure transactions are instantaneous). A client-server model can be used, and in some contexts, equations from fluid mechanics are even used to describe how packets of information are moved around.
The finished board stateWith that, we finally managed to satisfy Christian’s mathematical interestingness quotient and successfully connected the opposite sides of the board.
If you’d like to rewatch this or any other part of the 24 Hour Maths Game Show, you can find links to each segment on the website, and you can still donate to our charities by visiting 24hourmaths.com/donate.
Here’s a roundup of the maths news we missed in December 2022.
Maths NewsThe leap second, referred to in this Independent article as a ‘devastating time quirk’, is finally being abolished. This has been covered in a bunch of places, mostly being quite rude about the leap second, including a writeup in the New York Times where it’s referred to as ‘a kludge, a bain, a pain in the little hand’ (£), and this Live Science article (‘pesky’). A committee at the International Bureau of Weights and Measures apparently nearly unanimously voted in support of Resolution D, meaning there won’t be any leap seconds from 2035 until at least 2135.
Anti-maths news! Princeton mathematician Rachel Greenfield (pictured left – photo by Dan Komoda/Institute for Advanced Study), working with Fields Medalist Terry Tao, has posted a disproof of the periodic tiling conjecture. A preprint titled ‘A counterexample to the periodic tiling conjecture‘ is now on the ArXiv, and if it’s correct, means that any finite subset of a lattice which tiles that lattice by translations, must tile it periodically. There’s a nice explanation in the Quanta writeup!
Meanwhile there’s been a new claimed proof of the 4-colour theorem, which is non-constructive (meaning it doesn’t rely on finding a colouring for every possible map, but proves the theorem generally). Some people have been skeptical about the proof, including in this statement from Noam Zeilberger, which links to a Mastodon discussion with John Carlos Baez. (via Neil Calkin on Mastodon)
Another claimed proof – this time of the sunflower conjecture. A k-sunflower is a family of k different sets with common pair-wise intersections, and the conjecture gives conditions for when such a thing must exist.
ArXiv has posted a framework for improving the accessibility of research papers on arXiv.org – their plan is to offer html as well as PDF versions of papers. (via Deyan Ginev)
EventsBright-trouser-wearer and mathematician Marcus Du Sautoy is offering a free OU online course, entitled ‘What we cannot know’. Find out how he manages to break the rules of reality by facilitating you knowing something that it’s by definition impossible to know, by signing up online for the 8-week course (which can also be accessed without signing in but then you don’t get a badge).
Any excuse to include a photo of HF ❤️As part of their Elevating Mathematics video competition, the National Academies Board on Mathematical Sciences and Analytics (BMSA) invites early career professionals and students who use maths in their work to submit short video elevator speeches describing how their work in mathematics is important and relevant to our everyday lives, with a $1000 Prize for the best video.
And finally, in a rare instance of us linking to the Hollywood Reporter, Hannah Fry is to front a science and tech series for Bloomberg, entitled The Future With Hannah Fry. Sounds great! It’ll be available on Bloomberg’s Quicktake streaming service and will explore breakthroughs in artificial intelligence, crypto (not clear if -graphy or -currency), climate, chemistry and ethics.
It’s that time of year when we take a look at the UK Government’s New Years Honours list for any particularly mathematical entries. Here is the selection for this year – if you spot any more, let us know in the comments and we’ll add to the list.
- Paul Glaister, Professor of Mathematics and Mathematics Education, University of Reading. Appointed CBE for services to education.
- Dan Abramson, headteacher of King’s College London Maths School. Appointed OBE for services to education.
- Kanti V. Mardia, Senior Research Professor, Leeds University. Appointed OBE for services to Statistical Science.
- Charlotte Francis, maths teacher and entrepreneur. Appointed Medallist of the Order of the British Empire for services to education.
Get the full list from gov.uk.
This week and last I hosted a series of public maths talks featuring disabled presenters. I’ll post about how that went later, but for now I just want to share this clip of me ~~filling time~~ spreading Christmas joy.
This is a party trick that Katie Steckles showed me: you can fold a piece of paper and then make a single cut to produce a five-pointed star. I showed how to do it by following the instructions I’d been told, and then recreated the steps just starting from the insight that when you make the cut, all the edges of the shape need to be on top of each other.
Maybe you’ll show someone else how to do it during the Christmas holiday?
This doesn’t only work for stars: there’s a theorem that you can make any polygon by folding and a single cut. Erik Demaine has made a really good page about the theorem, with some examples to print out and links to research papers. Katie can cut out any letter of the alphabet on demand, which is impressive to witness!
We spoke to Coralie Colmez, mathematician and author of Math on Trial, about her genre-busting new Young Adult novel for mathematically minded teenagers: The Irrational Diary of Clara Valentine.
Tell us about your new book.The Irrational Diary of Clara Valentine is a fun novel aimed at readers aged 15-19. It’s got all the good things in it: a mystery, a sharp-witted narrator, an idiosyncratic best friend, a couple of charming potential boyfriends, a mostly-loveable family, and some maths!
What inspired you to write it?I first had the idea for the book around 10 years ago. My mother (a mathematician) and I had just written a popular maths book called Math on Trial, which was really fun to do, but it made me realise that I wanted to find a way to write about maths which was closer to the things I like to read myself, and I mostly read fiction. Following the release of Math on Trial, I got the chance to talk about it at events for students, which I really enjoyed, so I decided to write for that age group.
I found it easy to decide which maths topics I wanted to cover – they are all my own favourites, and the things that made me love maths when I was a teenager! The topics are quite abstract and high-level, like countable infinity and logic – things that would normally only be introduced at university, even though they don’t require much prior knowledge and I think high-school students would really enjoy learning about them.
I was also interested in writing YA because I felt that, while there is a lot of great YA literature, none of it looked like my own experience of teenagerhood. Characters are either off dealing with a fantasy world, with major emotional trauma, living an extreme life (Euphoria-style), or on the contrary behaving in a totally PG way. There is a space and a need for all of these types of characters, but I wanted to try and write ones that felt more real, and could have been my friends and me.
You’ve done a great job of including the maths naturally, as part of the story – did you find this difficult?Thank you, and I am glad you think so! Because the topics I wanted to cover are quite high-level, I had to find some creative ways to include them. I got a lot of well-meaning publishing professionals suggesting that I have Clara solve problems involving measuring angles, calculating the length of a rope and that kind of thing, but I was really set on sticking with more abstract concepts.
I definitely wanted to make sure there were a few different ways that the maths became part of the story: some of it happens via the characters in the book that know maths at a high level, but we also see moments like Clara teaching her little sister something, or Clara’s best friend learning a bit of mathematical history in her philosophy class. That’s what it’s like in my family (which is made up of 50% mathematicians, so not entirely representative): maths is just a part of normal life.
I also wanted to include what it’s like to think about maths, so I really liked getting in Clara’s mind when she is solving a question: how she approaches problems from different angles – how some of these angles sometimes don’t work at all – and how amazing it feels to crack a problem.
Who do you hope will read the book?I really think that anyone could enjoy the book. As a novel, it’s a pretty exciting read, it’s funny, and hopefully I’ve managed to capture a bit of today’s really exciting generation of young people, who are so sharp, aware and witty.
When it comes to the maths, it’s written so that readers of different levels can take what they want. Someone who has quite a high level of maths might even try to solve some of the problems along with Clara, whereas someone who only has a basic interest in maths might simply enjoy the overall concepts that are introduced, like realising how different ‘Infinity’ is to just ‘A really big number’. In terms of the level of the maths, I would say that an 18-year-old reader who already has an interest in maths might already have heard of 2 or 3 of the 8 topics covered, but hopefully they’ll see even the ones they already know in a fresh way, with some new anecdotes attached! A couple of the topics are included in the A Level syllabus, though most aren’t.
The book is aimed at readers all of all genders, but as a woman maths graduate – the only one in my year at my college – with a mathematician mother, it’s really important to me to encourage more girls into maths and science. I hope that having an awesome (if I do say so myself) female narrator like Clara can help with that.
Finally, age-wise, I wouldn’t recommend the book to younger readers, because there are some themes that might be too old for them, and there is some sex (which is entirely age-appropriate, at least if you are French, and also very positive for ages 15+) I’ve had some very lovely comments from older readers though, so I’m going to say that there is no maximum age to enjoy the book!
What made you decide to self-publish the book?When I finished my first draft of the book, I actually found an agent very quickly, and there was immediate interest from publishers when she sent out the manuscript. I’ll be honest, at the point when I was talking to three big publishers at once, I thought I was about to be famous! But in the end, none of them wanted to publish exactly the book I wanted to write. One wanted it for a younger audience, another wanted me to focus just on the mystery, the third wanted me to first write a novel with no maths, as they were nervous about Clara being a debut. There was a lot of talk about which ‘shelf’ my novel would fit in, and I realised I didn’t want to write a book that just fitted on one shelf, because that’s not what life looks like! Clara cares about maths, about her relationships, about solving a mystery, about politics, about doing well at school… just like we all do (well, apart from the mystery maybe).
How can people get hold of a copy?You can get a copy on pretty much any online retailer, like Amazon – or if you are avoiding Amazon, it’s on Waterstones if you are in the UK and Bookshop.org if you’re in the US, for example. I’ve also put the PDF on my website, so anyone can read it for free. The book is pretty though, and I’m quite proud because I designed the cover myself, so I’d recommend getting a copy!
Looking for small/inexpensive items to put inside a piece of festive footwear (or, to keep for yourself)? Here’s a selection of things we’ve seen lately that you might want to buy! All the items we’re showing are under £20, and range from slightly mathematical to very mathematical.
JewelleryFriends of the site Maths Gear have their usual selection of excellent gifts, including a new range of Maths Icons earrings (£14.99) including the Mandelbrot set, and a set of nested polygons. They also have a range of other jewellery and cufflinks which includes the wonderful mug/donut earrings (£17.89) and π cufflinks (£6.91).
Pythagorean earrings from CofactorThere’s also some great Impossible Shape jewellery (from £3) – including Borromean Rings and Necker cubes – available from Earth Symbols, and there’s of course the classic Cofactor Pythagorean theorem earrings ($15 in 3D printed nylon).
DiceMaths Gear Go First DiceMaths Gear also have a wide collection of different types of dice, as do scholastic suppliers Tarquin (including class sets and some individual items). We particularly like Maths Gear’s set of polyhedral dice in shapes they don’t usually come in (£10.97), and Tarquin’s set of blank rewriteable dice (£5.99), for when you want to make your own rules.
More generally, Tabletop Supply are a good go-to for many different types of dice, as well as replacement (or upgrade!) pieces for existing games, or games you’ve invented yourself.
Games and toysPlenty of mathematically interesting board games come in travel versions which can fit into a small space – some of our favourites include number-based favourites Red 7, 6 Nimmt! and The Game, and if shapes is more your think we can also recommend travel Blokus and travel Qwirkle.
Oink Games also do some visually stunning and simple games with a mathsy vibe, and our favourites include Troika, Deep Sea Adventure and The Pyramid’s Deadline. We also love OK Play which is very compact.
Oink Games’ Deep Sea AdventureSimple classic board games with a mathematical twist include Shut The Box, and you can’t go wrong with a deck of cards (this Math Stack variant (£10.49) from Maths Gear is pretty, but any will do!)
Site editor Christian recommends what he calls a Nobbly Wobbly, but tends to be sold under the name ‘woven bouncy ball’ or ‘rainbow spaghetti ball‘ (£5.95 for 5) depending on who you ask. The underlying geometry of the shape makes it mathematically interesting, but dogs and small children alike will have fun with it.
PuzzlesCat Stax, by HuchHappy Puzzle Co have IQ Minis which fit in your hand, and a range of other pocket puzzles. We also love Edmund Harriss’ Curvahedra (£11.93), which are available from Maths Gear. Paper-folding puzzle Manifold was a big hit with us a few years ago, and it looks like Manifold 2 is now available.
There’s also Rush Hour (£16.49), and we’ve had a recommendation for the STAX games from Huch’s range of puzzles, which comes in Cat, Dog and Sea versions. Don’t forget about the Rubik’s cube and other twisty puzzles too!
Maths ParaphernaliaMetal protractors, from Present and CorrectFor a more refined mathematical palate, you can pick up some elegant vintage maths gifts from Present and Correct, including a 1970s desk abacus (£19.50), gorgeous metal protractors (£4), this ruler sticker tape (£3) and a classic geometry puzzle (£12). Or why not pick up a Golden Mean Compass (£14.99) from Grand Illusions?
If you have any suggestions of your own, feel free to include them in the comments below!
Here’s a roundup of things that happened online in November that we didn’t cover here at the time!
Maths Research NewsAccording to an article on philosophy news site Daily Nous, an international symbolic logic journal printed then shortly retracted two articles, one entitled “The Twin Primes Conjecture is True in the Standard Model of Peano Arithmetic: Applications of Rasiowa–Sikorski Lemma in Arithmetic” and the other “There are Infinitely Many Mersenne Prime Numbers. Applications of Rasiowa–Sikorski Lemma in Arithmetic“. After a discussion on MathOverflow, mistakes were found in both papers, and the journal’s editor posted:
Recently two articles on the applications of the Rasiowa-Sikorski Lemma to arithmetic were published online in Studia Logica without proper examination and beyond reasonable standards of scholarly rigor. As it turned out, they contained an irrrepairable mistake and, consequently, have been retracted from the journal’s website. The papers will not appear in print.
Studia Logica editor-in-chief Jacek Malinowski
(via Catarina Dutilh Novaes on Twitter, whose thread includes some clarifications.)
Gliders producing decimal digitsAccording to Conway’s Life, a blog which documents developments in research around Conway’s Game of Life, on November 9, 2022 Pavel Grankovskiy discovered that 15 gliders can make any pattern in Conway’s game of life. Given a particular shape, the gliders can be set up to create it (eventually) beating a recent record of 16 gliders. (via Oscar Cunningham on mathstodon,xyz:)
Fields medalist Terry Tao reports some progress on the union closed sets conjecture, an open problem in combinatorics, which has seen rapid developments thanks to (in Tao’s words) ‘maths at internet speed’.
Other NewsAs of 11th November, applications for Young Researchers for the Heidelberg Laureate Forum 2023 are open. If you or someone you know is a researcher in maths or computer science at undergrad or postgrad level, and would like to spend a week next September in a lovely town in Germany meeting the world’s most decorated mathematicians and computer scientists, you should consider applying!
The latest issue of The Mathematics Enthusiast is a special issue collecting 29 reviews of popular maths books by maths educators, including Matt Parker, Hannah Fry, Eugenia Cheng, Simon Singh and Jordan Ellenberg among many others. If you’re looking for new pop maths book recommendations, it’s a good place to start!
Check out these absolute units (Image: NASA/Brian0918/ Wikipedia Commons)It was announced earlier this month that having discovered sufficiently many very big and very small numbers, it’s time for some new SI prefixes: ronna-, ronto-, quetta- and quecto- have joined the ranks of things that make numbers bigger and smaller, allowing you to describe itty bitty quantities as small as $10^{-27}$ (ronto) and $10^{-30}$ (quecto), as well as chonky numeros in the region of $10^{27}$ (ronna) and $10^{30}$ (quetta). The earth weighs 6 ronnagrams, and Jupiter is about 2 quettagrams.
“‘R’ and ‘Q’ were the only letters left in the English alphabet that hadn’t been used by other prefixes.”
Richard Brown, National Physical Laboratory
And in computer news, Google Chrome now supports MathML core, a language for describing mathematical notation embeddable in HTML and SVG. (via axel rauschmayer)
Looking for something mathematical to amuse you, give your puzzling brain a workout or otherwise satisfy your mathematical curiosity every day for the next month? Look no further – here’s a round-up of our favourite mathsy advent calendars for 2022. NRICH Advent Calendars As has become traditional, NRICH are treating us to two advent calendars…
Does this picture make you think of Srinvasa Ramanujan? I’m always fascinated by the pace and range of little conversations with my seven-year-old son that wander in and out of maths. Let me tell you how we got there during a five minute chat while leaving the house and walking to school this morning. He…
Some thinking aloud about what’s happening on social media in my world, I hope you don’t mind. Me and Twitter I joined Twitter in February 2009, having considered doing so for about a year. I wrote on this blog at the time that “Now it is really taking off I have decided to give it…
This month’s Carnival is hosted right here at The Aperiodical, and rounds up interesting internet maths content from the month of October 2022. The Carnival of Mathematics is a monthly blogging round up hosted by a different blog each month, and collects blog posts, videos, social media posts and other internet content from the month…
I’ve put together a series of online public maths presentations, to take place in the last couple of weeks of term before Christmas. This came about after a few people on the Talking Maths in Public WhatsApp group complained that we can hardly ever take up requests for a speaker to deliver a fun maths…
Research AI research company DeepMind said that their AlphaTensor system has discovered a new way to multiply matrices, citing this as the first such advance since the Strassen algorithm was proposed in 1969. AlphaTensor found thousands of algorithms for multiplying matrices of different sizes, but most were not better than the state of the art.…
Here’s a roundup of mathematical news stories we didn’t get round to writing about yet this month. Events From 11th October onwards, the ICMS are organising a series of talks titled Diverse Voices of Maths, all taking place at the Bayes Centre in Edinburgh and being streamed online, with free tickets. Speakers include friends of…
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of September and hosted by Jeremy Kun, is now online at Math Intersect Programming. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of August and hosted by JamesA, is now online at alephjamesa.co.uk. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Next week, I (Aperiodical team member Katie Steckles) and Sophie Maclean (Chalkdust team member and cool maths person) are off to Heidelberg to cover the Heidelberg Laureate Forum. The HLF is an annual conference bringing together respected maths and computer science laureates (including Fields medalists, Abel Prize winners and others) to meet each other and…
Not much going on in the world of maths this month (or, we’re on holiday so we haven’t been paying attention), but here’s a round-up of a few stories we saw this month. The next Black Heroes of Mathematics Conference is scheduled for the 4th and 5th October, taking place online and featuring speakers including…
Peter suggested it, so Katie had to do it: here’s a video of Katie and fellow maths/Marvel fan Jimi watching through the end credits to Spiderman: No Way Home (warning: contains spoilers for the film) and talking about the mathematical things found therein.
It’s nine years since the first integer sequence review, and six years since the last one. We’ve grown as people, and in CLP’s case, grown people. The world has changed, but our love for the Online Encyclopedia of Integer Sequences hasn’t. A101544 Smallest permutation of the natural numbers with $a(3k-2) + a(3k-1) = a(3k)$, $k…
DALL·E is an Artificial Intelligence (AI) system that has been designed to generate new images given a text prompt. It’s very much like doing a Google image search with one very important difference: DALL·E doesn’t try to find existing images to match your query, but creates a handful of new ones that it hopes will…
I made a new LaTeX package for drawing dice, customdice. I’ve long struggled to find a dice package I fully like. I have been using epsdice, but it only offers white or black standard dice. I found myself in a situation where I wanted to draw dice with other text on the faces. This package…
Guest author David Benjamin shares some of his favourite ways to use sequences in a teaching context. As a maths teacher, I’ve found that sequences are a great way to engage and inspire mathematical reasoning. I thought I’d share some examples of sequences, and sequence-related activities, I’ve used with success in the past. A simple…
Here’s a roundup of some mathematical news we didn’t yet report from the last month. The makers of documentary film ‘Olga Ladyzhenskaya’, detailing the life of the Russian mathematician, have released a five-minute trailer giving a flavour of the film. (via ICM Intelligencer) Research According to a new ArXiV paper, the triple bubble conjecture (a…
A conversation about mathematics inspired by some fingers. Presented by Katie Steckles and Peter Rowlett, with special guest Ben Orlin. Ben’s new book is Math Games with Bad Drawings.
A conversation about mathematics inspired by the game Quarto. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a slinky. Presented by Katie Steckles and Peter Rowlett.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of May and hosted by Rob, is now online at Rob Eby’s Math Blog. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
A conversation about mathematics inspired by the nodal cubic. Presented by Katie Steckles and Peter Rowlett. We go closer to the cutting edge of research than usual in this chat with Angela Tabiri about her PhD research.
Here’s a game I’ve been trying to make for a while. For a while I’ve had a hunch that there’s fun to be had in moving between numbers by using something related to the prime numbers. Over the years I’ve tried out a few different ideas, but none of them ever worked out – they…
Every now and then a phrase pops into my head and won’t leave until I write it down or tell it to someone else. One day the little voice in my head suggested putting “Didn’t” before the classic series of maths textbooks, Graduate Texts in Mathematics. So I found a cover of a GTIM book,…
A conversation about mathematics inspired by the PageRank algorithm. Presented by Katie Steckles and Peter Rowlett.
Here’s a round-up of the mathematical and maths-adjacent news stories we saw in the month of April. Proof News The Kahn-Kalai conjecture, a result from graph theory, has been proved in this ArXiV paper by Stanford mathematicians Jinyoung Park (a former postdoc of Abel prize winner Avi Widgerson) and Huy Tuan Pham. Here’s the writeup…
A conversation about mathematics inspired by a joke. Presented by Katie Steckles and Peter Rowlett, with special guest Bec Hill.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of April and hosted by Sophie, is now online at Sophie The Mathmo. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Recently I came across an interesting idea about little mistakes in counting problems that actually don’t amount to much. In A Problem Squared 030, Matt Parker was investigating the question “What are the odds of having the same child twice?” and made some simplifying assumptions when thinking about DNA combinatorics. He justified leaving out a…
In this series of posts, we’ll be featuring mathematical podcasts from all over the internet, by speaking to the creators of the podcast and asking them about what they do. We spoke to Michael Brooks, who co-presents the Eureka! podcast with Rick Edwards. Podcast title: Eureka!Website: shows.acast.com/eureka Links: Spotify, Apple Podcasts, Google Podcasts, RSSAverage episode…
Not that we’re overly consumed with numerical coincidences, but it’s perhaps nice to note that ten years ago today we made a little fuss of launching a new blog site with our first post, a post marking Felix Klein’s 163rd birthday, and a video about the Klein Bottle featuring Matt Parker and Katie Steckles. We…
A conversation about mathematics inspired by a hairy ball. Presented by Katie Steckles and Peter Rowlett.
When teaching moved online due to COVID-19, we had to quickly work out how to deliver our modules online. The main options used to replace in-person classes were: pre-recorded videos followed by live online tutorials for students to get support while completing exercises; live online classes offering a mixture of lecturer delivery and student activity.…
A conversation about mathematics inspired by a superegg. Presented by Katie Steckles and Peter Rowlett, with special guest Hannah Fry.
Friend of the site and good writer/bad drawer Ben Orlin has recently released a new book, and we were kindly sent a copy to play with and review. The full title is “Math Games with Bad Drawings: 75¼ simple, challenging, go-anywhere games – and why they matter”, and it’s a sizeable collection of fun pen-and-paper…
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of March and hosted by Ben, is now online at Math Off The Grid.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Earlier today, I tweeted about my exciting new Pi search website, which lets you search for any string of digits within the infinite decimal expansion of π. If you haven’t seen it, go and check it out now.
I was inspired by seeing a lot of people getting excited about π for Pi Day, which took place a few weeks ago – all over the internet, memes like the ones above were breathlessly shared by excited universe-beauty-lovers and, hopefully, inspired some people to appreciate slightly better what it is about mathematics we mathematicians find so fascinating.
Unfortunately (and for those of you who’ve noticed the date, appropriately), the website I’ve built doesn’t actually do what it, and these lovely memes, claim to do. Yes, if you search for a string of digits it’ll return the position of that string within π, along with a chunk of digits either side:
The problem comes if you try to search for strings longer than about 4 digits. In that case, you’ll still get a result from the site, and it’ll tell you where to find it in the infinite decimal expansion of π:
But if you’ve taken the time to look up my JavaScript, which can be easily found in the source code (and anyone doing so should bear in mind that this is about the third thing I’ve ever written in JavaScript, and yes I know it’s terrible) you’ll notice that it’s doing something slightly different to what I’m claiming.
What Have You Done, Steckles
The code reads in the file pi_const.js, which contains the variable pi, which contains the digits of π. Impressively, my data storage optimisation is so efficient that this file, which contains all of the digits of an infinite non-repeating decimal (and based on the memes above, a good deal more besides) only takes up about 100KB of space. Suspicious!
Further digging might reveal that this file only actually contains about the first 100,000 digits of π, and if the number you’re searching for is found there, it’ll return the correct position and chunks of π from either side.
If your number isn’t found within that range, I’ve made use of a seeded random number generator – one which takes in an input and returns a randomly generated number, but in such a way that the same input will always give the same random number. (Earlier iterations of the site would find the long string of digits at different positions in π each time you refreshed the page, which I thought would be less convincing).
So unfortunately, if the position my site finds your number at in the digits of π is more than about 100,000, the chances are you’re being fed a line – the position, and the digits either side, are randomly generated and not actually anything to do with the digits of π. But you couldn’t tell, could you?!
Fool me π times, shame on… Euler? In the classic April Fools’ tradition, the beauty of this trick is that on the face of it, you’d have no reason to suspect this isn’t just a genuine π search website. If you’ve recently shared a meme like the one above, you might even be pleased that someone’s bothered to implement something to let you find out exactly where the complete works of Shakespeare can be found in the circle constant.
The main reason someone might have to be suspicious of a website like this is that for several reasons, what I’m doing here isn’t possible.
- Reason #1: This is literally, as I said, like the third JavaScript programme I’ve ever written, so the chances of me being able to pull something like an infinite π searcher off, without a large dev team or loads of time, are slim at best. It’s also SUSPICIOUSLY quick to return a result.
- Reason #2: We don’t actually know enough digits of π. The longer the string you’re searching for, the less likely it’ll appear within a given chunk of π. From some brief investigations, the 100,000 digits I’ve given it do contain all 4-digit numbers (but ‘10000’ isn’t found in there!) and generally, if you’re looking for an n-digit string, you need an n-digit number of digits of π to be likely to find it. So, given that the current world record for computation of digits of π is about 62 trillion digits (a 14-digit number), searching for anything over 14 digits is likely to require me to literally bust the record for the number of digits that humanity as a species has ever calculated, which I again lack the computational resources to do.
- Reason #3, and the one which some people might have been getting annoyed by for a while now: the ‘fact’ so excitedly shared by the memes above isn’t actually known to be a fact. As nice as it would be, we haven’t actually proven mathematically that all possible strings of digits are present in the digits of π.
Disappointing as this sounds, the more you think about it, the less sense this assumption makes. Going from ‘pi is an infinite decimal which never repeats’ via a casual ‘means that’ to ‘it contains every possible finite string of digits’ is sloppy maths, easily disproven.
What if I were to take the digits of π and remove every occurrence of the number 7? I’d still have an infinite non-recurring decimal, but it wouldn’t contain every possible string of digits! The property we’re looking for here is something more subtle, which doesn’t necessarily fall out from just being an infinite non-repeating string of digits.
Normal service will be resumed If a number contains every possible string of digits (or, it turns out to be equivalent: each digit or string of digits of a given length occurs equally frequently in the digits), it’s called a normal number. This is a beautiful and clever thing for a number to be, and it’s been proven that in the uncountably infinite space of real numbers, almost every number has this property.
Annoyingly though, we don’t actually know what any of them are. Actually, that’s not true – there’s Champernowne’s constant, which is tediously defined as:
[ 0.1234567891011121314151617181920\ldots]
That is, the number made up by writing out every whole number in order one after the other and sticking them together as an infinite decimal. It’s normal, but doesn’t have any use beyond being an example of a normal number.
To further complicate things, the property of being normal depends on which number base you write the number in – since it’s about the digits. If a number is normal in every base, it’s called absolutely normal. If you’re only bothered about one number base, you can find some examples of rational numbers that have an equal proportion of each digit (called simply normal), a nice but obvious example being:
[ \frac{123456789}{9999999999} = 0.\overline{0123456789} ]
But this won’t have all the other properties of a normal number, since the equal-proportions rule doesn’t extend to two-or-more digit strings.
There are some numbers that have been shown to be normal, but they are inconveniently not computable, which means we can define them and say what they mean or are the solution to, but not actually work out what the digits of them are.
This is a reference to the Riemann hypothesis – at time of writing, still unproven.
It’s also strongly conjectured that various other irrational numbers, like π, e and the square root of 2 are normal – but a proof hasn’t yet been produced. So while it’s a lovely and elegant thought, the memes above might as well be saying ‘All the Riemann zeroes are on the critical line – isn’t that nice?’
I hope that my daft π search page has at least provided you with a small amount of amusement – and maybe this result will be proved in the future, in which case you’re fine to go ahead and share the memes and bask in the glory of clever numbers. For now, we’ll just have to go back to crunching our π calculations and wondering if some day we’ll discover that there stop being any 4s at all beyond about a squillion digits. Who knows?
Here’s a roundup of mathematical things that have happened in March 2022.
Politics
Mathematician Yulia Zdanovska was killed by Russian shelling in Kharkiv. Yulia was a silver medallist at the European Women’s Mathematical Olympiad in 2017.
- The International Mathematical Union’s Committee for Women in Mathematics has published an expression of sympathy.
- The Mathematical Oympiad in Kyiv has passed on a request from her parents for donations to the Kharkiv Station project, where Yulia had been active.
- MIT has set up Yulia’s Dream, “a free math enrichment and research program for exceptional high school students (grades 9-11) from Ukraine”.
A bill has been introduced to the US congress to honour Bob Moses, a mathematician and civil rights leader. (via Dave Kung on Twitter)
There was a passing reference in the UK Spring Statement that businesses will be able to claim tax relief on R&D supported by pure maths.
A second period of Azat Miftakhov Days will take place on 5th and 6th July this year, in solidarity with the Russian maths graduate student who has been detained by Russian authorities since Febraury 2019.
Awards The 2022 Abel Prize went to Dennis Parnell Sullivan, “for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects”.
The Royal Statistical Society has announced the recipients of its 2022 honours.
Other news π day happened again. UNESCO, still trying to rebrand it as the International Day of Mathematics, has published a toolkit for teachers, Mathematics for Action: Supporting Science-Based Decision Making, which “deciphers the role of mathematics in achieving the Sustainable Development Goals to 2030 that were adopted by the global community in 2015”. The toolkit seems to only be published as a massive PDF.
The inclusion/exclusion blog, formerly hosted by the AMS, is back, and striking out on its own.
Following on from the series of ‘Pascal’s Triangle and its Secrets‘ posts, guest author David Benjamin shares another delightful piece of mathematics – this time relating to prime numbers.
At the time of writing the largest known prime number has $24862048$ digits. The number of digits does not reflect the true size of this prime but if we were to type it out at Times New Roman font size 12, it would reach approximately $51.5$ km, or about $32$ miles. Astonishing!
Patrick Laroche from Ocala, Florida discovered this Mersenne prime on December 7, 2018. I was surprised to discover that it’s exponent $82589933$ is the length of the hypotenuse of a primitive Pythagorean triple where $82589933^{2} = 30120165^{2} + 76901708^{2}$ as indeed are 8 of the exponents of those currently ranked from 1 to 10.
The Greek mathematician Euclid of Alexandria ($\sim$325 BC-265 BC) was arguably the first to prove that there are an infinite number of primes – and since then, people have been searching for new ones. Some do it for kudos, for the prize money, to test the power of computers and the need to find more of the large primes used to help protect the massive amount of data which is being moved around the internet.
Mersenne primes, named after the French monk Marin Mersenne, are of the form $2^{p} -1$, where the exponent $p$ is also prime. Mersenne primes are easier to test for primality, which is one reason we find so many large ones (all but one of the top ten known primes are Mersenne). When Mersenne primes are converted to binary they become a string of $1$s, which makes them suitable for computer algorithms and an excellent starting point for any search.
Marin Mersenne Since generally testing numbers for primality is slow, some have tried to find methods to produce primes using a formula. Euler’s quadratic polynomial $n^2+n+41$ produces this set of $40$ primes for $n = 0$ to $39$. When $n=40$, the polynomial produces the square number $1681$. Other prime-generating polynomials are listed in this Wolfram Mathworld entry.
The French mathematician LejeuneDirichlet proved that the linear polynomial $a+nb$ will produce an infinite set of primes if $a$ and $b$ are coprime for $n=0,1,2,3,4,…$. Then again, it also produces an infinite number of composite numbers! However, this gem: $224584605939537911 + 1813569659748930n$ produces 27 consecutive primes for $n=0$ to $n=26$ – and of course, all the primes are in arithmetic progression.
14 fruitful fractions The primes are unpredictable, and become less common as they get larger. Consequently there is no formula that will generate all the prime numbers. However, there is a finite sequence of fractions, that – given an infinite amount of time – would generate all the primes, and in sequential order.
They are the fruitful fractions, created by the brilliant Liverpool-born mathematician, John Horton Conway (1937–2020) who, until his untimely death from complications related to COVID-19, was the John von Neumann Emeritus Professor in Applied and Computational Mathematics at Princeton University, New Jersey, USA.
John Horton Conway (Photo: Denise Applewhite, Office of Communications) The fruitful fractions are
| $\frac{17}{91}$ | $\frac{78}{85}$ | $\frac{19}{51}$ | $\frac{23}{38}$ | $\frac{29}{33}$ | $\frac{77}{29}$ | $\frac{95}{23}$ | $\frac{77}{19}$ | $\frac{1}{17}$ | $\frac{11}{13}$ | $\frac{13}{11}$ | $\frac{15}{44}$ | $\frac{15}{2}$ | $\frac{55}{1}$ | | A | B | C | D | E | F | G | H | I | J | K | L | M | N |
The first time I encountered this set of fractions was in the wonderful book, The Book of Numbers, by Conway and Guy. I was so intrigued as to how Conway came up with his idea, I emailed him to ask. I was delighted to receive an outline of an explanation and even a second set of fractions, neither of which I can now find – it was 1996 and pre-cloud storage! But no worries… Conway explains everything in this lecture, which also demonstrates his passion for mathematics and his ability to express his ideas in a relaxed and humorous way, even when he searches for an error in his proof on 26 minutes. The lecture also includes an introduction to Conway’s computer language, FRACTRAN, which includes the statement:
‘It should now be obvious to you that you can write a one line fraction program that does almost anything, or one and a half lines if you want to be precise‘.
Using the fractions to find prime numbers Here’s how the fractions are used to generate primes.
- Start with the number $2$
- Multiply by each of the fourteen fractions until you find one for which the product is an integer
- Starting with this new integer, continue multiplying through the fractions until another integer is produced. (If this process reaches fraction $N=\frac{55}{1}$, the integer’s product with N is guaranteed to be another integer as N has a denominator of $1$; the process continues with this new integer being multiplied by fraction A)
- Continue multiplying through the set to create more integers
- When the integer is a power of $2$, its exponent will be a prime number.
The 19 steps needed to produce the first prime number are:
$2 \overset{ \times M}{\rightarrow} 15 \overset{ \times N}\rightarrow 825\overset{ \times E} \rightarrow 725 \overset{ \times F}\rightarrow 1925\overset{ \times K} \rightarrow 2275 \overset{ \times A}\rightarrow 425 \overset{ \times B}\rightarrow 390 \overset{ \times J}\rightarrow 330 \overset{ \times E}\rightarrow 290 \overset{ \times F}\rightarrow 770 \overset{ \times K}\rightarrow 910\overset{ \times A} \rightarrow 170\overset{ \times B} \rightarrow 156\overset{ \times J} \rightarrow 132\overset{ \times E} \rightarrow 116 \overset{ \times F}\rightarrow 308\overset{ \times K} \rightarrow 364\overset{ \times A} \rightarrow 68 \overset{ \times I}\rightarrow 4 \equiv2^{2}$
The number of steps needed to produce the first 7 primes are shown in the table below:
| Prime | 2 | 3 | 5 | 7 | 11 | 13 | 17 | | Steps | 19 | 69 | 281 | 710 | 2375 | 3893 | 8102 |
And here is the start and end of the sequence of fractions used to produce the next prime number from $2^{2}$:
$4 \overset{ \times M}{\rightarrow} 30 \overset{ \times M}\rightarrow 225\overset{ \times N} \rightarrow 12375 \overset{ \times E}\rightarrow 10875 \rightarrow \cdots \rightarrow 232 \overset{ \times F}{\rightarrow} 616 \overset{ \times K}\rightarrow 728\overset{ \times A} \rightarrow 136 \overset{ \times I}\rightarrow 8\equiv2^{3}$
The steps needed for the first 34 primes are given as OEIS A007547 and the first 8102 products in the B-list for A007542.
The successive primes are produced almost like magic – but the number of multiplications needed to produce each new prime becomes larger and larger, and so the method, though wonderfully inventive, is not at all efficient.
Further Reading on John Conway * Listen to this Numberphile interview with Conway on how he invented the Game of Life * Play the Game of Life * Aperiodical posts about John Conway * Conway’s publications on Scholia * Conway and knots: ‘I proved this when I was at high school in England’ * Graduate Student Solves Decades-Old Conway Knot Problem, in Quanta
A while ago I made myself a calculator. I don’t know if anyone else uses it, but for the particular way I like doing calculations, it’s been really good. You’d think that if a calculator does anything, it should perform calculations correctly. But all calculators get things wrong sometimes! This is the story of how I made my calculator a bit more correct, using constructive real arithmetic.
One thing you need to think about when making a calculator is precision. How precise do the answers need to be? Is it OK to do rounding? If you do round, then it’s possible that errors accumulate as you compose operations.
I’ve always wanted to make a calculator that gives exactly correct answers. This isn’t strictly possible: there are more real numbers than a finite number of bits of memory can represent, or a digital display can show, no matter how you encode them. But I’m not going to use every real number, so I’ll be happy with just being correct on the numbers I’m likely to encounter.
The errors that a calculator makes depend on its model of arithmetic: how it stores numbers, and how it performs calculations on them.
No calculators use the same model of arithmetic that mathematicians claim to use when proving facts about numbers. In fact, I’m not sure that even mathematicians really use the “standard” model of arithmetic when doing arithmetic.
Let’s say I’m a calculator. You can be one too, if you like. I represent numbers as strings of digits: an integer part, and a fractional part, separated by a symbol. I’ll use a dot; you might use a comma or something else. I start with the most significant non-zero digit of the integer part, or just a 0 if the number is smaller than 1. I write digits of the fractional part until they start recurring, or until I get bored.
I know how to compute addition, subtraction, multiplication and division of numbers in this form. Calculations look something like this:
You might already know that numbers don’t have unique representations in this form: $1$ can also be written as $0.999\ldots$, which I’d say are both valid.
There are plenty of everyday numbers that I’ve never written in this form, such as $\sqrt{2}$ or $\pi$. More on π later.
Practically, I either round off to an approximation with a manageable number of digits, or switch to a different way of representing a number. I might maintain an air of authority by saying I’m writing things “algebraically”. But if you catch me at a moment when I don’t have all of my wits about me and ask how I write numbers, I’ll reply with a description of the above.
Some early computers represented numbers this way too, after a fashion. They’d use a collection of binary switches to represent a single digit, and collections of digits to represent numbers. Plenty of other ways of representing numbers have been tried. My favourite is the balanced ternary used by the Russian Сетунь (Setun) computer.
A Setun simulator. Each pair of lights and each switch represents a balanced ternary digit. (I think. For all I say it’s my favourite computer, I’ve never worked out how to actually use it! My desire is unachievable, a dream, like that of a dog for a sausage on a high shelf.) The most common way of representing real numbers on a computer now is to use floating point arithmetic. The gist of it is that you use some fixed number of digits to represent a number in the form
[ m \times b^e ]
It’s like a binary version of scientific notation. There’s one binary bit for the sign – whether it’s positive or negative; then the other digits are shared between the mantissa $m$ and the exponent $e$, both integers. Separating out the exponent allows you to represent numbers to the same precision (measured in significant digits) at different orders of magnitude.
A calculator which stores numbers as 4-digit floating-point decimals might assign 3 digit for the mantissa, and 1 for the exponent. That would work like this:
| 2 | $200 \times 10^{-2}$ | | 416 | $416 \times 10^{0}$ | | 1024 | $102 \times 10^{1}$ | | 0.0001234 | $123 \times 10^{-4}$ |
In the last two examples, we had to round off, introducing an error. For each floating-point number, there’s more than one real number that it could represent.
Floating point is far from perfect. In fact, pretty much anyone who uses it eventually encounters a result that seems counter-intuitive or obviously wrong, prompting even proponents of floating-point arithmetic like William Kahan to write essays with titles such as “How Java’s Floating-Point Hurts Everyone Everywhere“.
There are a surprising number of different ways floating point arithmetic can go wrong. Mike Sebastian maintains a page of “calculator forensics”, recording the result of computing $\arcsin (\arccos (\arctan (\tan (\cos (\sin (9) ) ) ) ) )$ on as many different models of calculator as he or his pals can get their hands on. The value should be exactly 9, but Mike has seen calculators produce results as small as 0 and as large as 71.252182, and all sorts of different values in-between.
Or, more succinctly, you might see this while using a floating-point calculator:
```
1/5 + 1/5 0.4 1/5 + 1/5 + 1/5 0.6000000000000001 ```
This happens because $\frac{1}{5}$ doesn’t have a terminating representation in binary: the computer has to round off. Sometimes it can tell that this might have happened and gives you the shortest corresponding decimal number, but if you reuse several rounded-off numbers, the error might add up to something that is within the precision that the computer feels is safe, so it just shows you what it’s got. And what it’s got is not correct.
Or
```
2300/(2300-2000000) == 1 true ```
This happens because the computer only uses 50-ish bits for the mantissa, so even though I’ve subtracted two million from $2^{300}$, it doesn’t make a difference in the computer’s representation, so dividing one by the other gives exactly 1. The real value is a little bit more than 1.00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00009.
We just live with these errors and they accumulate constantly and yet on the whole things seem to work.
Thinking about this helps me when I worry about the universe.
Floating-point arithmetic is widely used because it’s efficient. But who cares about efficiency? Not me, I want the right answer!
The efficiency of floats comes from using a fixed number of bits to represent a number, so any operation always takes the same amount of time. So we might find correctness by dropping this constraint and using more bits for some numbers than others.
I’m going to try doing that now. You can try too. I will write down $1 \div 3$ in decimal.
[ 0.3333333333 \ldots ]
Bear with me, this will take a while.
We have a problem: if some numbers need more digits than others, then some numbers will take forever to write. That means we don’t have enough memory to store the whole result, and if we want to do anything else with it then we’d have to first wait forever.
There’s a joke about a finitist mathematician – someone who believes there aren’t infinitely many numbers:
“So there’s a biggest number, eh?”
“Yep”
“Do you believe 1 exists?”
“Yes”
“Do you believe 10 exists?”
“… Yes.”
“Do you believe 100 exists?”
“……… Yes.”
etc.
This joke might originate with Alexander Esenin-Volpin.
I suppose the implication is that eventually the delay before the finitist answers gets so long that the questioner gets bored and wanders off. As a parent of a four-year-old, all I have to say is that numbers might not be infinite, but the mathematician’s patience would have to be!
Matt has made a video of the attempt. Here’s a clip of me shortly after I realised the right way to do it
Recently I took part in Matt Parker’s attempt to calculate π by hand. He wanted to repeat local hero William Shanks’s calculation using Machin’s formula,
[ \frac{\pi}{4} = 4\arctan\left(\frac{1}{5}\right) – \arctan\left(\frac{1}{239}\right) ]
We, and Shanks, performed the calculation by rewriting the above formula using the Taylor series expansion for $\arctan$, to get
[ \pi = 4 \sum_{n=0}^{\infty} \frac{(-1)^n)}{2n+1} \left( 4 \left(\frac{1}{5}\right)^{2n+1} – \left(\frac{1}{239}\right)^{2n+1} \right) ]
Then it’s just repeatedly multiplying, dividing, adding and subtracting numbers: all things we know how to do on paper.
Each term is smaller than the last, so in order to have an answer that was definitely correct to $n$ decimal places, we only had to compute until we got to a term with $n$ zeros. We initially planned on computing everything to a fixed 100-ish digits, so that we’d have 100 definite digits of π by the end.
After a while, we noticed that since each term is computed using the previous term, if we had 10 decimal places of $\left(\frac{1}{239}\right)^n$, we could only compute 10 decimal places of $\left(\frac{1}{239}\right)^{n+1}$. But if we later get more digits of the first term, we can pick up the calculation of the next term where we left off, and compute more digits of that too.
So we could first compute 10 digits of π, then go back to the start and compute 10 more digits of each of the terms, to get a few more digits of π. We’d slowly build up the real value of π over time, just like William Shanks did.
A bell started ringing in my head. I faintly remembered reading a while ago that the Android operating system’s built-in calculator lets you swipe right to see as many digits of the result as you’d like. A search for “calculator” in my Interesting Esoterica collection produced the paper Small-data computing: correct calculator arithmetic by Hans-J. Boem, about how they achieved this.
The Android calculator app showing me lots of digits of $\sqrt{2}$. The representation is called constructive reals. It’s a really simple idea: rather than completing an entire calculation immediately, the calculator produces functions which can produce an approximation of the real value to any given number of digits. You store the construction, and only do real calculations with digits when you have to. When you do an operation like adding two numbers together, it stores those two functions and returns another one. If you want $n$ digits of $x+y$, it computes $n+2$ digits of $x$ and $y$, then adds those together and returns the first $n$ digits.
This is the same thing we noticed when computing π: if you want more digits later, you can just come back and compute more digits of all the bits you need. Like the finitist mathematician in the joke, this swaps precision now for time later on.
This works really well for a calculator interface: it can show you the first few digits, and only compute more if you ask for them. It wouldn’t work as well for a long-winded, automatic process, where you have to decide beforehand how much precision you want in the final result, so you might as well use that precision throughout.
Android, or at least this bit of it, is open-source software, so I thought it would be a nice project to translate Boehm’s Java code to JavaScript, to use in my calculator. It’s surprisingly short: less than 2000 lines in the end, to implement all the arithmetic functions, exponentials and logarithms, and trigonometric functions.
The exponential and trigonometric functions use exactly the same trick as Shanks did, almost 200 years ago: they compute terms of the Taylor series, and stop when they’ve got enough digits. In fact, the first version of the code used Machin’s formula exactly!
Translating the code took a day or two, and it worked immediately. This never happens! Thanks, Hans-J. Boehm!
I had to make a few decisions about how to present very long numbers. The Android calculator initially shows as many of the most-significant digits as will fit on the screen, then when you swipe to see more it appends an exponential like “Ennn” to show the number of digits that are now hidden to the left.
I don’t quite have it in me to make such a slick transition, so I went with something easier to achieve: the initial view shows at most 10 characters – either the whole number if it’ll fit, or scientific notation if not. When you tap the number, it expands to show all the digits in a box that can scroll, computing more when needed. I found it hard to spot the decimal separator while scrolling, so it draws the digits of the fractional part in italics.
You can use my calculator yourself, or if you’ve got an Android device you can enjoy the same level of accuracy by opening the built-in Calculator app.
My JavaScript translation of the constructive reals library is on GitHub.
Maria Gaetana Agnesi by Bianca Milesi Mojon (1836) and 祖冲之铜像.jpg by 三猎. I wrote a mathematics-themed competition for British Science Week, which is a UK-wide event lasting ten days taking place this month.
The competition calls for individuals or groups to research the life and/or work of a mathematician and produce a poster to share their findings. The six mathematicians available to choose from are:
- Maria Agnesi (1718-1799): Italian mathematician especially known for a famous cubic curve.
- Muhammad ibn Musa al-Khwarizmi (c.780-c.850): Persian mathematician influential in many areas including algebra and algorithms.
- Apollonius of Perga (c.250BCE-c.190BCE): Greek mathematician who introduced the conic sections – two-dimensional shapes formed by cutting through a cone.
- Brahmagupta (598-670): Indian mathematician who set out rules for arithmetic involving zero.
- Philippa Fawcett (1868-1948): English mathematician, first woman to score top in the Cambridge mathematical exams, but was not awarded the degree she earned by doing so.
- Zu Chongzhi (429-500): Chinese mathematician and astronomer who calculated the most accurate approximation of pi for most of the next 1000 years.
Posters will be judged according to the following criteria:
- Creativity of presentation – How well have you used the space to creatively present your ideas?
- Clarity of presentation – How did get your ideas across?
- Research – What do you know about your chosen mathematician’s life and work?
The deadline for entries is Friday 18th March 2022. More information is available from the SHU Science Week Challenges & Competitions page under ‘Mathematics Poster Competition’.
It’s been a while since we’ve seen a MathsJam recap, but having restarted the MathsJam in Leuven after a hiatus, Dieter was too excited not to share what they’d been up to.
The first (re)edition of the Maths Jam in Leuven (Historic university city in Belgium) was a tiny success. I brought a couple of physical copies of the single page worksheet called the MathsJam Meta Shout with $\sim$10 problems from different sub areas of maths which I received earlier from Katie who coordinates MathsJams internationally.
The problems on the sheet were ranging from simple (?) Fold-and-cut fun, Tangrams (geometry), to some number theory, a touch of Linear Algebra and an easy arithmetic problem, solvable with 12-yo-level calculations (i.e. arrange all numbers from 1-15 so that each adjacent numbers sum to a square of a whole number under 16). Nice to see how broad the difficulty space is on the worksheet. Creative problem solving for (nearly) all ages!
On the second to last Tuesday of February 2022 (and hopefully each month from now), we were 3 in total. Which is a good number, I guess, for the Leuven revival anno 2022. Plenty of room to go from there – and a prime number, naturally! :-) Unfortunately we got kicked out of the venue at 21h30 (we started at 20h GMT+1) because we were the only ones left and the venue closes at 22h on Tuesdays (something I didn’t check, nor expected really). But puzzle minded we were, we just overflowed to someone’s home -after a rainy bike intermezzo which refreshened our minds. This didn’t stop us continuing our puzzling until 23h.
One of the attendees (a mathematician by degree) aced all the problems in <3h all the while (attempting) to explain his rationale. It was quite impressive to see! And fun too, because I definitely learned quite some things that night. I was still attempting to fold-and-cut the necessary T-shape with the proper dimensions (3 unit squares on top & 4 from top to bottom) when others had already finished their second tangrams (with some clever area proportion estimates). I forgot to bring my edition of the Set game so we didn’t participate in the online inter-MathsJam set-hunt – being only three we were too eager to just dive into the puzzles first.
After that second to last Tuesday, I tried some of the puzzles I hadn’t completed that night myself and I still haven’t finished them all just yet. (Some of them really make the gears in my brain grind!) I really liked the mix of complexity and variation in type of problems (kudos/thx/merci to all those involved in the making of the Shout).
New Leuven MathsJam venue, at Opek I’m already eagerly looking forward to the next edition Shout and the next physical meetup by extension (Tuesday 22th of March), and have arranged a new venue for this month in the bar Café Entrepot of the local art center Opek. This seems very fitting for the subtle art of maths and I’ve got the guarantee that they will host us at length, yay! I sure hope to see you there on a second to last Tuesday soon. :-)
At the 2021 UK MathsJam Gathering, I gave a talk on a subject that has bothered me more than is reasonable: the graph-theoretic layout of the narrative of the baby’s book Each Peach Pear Plum, by Janet and Allan Ahlberg.
It’s one of my son’s favourite books to fall asleep to. It was his older sister’s favourite, and mine and my wife’s when we were little. I agree with the quote on the back cover, that it’s “the perfect first book”.
BUT
Every time I read it, a thought pops into my head that dampens my enjoyment: an incompleteness, like noticing your coat is missing a button, or one book on a bookshelf is taller than the others.
Basically, I had to give this talk in order to stop the quiet voice in my head that requires Order In All Things.
So, here’s the recording of my talk.
The slides are on my homepage, in case you want to look through them more slowly.
After I had the idea, I wasn’t sure if this was a maths talk. While making up my “improved” poem, I realised that it is.
In order to make the story follow the complete graph, I needed a page from character $x$ to character $y$, for each distinct $x$ and $y$.
So I opened a spreadsheet and typed them out. Each character will appear on eight pages: four where they are the “I spy”, and four where they do something rhyming with their name.
After writing out four rhymes for each character (actually three – I reused the original rhymes from the book) I needed to work out the order they appear.
It turned out this is harder than I thought! I assumed that I could just do this:
- Draw out a 9×9 grid, where rows represent the first character on a page, and columns the second character.
- Erase the diagonal, because you don’t have a page with the same character twice.
- Start at the first row, second column: $(i,j) = (2,1)$.
- Pick the smallest $k$ such that $(k,i)$ hasn’t been visited yet.
- Mark $(k,i)$ and $(i,k)$ as visited.
- Repeat steps 4 and 5 until every cell has been visited.
I tried this, and quickly got stuck with no valid options for the next page, but several unvisited cells.
So I did some thinking. I want to avoid ending up with no choices, so I thought I could put that off by picking the column with the most options left.. That worked! I don’t know if it’s guaranteed to, but I think it is. If I wasn’t spending so much time reading stories to babies, I could maybe write down a proof!
My completed grid. The tour starts at “Tom Thumb in the cupboard, I spy Mother Hubbard”, and ends at “Baby Bunting sat on his bum, I spy Tom Thumb.” I’ve made an interactive thingy to try filling out a grid. Have a go, and see if you can come up with an algorithm to fill it all.
Each time you click on a cell $(i,j)$, it marks that cell and $(j,i)$ as visited, and both cells are filled in with their position in the sequence. On your next move, you can only pick a cell from the row corresponding to the column the last cell was in.
It starts with the lines from the original book filled in, though you can undo those if you want to start a different way.
So hopefully, now I’ve scratched this mathematical itch, I can again enjoy this book as much as the baby does. Maths as therapy.
This is the final part in the Pascal pentalogy, a series of guest posts by David Benjamin exploring the secrets of Pascal’s Triangle.
Probability and combinations In Part 1 of this series we stated that Pascal is credited with being the founder of probability theory – but credit also needs to be given to other mathematicians, in particular the Italian polymath Girolamo Cardano.
The connection between probability and the numbers in Pascal’s triangle can be shown by looking at the outcomes when one or more coins are tossed. The table below, from row two, lists the outcomes for one, two and three unbiased coins.
| $1$ | | $1$ H | $1$ T | | $1$ HH | $2$ HT, TH | $1$ TT | | $1$ HHH | $3$ HHT, HTH, THH | $3$ HTT, THT, TTH | $1$ TTT | | $1$ | $4$ | $6$ | $4$ | $1$ |
Reading from the left: all possible outcomes, heads decreasing by one moving to the right. For four coins there is $1$ outcome for four heads, $4$ outcomes for three heads and one tail, $6$ outcomes for two heads and two tails, $4$ outcomes for one head and three tails and one outcome for $4$ tails.
Row four shows us that when three unbiased coins are tossed, the probability they will land showing two heads and one tail in any order is $\frac{3}{1+3+3+1}=\frac{3}{8}$.
As the sum of the $n^{th}$ row is $2^{n}$, the number of possible outcomes for four coins is $2^4=16$, $32$ for five coins, $64$ for six coins, …
Quincunx A Quincunx, or Galton Board, is named after the English explorer and anthropologist Francis Galton (1822-1911) – although this name is now less popular, because of Galton’s views on eugenics and racist attitudes.
A Galton Board The board is a triangular array of pegs. Balls are dropped onto the top peg and then bounce their way down to the bottom where they are collected in containers. Each time a ball hits one of the pegs, it bounces either left or right with an equal probability of $\frac{1}{2}$ and the balls collect in the containers to form the classic bell-shaped curve of the normal distribution.
Generated from Math Is Fun’s page on the Quincunx The Quincunx is like Pascal’s triangle with pegs instead of numbers. The number on each peg represents the number of different paths a ball can take to reach that peg. If there are $10$ rows and the last row contains the containers, then the probability of landing in the third container from the right can be calculated by using the formula for the Binomial distribution.
The probability of landing in the third bin from the right is $120\times(\frac{1}{2})^3\times(\frac{1}{2})^7=\frac{15}{128}=0.1171875$, where $120$ is the number of different paths to that bin.
Statistics and permutations The link between statistics and the triangle can be demonstrated using combinations. Consider these 5 mathematicians Euler, Pascal, Ramanujan, Hilbert and Conway and the possible teams for a three-legged race.
There are $10$ different teams of $3$:
EPR EPH EPC ERH ERC EHC PRH PRC PHC RHC
The formula to calculate the number of combinations is $_n{C}_r =\frac{n!}{r!(n-r)!}$ where $n$ represents the total we are choosing from, $r$ the number in the team and
[ n!=n\times(n-1)\times(n-2)\times(n-3)\times…\times1]
In our example $n=5$, $r=3$ and $\frac{5!}{3!(5-3)!}=\frac{120}{6\times2}=10$
$_n{C}_r$ can be used to calculate the rows of Pascal’s triangle as shown below for row $6$, where in the calculation of $_5{C}_0$, $0!=1$
| $_5{C}_0$ | $_5{C}_1$ | $_5{C}_2$ | $_5{C}_3$ | $_5{C}_4$ | $_5{C}_5$ | | $1$ | $5$ | $10$ | $10$ | $5$ | $1$ |
The animation film Of Dice and Men by John Weldon is a lovely way to introduce students to probability and statistics.
Pascal the polymath: mathematics, inventor, science and religion Pascal’s father was a tax collector and in 1642 Blaise invented a mechanical calculator to assist his father. It was called the Pascaline and had a wheel with eight movable parts for dialing. Each part corresponded to a particular digit in a number. Numbers could be added by turning the wheels located along the bottom of the machine. Subtraction was carried out by exploiting a method called nines’ complement representation, the use of which allows subtraction to be reduced to addition. Each digit in the answer was displayed in a separate window. The workings of the Pascaline are demonstrated here.
A Pascaline. An original is displayed in The Musée des Arts et Métiers in Paris, France A close-up of the dials which are rotated by inserting a spoke The Musée des Arts et Métiers in Paris has one of the original Pascalines. The invention was not a commercial success – it was very expensive and often only purchased as a novelty rather than for use. Essentially, it was an adding machine. Subtraction was turned into a form of addition, as was multiplication. Division was done by repeated subtraction. Nines’ complement representation is still used in modern digital computers by a similar technique called ones’ complement which is used to represent negative numbers and hence perform subtraction in the same way as addition. Pascal did not discover this method but his calculator is the earliest known device to employ it. He continued to make improvements to his design until 1652.
Conic sections – normally just called conics – are obtained when a mathematical cone is sliced by a plane. Depending on the angle of the slice, the intersections create a circle, an ellipse, a parabola and a hyperbola. Conics have many applications including the wheel of course, ophthalmic, parabolic mirrors and reflectors, telescopes, searchlights and projectile motion.
Pascal wrote a short treatise, Essai pour les coniques (Essay on Conics) when only 16. In it he included what is known as Pascal’s Theorem which states that if a hexagon is inscribed in a conic section then the three intersection points of opposite sides lie on a straight line – the Pascal line. The theorem [also referred to as Pascal’s Hexagrammum Mysticum Theorem] was his first important mathematical discovery and a breakthrough in the field of projective geometry.
A rare copy of the Essay pour les coniques which is kept in the National Library of France In 1647Pascal expanded on the work of the Italian physicist Evangelista Torricelli, the inventor of the barometer by writing Experiences nouvelles touchant le vide (New experiments with the vacuum) in which Pascal gave detailed rules to describe to what degree various liquids could be supported by air pressure. In 1971 the SI unit for pressure [equal to one newton per square metre] was named the pascal.
A pressure gauge reading in psi (red scale) and kPa (black scale) Also in 1647 he discovered Pascal’s Law of hydrostatics allowing for the development of the hydraulic press. Pascal himself used the principle to invent the syringe.
Pascal’s Law is the principle behind hydraulic lifting and pressing devices Pascal wrote an extremely influential theological work which was unfinished at the time of his death. It was posthumously called Pensées (Thoughts) and contained a detailed and coherent examination and defence of the Christian faith.
Pascal – Pensées, édition de Port-Royal, 1670 In 1655 Pascal was trying to invent a perpetual motion machine, a machine that continues to operate without drawing energy from an external source. The laws of physics now say this is impossible. Naturally he failed but he ended up inventing a basic roulette wheel, now upgraded and used in casinos as a game of chance.
The Swiss computer scientist Niklaus Emil Wirth, born in 1934, named one of his programming languages Pascal in honour of Blaise. Wirth along with Helmut Weber also designed the programming language named after another mathematician, Euler. [Recommended read: Euler: The Master of Us All ]
Pascal died in extreme pain at the young age of 39. He had a malignant growth in his stomach which had spread to his brain. Like many others, such as Évariste Galois and Franz Schubert, we are left wondering what else Pascal could have achieved had he lived longer. His work with Fermat into the calculus of probabilities helped the German mathematician Gottfried Leibniz [1646-1716] develop the infinitesimal calculus. Pascal is buried in the Saint-Étienne-du-Mont church in Paris and his death mask is held at the J. Paul Getty museum in Los Angeles, California.
The death mask of Blaise Pascal
To celebrate 14th March (π day), MathsCity in Leeds is hosting a competition to celebrate everyone’s favourite geometrical shape whose circumference is π times its diameter: the circle.
Visitors to the MathsCity maths discovery centre between Saturday 5th March and Sunday 13th March 2022 will be able to take part in the competition by drawing their best circle on a giant whiteboard. The person to most closely approximate a perfect circle will win £20 to spend in the gift shop!
If you want to hone your skills before you take part, MathsCity recommends this online game you can use to practice. They’ll also be running π-related craft activities through the week (from 8th March).
To visit, you can book tickets on the MathsCity website (£6.50 adult, £4.50 child 3-16/concession, £18 family ticket) and while you’re there you can check out all their other excellent hands-on maths exhibits, games and puzzles, as described by Peter in his review.
In recent days there have been calls for the International Mathematical Union (IMU) to not hold in the International Congress of Mathematicians (ICM) in Russia in July 2022 due to the developing situation in Ukraine.
This is in addition to previous complaints that Russia is not a safe place to host the ICM, particularly because of its laws affecting LGBTQ+ people.
The IMU announced today that the ICM and associated General Assembly of the IMU will not be held in Russia. Instead, the ICM will be a wholly virtual event – and free to attend. They are seeking an alternative location outside Russia for the General Assembly and prize-giving.
Some details:
- The programme will follow the St Petersburg time schedule.
- They considered postponing for a year (there’s precedent: the 1982 Warsaw Congress) or cancelling, but decided that in 2022 the solution to this problem is a virtual meeting.
- Speakers can choose to pre-record or present live.
- Further details are promised later.
The statement takes pains to point out the hard work of the St. Petersburg team who have been organising the event, and to express sympathy with the people of Ukraine.
IMU statement ‘Decision of the Executive Committee of the IMU on the upcoming ICM 2022 and IMU General Assembly’.
Here’s a roundup of mathematical things that have happened in February 2022.
Ukraine The deeply troubling and developing situation in Ukraine has implications for the 2022 International Congress of Mathematicians (ICM) due to take place in St. Petersburg, Russia in July. A group of Ukrainian mathematicians has issued a call for mathematicians to boycott the event. National organisations around the world have been issuing statements setting out their positions, standing down their participation and calling on the International Mathematical Union to not hold the event as planned. Here are some we spotted:
- American Mathematical Society
- London Mathematical Society, UK
- Institute of Mathematics and its Applications, UK
- Société Mathématique de France
- Australian Mathematical Society
- Canadian Mathematics Society
The International Mathematical Union (IMU) itself wrote to its member organisations expressing its deep concern, acknowledging the calls and saying it is assessing the situation.
Other news The organisers of the Gathering 4 Gardner recreational maths conference have announced that this year’s event, taking place in April, will be a hybrid event with 50% discount for online-only places, making them a snip at $200. Registration is restricted to previous attendees and invitees, but it is possible to nominate yourself or someone else for an invitation.
Casualties of the recent storms in the UK apparently also include Newton’s apple tree – not the actual tree an apple fell on his head from, but scions of the original are planted all over the UK and one of the ones at Cambridge, which was planted in 1954, hasn’t survived the combined effects of Storm Eunice and gravity. More info in this excellent Twitter thread.
The Royal Statistical Society has released a report entitled Behind the numbers: The RSS puts the statistical skills of MPs to the test, in which they report the results of asking an anonymous unspecified group of Labour and Conservative MPs a series of simple stats and probability questions. The survey concluded that while MPs performed better than they did in a similar test ten years ago, their stats skills were still sub-par. It may not be as unambiguous as the research seems to claim though – Rob Eastaway has thoughts about the questions used.
Prizes Dr. Matilde Lalín (photo: CMS) Canadian number theorist Dr. Matilde Lalín is to receive the Krieger-Nelson prize, awarded since 1995 by the Canadian Mathematical Society to recognise outstanding contributions in the area of mathematical research by a female mathematician. (via Jordan Ellenberg)
The winners of the 2022 Mathical book prize, an annual award for fiction and nonfiction books that inspire children of all ages to see maths in the world around them, have been announced. The winners look to include some lovely titles, including Maryam’s Magic – the story of mathematician Maryam Mirzakhani – and the fantastic-sounding Uma Wimple Charts Her House. (via Jordan Ellenberg)
And finally If you like that kind of thing, you can buy a bunch of cheap maths puzzle book PDFs in a Humble Bundle (via Adam Atkinson). And if you like proof assistants, there’s now a Proof Assistants Stack Exchange.
Chris Sangwin and I wrote a LaTeX package for drawing Hex boards and games called hexboard. It can produce diagrams like this.
First: why? Then: how do you use it?
Why? Recently I noticed Chris Sangwin’s book review of Hex: The Full Story in AMS Notices had some nice-looking Hex diagrams.
I have been struggling to find a nice way to represent Hex board in my classes for our final year module Game Theory and Recreational Mathematics. I show Hex as an example of a combinatorial game (while mostly talking about Nim), and later use a 2×2 Hex game for a simple Minimax game tree search exercise for students (while mostly covering Noughts and Crosses). There are two LaTeX packages I’m aware of, havannah and hexgame – neither of which quite look right to me. What really appealed about Chris’ diagrams is that they look quite similar to the Hex board in our Maths Arcade, which came from Nestor Games.
I asked Chris what he’d used to draw his diagrams. It turns out he’d written his own code to draw these, which he shared with me. Unfortunately it was written in PSTricks, which I don’t know, but Chris mentioned updating this to use TikZ, which I do. So I remade the code in TikZ and added a few bits. I was aiming to get to the point where the code would do everything in Chris’ intro to Hex worksheet and everything I needed for my class.
Anyway, our new package has led to what I consider to be an improvement in my Hex materials this year.
Getting started
First, do you have hexboard installed? Make a LaTeX file with \usepackage{hexboard} in the preamble and see if it runs. If you use MiKTeX, it should be that compiling a document that uses the hexboard package will install it for you. If you use TeXLive, you may need to update packages. On my Ubuntu machine with TeXLive 2021 installed manually, I ran tlmgr install hexboard and it installed. As a last resort, you can download the package from CTAN, unzip the contents and run latex hexboard.ins which will create hexboard.sty. Then put hexboard.sty in the same folder as your LaTeX file.
To use the package, include in your preamble \usepackage{hexboard}.
Basically, Hex diagrams exist in a hexpicture environment. Here is a simple example. It just draws a blank Hex game board.
\begin{hexpicture}
\hexboard{11}
\end{hexpicture}
The output should look like this:
You can place various objects on a board, including counters, dots, lines and general LaTeX code (e.g. text). Here’s a more elaborate example.
\begin{hexpicture}
\hexboard{3}
\hexcounter{b}{1}{A}
\hexcellshaded{b}{2}
\hexdot{a}{2}
\hexdot{b}{2}
\hexconnect{b}{1}{a}{2}
\hexconnect{b}{1}{b}{2}
\hexcontent{a}{3}{x}
\end{hexpicture}
Here \hexcounter{b}{1}{A} draws a counter for player A in cell b1, \hexdot{a}{2} draws a dot in cell a2, \hexconnect{b}{1}{a}{2} draws a line from cell b1 to cell a2, \hexcellshaded{b}{2} shades cell b2, and \hexcontent{a}{3}{x} puts an x in cell a3. This should appear like this:
There is also an environment to display games in progress. Actually two – hexgame and hexgamelabels. In these, you specify moves in a game and the package keeps track of whose turn it is and colours the counters accordingly. The difference is that hexgamelabels puts a number on each counter to show when it was played. Here is an example of hexgamelabels.
\begin{hexgamelabels}[5]
\hexmove{c}{2}
\hexmove{b}{4}
\hexmove{b}{3}
\hexmove{d}{3}
\hexmove{a}{5}
\hexmove{a}{4}
\hexmove{c}{3}
\hexmove{c}{4}
\hexmove{e}{3}
\hexmove{e}{2}
\end{hexgamelabels}
This should look like this:
There is some customisation (including size and colours), and other options such as drawing partial board diagrams (and the news that secretly hexpicture is a type of tikzpicture). Much more on the package can be found in a series of examples in the documentation.
The code and a place to log issues are available as hexboard on GitHub.
The next issue of the Carnival of Mathematics, rounding up blog posts from the months of December and January, is now online at Ganit Charcha.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
This is the fourth in a series of guest posts by David Benjamin, exploring the secrets of Pascal’s Triangle.
Triangles and fractals If we highlight the multiples of any of the Natural numbers $\geq 2$ in Pascal’s triangle then they create a pattern of inverted triangles.
The images above are evocative of the Sierpinski sieve (also known as the Sierpinski gasket or Sierpinski’s triangle), a fractal described in 1915 by the Polish mathematician Waclaw Sierpiński (1882-1969).
The Sierpiński Sieve
Medal of Waclaw Sierpiński in the Yale University art gallery
Fractals are beautiful geometric shapes. Small, even down to (theoretically) infinitesimal areas of a fractal are identical to the entire shape. The Koch snowflake, generated geometrically by successive iterations on an equilateral triangle, is an example of a fractal. Julia sets and Mandelbrot sets are examples of fractals generated using recursion on complex functions. Many examples of fractals appear in nature, and the Polish-born French-American polymath Benoit Mandelbrot (1924-2010) suggested that fully developed turbulent flows are fractals.
It is a lovely surprise to discover that a simple fractal can be found inside Pascal’s triangle. It is achieved by considering all the numbers in the triangle modulo 2 – equivalent to colouring in only the multiples of 2, as in the first diagram at the top of the post. In this version, every odd number becomes $1$ and every even number becomes $0$, and by considering sufficiently many lines of the triangle, the Sierpinski pattern emerges.
The areas containing the odd numbers have been shaded red and the areas containing the even numbers, black.
Pascal’s triangle modulo 2. The entries when concatenated can be read as binary numbers which are palindromic.
Number patterns in the triangle If we consider the first 32 rows of the mod$(2)$ version of the triangle as binary numbers: $1, 11, 101, 1111, 10001,…$ and convert them into decimal numbers, we obtain the sequence:
[1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, ]
[196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, ]
[50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295]
Interestingly, all members of this sequence are factors of the final term, $4294967295 = 2^{32} – 1$. Since this is one less than a power of two, it’s a Mersenne number. Why the first $31$ terms are all factors of the 32nd term is difficult to summarise here but there is a thread on StackExchange discussing what happens to the pattern after the $32nd$ term.
$4294967295$ has prime factorisation $3 \times 5 \times 17 \times 257 \times 65537$. These five prime factors are Fermat numbers – numbers of the form $2^{2^{n}}+1$ – in this case with $n = 0, 1, 2, 3$ and $4$. As of the time of writing these are the only known Fermat numbers which are also prime.
These patterns in the rows of the triangle are intriguing, and my own efforts to understand them have uncovered a few other interesting discoveries – notably, that while the 32nd term is not divisible by the 33rd, the 34th term is exactly 3 times the 33rd. The pairs of terms after that seem to alternate, as they do from the start of the sequence, between a non-integer ratio and a ratio of exactly 3, which I conjecture is a pattern that will continue.
Two welcome appearances $e$ and $\pi$ are two of the most used transcendental numbers. The Swiss mathematician Leonhard Euler (1707-1783) connected them with the most beautiful equation, called Euler’s identity:
[e^{i\pi}+1=0]
There are many approximations connecting $e$, $\pi$ and other irrational numbers to be found here.
In 2012 Harlan J. Brothers proved that
[\displaystyle\lim_{n\to \infty} \frac{\ \displaystyle\frac{s_{n+1}}{s_n}\ }{\displaystyle\frac{s_n}{s_{n-1}}}=e]
where $s_n$ is the product of the numbers on row $n$ of Pascal’s triangle. The proof can be found on Cut the Knot, part of the wonderful website of Dr Ron Knott.
In 2007 Jonas Castillo Toloza discovered a connection between $\pi$ and the reciprocals of the triangular numbers (which can be found on one of the diagonals of Pascal’s triangle) by proving
[\pi= 2 + \frac{1}{1} + \frac{1}{3} – \frac{1}{6} – \frac{1}{10} + \frac{1}{15} + \frac{1}{21} – \frac{1}{28} – \frac{1}{36} + \frac{1}{45} + \frac{1}{55} – \ldots]
Three proofs are given on Cut the Knot.
Harmony in the triangle The infinite sum of the reciprocals of the Natural numbers is called the harmonic series, $H_n$, where
$H_n = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} \ldots$
The series is divergent, but it crawls its way towards infinity, and takes $15092688622113788323693563264538101449859497$ terms just to pass a total of $100$.
The harmonic series can be used to create a version of Pascal’s triangle – the series itself is placed along the two leading diagonals, and the entries are then related by each being the difference of the fraction to its left, and the one diagonally above it and to its left. For example, $\frac{1}{30} = \frac{1}{5}-\frac{1}{6}$.
Dividing the first term in the $n^{th}$ row by every other term in that row creates the $n^{th}$ row of Pascal’s triangle. The table below shows the calculations for the $5^{th}$ row:
| $\frac{1}{5}$ | $\frac{1}{20}$ | $\frac{1}{30}$ | $\frac{1}{20}$ | $\frac{1}{5}$ | | $\frac{1}{5}\div \frac{1}{5} =1$ | $\frac{1}{5}\div \frac{1}{20} =4$ | $\frac{1}{5}\div \frac{1}{30} =6$ | $\frac{1}{5}\div \frac{1}{20} =4$ | $\frac{1}{5}\div \frac{1}{5} =1$ |
In our next post, we’ll talk about probability and statistics in Pascal’s triangle, and consider some of Pascal’s other contributions.
Here’s a roundup of some of the mathematical things that happened in the first month of the year.
Maths news Donald Knuth has published an amendment to his book Concrete Mathematics in which he accepts Peter Luschny’s definition of the Bernoulli numbers, under which (among other things) the value of $B_1$ should really be $+\frac{1}{2}$, not $-\frac{1}{2}$, as outlined in Luschny’s Bernoulli Manifesto. CL-P wrote about this way back in Aperiodical Round Up 10. (via Russ Cox)
A long-standing conjecture about the Game of Life has been settled, Adam Goucher reports: there is a still-life that can’t be constructed by gliders. Two postdocs at the University of Turku, in Finland, have found the configuration below, has the property that, if it occurs within a universe at time $T$, it must have existed in that same position at time $T-1$ (and therefore, by induction, at time $0$).
Jason Kottke reports that Charles and Ray Eames’ 1977 short film Powers of Ten, a classic piece of science communication which showed objects at every scale from a picnic by the lakeside in Chicago to the outer edges of the universe and zooming out by a factor of 10 every 10 seconds, has been updated to reflect another 45 years of scientific discovery.
Books news
Nathaniel Johnston and Dave Greene have published a book about Conway’s Game of Life, which aims to “demystify the Game of Life by breaking down the complex patterns that have been developed in it into bite-size chunks that can be understood individually”. It’s available to download for free as a PDF, but a print version is coming soon. (via Rudy Rucker)
There’s a Kickstarter for Ben Orlin’s new book, Math Games with Bad Drawings (right). The book includes over 75 pen-and-paper games anyone can play, and Kickstarter editions will be signed and come with some bonus game cards. It’s also available to preorder in a variety of places, and will be out on 7th April.
Events and other news Imperial College London is running the London Learning Lean seminar, aimed at formalising undergrad-to-research level mathematics in the Lean theorem prover. Sessions will aim to spend half of the time on describing some maths and the other half on formalising it in Lean. The seminar will take place in person at Imperial on Thursdays at 4pm (GMT) and be streamed live via Zoom. (via Kevin Buzzard)
Sadly, this month we lost algebraic topologist Fred Cohen (via Selman Akbulut) and statistician David Cox (via David Spiegelhalter).
We invited mathematician and wordplay fan Ali Lloyd to share his thoughts on hit internet word game phenomenon Wordle. If you’re not familiar with the game, we recommend you go and have a play first.
CC BY-SA ZeroOne When I first saw Wordle I said what I saw many other people subsequently say: “Oh, so it’s a bit like Mastermind but with words? That’s a neat idea”.
In formal language terms, Mastermind is played over an alphabet of six ‘symbols’ (represented by six differently coloured pegs) and a ‘word’ is any combination of those symbols of length 4. There are therefore $6^4=1296$ different words, all of which are well-formed – any combination of 4 symbols is a valid word.
In a game of Mastermind, a target pattern is fixed, and the player takes guesses by placing pegs in the board. Red and white pegs are used in response to indicate which colours in the guess are in the right place and which are present in the target pattern, but in a different place. The player then takes another guess. If you’re playing Consistent Mastermind, the next guess must incorporate all the clues given after previous guesses. The amount of allowed guesses before you lose is typically between 8 and 12.
The fact that all words are well-formed simplifies Mastermind strategy somewhat: it makes it completely uniform, in the sense that all combinations of colours are valid guesses and the strategy itself does not need to change depending on which colour and position you have information about.
As an illustration of this, Donald Knuth has an algorithm for Mastermind which is guaranteed to win in 5 guesses or fewer, and involves starting with a combination like ‘red red blue blue’ – however, this could equally be ‘yellow yellow green green’ or ‘red blue red blue’; the important thing is that there are two of each.
Wordle
Wordle is very similar to Mastermind, except it uses the alphabet as the ‘alphabet’, and English words as the ‘words’, and you only get 6 guesses. It also has a consistent version, where you have to use the previous clues marked as correct in subsequent guesses – in Wordle this is called ‘Hard mode’. Hard mode Wordle is arguably harder than Consistent Mastermind as your guess has to fit the pattern and also be a valid word. On the other hand, this pushes you to solve more directly.
Plainly the uniformity of Mastermind does not exist in Wordle – ARERE ((Spenser) backward, behind) is probably a better opening guess than XYLYL (a chemical compound found in coal-tar). Indeed, while it would be easier to come up with an algorithm for solving anything’s-a-word Wordle, it would not be much fun at all to actually play.
Wordle has both a ‘source’ word list and a ‘target’ word list. You can guess anything from the source word list (which is the 5 letter words from CSW19). Incidentally there is exactly one 5-letter word in the US Scrabble dictionary that does not appear in CSW19 – so bad luck if you really want to guess LUVED.
The target word list is a small hand-curated subset of less than 2500 of these words. This complicates potential strategy in two ways: firstly, it involves the consideration of whether the word has been deemed common enough to be fair by a specific other person; secondly, it opens up the possibility of guessing a word which will definitely not be correct, but will rule out enough to make it worthwhile.
One of the reasons a curated list is a good idea is that CSW19 is chock-full of potentially horrible traps, especially for hard-mode. If the target word is HILLS and you are unlucky enough to guess BILLS on your first go, you end up with B I L L S with any one of (B)CDFGHJKLMNPRSTVWYZ the correct first letter.
As it happens, in fact none of these words actually appears in the Wordle target word list, as most words ending in S did not make the cut. However the second-worst scenario of this kind in CSW19, ? I G H T, which has 15 possible starting letters (ABDEFHKLMNPRSTW), is still a potential Wordle nightmare, with 9 of those being included in the target word list
This was conveniently recently illustrated by Wordle itself.
Another theoretical pitfall is repeatedly getting yellows instead of greens. In mathematical terms, the worst-case scenario here would be akin to a cyclic permutation of length 5 (i.e. with no fixed points) – thankfully there is no such cycle in CSW19, but it comes close – PESTO, ESTOP (to hinder or preclude), STOPE (to excavate in layers) and TOPES (drinks liquor to excess) are all in there, meaning it’s theoretically possible to have three straight rows of yellows (of course if that did happen you would probably get the answer on the next go):
T O P E S
S T O P E
E S T O P
P E S T O
Frequency Analysis The fact that not all letters are equally likely to occur in the answer word can be used to your advantage when guessing. Everybody will likely be doing this already, knowing intrinsically that RATES is more likely to give you information about the answer word than RAZES. Of course if you get lucky and there is a Z, fair play to you. But in all likelihood you will discover that there isn’t a z, which doesn’t narrow down your options much. On the other hand, all three bits of information about that T are reasonably useful – if it goes green, you narrow the list of possible words down to 616. If yellow, 2417. If black, take consolation from the fact you’ve eliminated up to 3000 words from the list.
It is well known that when you order letters by frequency of appearance in English words, you get ETAOIN SHRDLU. This is especially well known to anyone who has solved a substitution cipher, or written a substitution cipher solver – for ciphertext of a reasonable length, where the original has not been constrained deliberately to make decoding harder, starting out by mapping letters by frequency in ETAOIN SHRDLU order is almost always close enough to the plain text that it only requires a few swaps to get there.
However, this is highly dependent on the word list you are analysing. If you take all 5 letter words in CSW19, you get SEAORY LTNUDY. If you instead take the 5 letter words in Google’s 10,000 common words list, you get ESAROT LINDCH.
If you instead take the words that have already been Wordle answers, you get ERAOTI LSCNUD. You can even dig into the source (not very deeply) and find the complete list of words that will be Wordle answers (this thread discussing this does not contain spoilers, unless you consider the letter distribution of the answers a spoiler).
Using this letter distribution has apparently yielded results for more than one person. The unusually high prevalence of the letter C and low prevalence of H can inform your strategy – if you’re thinking of trying SHRUB, maybe consider SCRUB instead.
Forget “Etaoin shrdlu”. For the 2315 words used in Wordle, the new coolness is: “Earot Lisncuyd”.
The full frequency table is:
e:53% a:42% r:39% o:33% t:31% l:31% i:29% s:29% n:25% c:21% u:20% y:18% d:17% h:17% p:16% m:14% g:13% b:12% f:10% k:9% w:8% v:7% z:2% x:2% q:1% j:1%
— Paul Lamere (@plamere) January 4, 2022
Frank Swain points out that it probably isn’t just a case of how frequent letter occurrence is, nor even just the frequencies for each word position, but how often letters appear together, and has produced this heat map:
This uses a corpus of 5757 words which I assume is Donald Knuth’s 5757 list (him again!) – a list which is often used as the basis for programming challenges.
Taking that idea and running with it gives you the basis of a potential strategy – try and guess things which are likely to appear in the same word – i.e. attempt to maximise positive information about the answer word. One way to do this is take every guessable word still in consideration and figure out how much information it gives you about the answer, for each of the other words in the list.
Solver I wrote a (hard-mode) solver along the above lines, and it does ok. There are many potential refinements that could be made – in particular it does nothing to avoid potential ? I G H T situations, and could probably do with a more carefully chosen initial word. The solver looks at all the remaining possibilities and for each, computes the sums of the number of greens and yellows that word would give if you guessed it, for all possible answer words.
It has four options: it either uses a 3/2 green/yellow weighting to build an overall score for each word based on the remaining possibilities, or it is greedy for greens – ordering lexicographically <# greens, # yellows>. It can either favour common words or not. If it favours common words, it promotes any words from Google’s 20,000 common words list to the top, if they are already in the top 20 options for next word.
The initial word picked by each strategy is as follows:
- green-greedy strategy that doesn’t favour common words – SORES
- green-greedy strategy that favours common words – CORES
- weighted strategy that doesn’t favour common words – TARES
- weighted strategy that does favour common words – CARES
Here is how each performs over the first 224 Wordle puzzles:
And here is how they stack up in terms of win percentage and average guesses:
| Max G | Max G + favour common | Weighted G/Y | Weighted G/Y + favour common | | Win % | 84.82% | 95.09% | 89.29% | 94.64% | | Average guesses | 4.589473684 | 4.056338028 | 4.375 | 3.900943396 |
You can experiment with my solver (christened Solvador D’Ali by a friend), and a ‘next word suggester’ I also made.
One thing about all of these strategies is that they don’t include any meta-information about previous Wordles. For example, it could be made to ‘notice’ that no previous Wordle has been a plural, and add in a weighting against selecting plurals.
As an alternative to trying to minimize your average number of guesses, you might want to just maintain your streak. Friend of the Aperiodical Andrew Taylor has implemented a strategy (where he summarises the first guess as “how few solutions would give the same clues in the worst case”) which always wins. It’s essentially a proof-by-cases, and each case keeps going until the list of possible word is less than or equal to the number of guesses remaining. There is also this ‘easy mode’ strategy of ruling out as many letters as possible.
So even though you know the answer probably won’t be EEVEN (the latter part of daylight hours) or VOZHD (a supreme leader in Russia), whichever way you look at it, strategy is still just a numbers game. The question is, which numbers are important?
Ultimately, it seems very likely that some sort of ‘genetic algorithm’ would be the best Wordle solver, just like in Mastermind. Solving a Mastermind board is an NP-complete problem. If you’re interested in that, maybe you would enjoy playing Gödle instead?
Starting word As we’ve seen above, TARES, SORES, CARES, CORES, REAIS, BLAHS, SOLAR and many others potentially make good starting words. If you have a hard time deciding why not play ‘Wordle Legacy’?
Or if you fancy a real challenge, here is a list of some of the most useless starting words:
ZOPPO (having a syncopated or temporarily changing accent of a beat)
PHPHT (used as an expression of annoyance)
KIBBI (an Eastern dish of ground lamb and crushed wheat)
WHIZZ
QAJAQ (Inuit word for a kayak)
OXBOW
FLUFF
YUKKY
KUDZU (an ornamental papilionaceous plant of China and Japan)
FUZZY
JUGUM (a pair of opposite leaves)
FUFFY (Scots for light and soft)
HYPHY (a style of hip-hop music originating in the Bay Area of San Francisco)
IMMIX (archaic to mix in, commingle)
XYLYL
Sharing With the news of Wordle’s buyout by the New York Times, you might wonder what it is they have actually bought. After all the concept of the game long predates Wordle’s existence, notably in the game show Lingo.
Wordle’s primary innovation, which ensured it became a viral sensation, is its shareability. This arises from the communal effect of having a fixed daily word (so that everyone is doing the same puzzle), and the neat method of communicating your result via spoiler-free sequences of coloured squares. The shareable string of squares and Twitter were practically made for each other.
As well as people sharing their results every day, this leads to all sorts of Twitter shenanigans. The account WordleStats trawls Twitter looking for Wordle results and summarising them. There have been Pokémon references, and recreations of the loss meme; even Kyle MacLachlan getting a Twin Peaks reference in; you can make a town out of your pattern of squares, or you can do something else entirely.
Incidentally, the unicode codepoints used for the sharing string are: White large square (U+2B1C) or Black square (U+25A0), Large yellow square (U+1F7E8) and Large green square (U+1F7E9). These last two are rather excitingly (if you like that sort of thing) in the second plane of unicode characters, the Supplementary Multilingual Plane (SMP), and they are codepoints consisting of two UTF-16 codeunits.
One interesting consequence of this is that the exact format of Wordle shareable results wouldn’t have been possible (or at least widespread) a few years back where support for display of SMP characters was nowhere near universal!
Variants Some other variants you may be interested in:
- Primel: prime number wordle (also a version by Andrew Taylor)
- Letterle: single letter wordle
- Absurdle: adversarial wordle
- Seven Wordles: speed wordle
- Lewdle: rude wordle
- 15 letter word wordle
- Nerdle: a maths one
- Wordles of the World: a comprehensive list of Wordles in different languages, and other variants
My daily regulars are Wordle, Lewdle and Facle, a Scottish Gaelic version of Wordle.
In this series of posts, we’ll be featuring mathematical podcasts from all over the internet, by speaking to the creators of the podcast and asking them about what they do.
We spoke to Della Dumbaugh and Deanna Haunsperger about their podcast, Count Me In with Della and Deanna.
Podcast title: Count Me In with Della and Deanna
Website: Math Values Blog
Links: Spotify, Apple Podcasts
Average episode length: 57 minutes
Recommended episode: Season 1 Episode 1: Talithia Wiliams
How did your podcast start? Della Dumbaugh (left) and Deanna Haunsperger (right) In the spring of 2021, we finished editing our book Count Me In: Community and Belonging in Mathematics (to appear in spring 2022) which highlights the stories of twenty-six diverse communities in mathematics. These communities show the power of belonging to a community to help anyone, particularly women and members of groups underrepresented in mathematics, to see themselves as mathematicians. One day we were out for a walk when Della insisted that Deanna hear her out on her grand idea that grew out of the book: a podcast which would show the humanity of mathematics by interviewing mathematicians and showing them as complete human beings who are vulnerable and had obstacles to overcome on their way to becoming mathematicians. Deanna thought it was brilliant.
Two months later we had found someone to explain podcasting to us and who would do production, and we started interviewing mathematicians from all walks of life and stages of their careers. We asked them about growing up, times they didn’t feel like they belonged, hardships they had overcome, how they found their community, how they take care of themselves, and a few fun lightning questions at the end.
Who is the intended audience for the podcast? Who should listen? The podcast is for anyone who loves mathematics. It’s for anyone who has ever wondered if they could be a mathematician. It’s for anyone who has ever assumed that for some the road to becoming a mathematician was completely smooth.
Mathematicians can learn that they are not alone in their joy of mathematics or in some of the struggles they have encountered along their journey. Teachers can ask their students to listen and write about experiences the students have had which are similar to the mathematicians’ so that they can begin to see mathematics is a possibility for them, too.
What are some of the stories you have found compelling so far? Colleagues recounted stories about the strength of their family as they transitioned from a two-room adobe hut in Mexico to a farming community in western Nebraska, how mathematics itself or important people in their lives helped them overcome struggles, how they found others who shared their joy in mathematics, how they came up with a clever way to make themselves invaluable to a study group in graduate school, how their cats or hockey or their faith keep them grounded, how potato chips helped them compete in a Mathematical Olympiad, and how all teachers can help build community in their classrooms. They were vulnerable and allowed us to see their very human selves.
What are your plans for the future? Our eight-episode Season 2 is being audiotaped right now. We’re very excited about the stories of the people whom we are interviewing!
In this series of posts, we’ll be featuring mathematical podcasts from all over the internet, by speaking to the creators of the podcast and asking them about what they do.
We spoke to Nathalie Vega-Rhodes, host of the Infinitely Irrational podcast.
Podcast title: Infinitely Irrational
Website: infinitelyirrational.com
Links: Podbean – Apple Podcasts – Spotify – Google Play
Average episode length: about 35 minutes
Recommended episodes:
- The Strange Case of Sir Newton and Mr. Leibniz
- Fermat: The Adventure of the Mathematical Magistrate
- Erdös: Mammas Don’t Let Your Babies Grow Up to Be Mathematicians
- Choose Your Own Hypatia: Portrait of a Lady
What is your podcast about, and why did you start making it? Up until recently, I taught math at the college level, from developmental math through Calculus. But that wasn’t where I started. As a student, I was a theatre major with a passion for good stories. When I changed my major to math, I discovered that the mathematics field is filled with figures and stories that would make Shakespeare (or George R.R. Martin, anyway) jealous. There are cunning witches, golden gods, ruthless murderers, epic duels, windswept romances, and so much more.
When I started teaching, it wasn’t long before I started sharing some of these stories with my students. Not only did my students get more interested in the math itself, the stories had the unintended benefit of making my students more open with me and more willing to go out of their comfort zones (especially with the more challenging word problems!). This planted the seeds for the Infinitely Irrational podcast. I knew these were good stories, but I needed to dig in deeper to get all the facts. These histories are woven with myth and intrigue and should be shared with the world!
Who is the intended audience for the podcast? Anyone! While math aficionados and history buffs alike will enjoy learning about the history of mathematicians and how some of the concepts came into existence, anyone who loves stories will enjoy the podcast.
When I asked people to think back to their college math class, so many of them described a blackboard, or several, filled with intimidating equations. Or a dust-covered professor droning on about how to solve for x. Some mentioned classmates frantically scribbling or struggling to stay awake, no matter how much coffee they’re drinking along with a feeling of despair that they’d never “get it” because they had no idea what was being discussed.
When I started college as a theatre major, this was my experience in a math class. And as a math professor, I heard it over and over again: “Math is for ‘smart’ people.” “I’ll never get it. I’m not a ‘math person’.” “Why do I have to learn this?” “Thank goodness! This is my LAST math class!” In fact, I’ve had many of these same thoughts myself over time. What changed? I discovered that there is more to math than solving equations: the men and women behind the math have some fascinating stories.
Learning their stories made math more accessible. I believe that math should be accessible to as many people as possible; it isn’t something that only a few privileged people can understand. But since so many people incorrectly believe this to be fact, how can I both share my love for and pique interest in math? I hope to inspire curiosity and maybe be the catalyst in realizing that mathematics can be interesting and fun – it turns out that dust-covered math professor isn’t boring; he’s Indiana Jones.
What is a typical episode like? Since the podcast’s inception, I share each mathematician’s in a trilogy (though Cardano was a special case and required four episodes to adequately summarize his drama-filled life). Episodes range from 20-40 minutes, but I try to keep them around 30 minutes. Most episodes have an Easter egg, but I’ll leave that to the listener to discover.
Why should people listen to Infinitely Irrational? Everyone loves a good story and it’s a bonus if you learn something! When I started looking for math podcasts, I found that most of them focused around either teaching specific concepts such as “how to factor” or were high-level discussions of mathematical concepts, for a more advanced audience but I wanted to learn more about the people behind the math. I also wanted to make math more accessible to everyone, starting with my students, while having fun along the way.
What are some highlights of the podcast so far? Highlights for me have been realizing how truly interconnected everything is. With Fermat’s Last Theorem, it was really cool to talk about how excited mathematicians were to break it open, even just a little bit. Seeing the influence that one mathematician can have on so many other people – how Erdös was able to make connections with so many mathematicians and be beloved by all – is just awe-inspiring!
It’s fascinating to me to learn about concepts first coming into being, such as probability, which has changed our whole life, simply by virtue of someone asking a question. Every time someone asks me which is my favorite mathematician from the podcast, I start off with one and realize soon enough that I’ve mentioned everyone we’ve covered.
But one of my favorite trilogies was my collaboration with Ben Orlin of Math with Bad Drawings. In his recent book Change is the Only Constant: The Wisdom of Calculus in a Madcap World, he shares some of the history and controversy about the origins of calculus. He collaborated with me on a trilogy where we talk about Sir Isaac Newton and Gottfried Leibniz and it truly was one of the most fun experiences. The first episode of this trilogy can be found on all the podcast players.
What exciting plans do you have for the future? By listener request, I am working on women mathematicians for the next few episodes. I’m currently finishing up Mary Fairfax Somerville but it’s taking longer than I’d like. Since changing careers in mid-2020, it has been challenging to dedicate the time needed to do justice to these wonderful mathematicians with the current format, but I’m hopeful that I will be able to get some new episodes out soon!
I’ve made a little game.
David Butler, of the Maths Learning Centre at the University of Adelaide, runs an event called One Hundred Factorial. He invites anyone passing by the MLC to join in with maths games and puzzles. David often tweets about what happens at One Hundred Factorial. It seems lovely, and the tweets always make me wish I could join in too.
A couple of weeks ago, I think partly prompted by the imminent arrival of a new lockdown in Australia, David decided to run a virtual One Hundred Factorial session at a time that would be convenient for people in the UK and USA. I signed up immediately!
I had a lovely time, chatting to David in real-time for the first time, and meeting a few other of David’s fans, who’d come to play.
One of the games that David had prepared was called Imparium. I’ve since found out it was invented by Walter Joris, a Belgian artist and prolific inventor of games.
There’s a nice interview with Walter by Ben Orlin. Ben is writing a book about mathematical games, which I’m going to buy as soon as it’s ready!
Here’s a brief version of the rules: you start with a 6×6 grid of boxes. You and another player take turns removing pairs of adjacent boxes. Whenever there’s a group made up of an odd number less than 10 contiguous boxes, the last player to move can claim them. Once all boxes have been either claimed or removed, the game is over and the person who has claimed the most boxes wins.
David had set up a board on the virtual canvas miro.io for the virtual One Hundred Factorial session, and we played Imparium by setting up a grid of square blocks, and to take a turn you had to drag some blocks out of the way. It worked quite well, but because the Miro interface isn’t designed for games we quite often made mistakes like moving too many pieces, or accidentally losing pieces underneath other bits of the canvas.
This seemed like a nice opportunity to make another interactive maths thing. I set about coding up the game’s rules in Elm, a nice programming language I’ve used in the past for things like this.
Once I’d got the basics of the game working, I thought about how to present it. Walter Joris’s original version of the game has you filling in boxes on a paper grid. David’s version involved moving boxes out of the way. I thought that the remaining pieces that are claimed by each player look like islands, so I came up with the idea that the players are trying to divide up a big island between each other, but they can only do it destructively, by digging out rivers and coastlines.
I think this works quite well: it justifies the odd-size rule, because an even-sized island can still be shared fairly between the two players.
Before the start of the game, I made a short and whimsical explanation of the rules. The condition for claiming islands is quite tricky to state succinctly, but showing some islands as they’d appear in the game alongside the text makes it much clearer.
You can play the game on my website. If you’re interested in seeing the source code, it’s on GitHub.
This is the third in a series of guest posts by David Benjamin, exploring the secrets of Pascal’s Triangle.
Leonardo Pisano (1170-1250), now universally known as Fibonacci, was born in Pisa, Italy, where he was also living at the time of his death. He was educated in north Africa as his father worked there, representing the merchants of the Republic of Pisa when they were trading in Bugia, now called Béjaïa, a Mediterranean port in Algeria.
The Fibonacci statue by Giovanni Paganucci preserved in the monumental Cemetery of Pisa Fibonacci returned to Pisa in about 1200 where he wrote a number of important books. His book Liber abaci introduced the Hindu-Arabic place-valued decimal system and the Arabic numerals we now use. Books and any copies had to be handwritten, as it predated the printing press. Fibonacci is now mostly remembered for introducing the Fibonacci numbers and sequence which appeared in the third section of Liber abaci as a problem about rabbits:
A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?
The resulting sequence is $1, 1, 2, 3, 5, 8, 13, 21, 34, 55…$ (although Fibonacci did not include the first term in the book).
Fibonacci’s rabbits The ratio of successive terms converges on the Golden Ratio, $\phi$.
$\phi = \displaystyle\frac{1 + \sqrt5}{2} \approx 1.618033988749. . .$
$\phi$ is an irrational number and is the positive solution of the quadratic equation $x^2 – x – 1 = 0$ Hence, since $\phi$ is the root of an integer polynomial, it is not transcendental, unlike $\pi$.
[ \frac{1}{1} = 1 \qquad \frac{2}{1} = 2 \qquad \frac{3}{2} = 1.5 \qquad \frac{5}{3} = 1.666 \ldots \qquad \frac{8}{5} = 1.6]
[ \frac{13}{8} = 1.625 \qquad \frac{21}{13} \approx 1.615384 \qquad \frac{34}{21} \approx 1.619047 \qquad \frac{55}{34} \approx 1.617647 \qquad \ldots ]
Indeed, convergence to $\phi$ remains true if we start with any pair of Natural numbers and follow the same pattern where any term after the second is the sum of the previous two terms.
| Terms | Ratio | | 3 | 2.33333… | | 7 | 1.428571… | | 10 | 1.7 | | 17 | 1.58823… | | 27 | 1.62962… | | 44 | 1.61363… | | 71 | 1.61971… | | 115 | 1.61739… | | 186 | 1.61827… | | 301 | 1.61794… | | 487 | 1.61806… | | 788 | 1.61802… | | 1275 | 1.61803… |
Convergence when the first term is smaller than the second term
| Terms | Ratio | | 5 | 0.6 | | 3 | 2.66666… | | 8 | 1.375 | | 11 | 1.72727… | | 19 | 1.57894… | | 30 | 1.63333… | | 49 | 1.61224… | | 79 | 1.62025… | | 128 | 1.61718… | | 207 | 1.61835… | | 335 | 1.61791… | | 542 | 1.61808… | | 877 | 1.61801… |
Convergence when the first term is larger than the second term
| Terms | Ratio | | 2 | 0.5 | | 1 | 3 | | 3 | 1.33333… | | 4 | 1.75 | | 7 | 1.57142… | | 11 | 1.36363… | | 18 | 1.61111… | | 29 | 1.62068… | | 47 | 1.61702… | | 76 | 1.61842… | | 123 | 1.61788… | | 199 | 1.61809… | | 322 | 1.61801… |
This is called the Lucas Sequence.
In Liber abaci, Fibonacci included other numeracy problems – on perfect numbers, the Chinese remainder theorem and on the sum of arithmetic and geometric series. He wrote a book on geometry, Practica geometriae, and perhaps his most impressive work was Liber quadratorum in which he included methods for finding Pythagorean triples. But it is for his sequence for which he is mainly remembered.
The Fibonacci Sequence in Pascal’s triangle Finding out that the Fibonacci sequence can be found in Pascal’s triangle was a delight for me and I find it hard to think it is just a coincidence. To view Fibonacci’s sequence we can display the triangle as a right-angled triangle.
Fibonacci’s sequence is hidden in the triangle The Golden ratio in art, music and architecture My interest in mathematics began when the film Donald Duck in Mathmagic Land was shown to our class in my first year at secondary school in Burnage, Manchester, England and as a teacher of mathematics I showed it in the lesson before Christmas to many year 7 groups.
The film illustrates how the Golden Rectangle has been used by artists and architects throughout history as well as connections between the golden ratio and music. The film mimics some of the novel Alice in Wonderland by Lewis Carroll, the pseudonym of the mathematician Charles Lutwidge Dodgson.
Further connections between the golden ratio and music can be found here and between the ratio and a Stradivarius violin here:
The Lady Blunt shown above shows the measurements connected to the golden ratio:
[ \frac{a_1 +a_2}{a_2}=\frac{a_2}{a_1}=\frac{b_2}{b_1}=\frac{b_2}{c_2}=\frac{c_2}{c_1}=\phi ]
Below is a geometric interpretation of the golden ratio and the golden rectangle:
The Lucas numbers in Pascal’s triangle François Édouard Anatole Lucas The French mathematician François Édouard Anatole Lucas (1842-1891) served as an artillery officer in the Franco-Prussian War, and subsequently became professor of mathematics at the Lycée Saint Louis and then professor of mathematics at the Lycée Charlemagne, both in Paris. Lucas did a lot of work on number theory and was particularly interested in the Fibonacci sequence and devised the test for Mersenne primes which is still used today.
Lucas died of erysipelas (a bacterial skin infection) a few days after a freak accident. He was at a banquet when a fragment of a dropped plate flew up and cut his cheek.
His sequence, the Lucas sequence, begins with the pair of numbers $2$ and $1$ and its terms are generated in the same way as for the Fibonacci sequence.
$2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521…$
There are a number of connections between the Fibonacci sequence and the Lucas sequence. The $3^{rd}$ Lucas number is the sum of the $1^{st}$ and $3^{rd}$ Fibonacci number, the $4^{th}$ is the sum of the $2^{nd}$ and $4^{th}$, the $5^{th}$ is the sum of the $3^{rd}$ and $5^{th}$, the $6^{th}$ is the sum of the $4^{th}$ and $6^{th}$,…
Division of the Fibonacci terms $2n$ and $n$ beginning with the $2^{nd}$ term yields the Lucas terms
$2^{nd} \div 1^{st} = 1 \div 1 = 1$
$4^{th} \div 2^{nd} = 3 \div 1 = 3$
$6^{th} \div 3^{rd} = 8 \div 2 = 4$
$8^{th} \div 4^{th} = 21 \div 3 = 7$
$10^{th} \div 5^{th} = 55 \div 5 = 11$,..
With some manipulation of Pascal’s triangle and some basic arithmetic, we can find the Lucas numbers in the triangle. We begin by setting out the triangle as below and sum the columns to obtain the Fibonacci sequence
The Fibonacci numbers revealed as the column sums We now multiply each Pascal number by its column number and divide by its row number, starting with row $1$ column $1$ and then sum the new entries in each column. The first few calculations are shown below:
The Lucas numbers revealed as the column sums Generally, $\displaystyle\frac{\phi^n -(\frac{1}{\phi})^n}{\phi -(\frac{1}{\phi})}$ is the formula for the $n^{th}$ Fibonacci number, $\displaystyle\frac{\phi^n +(\frac{1}{\phi})^n}{\phi +(\frac{1}{\phi})}$ is the formula for the $n^{th}$ Lucas number and $\phi^n =\displaystyle \frac{L_n+ \sqrt5 \times F_n}{2}$, where $L_n$ and $F_n$ represent the $n^{th}$ Lucas and Fibonacci numbers respectively.
In the next part, we’ll consider some more connections between the triangle and particular numbers, and types of numbers.
In this series of posts, Katie investigates simple mathematical concepts using the Google Sheets spreadsheet app on her phone. If you have a simple maths trick, pattern or concept you’d like to see illustrated in this series, please get in touch.
Since apparently I’m now a maven for interesting fun things built using Google Sheets, someone tagged me in to suggest I might like to see this Truchet Tiling Generator, built in Google Sheets using images generated in Google Drawing.
Pretty excited about this Truchet Pattern Generator in a Google Sheet. Strongly inspired by @divbyzero's @RandomTiling #playwithmath here and make your own #mathart!https://t.co/CL33e0rcug pic.twitter.com/CgWA52GywJ
— Mark Kaercher (@shskaercher) January 1, 2022
Truchet tilings consist of square tiles which have a design that isn’t rotationally symmetrical, so each tile can occur in one of two or four visually distinct orientations. Conventionally the designs are fairly simple, geometric patterns using two colours. The design of the tile is such that when tiles are placed in a grid, the edges of the tiles match up in some way – the position of the point where the colour changes is usually at a corner or mid-way along an edge, so that the tiles create pleasing designs.
Truchet tiles were first described in a paper by Sébastien Truchet, a French Dominican priest, entitled “Mémoire sur les combinaisons” which was printed the 1704 edition of Histoire de l’Académie Royale des Sciences. Including a large number of triangle-based patterns, this was the first text to write about Truchet tilings.
In 1987, the tilings were popularised by science historian Cyril Stanley Smith, who wrote a piece for the MIT journal Leonardo (JSTOR login required) in which he described Truchet’s tilings, compared them to historical Islamic and Celtic tiling patterns, as well as discussing them in the context of combinatorics, topology and crystallography (presumably inspired by Smith’s own background as a metallurgist). The paper also included Pauline Boucher’s translation of the original text by Truchet. Smith said:
It embodies an early representation of the principles of combinatorial theory and of crystallographic symmetry including color symmetry. Simple rules of the topology of separation and junction are used to extend Truchet’s concept of directional choice and, by relaxing symmetry rules, to generate diagrams illustrating field/ground relations, the hierarchy of structural freedom and the origin and nature of structural order and disorder in general.
The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy, Cyril Stanley Smith (1987)
The good news is, you too can now explore the hierarchy of structural freedom (and make pretty pictures), using a spreadsheet! New York-based math(s) teacher Mark Kaercher has built a magically updating Google Sheet which generates randomised tiling patterns. By generating four different orientations of your chosen tile and creating cells in the spreadsheet containing those as images, you can combine them randomly to make beautiful tilings, and ticking or unticking a checkbox in one of the cells, force the spreadsheet to recalculate (generating new random numbers using the =randbetween() function) and generating a new pattern.
Mark’s sheet, which you can make your own copy of with a single click, has tabs with a variety of designs, including triangles, quarter circles, diagonal lines, Smith curves (as introduced by Smith in the 1987 paper) and a couple of different types of hexagonal pattern. And yes, it does work on a phone!
If you’d like to read more about how the spreadsheet and tiles were created, you can read Mark’s writeup in a Google Doc.
Here’s a roundup of news stories from December 2021 that we didn’t cover at the time.
Maths results Firstly, some nice news of a proof of a result on the density of unit fractions – a set of integers of positive density must contain distinct $n_1,\dots,n_k$ such that $\frac{1}{n_1}+\ldots+\frac{1}{n_k}=1$. (via Thomas Bloom)
According to this post on Gil Kalai’s blog, Ringel’s circle problem has been solved. The problem states:
Consider a finite family of circles such that every point in the plane is included in at most two circles. What is the minimum number of colors needed to color the circles so that tangent circles are colored with different colors?
Turns out, you might need all the colours – the authors of a new ArXiV paper have found ways to construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number.
A portion of the analytical engine built by Charles Babbage, at the Science Museum
Photo: CC BY-SA Mrjohncummings
Plan 28, a project aiming to collect and understand Babbage’s notes about the analytical engine (and possibly finish building it) has issued a statement to the effect that they now think they understand all of the designs – an exciting step forward.
We have for the first time both an aerial view that integrates partial and seemingly unrelated developments, as well as the most detailed analysis yet of the specifics of implementation.
The group are hoping to be able to rewrite these notes into a format that can be used to build a physical implementation of the machine, as Babbage’s original notes didn’t include a design for a complete engine, and the work so far has taken five years. This is exactly the kind of unnecessary nerdery I love to see.
Prizes Per Nalini Joshi on Twitter, Serena Dipierro has been awarded the Australian Mathematics Society medal for 2021, which is given within 15 years of the award of someone’s PhD for distinguished research in the mathematical sciences. According to the AustMS citation,
Professor Serena Dipierro (University of Western Australia) has made outstanding contributions to the area of analysis and PDEs, with a special focus on the theory of nonlocal operators and free boundary problems. She is a prolific researcher with a large international network of collaborators and has become one of the leaders of her field. In the nine years since the award of her PhD, her publications have amassed over 1100 citations in the MathSciNet database; since moving to Australia in 2016 she has averaged one publication per month, including many in journals of the highest quality.
According to a blog post by Gil Kalai, Richard Stanley has won the Leroy P. Steele prize, awarded annually by the AMS for distinguished research work and writing in mathematics. According to the announcement,
Stanley has revolutionized enumerative combinatorics, revealing deep connections with other branches of mathematics, such as commutative algebra, topology, algebraic geometry, probability, convex geometry, and representation theory. In doing so, he solved important longstanding combinatorial problems, often reinvigorating these other fields with new combinatorial methods. Through his outstanding research; excellent expository works; and many PhD students, collaborators and colleagues, he continues to influence the field of combinatorics worldwide.
Bad news
In early December, the European Mathematical Society announced that Jacques Tits (pictured left) has died.
Jacques Tits was a highly influential group-theorist, proving the celebrated “Tits Alternative” (that every finitely generated linear group either has a solvable subgroup of finite index or contains a free subgroup of rank 2). Probably his most important contribution was the development of group-theoretic “Buildings”, a profound unifying idea which has subsequently had deep applications in diverse mathematical fields.
Following the publication of a fairly painful article in The Times just before Christmas entitled ‘Phwoar! Look at the vital statistics on these lads’ and listing the apparently increasingly attractive, and exclusively male, mathematicians and statisticians responsible for ‘crunching the data’ on the pandemic, the i newspaper published this excellent response pointing out the shocking news that some mathematicians who aren’t men have also been involved, and highlighting some of the top data experts who’ve been looking after us all with maths. The Times article includes a quote from “maths professor and author Hannah Fry — a woman” (that is literally actually what it says) who had correctly expressed on Twitter that mathematicians are hot – but I’m pretty sure she meant all of us and not just men.
Speaking of bad opinion pieces, what better way to sum up the year than this collection of terrible maths takes? Highlights include ‘How does misogyny impede a mathematician of doing a good job?’ [sic] and the wonderful ‘Physics is not math.’
The American Mathematical Society has cancelled this year’s Joint Mathematics Meetings, scheduled to take place in Seattle on 5-8 Jan, and will be refunding tickets and organising an online event instead. Unfortunately, they initially failed to notify attendees of this by email, and many found out via Twitter.
The AMS also announced in mid-December that they were shutting down all their blogs with two weeks’ notice. The AMS Blogs site has been replaced with an archive collecting all the past posts, but those who used it as a regular blogging outlet will have to find somewhere else to do that. (Hi!)
And finally Dynamic geometry powerhouse Geogebra has been bought by an online tutoring company called BYJU’S, run by a group of former maths teachers from India. They’ve stated that all current employees, contracts, agreements and software licenses will remain in place, and the software and online resources will continue to be free to use. (via Geogebra on twitter)
PROMYS Europe is a programme designed to encourage mathematically ambitious secondary school students to explore the creative world of mathematics. Competitively selected pre-university students from around Europe gather at Wadham College, Oxford for six weeks of rigorous mathematical activity. This summer it will run from 10th July – 20th August, and applications open on 11th January.
Gathering 4 Gardner’s 2022 Wall Calendar is now available to download and print, and some print copies are also available. Including important dates of huge mathematical significance (my birthday, among others) and a selection of bios, sketches, photos and puzzles any maths fan would enjoy, it’s the perfect solution if you forgot to get a calendar and like maths.
The Geometry Center videos, which brought brought concepts from geometric topology to general audiences through computer-generated visualisation in the early 1990s, have been remastered and are available for free. (via Robin Houston)
If you spot something you think should be in a future Aperiodical News Roundup, send it our way!
A conversation about mathematical jokes, humour and folklore inspired by a sheep, at least one side of which is black. Presented by Katie Steckles and Peter Rowlett. The jokes sent to Peter on Twitter that we mention can be found in the replies to this tweet.
A conversation about mathematics inspired by a plate of biscuits. Presented by Katie Steckles and Peter Rowlett, with special guest Alison Kiddle. What do you notice? What do you wonder? Alison’s Noticing and wondering page. We also mentioned A Problem Squared Episode 014 = Final Cheese Drama and Quick-Fire-O-Rama. You can see Peter’s kitchen floor…
A conversation about mathematics inspired by a Spirograph set. Presented by Katie Steckles and Peter Rowlett. Katie’s Spirograph GeoGebra file.
A conversation about mathematics inspired by a balancing bird. Presented by Katie Steckles and Peter Rowlett, with special guest Alom Shaha. Alom’s video and template about the balancing bird.
A conversation about mathematics inspired by UUID 0412a969-5b27-4c28-9662-85ef2c201e0c. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by an auctioneer’s hammer. Presented by Katie Steckles and Peter Rowlett, with special guest Tim Harford.
A conversation about mathematics inspired by cards from the game Dobble. Presented by Katie Steckles and Peter Rowlett. You can read more about Katie’s adventures in golfing combinatorics.
A conversation about mathematics inspired by an arbelos. Presented by Katie Steckles and Peter Rowlett, with special guest Catriona Agg. Catriona mentions this proof without words, which is taken from Proof Without Words: The Area of an Arbelos by Roger B. Nelsen in Mathematics Magazine.
A conversation about mathematics inspired by a box of Christmas crackers. Presented by Katie Steckles and Peter Rowlett. Merry Christmas!
A conversation about mathematics inspired by an Enigma machine. Presented by Katie Steckles and Peter Rowlett, with special guest Tom Briggs.
A conversation about mathematics inspired by some solids of constant width. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a ball of wool (yarn). Presented by Katie Steckles and Peter Rowlett, with special guest Pat Ashforth.
A conversation about mathematics inspired by a lottery machine. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a Klein bottle and Mathsteroids. Presented by Katie Steckles and Peter Rowlett, with special guest Matthew Scroggs. Play Mathsteriods!
A conversation about mathematics inspired by a hat. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics and education inspired by a hundred square. Presented by Katie Steckles and Peter Rowlett, with special guest Susan Okereke. In the episode, we mentioned the original Prime Climb colouring sheet and Peter’s Prime Climb colouring sheet on GitHub as drawing-primes.
A conversation about mathematics inspired by a Twenty Pence coin. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a vehicle. Presented by Katie Steckles and Peter Rowlett, with special guest Christopher Danielson.
A conversation about mathematics inspired by a Möbius band. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a mandala. Presented by Katie Steckles and Peter Rowlett, with special guest Hana Ayoob.
A conversation about mathematics inspired by acoustic mirrors. Presented by Katie Steckles and Peter Rowlett, with special guest James Grime. Image: WW1 Acoustic Mirror, Kilnsea; cc-by-sa/2.0 – © Paul Glazzard.
A conversation about mathematics inspired by number block cubes/snap cubes. Presented by Katie Steckles and Peter Rowlett. Peter’s blog post: Mathematical play with young children. Mike Lawler’s three-tweet thread of more advanced ideas starts here:
A conversation about mathematics inspired by a Rubik’s Cube. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a set of D&D dice. Presented by Katie Steckles and Peter Rowlett.
Katie and Peter give a little update on the podcast, life in lockdown and the upcoming season/series 3.
A conversation about combinatorics, the mathematics of counting, inspired by a robot caterpillar. Presented by Katie Steckles and Peter Rowlett.
A conversation about the mathematics of chemistry inspired by a pencil, plus a chat about approximation. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics including fractals inspired by a Romanesco Broccoli. Presented by Katie Steckles and Peter Rowlett, idea suggested by John Read (thanks John!).
A conversation about mathematics inspired by a deck of Set cards. Presented by Katie Steckles and Peter Rowlett. We mentioned an implementation of Set in Python by Ben Nuttall and a retro NES version by Katie.
A conversation about mathematics inspired by the game Ox Blocks. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a Correntator, a mechanical adding machine. Presented by Katie Steckles and Peter Rowlett, with special guest Christian Lawson-Perfect.
A conversation about mathematics inspired by the pseudorhombicuboctahedron. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a pair of skipping ropes. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a thermometer. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a Noughts and Crosses (Tic Tac Toe) board, covering Noughts and Crosses, a surprising number of variants, with a bit of higher dimensions and topology for good measure. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a tangerine (no, really!). Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a pile of matchsticks. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a set of Tantrix tiles, a beaded necklace and some juggling balls. Presented by Katie Steckles and Peter Rowlett, with special guest Alex Corner.
A conversation about mathematics inspired by a stick of chalk. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by a t-shirt featuring Pythagoras’ theorem. Presented by Katie Steckles and Peter Rowlett.
A conversation about mathematics inspired by the Towers of Hanoi puzzle. Presented by Katie Steckles and Peter Rowlett. Update: Here’s a lovely knitted Towers of Hanoi, tweeted in response to this episode by Pat Ashforth.
Presented by Katie Steckles and Peter Rowlett, episodes of Mathematical Objects will take an object, real or abstract, as inspiration to chat about a mathematical topic. This introduction explains the idea ahead of the first episode, coming soon.
As part of our special Apéry takeover today, I chatted to mathematicians Ben Sparks and James Grime, to find out what we know about the mathematics Apéry did – it’s an enjoyable 10-minute listen.
Cushing was injured in a serious maths accident recently (he fell out of the bath) so I wanted to assess the damage to his number-wrangling faculties. Fortunately, there’s the National Numeracy Challenge, which begins with a test to pinpoint your weak areas. National Numeracy is a charity that wants every adult in the UK to “reach a level of numeracy skills that…
MathsJam is an annual conference in the UK, and a monthly night in pubs around the world, organised respectively by mathematician and juggler Colin Wright, and stand-up mathematician Matt Parker. We cornered Matt and Colin at the MathsJam conference last November, and talked to them for just over half an hour about the conference, the…
David and I sat down again and talked about maths a bit more. I’m calling this number 1 because it suits both our counting systems: David can call this the first podcast of a new series, and I can say the one we put out under All Squared was number 0. Everyone wins! Here follows a…
We haven’t done one of these for absolutely ages. Since all three of us were at the big MathsJam conference a couple of weekends ago, we decided to introduce a local minimum into the fun curve by sitting down and talking about how this site’s doing. Actually, we ended up talking about the MathsJam baking…
Evelyn Lamb is a professional mathematician who has taken up journalism on the side. She received the AAAS Mass Media Fellowship last year, and spent the summer writing for the magazine Scientific American. We talked to her about maths journalism, the challenges involved in making advances accessible to a wider audience, and the differences between blogging and…
We have an unusual All Squared podcast for you this time. My good friend David Cushing has been asking to do a podcast for absolutely ages. We couldn’t decide on a single topic to talk about, so instead I suggested we just sit down and chat about maths in general, like we do when there isn’t…
This is the second and final part of our interview with Colm Mulcahy. Last week we talked about card magic; in this part we moved on to the subject of Martin Gardner and the gatherings of interesting people associated with his name. We’ve tacked on some blather we recorded about the British Science Festival in…
Colm Mulcahy is an original Aperiodical contributor (Aperiodicontributor?) and friend of the site. He’s spent the last year and a bit writing his new book, Mathematical Card Magic: Fifty-Two New Effects. It came out a few weeks ago, so we thought it was a good opportunity to talk to him and find out just what’s so…
This number of the All Squared podcast contains the final third of our interview with the inestimable David Singmaster, and then a bit from CP about his favourite book, “A treatise on practical arithmetic, with book-keeping by single entry“, by William Tinwell. The first part of the interview, and plenty of links to go with…
Good maths books are simultaneously plentiful and rare. While there are a few classics almost everyone knows about and has copies of (Gardner, Hardy, etc.), the trade in lesser-known maths books is considerably less well-organised. Very few bookshops have well-stocked maths sections, and insipid pop maths books dominate. Unless you hear about a good maths…
It’s a repeat booking for the Festival of the Spoken Nerd in number 4 (or 16 if you belong to Team All Squared) of our podcast. Standup mathematician Matt Parker joined us to talk about interesting coincidences. Here are some links to the things we referred to in the podcast, along with some bonus extras: The…
Remember, remember, The fourteenth of March. While the previous number of All Squared failed to achieve topicality by appearing several weeks after the event it was about, this time we’ve hit the nail bang on the head with a podcast all about π day… on π day! We chatted to Festival of the Spoken Nerd’s Steve Mould about remembering π…
Here’s the second edition of our new podcast, All Squared. This time we talked to Dr Andrew Taylor, PhD, about nonsense formulas in the news. In particular, since we recorded very close to pancake day, we took a close look at the various “formulas for the perfect pancake” printed in UK newspapers. Here are some links…
We’ve been quietly making plans and gathering material for a new project over the past couple of weeks, after noticing that there’s an unusual paucity of maths podcasts at the moment. Well, that exciting new project is now happening, and it’s a half-hour podcast featuring maths, guests, puzzles and links from the internet. It’s called…
Two days late, because that is the way we rotate here, it’s another episode of our sporadic navel-gazing podcast. In this episode we talked about: Our piece on the Invariant Subspace Problem (and the more recent news) Log-log! Who’s there? Not a power law! Our coverage of the new Mersenne Prime news, and our meta-coverage of everyone…
After two months we’ve finally done another podcast! We completely forgot even the most rudimentary things about how to do a podcast. Sorry. In this episode, we talked about: Mathematical Christmas cracker jokes Fractal Christmas trees Posts from MathsJam speakers – Tom Button on Radii of Polyhedra and Phil Harvey on AS Results and Batting…
We took the opportunity of us all being in the same small slice of space and time (MathsJam, last weekend) to record another episode of our continuing audio part-work, The Aperiodcast. We talked about: Christian’s Recreational Maths Seminar Dara O Briain: School of Hard Sums to return; maths students sought to take part Matt Parker’s…
Here’s another episode of our irregular podcast about what’s been happening on the site. This time, we talked about: Advances in pure nonsense Robert Schneider, Mathematical Musician/Musical Mathematician #MTT2K: Teachers critique Khan Academy Surds: what are they good for? Calculus of the Nervous System The new fonts on the site Christian’s new Aperiodical Round Up…
Leaves are falling, a chilly wind is blowing and I can hear the distant thunder of undergrads’ hooves as they stampede towards my department. Yes, Summer is giving way to Autumn, so it’s time for another Aperiodcast. If you had “42 days” in the “when will the next Aperiodcast appear” sweepstake, report to the comments…
Here’s the fifth Aperiodcast, covering what’s happened on the site basically since the start of Summer. Peter is busy doing work, so it was just Katie and me blathering on about a variety of things. The posts discussed in this episode were: Telegraph’s open letter to Michael Gove and Vince Cable on numeracy (presented with…
Here’s another Aperiodcast, covering things that happened on the site between the 4th and the 20th of June. Posts discussed in this episode Ask a mathematician: “Where should we live?” by Alistair Bird The mathematics examinations faced by school leavers in the Republic of Ireland by Colm Mulcahy P-Value Extravaganza posted by Christian The Super Subtraction…
After an unexpectedly long wait of over three weeks, here’s the third Aperiodcast, discussing what’s happened on the site between 13/5/2012 and 3/6/2012. You’ll notice that we recorded this podcast four days ago – we were all having too much jubilee fun to find time to upload it! Anyway, we had lots to talk about,…
In true Aperiodical fashion, we left 13 days before recording another Aperiodcast, so here’s what we think about the last almost-two-weeks on the site. We talked about: “Futurama theorem” slightly improved The number line is not an intuitive concept Grow Your Own Food Puzzlebomb – May 2012 Carnival of Mathematics 86 Charlotte Hillebrand’s post with…
Here’s the very first edition of what we’ve cleverly decided to call The Aperiodcast. The plan is to record a short podcast every week or week-and-a-bit (this is the Aperiodical after all) talking about what’s been happening on the site, and pointing out posts that we found particularly interesting or have generated a lot of…