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!

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.

David Smith has published a ‘scrapbook’ on his blog giving some of the story of how the shapes were conceived. David notes that they actually discovered two monotiles: the ‘hat’ (or t-shirt) above, and a shape made of 10 kites that looks like a turtle:

Think 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.

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.

The 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.

Is 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 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.

It’s also possible to tile the plane non-periodically 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.

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!

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.

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.

The 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:

Really, 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 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.

]]>Baroness 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.

The 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.

]]>The British Prime Minister Rishi Sunak has announced that all students will study maths to age 18. The response has been varied, with commentators from both within mathematics and from non-mathematical backgrounds weighing in (with varying degrees of nuance).

However, this isn’t planned to happen soon – only to start the work to introduce this during this Parliament, with actual implementation to happen at an unspecified point in the future.

It’s worth noting that there is a shortage of maths teachers, with nearly half of schools currently using non-specialist maths teachers, according to the *TES*.

The fact this might make maths a political football is a bit of a problem – the opposition Labour party say “they’ve nothing to offer the country except double maths”. (As much as we love maths, we’ll agree there are more important things to worry about at the moment).

The Chrome browser, and eventually other browsers built on it such as Edge and Opera, can now render MathML without any additional libraries as of version 109. Chrome briefly had some support for MathML, which was removed in 2013 due to lack of interest from Google. The developers who were working on it have kept plugging away, funded by the open source software consultancy Igalia.

Until now, the only reliable way to display mathematical notation on the web has been to use a JavaScript library such as MathJax or KaTeX, which do all the work of laying out symbols using generic HTML elements.

Now, you can just put MathML code in a page and expect most browsers to display it, like this:

$$\int \frac{1}{{x}^{2}+1}$$

There’s still a need for MathJax and the like: writing MathML code is no fun, so they’re still useful for translating LaTeX code, and MathJax adds a range of annotations that help with accessibility. But this is a step towards mathematical notation being much easier to work with on the web!

The US National Academies have released a series of posters “Illustrating the impact of the mathematical sciences”.

CLP’s place of work still has some Millennium Maths Project posters clinging on to the walls, older than almost all of the students, so maybe it’s time for a refresh! *(via Terence Tao)*

Tim Austin is the new Regius professor mathematics at Warwick. *(via Warwick Mathematics Institute)*

A bit of bureaucracy news: the Council for the Mathematical Sciences, comprising the five learned societies for maths and stats in the UK, is creating a new Academy for the Mathematical Sciences. It looks like the societies for the different sub-disciplines have acknowledged they need to work together, though this gives off a “now you have n+1 standards” smell. They’ve got a nice logo, though.

The Financial Times style guide changed so that ‘data’ is always singular, pragmatically following common usage. FT writer Alan Beattie said it best: “For anyone opposed, I’d like to know what your agendum is.“

The London Mathematical Society will hold a ceremony in London on 22nd March to officially award the Christopher Zeeman medal to the 2020 and 2022 medallists, Matt Parker and Simon Singh.

The ICMS in Edinburhgh has launched a “Maths for Humanity” initiative, which will be “devoted to education, research, and scholarly exchange having direct relevance to the ways in which mathematics, broadly construed, can contribute to the betterment of humanity.” *(via Terence Tao)*

Yuri Marin has died. The Max Planck Institute has posted an obituary describing his life’s work. One of his PhD students, Arend Bayer, collected some memories in a Mastodon thread.

William ‘Bill’ Lawvere has died. There is a page on the nLab describing his life’s work.

]]>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 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.

]]>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 together with some original new content written for the book, 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!

]]>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.

Goodwin 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 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.

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