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Mobile Numbers: Truchet Tiling

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.

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!

Aperiodical News Roundup – December 2021

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.

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.

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!

Mathematicians on TV at Christmas, 2021

A surprising number of mathematicians (including some friends of the Aperiodical) have been on UK telly this Christmas!

Nira Chamberlain on University Challenge

Winner of the Big Internet Math-Off 2018, Nira Chamberlain, captained the Portsmouth University team in an episode of the University Challenge Christmas special (on iPlayer, and the episode on YouTube). We’re pleased to see that since his Math-off victory, Nira continues to introduce himself wherever he goes as ‘The World’s Most Interesting Mathematician’.

Paul and Katie on Only Connect again

Aperiodical editor Katie Steckles and site regular Paul Taylor, accompanied by fellow mathematician Ali Lloyd, appeared in the Only Connect Champion of Champions special as part of their team the Puzzle Hunters. Having won series 16 this year, they were invited back to take on the winners of series 15, the 007s. Watch out for at least one maths question! The episode is on iPlayer, and on YouTube.

Ri Christmas Lectures

This year’s lectures were on the science behind virology and the pandemic, and hosted by Professor Jonathan Van-Tam. As part of the second episode, mathematician Professor Julia Gog joined to explain how mathematical modelling can be used to study the spread of viruses, around 38 minutes in. (Slightly worryingly, JVT claims he isn’t any good at maths, so he had to get someone in to help explain it).

All three episodes are available on iPlayer, and will be added to the Ri YouTube channel once they come off there (around the beginning of Feb).

The Mathematics of Spirograph

If you’re the kind of person who’s interested in doodling and/or fun toys, you might have encountered the fun doodling toy Spirograph, or some unbranded equivalent. It sits somewhere on the continuum between an artistic drawing tool and a neat mathematical gadget.

Between the three Aperiodical editors (myself, Christian Lawson-Perfect and Peter Rowlett), there’s a developing tradition of excellent mathematical gift-giving. This year, Christian has excelled himself by designing and creating a brilliant mathematical hoodie, which features a meme about an in-joke (and who can resist either a meme or an in-joke?)

Best Illusion of the Year Contest – Final

Since we know you’ll enjoy looking at some weird and wonderful illusion videos, we thought we’d share that the Best Illusion of the Year Contest has posted its 2021 finalists, and you can vote for the winner on their website.

Aperiodical News Roundup – November 2021

Here’s a roundup of some of the news stories from the world of maths in the month of November.

Events

Eugenie Hunsicker and collaborators have produced a film entitled “Words of Women in Mathematics in the Time of Corona”, which raises awareness of the impact of the pandemic on women in mathematics.

The QE Prize for Engineering’s ‘Month of Making’, as featured previously in a post announcing the start, is well under way and continues until 12th December, with scientific, mathematical and engineeringy ideas for make-it-yourself gifts every day.

This month saw the official launch of MathsCity Leeds, as previously covered in this review by hands-on discovery centre correspondent Peter and his son.

Leading mathematicians, council members, and key professionals from tourist attractions and universities across the country were just some of the guests that attended the bustling launch party last night for the UK’s first maths discovery centre. […] Celebrating the milestone achievement by the pioneering charity MathsWorldUK, the MathsCity launch was an opportunity to show donors, supporters, and future investors why the innovative new attraction that opened its doors in Leeds City Centre last month is so important for the future.

North East Post

Attempts to start a proof assistants stack exchange have been successful, and the Stack Exchange team are “are preparing for its launch and expect to create it soon”. (via Andrej Bauer)

A new paper has been published in Nature about the use of machine learning in pure maths research. This isn’t machine learning making new maths, but rather it’s pitched as a collaboration between mathematician and machine – the authors argue that machine learning can be used “to guide intuition and propose conjectures”. The paper gives some examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning.

Open Calls

The IMA has launched a poster competition called How Maths Helps People, in which high school students are asked to design an A4 “persuasive poster which shows how maths can be used to help people”. The poster should be aimed at high school students, and students with winning posters in each age group will receive an Android tablet. The closing date is 31st January 2022.

The LMS has announced its annual call for nominations for its 2022 prizes, which are awarded in various categories for mathematical research, innovation and exposition.

Recreational Maths Magazine has issued a call for proposals for its upcoming π-themed issue, which will be their first specially themed issue. Calls close on 14th March 2022 (obviously).

The Heidelberg Laureate Forum, which takes place in September in Heidelberg, Germany, and brings together top-level maths laureates with young researchers for a week of lectures, workshops and networking has announced that applications for young researchers to attend HLF 2022 are now open. If you know any PhD or postdoc mathematicians who would like a chance to meet some cool people and have a great trip to Germany, encourage them to apply!