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.

]]>According 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).

- 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.

The 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.

It’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.

The 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.

There 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.

]]>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.

Day 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 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 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!

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.

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