# You're reading: Posts Tagged: topology

### 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!

### Clopen Mic Night – new online maths variety show

The team that brought you the 24 Hour Maths Magic Show last October are at it again, and are planning a semi-regular evening YouTube variety show called Clopen Mic Night, with short segments from a selection of mathematical guests, including comedy, music, demonstrations, magic, puzzles and art, showcasing some top maths communicators and hopefully providing a fun night in. The event is supported by Talking Maths in Public, a network for maths communicators based in the UK, and this first show will take place alongside their 2021 conference event.

It’s called Clopen Mic Night because it’s both an open mic night (in the sense that you’ll see a variety of different people doing different things) and a closed mic night (in that the organisers curate the line-up to ensure a variety of quality acts). If you’ve not encountered the concept of a clopen set, it describes a set that is both open and closed. Usually things are clopen for tedious technical reasons – the empty set and the whole set are both trivially clopen, and most interesting examples crop up in awkwardly-defined sets with non-standard topologies and distance metrics.

The first event is taking place on Thursday 26th August, from 8-9pm, on my YouTube channel, and you can watch along for free, join in with the chat, and drop a coin in our virtual tip bucket if you like what you see. This will hopefully be the first of many such shows (assuming it all goes well!) and for future shows we’ll be looking for acts to join us – anyone participating will also be able to get advice and feedback on their bit in various ways, and we’re hoping it’ll be a chance for people to try out fun new material and showcase the best maths communication has to offer.

For more information about the show, including the lineup for this first event, you can visit the Clopen Mic Night website and sign up for a reminder before each show so you don’t miss it. For updates on future events and how to apply to perform (once that becomes a thing), check the @ClopenMic twitter account.

### Mathematical Objects: Klein bottle with Matthew Scroggs

A conversation about mathematics inspired by a Klein bottle and Mathsteroids. Presented by Katie Steckles and Peter Rowlett, with special guest Matthew Scroggs.

### Mathematical Objects: Möbius band

A conversation about mathematics inspired by a Möbius band. Presented by Katie Steckles and Peter Rowlett.

### Mathematical Objects: A pair of skipping ropes

A conversation about mathematics inspired by a pair of skipping ropes. Presented by Katie Steckles and Peter Rowlett.

### HLF Blogs – Michael Atiyah’s Favourite Manifold

This week, Katie and Paul are blogging from the Heidelberg Laureate Forum – a week-long maths conference where current young researchers in maths and computer science can meet and hear talks by top-level prize-winning researchers. For more information about the HLF, visit the Heidelberg Laureate Forum website.

As part of the HLF, the Laureates are participating in press conferences throughout the week, and being bombarded with questions by well-meaning journalists and bloggers. Unlike most press conferences, where participants often have a specific topical thing they’re there to speak to the press about, the Laureates can be asked about any of their past projects, on any area of maths they’ve worked on, and many of them have a very long and illustrious career to speak of.

It can be difficult then, to be put on the spot by a taxing question, especially if you’re not expecting it. I’ve been surprising the topologists whose press conferences I’ve attended with a deceptively deep but simple question: What’s your favourite manifold?

### Best way to explain topology: now officially ‘using baked goods’

Nobel Prize news!

The 2016 Nobel Prize in Physics has been awarded to a trio of physicists: Michael Kosterlitz, Duncan Haldane and David Thouless“for theoretical discoveries of topological phase transitions and topological phases of matter”.

And here’s the maths angle – their work is in the field of topological physics, which relates strange matter (superconductors, superfluids and the like) to topology, via the interesting way the properties of the materials change in phases, like the different fundamental shapes of objects in topology. None of the material we’ve taken a cursory glance at so far yields a simple explanation of how these two things are linked, but they have explanatory PDFs on the Nobel website if you’d like a dig around: Popular (PDF) and Advanced (PDF).

Also, impressively many newspaper headlines seem to have failed to notice that ‘strange matter’ is actually a thing in physics, and consequently mangled it in their explanations.

Cue of course an amazing press conference in which Nobel Committee for Physics member Thors Hans Hansson holds up a bun, a bagel and a pretzel to explain the difference. Classic topology.