### 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!

### 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?)

### Equatum puzzles – a chat with Justin Roughley

Equatum is a new puzzle format invented by mathematician Justin Roughley, and is now available in the form of a book. We chatted to Justin about his life and his puzzles.

### Mobile Numbers: Hitomezashi Stitching

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.

It’s been a while since we’ve had an entry in this column, but the other day I was sent a link to a very interesting spreadsheet (which I, of course, opened using the Google Sheets app on my phone). The initial view was a pleasing pattern of squares, in two colours:

### Maths books for children

We’ve noticed a lot of great books that have been released recently aimed at primary age children (under about 11). We thought it might be useful, for those who know children of those ages, to put together a list of these titles, and some classics, in case you might be looking for some gift ideas around now.

### Aperiodical’s Mathematical Seasonal Gift Guide, 2020

Given that it’s conventional to give objects to other people around this time of year, we thought we’d collect together some suggestions for things we think you, a mathematically interested person, might like to buy for your mathematical friends (or add to your list before you send it off to Santa).