Mathematics is often about patterns, symmetry and the rich detail you find in complex systems. But that doesn’t mean it has to be complicated – sometimes simple ideas can lead to complex and beautiful results. One of my favourite examples of this is truchet tilings.
Originally described in 1704 by a French priest and mathematician called Sébastien Truchet, truchet tilings are made from very simple units which can be combined in different ways. Inspired by seeing a pile of floor tiles awaiting installation, Truchet’s original document was called ‘Mémoire sur les combinaisons’ (roughly, ‘A study on combinations’), and considered the patterns you can make by repeating and combining different versions of a simple tile.
Truchet’s Own Tiles
Truchet’s original tile was a square, divided diagonally into two right-angled triangles, with one shaded black and the other white. Since the tile is square, it can be rotated and remain square, and doing so will produce four different versions of the tile:
If these tiles are used to cover a plane, the results are surprisingly beautiful. Truchet included a series of diagrams showing some possible simple tilings – using all the same tile, or alternating between multiple orientations, as well as some more complex layouts with a larger repeating section. Given just these four simple tiles, the possibilities for different combinations are huge.
Truchet’s original paper didn’t get much attention, but was later popularised by science historian Cyril Stanley Smith, who found and wrote about the original Memoire in a 1987 paper for MIT’s journal Leonardo, in which he described Truchet’s tilings, comparing them to historical Islamic and Celtic tiling patterns.
Smith also discussed them in the context of combinatorics, topology and crystallography (presumably inspired by his own background as a metallurgist). The paper also included Pauline Boucher’s translation of the original text by Truchet.
Smith also introduced some alternative designs for tiles that could be used in this way – notably, a tile containing two quarter-circles centred at opposite corners of the square. Such a tile has only two possible orientations, but when arranged in a grid can create beautiful wandering paths.
In Smith’s words, the tiles have ‘substructural lines that join to produce continuous closures and exclusively quadrivalent internal vertices, and to generate a hierarchy of closures and continuity’.
Such a tile was already known – in 1960, MIT undergraduate Larry Black invented what became known as the Black Path Game, in which these two tiles and a third design featuring a cross of two straight lines could be combined by players to fill a square grid. Players start at the board edge and each tile played must always extend an existing path, with the winner being the first to force their opponent to join the path back to the board edge.
There are many different possibilities for tiles to use to create Truchet tilings – even just using simple lines and shapes. For example, a tile with a single diagonal line across it can be seen to generate a labyrinth (above). Also pictured above are a design with hexagonal regions, and a variant on Truchet’s original using part of a circle instead of a triangle.
Tiles like Smith’s quarter-circle design but coloured in two ways, giving four possible square tiles, create similar looping patterns but with a colouring of the regions; this design is a great addition to the ‘Truchet blocks’ used at the MathsCity discovery centre in Leeds, UK where they can be built into a 3D shape with a truchet design on all faces (as long as you orient the blocks so the pattern matches).
Designs produced using these simple tilesets can be mesmerising, and can contain repeating patterns or be non-repeating – in fact, a common way people like to introduce more mathematics into this is to use some kind of mathematical algorithm to decide on the arrangement of tiles, rather than looking for an arrangement that’s aesthetically pleasing or follows a pattern. Whether that’s through a random number generator, a pattern of digits from a number like π, or encoding a secret message, the results are part of a growing movement of algorithmic art – an expression of creativity built around mathematical structure.
Tilings like this seem to crop up everywhere – as part of a recent conference I ran, we commissioned a mathematical greetings card to include in the conference pack for all attendees as a free gift, for them to display or send. One of the designs we used, created by artist and maths teacher Ayliean MacDonald, involved a simple pair of truchet tiles featuring a kinked line in two orientations – and Ayliean’s design included a key that hinted that the design encoded a secret message, as well as space inside for you to design your own tile or create a pattern of your own.
I’ve also seen a blog post from my colleague Peter Rowlett, in which he used truchet tiles to encode Braille letters. Since I’m a big fan of spreadsheets, people often send me links to interesting things people have made with spreadsheets, including this writeup of building a truchet tile generator in Google sheets, by maths teacher Mark Kaercher. And of course there’s the Random Tiling Twitter bot, which tweets a new randomly generated truchet tiling each day.
If this has inspired you to create your own beautiful tilings from mathematical ideas, you can always 3D print or laser cut and etch a set of tiles, or make them by hand from card; and you don’t have to stop with squares. You can extend the tiling to hexagonal tiles, or combine multiple shapes using a semiregular tiling. There are so many possibilities for beautiful and pleasing designs – in the words of Smith, ‘the eye enjoys the alternating perception of boundaries that enclose or exclude or are themselves regions’.