# You're reading: Posts Tagged: truchet

### 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!

### Truchet, Braille and Euler

In going through a hard drive I came across some playing around I did a couple of years ago with Truchet tilings which I thought I would share with you here.

I came across Truchet tilings in a talk a couple of years ago by Cameron Browne to the London Knowledge Lab‘s Maths-Art Seminar Series. (Sébastien Truchet is the first person I have thought would be in but found not to be in The MacTutor History of Mathematics archive).

To cut an interesting story short, Truchet tiles come in two forms which can be represented as:

These are then combined to form interesting curves, thus:

You can colour these tiles, making four visually different tiles:

So the pattern becomes like:

Or the inverse.

So. if you think about it, the two types of tile can be used to represent two points – for example binary data – using the two symbols and .

At the time I was playing around with Braille notations for mathematics. Braille characters are made up of 3 rows of 2 cells each (or, in some advanced forms, 4 rows of 2 cells). Cells contain raised dots or don’t and the pattern is used to feel which character is which. Representing raised dots as black dots and, then the Braille character for, say, “m” is:

If we take this as a pattern of 1s (raised dots) and 0s (absence of dots) then this is:

1 1
0 0
1 0

If we take one Truchet symbol to be a raised dot and the other to be the absence of such a dot then we can represent a pattern of dots as a Truchet tiling. So, for example take to be a raised dot and to be the absence. Then a British Braille “m” is:

Or a coloured version:

Okay, so then I looked for something suitable to encode this way. I chose Euler’s identity. Taking to be a raised dot and to be the absence I encoded the identity using the BAUK Braille Mathematics Notation and coloured this using one colouring or the other, chosen aesthetically. This gives (click to enlarge):

Finally, I gave this a bit of a colouring and eroded the shapes in a way I thought looked appealing. Please click on the following to enlarge:

Really this is all very contrived but I quite like it. Of course, there is something contrary to intentions about generating a visually appealing image from a tactile representation. I think the pattern is attractive and the hidden meaning, and particularly of such a beautiful formula, I think adds something to the effect.