Regular Truchet Tilings

I recently made my first piece of math art for my apartment: a 30″×40″ canvas print based on putting Truchet tiles on the truncated trihexagonal tiling.

A photo of the canvas print
The original image

I first became interested in these sorts of patterns after my former colleague Shane sent me a YouTube video of the one-line Commodore 64 BASIC program:
10 PRINT CHR$(205.5+RND(1)); : GOTO 10

I implemented a version of this program on my website, with the added feature that you could click on a section to recolor the entire component, and this idea was also the basis of Problem 2 and Problem 31 in my Open Problem Collection.

I saw this idea generalized by Colin Beveridge in the article “Too good to be Truchet” in Chalkdust Magazine. In this article, Colin counts the ways of drawing hexagons, octagons, and decagons with curves connecting the midpoints of edges, and regions colored in an alternating fashion. In the case of the hexagon, there are three ways to do it, one of which looks like Palago tiles.

An example of a hexagonal Truchet tiling. Cameron Browne would call this particular example a Palago “creature“, which are counted by Arnauld Chevallier‘s brilliant program.

It turns out that if you ignore the colors, the number of ways to pair up midpoints of the sides of a 2n-gon in such a way that the curves connecting the midpoints don’t overlap is given by the n-th Catalan number. For example, there are C_4 = 14 ways of connecting midpoints of the sides of an octagon, where different rotations are considered distinct.

Two distinct rotations
Eight distinct rotations
Four distinct rotations

There are three regular tilings of the plane by 2n-gons, the square tiling, the truncated square tiling, and the truncated trihexagonal tiling. Placing a Truchet tile uniformly at random over each of the 2n-gons, results in a really lovely emergent structure.

Square tiling
Truncated square tiling, randomly colored

If you find these designs as lovely as I do, I’d recommend taking a look at the Twitter bots @RandomTiling by Dave Richeson and @Truchet_Nested/@Trichet_Nested by @SerinDelaunay (based on a idea from Christopher Carlson) which feature a series of visually interesting generalizations of Truchet tilings and which are explained in Christopher’s blog post “Multi-scale Truchet Patterns“.

Edward Borlenghi has a blog post “The Curse of Truchet’s Tiles” about how he tried—mostly unsuccessfully—to sell products based on Truchet tiles, like carpet squares and refrigerator magnets (perhaps similar to “YoYo” magnets from Magnetic Poetry). The post is filled with lots of cool, alternative designs for square Truchet tiles and how they fit together. Edward got a patent for some of his ideas, and his attempt to sell these very cool products feels like it could have been my experience in another life.

If you want to see more pretty images and learn more about this, make sure to read Truchet Tilings Revisited by Robert J. Krawczyk! If you want to see what this looks like on a spherical geometry, check out Matt Zucker’s tweet. And if you want to try to draw some of these patterns for yourself, take a look at @Ayliean’s Truchet Tiles Zine.