Truchet tiles

Truchet tiles are square tiles divided along one diagonal into two colours. Father Sébastien Truchet (1657–1729) described them in 1704 in a paper for the Académie Royale des Sciences in Paris, "Mémoire sur les combinaisons", and showed how a single tile, turned in different directions, makes a large number of patterns.¹

The tile

Truchet wrote that at a château called La Motte S. Lyé, four leagues from Orléans, he had found "several square fayence tiles divided into two colors by a diagonal line", intended to tile a chapel and several other rooms.¹ One half of each tile is dark, the other light. Because a square can be turned four ways, the dark triangle can lie in any of the four corners.

Counting the combinations

Truchet began with two tiles placed side by side. The first can take four orientations and so can the second, which can also be placed against any of the four sides of the first, giving 4 × 4 × 4 = 64 combinations.¹

In 1722 Dominique Doüat published Méthode pour faire une infinité de desseins différens, with 72 tilings made from the same tile.¹ In 1987 the historian of science Cyril Stanley Smith and Pauline Boucher returned to the memoir in the journal Leonardo, calling it "in essence a graphical treatment of combinatorics".¹

Tiles at random

Truchet's and Doüat's patterns are all regular and symmetrical. Smith tried the opposite: he arranged the tiles at random orientations. He then removed the solid colour and kept only the diagonal line, which leaves two orientations instead of four. The result looks like the maze printed by the one-line Commodore 64 program 10 PRINT, published five years earlier.²

A later variant replaces the diagonal with two quarter-circle arcs, centred on opposite corners, each with a radius of half the tile's edge.³ It too has two orientations. Wherever two tiles meet, an arc continues into the next one, so a random grid of them forms unbroken curves, some of which close into loops.

New drawing

The tile is a small example of a rule that makes many images: one element, a few permitted states, and a way of choosing between them. The choice can be a plan, as in Truchet's plates, or chance, as in Smith's grid and in Vera Molnár's collage Quatre éléments distribués au hasard (1959), which the authors of 10 PRINT place beside the maze.² See also Generative art.