Douat: An Alphabet of Tiles

From Truchet to Douat

The story begins with Sébastien Truchet, a Dominican friar, royal engineer, and member of the Académie Royale des Sciences. Around 1704, while inventorying ceramic tiles, he noticed that a square tile divided diagonally into a dark and a light half is far richer than it looks: rotate it and it becomes four different tiles, and laid edge to edge those tiles produce an astonishing variety of patterns. He wrote up the observation in a short mémoire for the Academy.

Truchet never expanded the idea into a book. That was done by Dominique Douat, a Carmelite of the Province of Toulouse, who took Truchet's mémoire as a starting point and pushed it as far as it would go. His 1722 Méthode pour faire une infinité de desseins différens is, in effect, the first full combinatorial treatise on what we now call Truchet tiles — complete with 72 engraved designs, tables of permutations, and a recipe for building any pattern from scratch.

The book carries approbations from the leading mathematicians of the day — Fontenelle and Varignon among them — and is dedicated to a president of the Montpellier Academy. Two friars, a single diagonal line, and the whole machinery of 18th-century combinatorics: that is the unlikely origin of a system that generative artists still reach for today.

One tile, four orientations

Everything is built from a single square tile, mi-parti — split in half by a diagonal — one half colored, one half white. Because a square has four corners, the colored right-angle can point to any of them. Douat gives each of these four orientations a letter, named by the corner where the colored angle sits:

That is the entire alphabet. A is the colored corner at bottom-left, B at top-left, C at top-right, D at bottom-right. Once you can tell A, B, C and D apart, you can read — and write — every design in the book without ever looking at a drawing.

An alphabet of patterns

Douat's insight is that the letters are a notation. Take the four tiles four at a time, allowing repeats, and you get 4 × 4 × 4 × 4 = 256 little two-by-two arrangements — his fourth table. Repeat and combine those, row after row, and the count explodes: he patiently works out the number of designs you can make taking the 256 two, three, four… at a time, and the totals run to dozens of digits.

He loved that disproportion between tiny means and limitless results, and reached for analogies: mathematics grows from a single point, arithmetic from nine digits, music from seven notes, and the twenty-four letters of the alphabet spell more words than there are moments since the creation of the world. Four tiles, he argues, are no different — they spell an infinity of designs.

And because the patterns are written in letters, you don't need the engravings at all. Knowing only A, B, C and D, you can lay out cardboard tiles and reproduce any design — "sans étude", without study, as he puts it.

Read the grid of letters on the left; build the tiles it names; and you get the picture on the right. This is design 1 of his final table:

D D A A
D D A A
C C B B
C C B B

Opposites: diagonal, horizontal, perpendicular

The letters also make the symmetries easy to compute. Douat notes that A and C (and B and D) are diagonal opposites — they swap black for white. A and D (and B and C) are horizontal opposites — left for right. A and B (and C and D) are perpendicular opposites — top for bottom.

So from any one design you instantly get three more, just by swapping letters. A whole page of his book is given over to these oppositions, turning one pattern into four with no drawing required.

The 72 designs, written in letters

Below are Douat's seventy-two engraved designs — but not the engravings. Each one has been transcribed from his letter grids and rendered live, in your browser, from nothing but A, B, C and D. Use the arrows (or the keyboard) to leaf through them, exactly as a reader in 1722 could have built them tile by tile.