Gallery 4 - Douat's Method of Permutation

Born in 1681, Father Dominique Douat was a French Carmelite monk and mathematician who completed   a systematic study of the theory of pattern based on the quite simple observation that a square diagonally divided in two triangles of different colors, can be rotated 90 degrees in four different positions resulting in many thousands of permutations. He used a square divided about one diagonal (as displayed in the tables shown) and then coloured using two different colours, in this case, black and white.

He showed that by rotating the divided two-coloured square through 90 degrees, four variants are produced; these are marked with the first four letters of the alphabet and by numbers in the first table. In the second table,  sets of two such variants are shown to result in 16 possibilities; going on from that sets of three result  in 64 permutations and sets of four result in 256 possible permutations.

Douat’s system goes on to set out in more tables, further permutations of his two-colour, diagonal squares by using the first four letters of the alphabet which is a reminder of the genetic code discovered in the ‘double helix’ model of DNA. Taking each of these as potential units to be further combined, we arrive by doubling them at 256 to the power of 2 which is 65,536 and so on ad infinitum.

This tremendous amount results in correspondingly infinite possible variations in patterns, to be used for all kind of decorative purposes and basically grounded on…just a two-colour square! Using more colour combinations, as I have shown, leads to millions more permutations. On a few of my designs   I have left in the square tile grid base to show how they were constructed.