Douat Designs 

Notes on Jan. 29, 2004

From http://www.papyrusrarebooks.com/CatWeb4.htm 

- Mathematics in Decorative Patterns -

Dominique DOUAT Méthode pour faire une infinité de desseins differens, avec des carreaux mi-partis de deux couleurs par une ligne diagonale: ou observations du Père Dominique Doüat Religieux Carme de la Province de Toulouse sur une mémoire inseré dans l'histoire de l' Académie Royale des Sciences de Paris l'année 1704, présenté par le Reverend Père Sebastien Truchet, Religieux du même ordre, Academicien honoraire. Paris chez De Laulne, Jombert et Cailleau, 1722.

4to; Engraved title-page, 8 unn. leaves, 189 pages, 28 engraved plates. Contemporary calf, seamlessly rebacked. Marginal waterstain on a few pages; most of the pages inner lower margins carefully restored, never affecting printed or engraved surface. Good copy.
Only edition. “There are few places where the approaches of the artist and the scientist intersect more intimately than in the production and analysis of tiling patterns.” (C. S. Smith), and certainly in not many works is this intersection more evident than in this book. Dominique Douat, French Carmelite mathematician, was born in 1681, but it is quite difficult to find more information: the present work remained largely ignored until Gombrich re-discovered it. Douat’s work is largely based, as also acknowledged in the title, upon a previous study published by his fellow Carmelite (and not Dominican, as wrongly stated by C. S. Smith - possibly on the basis of Douat’s Christian name, Dominique; a mistake often present on following papers citing Smith’s) Sebastien Truchet (Lyons 1657- Paris 1729), an ‘engineer interested in mathematics and art’ (C. S. Smith) but also an expert in hydraulics, an inventor and the designer of the ‘Romain de Roy’, the first mathematically determined typeface. The mathematical background of Douat’s theory is based on the quite simple observation that a square diagonally divided in two triangles of different colors, can be rotated (90°, clockwise) in four different positions: these four variants (indicated by Douat as A, B, C and D) represent the basis for any further development. Sets of two of such variants can be arranged in 16 different ways; sets of three variants give 64 permutations; sets of all four variants result in 256 permutations. Taking each one of the 256 possibilities to be further combined with each other one we obtain 2562 = 65536 permutations. If we take each of the 65536 … 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-color square! The book is divided in four parts: the first giving the necessary background to produce the different variants; the second one gives several different patterns as example; the third one provides the necessary explanations for the previous part and the fourth one gives instructions to obtain different patterns without the use of the permutations table and without a previous drawing of the desired design. Douat’s work is much enlarged with respect to the study published by Truchet, and his purpose quite an ambitious one:

En effet de tous ceux qui ont écrit de l’ Architecture, très peu ont parlé du pavé ou carrelage, ou s’ ils ont traité cette matiere, ç’á été fort sucintement. Dans ce livre vous trouverez une source intarissable pour paver les Eglises & autres Edifices, carreler les planchers, & y faire des très beaux compartimens. Le Peintre y puisera des idées, les ouvriers en marqueterie, les Ebenistes, les Menusiers, les Vitriers, les Marbriers, les Tailleurs de pierres, & autres Ouvriers s’en servirons très utilement; les Brodeurs, les Tapissiers, les Tisserands, ceux qui travaillent sur les canevas, en un mot tous ceux qui se servent de l’aguille, y apprendront à faire de très beau ouvrages: & les Doctes Curieus qui s’adonnent à la Physique pourront aussi en tirer un grande avantage pour arriver à la connoissance de cette varieté incomprehensible qu’on voit dans les effets de la nature.

(Introduction).

Until recently, the importance of this book was not widely recognized and certainly not fully appreciated, perhaps because it falls rather uncomfortably between a treatise on mathematics and a treatise on design. Guilmard dismissed this book as an "ouvrage curieux à consulter". Gombrich, however, describes Doüat's book as "the earliest (and perhaps the rarest) treatise on the theory of design ...".

Cyril Stanley Smith, in his exhaustive analysis (including a full translation in English) of Truchet’s work, states:

“The book (Douat’s) had some influence on European decorative art in the eighteenth century and inspired illustrations in such works as Jeaurat’s Traité de perspective (Paris 1750) and Diderot’s Encyclopédie”, and: “Truchet’s treatise is of considerable importance for it is in essence a graphical treatment of combinatorics, a subject that, under the influence of Pascal, Fermat and Leibniz, was at the forefront of mathematics at the time. Truchet says that he got the idea when he saw a supply of tiles for paving apartments in a château near Orléans. ..."

Truchet ("Mémoire sur les combinaisons" in: Mémoires de l'Académie Royal des Sciences, 1704, pp. 363-372);

Guilmard (Les Maitres Ornamentistes, 1880) pages 122-123;

Gombrich (The Sense of 0rder. A study in the psychology of decorative Art, 1979) page 70;

Cyril Stanley Smith ("The Tiling Pattems of Sebastien Truchet and the Topology of Structural Hierarchy" Leonardo 20 n. 4, pp 373- 385, 1987);

Stuart Durant (Ornament from the Industrial Revolution to Today, New York, 1986) page 68; Berlin Catalogue 377;

Poggendorff II, 1141, has an entry for Truchet but (as apparently almost everybody else) totally disregards Douat.

 Page created Jan. 29, 2004