Regularity

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Imagine a square tile, bisected into two triangles, one black, one white. Imagine two such tiles, placed next to one another. Out of this simple configuration emerges a surprising number of possible arrangements. Sixty-four, to be precise, as each tile can be oriented four ways, shifting the apex of the black triangle from corner to corner. Now imagine a whole floor or tapestry covered with a regular pattern of these repeating tiles. This is the power of the Truchet Tile, so named because the Dominican priest Sebastien Truchet first described what he called the "fecundity of these combinations" in 1704, after experimenting with some ceramic tiles he came across at a building site for a chateau near Orléans (Smith 374).

−

Matching a single Truchet tile with another, and another, and another, and so on, a designer is able to create an incredible array of patterns. The interplay between the direction of each tile and the varying repetition of black and white -- of positive and negative -- produces designs that can range from grid-like patterns to mesmerizing, almost three-dimensional mazes. Unlike earlier Islamic patterns or Celtic designs, which both relied on multiple-sized shapes, the Truchet tile uses only a single size and a single shape (Smith 378). In his original 1704 essay, Truchet provided examples of thirty different patterns, barely evoking the aesthetic possibilities of his tiles, though he notes that he "found too great a number to report them all" (Smith 374). Truchet's work would be the inspiration for a latter book, Douat's modestly named ''Methode pour faire une infinite de desseins differents,'' which in turn had a considerable impact on 18th century European art (Smith 373).

+

Matching a single Truchet tile with another, and another, and another, and so on, a designer is able to create an incredible array of patterns. The interplay between the direction of each tile and the varying repetition of black and white -- of positive and negative -- produces symmetrical designs that can range from grid-like patterns to mesmerizing, almost three-dimensional illusions. Unlike earlier Islamic patterns or Celtic designs, which both relied on multiple-sized shapes, the Truchet tile uses only a single size and a single shape (Smith 378). In his original 1704 essay, Truchet provided examples of thirty different patterns, barely evoking the aesthetic possibilities of his tiles, though he notes that he "found too great a number to report them all" (Smith 374). Truchet's work would be the inspiration for a latter book, Douat's modestly named ''Methode pour faire une infinite de desseins differents,'' which in turn had a considerable impact on 18th century European art (Smith 373).

Yet all of Truchet's and Douat's examples are regular patterns, symmetrical and repetitive. The mathematician Cyril Stanley Smith observes that even more compelling designs can be generated from Truchet tiles if "dissymmetries" are introduced. What happens when the regularity of a Truchet pattern is interrupted by randomness? Smith provides one example, a block of Truchet tiles arranged at random:

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[[File:Truchet-Tiles-Randoms.jpg|500px|center|Smith, Cyril Stanley, and Pauline Boucher. “The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy.” Leonardo 20.4 (1987) : 373-385.]]

−

The latices of a grid are still visible, but randomness has made its mark, leaving imperfections that disrupt any nascent pattern. Unlike the examples Truchet and Douat give, there is no resolution to the structure. Smith continues to push the limits of the Truchet tiles' regularity by omitting colors from the tiles, leaving only the black diagonal line. Suddenly the four orientations of any given tile have been reduced to two. Smith does this trick with the random array of Truchet tiles pictured above:

+

The latices of a grid are still visible, but randomness has made its mark, leaving imperfections that disrupt any nascent pattern. Unlike the symmetrical examples Truchet and Douat give, there is no resolution to the structure. The center cannot hold, and neither can the margins. Smith next pushes the limits of the Truchet tiles' regularity by omitting colors from the tiles, leaving only the black diagonal line. Suddenly the four possible orientations of any given tile have been reduced to two. Smith does this trick with the random array of Truchet tiles pictured above:

[[File:Truchet-Tiles-NoColor.jpg|500px|center|Smith, Cyril Stanley, and Pauline Boucher. “The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy.” Leonardo 20.4 (1987) : 373-385.]]

−

Suddenly the Truchet tiles have generated a design that looks unmistakably like the 10 PRINT maze. The grid still remains -- indicating the edges of each tile -- but the diagonals no longer seem to bound positive or negative space. Instead, they appear to be walls. In this Truchet tile-produced artifact the dynamic between regularity and its opposite come into play, suggesting that regularity is not an aspect of design that exists in isolation, but rather can only be defined by its limitations, by those moments when the regular becomes irregular.

+

Suddenly the Truchet tiles have generated a design that looks unmistakably like the output of 10 PRINT. The grid still remains -- indicating the edges of each tile -- but the diagonals no longer seem to bound positive or negative space. Instead, they appear to be the walls of a maze, twisty little passages, all different. In this Truchet tile-produced artifact the dynamic between regularity and its opposite come into play, suggesting that regularity is not an aspect of design that exists in isolation, but rather can only be defined by its limitations, by those moments when the regular becomes irregular.

== Textile Grids ==

From Maze

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@@ Line 7: / Line 7: @@
 Imagine a square tile, bisected into two triangles, one black, one white. Imagine two such tiles, placed next to one another. Out of this simple configuration emerges a surprising number of possible arrangements. Sixty-four, to be precise, as each tile can be oriented four ways, shifting the apex of the black triangle from corner to corner. Now imagine a whole floor or tapestry covered with a regular pattern of these repeating tiles. This is the power of the Truchet Tile, so named because the Dominican priest Sebastien Truchet first described what he called the "fecundity of these combinations" in 1704, after experimenting with some ceramic tiles he came across at a building site for a chateau near Orléans (Smith 374).
-Matching a single Truchet tile with another, and another, and another, and so on, a designer is able to create an incredible array of patterns. The interplay between the direction of each tile and the varying repetition of black and white -- of positive and negative -- produces designs that can range from grid-like patterns to mesmerizing, almost three-dimensional mazes. Unlike earlier Islamic patterns or Celtic designs, which both relied on multiple-sized shapes, the Truchet tile uses only a single size and a single shape (Smith 378). In his original 1704 essay, Truchet provided examples of thirty different patterns, barely evoking the aesthetic possibilities of his tiles, though he notes that he "found too great a number to report them all" (Smith 374). Truchet's work would be the inspiration for a latter book, Douat's modestly named ''Methode pour faire une infinite de desseins differents,'' which in turn had a considerable impact on 18th century European art (Smith 373).
+Matching a single Truchet tile with another, and another, and another, and so on, a designer is able to create an incredible array of patterns. The interplay between the direction of each tile and the varying repetition of black and white -- of positive and negative -- produces symmetrical designs that can range from grid-like patterns to mesmerizing, almost three-dimensional illusions. Unlike earlier Islamic patterns or Celtic designs, which both relied on multiple-sized shapes, the Truchet tile uses only a single size and a single shape (Smith 378). In his original 1704 essay, Truchet provided examples of thirty different patterns, barely evoking the aesthetic possibilities of his tiles, though he notes that he "found too great a number to report them all" (Smith 374). Truchet's work would be the inspiration for a latter book, Douat's modestly named ''Methode pour faire une infinite de desseins differents,'' which in turn had a considerable impact on 18th century European art (Smith 373).
 Yet all of Truchet's and Douat's examples are regular patterns, symmetrical and repetitive. The mathematician Cyril Stanley Smith observes that even more compelling designs can be generated from Truchet tiles if "dissymmetries" are introduced. What happens when the regularity of a Truchet pattern is interrupted by randomness? Smith provides one example, a block of Truchet tiles arranged at random:
@@ Line 13: / Line 13: @@
 [[File:Truchet-Tiles-Randoms.jpg|500px|center|Smith, Cyril Stanley, and Pauline Boucher. “The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy.” Leonardo 20.4 (1987) : 373-385.]]
-The latices of a grid are still visible, but randomness has made its mark, leaving imperfections that disrupt any nascent pattern. Unlike the examples Truchet and Douat give, there is no resolution to the structure. Smith continues to push the limits of the Truchet tiles' regularity by omitting colors from the tiles, leaving only the black diagonal line. Suddenly the four orientations of any given tile have been reduced to two. Smith does this trick with the random array of Truchet tiles pictured above:
+The latices of a grid are still visible, but randomness has made its mark, leaving imperfections that disrupt any nascent pattern. Unlike the symmetrical examples Truchet and Douat give, there is no resolution to the structure. The center cannot hold, and neither can the margins. Smith next pushes the limits of the Truchet tiles' regularity by omitting colors from the tiles, leaving only the black diagonal line. Suddenly the four possible orientations of any given tile have been reduced to two. Smith does this trick with the random array of Truchet tiles pictured above:
 [[File:Truchet-Tiles-NoColor.jpg|500px|center|Smith, Cyril Stanley, and Pauline Boucher. “The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy.” Leonardo 20.4 (1987) : 373-385.]]
-Suddenly the Truchet tiles have generated a design that looks unmistakably like the 10 PRINT maze. The grid still remains -- indicating the edges of each tile -- but the diagonals no longer seem to bound positive or negative space. Instead, they appear to be walls. In this Truchet tile-produced artifact the dynamic between regularity and its opposite come into play, suggesting that regularity is not an aspect of design that exists in isolation, but rather can only be defined by its limitations, by those moments when the regular becomes irregular.
+Suddenly the Truchet tiles have generated a design that looks unmistakably like the output of 10 PRINT. The grid still remains -- indicating the edges of each tile -- but the diagonals no longer seem to bound positive or negative space. Instead, they appear to be the walls of a maze, twisty little passages, all different. In this Truchet tile-produced artifact the dynamic between regularity and its opposite come into play, suggesting that regularity is not an aspect of design that exists in isolation, but rather can only be defined by its limitations, by those moments when the regular becomes irregular.
 == Textile Grids ==