The Truchet Tile Grammar
A generative system for designing two-dimensional and three-dimensional tiles and patterns.
The Truchet Tile Grammar presented in this research is a generative system for designing two-dimensional and three-dimensional tiles and patterns. Based on the Truchet Tile, set forth by Sebastien Truchet in his paper “Mémoire sur les Combinaisons” (1704) and later expanded upon by C.S. Smith in his paper “The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy” (1987), this grammar defines shape rules to translate Truchet’s logic into a rule-based specification. The Truchet Tile Grammar builds on Truchet’s visual notations through two-dimensional pattern rules, but also expands it to include more variation in terms of tile geometry, shape, and three-dimensionality. The grammar consists of three rule sets: 1) Two-Dimensional Tile Rules, 2) Two-Dimensional Pattern Rules, and 3) Three-Dimensional Panel Rules. When applied recursively, this grammar generates three dimensional panels; the rule sets can also be employed independently depending on the design intent.
Publications:
Walter, N., Ligler, H., & Gürsoy, B. (2023). From Graphical Treatment of Combinatorics to Tiling Grammars. Nexus Network Journal, https://doi.org/10.1007/s00004-023-00715-2
Sebastien Truchet’s Tile
“Mémoire sur les Combinaisons” - Sebastien Truchet (1704)
In 1704, Sebastien Truchet published a short paper “Mémoire sur les Combinaisons” in which he examined the patterns attainable from a set of square tiles, each bisected diagonally into two colored triangles (Truchet, 1704; Reimann, 2009). He studied the “graphical treatment of combinatorics”, which was largely revolutionary in mathematics at the time (Smith, 1987).
Truchet’s bicolored tiles. Image published in Smith’s “The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy” (1987) (Courtesy of Burndy Library, Norwalk, Connecticut)
Smith’s Variation
“The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy” - C.S. Smith (1987)
In 1987, C.S. Smith published “The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy”, in which he analyzed Truchet’s tiling pattern and introduced the now recognized variation of Truchet’s original tile (Smith, 1987). In this variation, Smith used quarter-circle arcs in diagonal corners, replacing the two triangles of Truchet’s original tile.
Shape Grammar Theory
Shape rules are the fundamental building blocks of shape grammars, one of the most powerful formalisms for the generative description of designs (Stiny & Gips, 1972; Knight, 1994; Stiny, 1980; Stiny, 2006). The visual computations described in shape rules operate directly with points, lines, planes, and/or volumes. Shape rules are written using the general form A → B, where the shape A, sometimes embedded within the design context where the rule is applied, is replaced with the shape B. A shape grammar is a set of such shape rules applied consecutively to generate designs. The ice-ray grammar (Stiny, 1977) demonstrates how shape grammars have been applied to study the parametric structure of ornamental lattice designs . The key design operation for both Stiny’s simple lattice patterns and Truchet’s combinatorial tiling system is the orientation of a single motif and/or a single tile, suggesting a starting point for translating Truchet’s logic into a rule-based specification. This research takes on this effort by employing shape grammars to generate variations of the Truchet tile to characterize a two-dimensional and three-dimensional generative tiling system.
Two-Dimensional Tile Rules
The first set of shape rules are used to generate two-dimensional tiles. Although Smith’s original Truchet Tile is exclusively based on squares, the generic rules in the Truchet Tile Grammar expand Truchet’s tiling logic to include other regular and irregular polygons.
The grammar begins with an initial shape rule to place a labeled polygon in the cartesian plane.
Initial shape rule placing a labelled square, hexagon, or equilateral triangle in the cartesian plane
Polygons with an even number of sides:
After the initial shape rule “i”, shape rules consisting of sum operations and general transformations are then applied recursively to generate a single tile. Each tile generated is a closed shape with colored arcs in diagonal corners. The degree of the arc is controlled by parametric variables. A sample derivation using shape rules to generate two-dimensional tiles can be seen below.
R1: Bisect polygon edge with label
R2: Connect labels around polygon corner with a line segment and color field
R3: Change corner line segment to arc
R4-5: Remove labels
R6: Invert color fields
R7: If two color fields intersect, they combine to make a separate color field.
Polygons with an Odd number of sides:
Polygons with an odd number of sides have very similar rules as the rule set described above. The main difference is that for polygons with an even number of sides, the colored shapes are applied to every-other corner. For odd-numbered sided polygons, the colored shapes can be applied to all corners of the polygon. The only difference for this rule set is the absence of the black square label.
R1: Bisect polygon edge with label
R2: Connect labels around polygon corner with a line segment and color field
R3: Change corner line segment to arc
R4: Remove labels
R5: Invert color fields
R6: If two color fields intersect, they combine to make a separate color field.
A sample derivation using shape rules to generate two-dimensional tiles can be seen below:
Two-Dimensional Pattern Rules
The two-dimensional pattern rules are used to create basic tiling configurations and then to place specific tile motifs within a basic pattern. Based on the nature of the Truchet Tile, the shape rules make use of rotation and reflection while generating the two-dimensional tiling configurations. These rules similarly make use of weights and labels to specify the orientations and to distinguish between the two pairs of tiles generated with the previous rule set.
The two-dimensional pattern rules can be defined by the schemas x → x + t(x) and x → x + t(x’), in which a translated version of a tile or a translated and inverted version of a tile is copied adjacently to that tile. ”x” is depicted as a blank square while “x’” is depicted as a patterned square. The schema x → y is then employed to substitute the representative squares with the tiles designed in the previous rule set. Rule 17 is written as a conditional statement, stating that if x is one tile, then x’ is its inverted pair.
Shape rules for two-dimensional patterns
Sample two-dimensional tiling configuration with different two-dimensional Truchet tiles
Three-Dimensional Panel Rules
Three-dimensional tile rules transform the two-dimensional tiles into three dimensional panels in various ways. Parametric shape rules are used to control the depth of different regions of the individual tiles, which were previously specified with the color weights assigned. There are different rule sets within this stage depending on desired panel shape. The following rules are to generate extrusion-based panels.
Shape rules for three-dimensional extrusion-based panels
Sample three-dimensional patterns with different three-dimensional Truchet tiles.
In his “graphical treatment of combinatorics,” Truchet defined visual relations for combining two-dimensional tiles to generate various abstract patterns (Smith, 1987). The Truchet Tile Grammar builds on Truchet’s visual notations while defining the two-dimensional pattern rules, but also expands it to include more variations in terms of tile geometry, shape, and three-dimensionality. The grammar provides an open-ended framework to generate two-dimensional and three-dimensional tiles and patterns.
The Truchet Tile Grammar can be useful for various design scenarios. The grammar can be used for designing and manufacturing floor tiles and wall panels that can generate visually interesting and customizable patterns with only a small subset of unique components. On the other hand, the use of the Truchet Tile Grammar is not limited to designing architectural tiles and panels. As Lionel March has discussed in “Mathematics and Architecture since 1960”, these kinds of formalisms can also be used in urban planning and spatial design (March, 2015). The recent conceptual Tile House project by Matsys, for instance, makes use of Truchet tile logic in the generation of plan layouts (Matsys, 2021).
References
Knight, T.W. (1994). Transformations in Design: A Formal Approach to Stylistic Change and Innovation in the Visual Arts. Cambridge ; New York: Cambridge University Press.
March, L. (2015). Mathematics and architecture since 1960. In Architecture and Mathematics from antiquity to the future volume II, eds. K. Williams and M.J. Ostwald, 553–578. Birkhäuser. DOI https://doi.org/10.1007/978-3-319-00143-2_41
Reimann, D. (2009). Text from Truchet Tiles. Bridges Conference Proceedings, Banff, Canada, 325–326
Smith, C.S., & Boucher, P. (1987). The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy. Leonardo, 20(4), 373-385. https://muse-jhu-edu.ezaccess.libraries.psu.edu/article/600574/pdf
Stiny, G. (1977). Ice-Ray: A Note on the Generation of Chinese Lattice Designs. Environment and Planning B: Planning and Design, 4(1), 89-98. doi:10.1068/b040089.
Stiny, G. (1980). Introduction to Shape and Shape Grammars. Environment and Planning B: Planning and Design, 7(3), 343-351. doi:10.1068/b070343
Stiny, G. (1992). Weights. Environment and Planning B: Planning and Design, 19(4), 371-488
Stiny, G. (2006). Shape: Talking About Seeing and Doing. Cambridge, MA: MIT Press.
Stiny, G. & Gips, J. (1972). Shape Grammars and the Generative Specification of Painting and Sculpture. Paper presented at the Information Processing 71, Amsterdam: North-Holland.
Tile House. Matsys Design. (2021). Retrieved November 10, 2022 from: https://www.matsys.design/tile-house
Truchet, S. (1704). Mémoire sur les Combinaisons. Mémoires de l’Académie Royale des Sciences, 363-372.