Chaos and GraphicsTruchet curves and surfaces
Introduction
French clergyman Sébastien Truchet (1657–1729) is best known for two mathematical innovations; the invention of the point system for measuring the size of typographic characters, and his description of a certain type of artistic tiling. In a short paper titled “Memoir sur les Combinaisons” published in 1704, Truchet described a ceramic tile he had recently seen consisting of a square divided by a diagonal line between opposite corners into two coloured parts [1], and went on to provide a combinatorial analysis of the many aesthetically interesting patterns that can be created with the four possible orientations of such a tile (Fig. 1, top left). Although Truchet's analysis focussed on structured nonrandom designs, tilings formed by filling a square grid with random integers from {0, 1, 2, 3} and substituting the appropriate tile can still be aesthetically interesting (Fig. 1, bottom left).
In his 1987 description of Truchet's tiles, metallurgist and historian Cyril Stanley Smith described the tile variation shown in Fig. 1 (top right) consisting of two circular 90° arcs with radius equal to half the tile width and centred at opposite corners [1]. Random tilings of this modified tile in its two possible orientations form visually interesting results [2], [3]. The term Truchet tiling has been popularly adopted to refer to this modified form of tiling.
Truchet tiles have been used by many artists and rendered in various styles over recent years, some of which are shown Fig. 2. The leftmost design is a duotone pattern consisting of two unique tiles each of which have two orientations. Each tile placement is a free choice but must be oriented so that its edge colours match those of existing neighbours. Fig. 2 (middle) shows a tritone pattern consisting of three tiles with four orientations each. This design's regularity is no coincidence as placement is much more limited in this case—for each tile to be placed adjacent to two existing tiles, only one particular tile in one particular orientation will match edge colours with both neighbours. Fig. 2 (right) shows colour-coding of the continuous paths formed by a Truchet tiling. Each path is assigned a random colour for decorative purposes.
Section snippets
Truchet contours
A number of disjoint simple closed curves will tend to form in Truchet tilings, such as the red and gold curves in Fig. 2 (right). Such curves are described as Truchet contours by Gale et al. [4] and, like any other simple closed curve, will have clearly defined inside and outside regions in accordance with the Jordan Curve Theorem [5]. Fig. 3 shows the three simplest Truchet contours: the circle and the left and right dumbbell shapes.
Alan Tonisson, a researcher at Canon Information Systems
Spanning tree (ST) contours
It is sometimes desired to fill an area of a specific shape with a single uninterrupted curve. This may be for practical reasons, such as path-planning for automated embroidery equipment, or for purely artistic reasons. Truchet contours provide a convenient and attractive way to achieve this goal.
Listing 1 describes an algorithm for generating the minimum number of Truchet contours that will fill a given area to a specified resolution. This minimum number will be one unless the area is too
Contour characteristics
Given r, the radius of the contour's 90° arcs (which will be half the unit cell size), the perimeter length and area of a given ST contour can be calculated directly from V the number of vertices in its underlying graph.
Notice that the rightmost four replacement rules for cells with two vertices in Fig. 5 generate arc pairs of equal length regardless of whether the two vertices are connected by an edge or not. Cutting all edges of the spanning tree as shown in Fig. 8 (left) will not affect the
Multiresolution ST contours
So far the contours have been based upon grids of uniform resolution. Fig. 12 demonstrates how the algorithm can be adapted to generate more complicated contours based upon multiresolution grids.
Starting with a quadtree formed by a basic 2×2 square grid (left) the top right cell is subdivided into a nontree of nine equal sub-cells and the 90° arc for this cell reshaped into eleven 90° sub-arcs to fit the finer grid (middle). This process is then repeated to generate sub-sub-arcs to fit a
Truchet tiling in three dimensions
The Tangle, a manipulative puzzle from the 1970s composed of plastic 90° arcs that may be clipped together at various angles to create closed space loops [12], suggests a natural way to extend Truchet tiling to three dimensions. Fig. 15 (left) shows two unit cubes with the midpoints of adjacent faces joined by such 90° space arcs, as opposed to the 90° plane arcs that join adjacent sides of square two-dimensional Truchet tiles. Fig. 15 (right) shows a packing of such unit cubes, randomly chosen
Truchet surface construction
Fig. 18 (left) shows the Truchet surface formed by two spheres joined by a neck. This surface is similar to a shape called the Cassini oval, but is actually the surface of revolution formed by the ubiquitous dumbbell shape, and is the building block of the Truchet surface. Fig. 18 (right) shows its component parts: two spheres, a neck and the two spherical caps sliced off where the neck joins each sphere. The sphere centres must lie 2√2r units apart to ensure that the neck's maximum curvature
Surface characteristics
As with the two-dimensional case, Truchet surfaces are simple, smooth and closed, have clearly defined inside and outside regions, no holes or loops (they are genus 0) and contain no smaller Truchet surfaces within their volume. No point contained within the surface's volume will ever be more than r distance from the surface and the curvature of the surface will never exceed 1/r at any point. Unlike the two-dimensional case, however, the surface's complement does not share all of these
Other dimensions
In closing, we consider the extrapolation of Truchet tiling to other dimensions. One-dimensional Truchet tiles might consist of two points which may or may not be joined by a line segment, yielding two tile states. A composite shape randomly built from such tiles would form a series of line segments (a Truchet dash?) for which the distinction between inside and outside might depend on whether a given point falls within a line segment or not.
The four-dimensional case is more interesting but
Conclusions
This paper presents an algorithm for generating ST contours, a special class of Truchet contour based on a random spanning tree of a Truchet tiling's underlying graph. Such contours are well defined and have some attractive properties, especially known minimum curvature, known maximum distance to the boundary for any contained point, and good ratios of perimeter length to area.
The technique is extended to multiresolution contours that may be used for image halftoning, and to three dimensions to
Acknowledgements
Many thanks to Alan Tonisson for his input regarding the combinatorial aspects of Truchet tilings, and to Ken Brakke for his generous time and advice in relation to the diagonal neck intersection problem. Thanks to Bill Taylor and the anonymous reviewers for corrections and useful suggestions.
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2013, Computers and Graphics (Pergamon)Citation Excerpt :An extension of Truchet tiling that is related to this work is introduced by Clifford A. Pickover [22] as a single tile consisting of two circular arcs of radius equal to half the tile edge length centered at opposed corners. The two possible orientations of this tile, and tiling the plane using tiles with random orientations gives visually interesting curves called Truchet curves [23,24]. Truchet curves are not necessarily single curves, but they guarantee to separate the plane into two regions and therefore they are also used to create duotone planar artworks [14].
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