Ladies and gentlemen, young and old! In the next zhang big less.
Dense patterns dominate our visual and material worlds. Cyril Stanley Smith is very sensitive to this in every aspect of his work. As a pastime, he recreated the 1704 Trusseau Shops and added to their richness and diversity.
1. Introduction
Cyril Stanley Smith was very erudite. He was one of the most creative metallurgists of the 20th century and a prominent historian of science and technology. The aesthetic aspects of science and the relationship between art and science aroused his great interest.[1] It was out of this latter interest that he reproduced an extraordinary work by Sebastian Truchet of 1704 on the dense paving motif, and saw some of the deeper implications of Truchet's thought.
2 Sebastian Truchet and his shop
Sebastian Truchet was a Carmelite priest who, by the way, was also the inventor of the "pound system" used to indicate font size – now familiar to all PC users!
His approach of producing an infinite variety of densely paved designs by combining four-letter codes was a surprising early beginning for the idea of encoding visual patterns (see [2]).
Truchet's idea is simple, but the possibilities for its implications and concealments are enormous. A Truchet puzzle is a square block with a simple diagonal two-tone decoration (Figure 1). In a plane laid with Trussie tiles, each piece can have 4 possible directions, which can be called A, B, C, and D. For example, Figure 2 shows the pattern encoding of four cells
BDCA...
DBAC...
ACDB...
CABD...
Figure 1 (left): Truchet's block.
Figure 2 (right): A simple Trusseau pavement pattern of 4 units.
Truchet's book contains a variety of examples. This representative example is sufficient, and more examples can be seen through search engines.
The possible symmetry of a periodic Trusseau pavement pattern could be one of 12 of the 17 "wallpaper groups" (i.e., those without 3 or 6 symmetries). An interesting exercise in combinatorials is to enumerate and classify Truchet patterns with given symmetry and a given number of bunks per cell. Obviously, the 2- and 4-fold symmetry, reflection, and slip symmetry of a pattern correspond to the combined properties of its generating code. The Truchet pattern may also exhibit "bichromatic" symmetry when the black and white areas of the pattern are consistent, as shown in Figure 2.
Truchet's work, which appeared at a time when famous mathematicians such as Fermat, Leibniz, and Pascal developed probability theory and the related mathematics of permutations and combinations, can be seen as a manifestation of this zeitgeist. Leonardo Cyril Stanley Smith put it this way in his essay "Smith's Application of the Trussie Shop": "Trussie's paper is quite important because it is essentially a graphical treatment of combinatorics, which, under the influence of Pascal, Fermat and Leibniz, was at the forefront of mathematics at the time. Truchet said he came up with the idea when he saw tiles from an apartment in a castle near Orléans. ”
Douat further developed Truchet's combinatorial view, which was published in 1722 to introduce it to a wider audience. Art historian E. H. Gombrich rediscovered the obscure book and copied some of its pages in The Sense of Order.
Symmetry: In addition to translation, acyclical pattern can have 2, 3, 4, 6 weight symmetries, which means that the pattern is unchanged when the pattern is rotated 180°, 120°, 90°, or 60° around a point, respectively.
It may also have reflective symmetry or slip symmetry. Slip is a reflection on a straight line, followed by a translation in the direction of a straight line.
"There are few places where the methods of artists and scientists can be more closely integrated than the production and analysis of densely paved patterns."
—C. S. Smith
3. Theme changes
In Leonardo Cyril Stanley Smith's article, he introduces two options for basic Trussie's pieces. By omitting black and white shading and keeping only diagonal lines, a tile with only two possible directions is given instead of four; the resulting pattern can be encoded in binary notation. The final pattern has an interesting "labyrinthy" appearance (Figure 3). While Truchet and Douat only considered the periodic pattern, Smith's interest was in extending the method to illustrate the principles of hierarchy, which could be generated by iterative rules applied to binary encoding. Another variant of the Truchet puzzle introduced by Smith is decorated by two arcs. This piece also has only two possible directions and produces patterns with strange curved structures (Figure 4).
Figure 3: One variation of Smith's Trussie Pavement pattern. This type of pattern can be specified by binary encoding.
Figure 4: Another variant of Smith.
Pickover [5] takes into account random constructs. Figure 5 shows part of this random pattern, which was constructed from a variant of Smith's Trusseau tile. One feature of the patterns built with this block is that curved lines divide the plane into two areas; the shades of gray in the figure emphasize this. In this way, binary sequences are converted into visual patterns that can reveal interesting aspects of chaotic behavior. Observe the small circles that appear in the pattern. Pickover deduces that if the tile orientation is indeed random, the number of circles divided by the number of tiles should be about 0.054. Similarly, the "dumbbell" (two combined circles) has a score of 0.0125. The more complex statistical aspects of random patterns can also be calculated. Pickover recommends using a Trussie Pavement to suggest testing for true random behavior, as well as identifying subtle deviations from randomness in chaotic systems.
Figure 5: The appearance of a random pavement of the tiles shown in Figure 4.
An interesting modification of the Truchet puzzle is the 60° diamond on the left side of Figure 6. This allows for an infinite variety of densely paved patterns with triple and six symmetries. In a flat pavement using these diamond-shaped tiles, individual tiles can appear in 6 different directions. To the right of Figure 6 is a set of 6 tiles illustrating the expansion rules that will produce a quasi-periodic pattern (the concept of inflation in the context of Penrose tiles is described in detail by Grumbaum and Shephard[6]). Figure 7 shows the dense pavement obtained after four iterations.
Figure 6: A decorated diamond-shaped tile and an "expansion rule" used to generate an aperiodic pattern.
Figure 7: The appearance of the dense pave resulting from the 4 iterations of the expansion rule.
Periodic pattern: The pattern covering the plane can be cyclical or non-cyclical. A double-cycle pattern has unit elements (i.e., tiles) that are repeated in both directions by panning without changing orientation to produce the entire pattern. Periodic patterns may have other symmetries than translation. The double-cycle patterns on the plane can be classified according to their symmetry, and there are only 17 types, and these 17 sets of symmetries are called "wallpaper groups". Penrose shops are non-cyclical.
4. 3D generation
A question that comes naturally to mind is: Is there a three-dimensional model similar to the Trussie pavement? We give two examples of "3D Trussie patterns". Look at the shape of the box in Figure 8 – shown in 4 possible directions – and each side of the box is connected in a pair of arcs, like the decorations on Smith's Trusseau tile variant. By paving the three-dimensional space with these "pieces", we have obtained an infinite variety of surfaces. However, natural limitations arise, and there are no similar restrictions in the two-dimensional case, where the puzzle can be completely arbitrarily oriented. Because patches in two consecutive boxes must match on the interface, there are actually only two possible directional relationships between a pair of contiguous tiles: they are related by translation or reflection on the common plane. In addition, if two adjacent tiles are linked together by reflection in their common plane, that plane must be the reflection plane of all the tile pairs in that plane that have their common faces. An example of part of this 3D Trussie pavement is shown in Figure 9.
Figure 8: 4 possible directions for 3D generation of a Truchet block.
Figure 9: The surface produced by densely paving the three-dimensional space with the mosaic shown in Figure 8.
The surface tiles in the basic tile have the shape of modular elements that form one-eighth of the smallest surface element, called "Schoen's Batwing" (see [7] and Ken Braque's Website[8]). Adjacent elements are relevant everywhere through reflections from their common faces. For the smallest surface,"pseudo-batwing" named by Braque, a similar module is the entire unit —all with the same orientation. It is very noteworthy that for the different minimal surfaces of the symmetry of these two completely different spatial groups, the difference in the shape of the mold surface piece is difficult to detect. Figure 9 was made by Alan McKay, who first realized that the Batwing module could be thought of as a 3D Trussie pavement, thus illustrating the possibility of an infinite number of possible minimum surfaces based on the 3D Trussie Pavement principle.
Another possible generalization of Truchet's idea for describing three-dimensional structures is its application to describing and classifying three-dimensional weaving patterns. 3D weaving has recently become increasingly important in composite fabrication, where fibers embedded in the matrix make up the reinforcement of the material. The 2D braid pattern has two orthogonal line directions (warp and weft). Three-dimensional analogues can consist of modular units containing three orthogonal threaded sections (Figure 10). Dirk van Svenberg studied the possibility of three-dimensional weaving patterns with three orthogonal line directions, provided that each two-dimensional layer is simply 2D down-down-up weaving (a kind of "mat"). He holds patent rights to these 3D weaving methods. Continuous elements like Figure 10 can be associated by rotating around axes in the cube surface, by spiral transformations, by slippage, reflection, and so on. However, the requirement that the final pattern should be stacked from simple 2D pads has strict restrictions in all three directions. In fact, adjacent cubes can only be associated by inversion on their common surface or by a 2x spiral transformation (0 and 1 in Van Svenberg's notation, respectively). According to the three-letter symbol listing the types of transformations applied in the three orthogonal directions, depending on whether the basic module is the module shown in FIG. 10 or a mirror image thereof, we obtain four weaving patterns OOO, OOI, OII and III (III) present in the right-handed form IIIR or the left-handed form IIIL. Figure 11 shows one cell of OOO (where the lines have a sinusoidal form, And Figure 12 shows a stereoscopic image of the IIIR, where all the lines are right-handed spirals.)
Figure 10: Basic 3D tiles that generate a 3D weave pattern.
Figure 11: A three-dimensional weave obtained by stacking the cubic blocks in Figure 10 so that adjacent tiles are related to each other by inversion.
5. Conclusion
The advent of high-speed computers has changed the way scientists think about the structure of materials. The computer age has not only opened up the possibility of solving previously intractable problems — new and hitherto unintended areas of problems have also opened up. Encoding a two- or three-dimensional pattern or structure into a string of alphanumeric symbols and reconstructing a simulation of a pattern or structure based on its abstract code—"inorganic genes"—is an important example of this change in perspective, which is likely to become increasingly important, especially in materials science. Truchet's idea, though simple, was very prescient; Cyril Stanley Smith recognized its potential, and that was to his credit. We hope that the possibility of further generalizing Truchet's point of view that we have presented here may inspire further exploration.
Figure 12: Three-dimensional woven stereoscopic image pairs in which the basic dense paving is linked together by a spiral transformation.
bibliography
[1] CS Smith, Search for Structure: Selected Essays in Science, Art and History, MIT Press, Cambridge MA, 1981
[2] S Truchet, Memoir sur les combinaisons, Memoires de l'Academie Royale des Sciences, pp.363-72, 1704.
[3] C S Smith, The tiling patterns of Sebastien Truchet and the topology of structural hierarchy, Leonardo, Vol.20, pp.373-385, 1987.
[4] E H Gombrich, The Sense of Order, Cornell Univ. Press, pp.70-72,1979.
[5] C A Pickover, Computers, Pattern, Chaos and Beauty, St. Martin's Press, New York, pp.329-332,1993.
[6] B Griinbaum and G C Shephard, Tilings and Patterns, W H Freeman, 1987.
[7] E A Lord and A L Mackay, Periodic minimal surfaces of cubic symmetry, Curr. Sci., V01.85, pp.346-62, 2003.
[8] Ken Brakke, http://www. susqu.edu/brakke/
[9] E A Lord and S Ranganathan, Truchet Tilings and their Generalisations
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