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Wang tiles (or Wang dominoes), first proposed by Hao Wang in 1961, are a class of formal systems. They are modelled visually by equal-sized squares with a color on each edge which can be arranged side by side (on a regular square grid) so that abutting edges of adjacent tiles have the same color; the tiles cannot be rotated or reflected. The following shows an example set of 13 Wang tiles: Hao Wang 王浩 (1921 – 1995) was a Chinese-American logician, philosopher and mathematician. ...
In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ...
The basic question about sets of Wang tiles is proving whether they can tile the plane or not. This means that copies of the tiles can be arranged one by one to fill an infinite plane, without any grid position where no tile in the set can match the side colors of already laid out adjacent tiles. Image File history File links Wang_tiles. ...
A tessellated plane seen in street pavement. ...
In 1961, Wang presented an algorithm to take any finite set of tiles and decide whether they tiled the plane. In his purported proof of the correctness of the algorithm, he assumed that any set that could tile the plane would be able to tile the plane periodically (with a pattern that repeats, like standard wallpaper).
However, in 1966 Robert Berger proved Wang's conjecture was wrong. He presented a set of Wang tiles that could only tile the plane aperiodically. This meant it could fill the plane without holes, but the tiling couldn't be a simple repetition of a finite pattern. This is similar to a Penrose tiling, or the arrangement of atoms in a quasicrystal. Although Berger's original set contained 20,426 tiles, he hypothesized that smaller sets would work, including subsets of his set. In later years, increasingly smaller sets were found. For example, the set of 13 tiles given above is an aperiodic set published by Karel Culik, II, in 1996. It can tile the plane, but not periodically. Robert Berger invented the first aperiodic set of tiles consisting of 20426 distinct tile shapes by using the rules of Penrose Tilling and the Golden Rule in 1966. ...
This article or section is in need of attention from an expert on the subject. ...
A Penrose tiling A Penrose tiling is an aperiodic tiling of the plane discovered by Roger Penrose in 1973. ...
Quasicrystals are aperiodic structures which produce diffraction. ...
Wang's algorithm for determining whether a given set of tiles can tile the plane was not correct. In fact, no such algorithm can exist. It is possible to translate any Turing machine into a set of Wang tiles, such that the Wang tiles can tile the plane if and only if the Turing machine will never halt. The halting problem is undecidable, therefore the Wang tiling problem is also uncomputable. In a sense, Wang tiles have computational power equivalent to that of a Turing machine. An artistic representation of a Turing Machine . ...
In computability theory the halting problem is a decision problem which can be informally stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. ...
The fact that Wang's procedure cannot theoretically work for arbitrary large tile sets does not render it useless for practical purposes. Using an optimized version of the original method Sergio Demian Lerner proved that no aperiodic set exists with 7 tiles or less. This lower bound leaves only a small gap for higher bound improvements.
Wang tiles can be generalized in various ways, all of which are also undecidable in the above sense. For example, Wang cubes are equal-sized cubes with colored faces and side colors can be matched on any polygonal tessellation. Culik and Kari have demonstrated aperiodic sets of Wang cubes. Winfree et al. have demonstrated the feasibility of creating molecular "tiles" made from DNA (deoxyribonucleic acid) that can act as Wang tiles. Mittal et. al have shown that these tiles can also be composed of peptide nucleic acid (PNA), a stable artificial mimic of DNA. A tessellated plane seen in street pavement. ...
The structure of part of a DNA double helix Deoxyribonucleic acid, or DNA, is a nucleic acid molecule that contains the genetic instructions used in the development and functioning of all known living organisms. ...
PNA can also refer to the Palestinian National Authority or Pakistan National Alliance. ...
Wang tiles have recently become a popular tool for procedural synthesis of textures, heightfields, and other large and nonrepeating bidimensional data sets; a small set of precomputed or hand-made source tiles can be assembled very cheaply without too obvious repetitions and without periodicity. In this case, traditional aperiodic tilings would show their very regular structure; much less constrained sets that guarantee at least two tile choices for any two given side colors are common because tileability is easily ensured and each tile can be selected pseudorandomly. Important papers about this new application include: Procedural synthesis is a technique in computer programming, specifically computer graphics and game design, wherein content is dynamically generated using algorithms, instead of being composed ahead of time. ...
- Jos Stam (1997), Aperiodic Texture Mapping introduces the idea of using Wang tiles for texture variation, with a deterministic substitution system.
- Michael F. Cohen, Jonathan Shade, Stefan Hiller, Oliver Deussen (2003), Wang Tiles for Image and Texture Generation introduces stochastic tiling and is very popular.
- Li-Yi Wei (2004), "Tile-Based Texture Mapping on Graphics Hardware" applies Wang Tiles for real-time texturing on a GPU
- Johannes Kopf, Daniel Cohen-Or, Oliver Deussen, Dani Lischinski (2006), Recursive Wang Tiles for Real-Time Blue Noise shows advanced applications.
The short story Wang's Carpets, later expanded to the novel Diaspora, by Greg Egan, postulates a universe, complete with resident organisms and intelligent beings, embodied as Wang tiles implemented by patterns of complex molecules. Diaspora is a 1997 science fiction novel by Australian writer Greg Egan. ...
Greg Egan (August 20, 1961, Perth, Western Australia) is an Australian computer programmer and science fiction author. ...
See also A tessellated plane seen in street pavement. ...
References - Wang, Hao (January 1961). "Proving theorems by pattern recognition—II", Bell System Tech. Journal 40(1):1–41. (Wang proposes his tiles, and hypothesizes there are no aperiodic sets).
- Wang, H. (November 1965). "Games, logic and computers" in Scientific American, pp. 98-106. (Presents them for a popular audience)
- Berger, R. (1966). "The undecidability of the domino problem", Memoirs Amer. Math. Soc. 66(1966). (Coins the term "Wang tiles", and demonstrates the first aperiodic set of them).
- Cohen, M. F., Shade, J., Hiller, S., and Deussen, O. 2003. "Wang Tiles for image and texture generation", In ACM SIGGRAPH 2003 Papers (San Diego, California, July 27 - 31, 2003). SIGGRAPH '03. ACM Press, New York, NY, 287-294.
- Culik, K. (1996). "An aperiodic set of 13 Wang tiles", Discrete Mathematics 160, 245-251. (Showed an aperiodic set of 13 tiles with 5 colors).
- Kari, J. (1996). "A small aperiodic set of Wang tiles", Discrete Mathematics 160, 259-264.
- Culik, K., and J. Kari (1995). "An aperiodic set of Wang cubes", Journal of Universal Computer Science 1, 675-686 (1995).
- Winfree, E., Liu, F., Wenzler, L.A., and Seeman, N.C. (1998). “Design and Self-Assembly of Two-Dimensional DNA Crystals, Nature 394, 539-544.
- Lukeman, P., Seeman, N. and Mittal, A (2002). “Hybrid PNA/DNA Nanosystems.” In 1st International Conference on Nanoscale/Molecular Mechanics (N-M2-I), Outrigger Wailea Resort, Maui, Hawaii.
Nature is one of the most prominent scientific journals, first published on 4 November 1869. ...
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