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Article Computer Generation of Penrose Tilings

Author(s): J. Rangel-Mondragon, S. J. Abas.
Article: Computer Graphics Forum, Vol. 7, No. 1, pp. 29--37, 1988.
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Tiling patterns have been of interest to artists, craftsmen and geometers for thousands of years. More recently, because of their applications in crystallography, in the machine shop for cutting and shaping of materials and in pattern recognition, they have also become of importance to chemists, physicists, engineers and workers in the field of Artificial Intelligence. Another reason for recent heightening of interest in the subject comes from the discovery in 1984 at the National Bureau of Standards in USA1 of a material whose diffraction pattern exhibits five-fold symmetry incompatible with a three dimensional space lattice. Such materials have since been called quasicrystals and it appears that their structure characterises an intermediate state between the structures of crystalline and amorphous substances. This discovery has the profoundest implications for material science. The theoretical explanation of the structure of quasicrystals has been given in terms of the mathematical theory of Penrose tiling2 Penrose tiles not only explain the order underlying quasicrystals but have mathematical properties of great interest.3 They also offer a new spatial structure for creating aesthetically pleasing designs in applied arts and because they generate packed structures with five-fold symmetries, the tilings may turn out to be useful in modelling of biological forms. Tiling theory comprises a vast body of knowledge which rather surprisingly has only very recently been brought together in a definitive treatise.3 Despite the explosive interest in the subject and the widespread references to Penrose patterns in the literature, only one early paper has appeared on their computer generation†. The object of our article is to describe Penrose tilings and develop an efficient algorithm for their generation. We will also give some examples of designs based on their structure.

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