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.
[BibTeX] 
Abstract:
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.