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It came as a surprise to mathematicians in 1964 when Robert Berger discovered a set of tiles that would only form a non-periodic tiling. Berger’s non-periodic tiling was composed from a collection of over 20,000 different tiles. Once it had been demonstrated that such non-periodic tilings were possible other mathematicians began to search for simpler examples. Raphael Robinson devised a collection of just six tiles that would only tile non-periodically.

Around the same time, the physicist Sir Roger Penrose independently constructed a set of six tiles with the same properties. Penrose found his tiles by dissecting and rearranging regular pentagons. The angles of regular pentagons are 108 and, as no multiple of 108 equals 360, it is not possible to form a tessellation solely with regular pentagons. In fact, there are no periodic tilings with fivefold rotational symmetry.

After more experimentation, Penrose realised in 1974 that he could reduce the number of different tiles still further to just two types of quadrilateral and the resulting tilings would still necessarily be non-periodic. The two quadrilaterals that Penrose found are known as kites and darts. Closely related to kites and darts are two types of rhombus that also force non-periodicity in their tilings. One of the rhombi has angles of 108 and 72, the other has angles of 144 and 36.

Penrose tilings are fascinating because they have an approximate five-fold symmetry and patterns can be found within the tilings that repeat in small regions, so they are almost periodic, but the repetition cannot be continued throughout the tiling. The combination of repetition with variation is a feature of many art forms, especially music, and it may be this aspect of Penrose tilings that gives them their aesthetic appeal.

The Geometry Centre, University of Minnesota
Web Link:
http://www.geom.uiuc.edu/apps/quasitiler/
Penrose Tilings can be constructed by taking a two dimensional section of a honeycomb of five dimensional cubes.
 

Diffraction pattern of a quasicrystal, exhibiting 5-fold or 10-fold rotational symmetry.
Web Link:
http://www.cmp.caltech.edu/~lifshitz/quasicrystals.html