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Penrose tiling

Invented by Roger Penrose, a Penrose tiling is a pattern of tiles which covers an infinite surface completely in a pattern which is always non-repeating (aperiodic).

There are many different possible sets of penrose tiles; the images below display one of the more commonly considered sets. It consists of two tiles, each having four sides with a length of one unit.

In other words, the angles are one tenth of a circle (36 degrees) times {2,2,3,3} and {1,1,4,4}.

The tiles are put together with one rule: no two tiles can be touching so as to form a single parallelogram. The tiles can be modified with bumps and dents around the perimeter to enforce this rule, but the tiling looks better if they have smooth sides.

Given this rule, there are many ways (in fact, uncountably many ways) to tile an infinite plane with no gaps or holes, but the tiling is always guaranteed to be aperiodic. This means that the pattern never repeats exactly. However, given a bounded region of the pattern, no matter how large, that region will be repeated an infinite number of times within the tiling (and, in fact, in any other Penrose tiling).

The Penrose tiling was first created as an interesting mathematical structure, but physical materials were later found where the atoms were arranged in the same pattern as a Penrose tiling. This pattern is not periodic (repeating exactly) but it is quasiperiodic (almost repeating), so the materials were named quasicrystals. See quasicrystal for more on these materials, and on the mathematics of quasiperiodic patterns.

The following picture shows an example of a Penrose tiling using the two tiles described above. There are a number of ways to generate such images; this was generated using an L-system.