Penrose tiles

A Penrose tiling is an example of non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Sir Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of prototiles implies that a shifted copy of a tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right.

A Penrose tiling has many remarkable properties, most notably:

Various methods to construct Penrose tilings have been discovered, including matching rules, substitutions or subdivision rules, cut and project schemes and coverings.

Penrose tilings are simple examples of aperiodic tilings of the plane. A tiling is a covering of the plane by tiles with no overlaps or gaps; the tiles normally have a finite number of shapes, called prototiles, and a set of prototiles is said to admit a tiling or tile the plane if there is a tiling of the plane using only tiles congruent to these prototiles. The most familiar tilings (e.g., by squares or triangles) are periodic: a perfect copy of the tiling can be obtained by translating all of the tiles by a fixed distance in a given direction. Such a translation is called a period of the tiling; more informally, this means that a finite region of the tiling repeats itself in periodic intervals. If a tiling has no periods it is said to be non-periodic. A set of prototiles is said to be aperiodic if it tiles the plane but every such tiling is non-periodic; tilings by aperiodic sets of prototiles are called aperiodic tilings.

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