Penrose Tiling

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

A Penrose tiling has many remarkable properties, most notably:

• It is non-periodic, which means that it lacks any translational symmetry.
• It is self-similar, so the same patterns occur at larger and larger scales. Thus, the tiling can be obtained through "inflation" (or "deflation") and any finite patch from the tiling occurs infinitely many times.
• It is a quasicrystal; implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold symmetry and the underlying long range order.

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

Periodic and Aperiodic Tilings

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.

Penrose Tilings and Art

The aesthetic value of tilings has long been appreciated, and remains a source of interest in them; here the visual appearance (rather than the formal defining properties) of Penrose tilings has attracted attention. The similarity with some decorativer patterns used in the Middle East has been noted and Lu and Steinhardt have presented evidence that a Penrose tiling underlies some examples of medieval Islamic art.

Drop City artist Clark Richert used Penrose rhombs in artwork in 1970. Art historian Martin Kemp has observed that Albrecht Durer sketched similar motifs of a rhombus tiling.

San Francisco’s new $4.2 billion Transbay Transit Center is planning to perforate its exterior's undulating white metal skin with the Penrose pattern.

The floor of the atrium of the Molecular and Chemical Sciences Building at the University of Western Australia is tiled with Penrose Tiles.

The Andrew Wiles Building, the location of the Mathematics Department at the University of Oxford as of October 2013 includes a section of Penrose tiling as the paving of its entrance.

http://en.wikipedia.org/wiki/Penrose_tiling