Penrose Tiling


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Sir Roger Penrose

Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. He is renowned for his work in mathematical physics, in particular his contributions to general relativity and cosmology. He is also a recreational mathematician and amateur philosopher.

Penrose Tiling

A Penrose tiling is a nonperiodic tiling generated by an aperiodic set of prototiles named after Sir Roger Penrose, who investigated these sets in the 1970s. Because all tilings obtained with the Penrose tiles are non-periodic, Penrose tilings are considered aperiodic tilings. Among the infinitely many possible tilings there are two that possess both reflection symmetry and fivefold rotational symmetry, as in the diagram above, and the term Penrose tiling usually refers to them. A Penrose tiling has many remarkable properties, most notably:

  • It is nonperiodic, which means that it lacks any translational symmetry. More informally, a shifted copy will never match the original exactly.
  • Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. This property would be trivially true of a tiling with translational symmetry but is non-trivial when applied to the non-periodic Penrose tilings.
  • It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction; the diffractogram reveals both the underlying fivefold symmetry and the long range order. This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called "deflation" or "inflation."

    • Robert Ammann independently discovered the tiling at approximately the same time as Penrose.
    Various methods to construct the tilings have been proposed: matching rules, substitutions, cut and project schemes and covering.


    The Penrose rhombuses are a pair of rhombuses with equal sides but different angles.

  • The thin rhombus t has four corners with angles of 36, 144, 36, and 144 degrees. The t rhombus may be bisected along its short diagonal to form a pair of acute Robinson triangles.
  • The thick rhombus T has angles of 72, 108, 72, and 108 degrees. The T rhombus may be bisected along its long diagonal to form a pair of obtuse Robinson triangles.

    There are 54 cyclically ordered combinations of such angles that add up to 360 degrees at a vertex, but the rules of the tiling allow only 7 of these combinations to appear. Ordinary rhombus-shaped tiles can be used to tile the plane periodically, so restrictions must be made on how tiles can be assembled. The simplest rule, prohibiting two tiles to be put together to form a single parallelogram, is insufficient to ensure aperiodicity. Instead, rules are made that distinguish sides of the tiles and require that only particular sides can be put together with each other.
    • An example of appropriate matching rules is shown in the diagram above. Tiles must be assembled so that the curves across their edges match in color and position. An equivalent condition is that tiles must be assembled so that the bumps on their edges fit together.
    The same rules can be specified with other formulations. There are arbitrarily large finite patches with tenfold symmetry and at most one center point of global tenfold symmetry where ten mirror lines cross. As the tiling is aperiodic, there is no translational symmetry—the pattern cannot be shifted to match itself over the entire plane. However, any bounded region, no matter how large, will be repeated an infinite number of times within the tiling. Therefore a finite patch cannot differentiate between the uncountably many Penrose tilings, nor even determine which position within the tiling is being shown. The only way to distinguish the two symmetric Penrose tilings from the others is that their symmetry continues to infinity. The rhombuses may be cut in half to form a pair of triangles, called Robinson triangles, which can be used to produce the Penrose tilings as a substitution tiling. The Robinson triangles are the isosceles 36º-36º-108º and 72º-72º-36º triangles. Each of these triangles has edges in the ratio of (1+\sqrt{5}):2, the golden ratio. The rules that enforce aperiodicity on a Robinson triangle tiling make the triangles asymmetric, and each triangle appears in conjunction with its mirror image to form a rhombus, kite, or dart.