page_title: Penrose tiling
section_title: Rhombus tiling (P3)
hierarchical_section_title: Penrose tiling / Penrose tilings / Rhombus tiling (P3)
context_page_description: A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and aperiodic means that shifting any tiling with these shapes by any finite distance, without rotation, cannot produce the same tiling. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s.
  There are several different variations of Penrose tilings with different tile shapes. The original form of Penrose tiling used tiles of four different shapes, but this was later reduced to only two shapes: either two different rhombi, or two different quadrilaterals called kites and darts. The Penrose tilings are obtained by constraining the ways in which these shapes are allowed to fit together. This may be done in several different ways, including matching rules, substitution tiling or finite subdivision rules, cut and project schemes, and coverings. Even constrained in this manner, each variation yields infinitely many different Penrose tilings.
context_section_description: The third tiling uses a pair of rhombuses (often referred to as "rhombs" in this context) with equal sides but different angles. Ordinary rhombus-shaped tiles can be used to tile the plane periodically, so restrictions must be made on how tiles can be assembled: no two tiles may form a parallelogram, as this would allow a periodic tiling, but this constraint is not sufficient to force aperiodicity, as figure 1 above shows.
  There are two kinds of tile, both of which can be decomposed into Robinson triangles.
  The thin rhomb t has four corners with angles of 36, 144, 36, and 144 degrees. The t rhomb may be bisected along its short diagonal to form a pair of acute Robinson triangles.
  The thick rhomb T has angles of 72, 108, 72, and 108 degrees. The T rhomb may be bisected along its long diagonal to form a pair of obtuse Robinson triangles; in contrast to the P2 tiling, these are larger than the acute triangles.
  The matching rules distinguish sides of the tiles, and entail that tiles may be juxtaposed in certain particular ways but not in others. Two ways to describe these matching rules are shown in the image on the right. In one form, tiles must be assembled such that the curves on the faces match in color and position across an edge. In the other, tiles must be assembled such that the bumps on their edges fit together.
  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 seven of these combinations to appear (although one of these arises in two ways).
  The various combinations of angles and facial curvature allow construction of arbitrarily complex tiles, such as the Penrose chickens.
caption: Matching rule for Penrose rhombs using parabolic edges to enforce the tiling rules