Rhombus Tiling (P3)
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.