Penrose Tiles

The Penrose tiles are a pair of shapes that tile the plane only aperiodically (when the markings are constrained to match at borders). These two tiles, illustrated above, are called the "kite" and "dart," respectively. In strict Penrose tiling, the tiles must be placed in such a way that the colored markings agree; in particular, the two tiles may not be combined into a rhombus (Hurd).

Two additional types of Penrose tiles known as the rhombs (of which there are two varieties: fat and skinny) and the pentacles (or which there are six type) are sometimes also defined that have slightly more complicated matching conditions (McClure 2002).

In 1997, Penrose sued the Kimberly Clark Corporation over their quilted toilet paper, which allegedly resembles a Penrose aperiodic tiling (Mirsky 1997). The suit was apparently settled out of court.

Penrose tilings are substitution tilings. Their replacement rules can be expressed as subdivisions of the triangular half-tiles described below (Frank 2008).

To see how the plane may be tiled aperiodically using the kite and dart, divide the kite into acute and obtuse tiles, shown above (Hurd).

Tile deflation replaces each triangular half-tile by smaller half-tiles. The rule illustrated above replaces an acute triangle by the union of two acute triangles and one obtuse triangle, and an obtuse triangle by an acute triangle and an obtuse triangle. The corresponding tile inflation rule enlarges the pattern before applying the subdivision, so the resulting tiles have the original sizes (Frank 2008).

When applied to a collection of tiles, tile deflation leads to a more refined collection. The rules do not respect tile boundaries, but do respect the half tiles defined above. There are two ways to obtain aperiodic tilings with 5-fold symmetry about a single point. They are known as the "star" and "sun" configurations, and are shown above (Hurd).

Higher order versions follow by repeated tile deflation. For example, the illustrations above depict the patterns after three applications of tile deflation (Hurd).

John Conway has asked if Penrose tilings are three colorable in such a way that adjacent tiles receive different colors. Sibley and Wagon (2000) proved that tilings by rhombs are three-colorable, and Babilon (2001) proved that tilings by kites and darts are three-colorable. McClure then found an algorithm that appears to three-color tilings by kites and darts, rhombs, and pentacles.

A three-dimensional analogue is the Ammann-Kramer-Neri tiling, whose tiles are the acute golden rhombohedron and obtuse golden rhombohedron (Dietl and Eschenburg 2017).