Penrose Tiling - Features and Constructions - The Golden Ratio and Local Pentagonal Symmetry

The Golden Ratio and Local Pentagonal Symmetry

Several properties and common features of the Penrose tilings involve the golden ratio φ = (1+√5)/2 (approximately 1.618). This is the ratio of chord lengths to side lengths in a regular pentagon, and satisfies φ = 1 + 1/φ.

Consequently, the ratio of the lengths of long sides to short sides in the (isosceles) Robinson triangles is φ:1. It follows that the ratio of long side lengths to short in both kite and dart tiles is also φ:1, as are the length ratios of sides to the short diagonal in the thin rhomb t, and of long diagonal to sides in the thick rhomb T. In both the P2 and P3 tilings, the ratio of the area of the larger Robinson triangle to the smaller one is φ:1, hence so are the ratios of the areas of the kite to the dart, and of the thick rhomb to the thin rhomb. (Both larger and smaller obtuse Robinson triangles can be found in the pentagon on the right: the larger triangles at the top – the halves of the thick rhomb – have linear dimensions scaled up by φ compared to the small shaded triangle at the base, and so the ratio of areas is φ2:1.)

Any Penrose tiling has local pentagonal symmetry, in the sense that there are points in the tiling surrounded by a symmetric configuration of tiles: such configurations have fivefold rotational symmetry about the center point, as well as five mirror lines of reflection symmetry passing through the point, a dihedral symmetry group. This symmetry will generally preserve only a patch of tiles around the center point, but the patch can be very large: Conway and Penrose proved that whenever the colored curves on the P2 or P3 tilings close in a loop, the region within the loop has pentagonal symmetry, and furthermore, in any tiling, there are at most two such curves of each color that do not close up.

There can be at most one center point of global fivefold symmetry: if there were more than one, then rotating each about the other would yield two closer centers of fivefold symmetry, which leads to a mathematical contradiction. There are only two Penrose tilings (of each type) with global pentagonal symmetry: for the P2 tiling by kites and darts, the center point is either a "sun" or "star" vertex.