Pinwheel tiling

From Wikipedia, the free encyclopedia

A Pinwheel tiling
A Pinwheel tiling

The pinwheel tiling is an aperiodic tiling proposed by John Conway and Charles Radin.

It is constructed with a right triangle which appears in infinitely many orientations. This is its most remarkable feature, which was expressly sought by Radin. The first example with this property was proposed by Filipo Cesi, who used four tiles (two squares with incommensurate sides, a rectangle, and a triangle).[1] Conway proposed a solution using just one triangular prototile with dimensions 1,2, \sqrt 5. If tile flipping is not allowed there should be right-handed and left-handed versions of the shape. The tiles do not match only edge-to-edge, but vertex-to-edge configurations occur. The full set of matching rules[2] is rather complicated, so the standard method to construct the tiling relies on substitution.

The pinwheel substitution
The pinwheel substitution

The figure shows how a single tile is recomposed from five smaller tiles. Their type, left (L) or right (R), is indicated in subscripts.

Radin introduced the notion of statistical symmetry to describe the distribution of tile orientations. For a domino tile there are just two possible orientations, in a Penrose tiling they are ten, and in the pinwheel they are an infinite set. This happens when the basic triangle has an angle which is not a rational fraction of π, e.g. arctan(2). The tiling is not a quasicrystal and it cannot be obtained as a projection from a simple higher dimensional lattice. However, all the vertices have rational coordinates. Being obtained from substitutions, the pinwheel tiling can also be seen as a fractal. If at each iteration step the middle triangle is discarded, a fractal object with Hausdorff dimension

d = \frac{\ln 4}{\ln \sqrt 5}\approx 1.7227

is obtained.

Radin and Conway proposed a three dimensional analogue which was dubbed the quaquaversal tiling.[3] There are other variants and generalizations of the original idea.[4]

[edit] References

  1. ^ Radin, C., Aperiodic tilings, ergodic theory and rotations, in The Mathematics of long-range aperiodic order, ed. by V.Moody, NATO ASI series vol.489 (1997) pp.499–519
  2. ^ Radin, C. (May 1994). "The Pinwheel Tilings of the Plane" (PDF/PostScript). Annals of Mathematics 139 (3): pp.661–702. doi:10.2307/2118575. 
  3. ^ Radin, C., Conway, J., Quaquaversal tiling and rotations, preprint, Princeton University Press, 1995
  4. ^ Sadun, L. (January 1998). "Some Generalizations of the Pinwheel Tiling" (PDF/PostScript). Discrete and Computational Geometry 20 (1): pp.79–110. 

[edit] External links