Pinwheel tiling

In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations.

The Conway tessellation

Conway triangle decomposition into homothetic smaller triangles.

Let ${\displaystyle T}$ be the right triangle with side length ${\displaystyle 1}$, ${\displaystyle 2}$ and ${\displaystyle {\sqrt {5}}}$. Conway noticed that ${\displaystyle T}$ can be divided in five isometric copies of its image by the dilation of factor ${\displaystyle 1/{\sqrt {5}}}$.

The increasing sequence of triangles which defines the Conway tiling of the plane.

By suitably rescaling and translating/rotating, this operation can be iterated to obtain an infinite increasing sequence of growing triangles all made of isometric copies of ${\displaystyle T}$. The union of all these triangles yields a tiling of the whole plane by isometric copies of ${\displaystyle T}$.

In this tiling, isometric copies of ${\displaystyle T}$ appear in infinitely many orientations (this is due to the angles ${\displaystyle \arctan(1/2)}$ and ${\displaystyle \arctan(2)}$ of ${\displaystyle T}$, both non-commensurable with ${\displaystyle \pi }$). Despite this, all the vertices have rational coordinates.

The pinwheel tilings

A pinwheel tiling: tiles can be grouped in sets of five (thick lines) to form a new pinwheel tiling (up to rescaling)

Radin relied on the above construction of Conway to define pinwheel tilings. Formally, the pinwheel tilings are the tilings whose tiles are isometric copies of ${\displaystyle T}$, in which a tile may intersect another tile only either on a whole side or on half the length ${\displaystyle 2}$ side, and such that the following property holds. Given any pinwheel tiling ${\displaystyle P}$, there is a pinwheel tiling ${\displaystyle P'}$ which, once each tile is divided in five following the Conway construction and the result is dilated by a factor ${\displaystyle {\sqrt {5}}}$, is equal to ${\displaystyle P}$. In other words, the tiles of any pinwheel tilings can be grouped in sets of five into homothetic tiles, so that these homothetic tiles form (up to rescaling) a new pinwheel tiling.

The tiling constructed by Conway is a pinwheel tiling, but there are uncountably many other different pinwheel tiling, they are all locally undistinguishable (i.e., they have the same finite patches). They all share with the Conway tiling the property that tiles appear in infinitely many orientations (and vertices have rational coordinates).

The main result proven by Radin is that there is a finite (though very large) set of so-called prototiles, with each being obtained by coloring the sides of ${\displaystyle T}$, so that the pinwheel tilings are exactly the tilings of the plane by isometric copies of these prototiles, with the condition that whenever two copies intersect in a point, they have the same color in this point.[1] In terms of symbolic dynamics, this means that the pinwheel tilings form a sofic subshift.

Generalizations

Radin and Conway proposed a three-dimensional analogue which was dubbed the quaquaversal tiling.[2] There are other variants and generalizations of the original idea.[3]

Pinwheel fractal

One gets a fractal by iteratively dividing ${\displaystyle T}$ in five isometrics copies, following the Conway construction, and discarding the middle triangle (ad infinitum). This "pinwheel fractal" has Hausdorff dimension ${\displaystyle d={\frac {\ln 4}{\ln {\sqrt {5}}}}\approx 1.7227}$.