Pinwheel Tiling

Visualization

@
$dim=2
s=$companion([1,0])
r=[-1,0,0,1]
g=2+s^3
h0=s^3*r*[2,0]
h1=[0,1]
h2=s^2*[2,-2]
h3=s^2*r*[0,-1]
h4=s^2*r*[-2,-2]
A=g^-1*(h0|h1|h2|h3|h4)*A

Overview

The Pinwheel Tiling was discovered by John Conway and formalized by Charles Radin (1994). Its prototile is the right triangle with legs 1 and 2 (hypotenuse $\sqrt{5}$), which decomposes into five congruent smaller copies scaled by $1/\sqrt{5}$ — an integer Moran ratio $5 \cdot (1/\sqrt{5})^2 = 1$.

The defining feature of the pinwheel tiling is rotational ergodicity: every orientation of the tile appears in the tiling with equal statistical frequency. No translational symmetry exists. This puts the pinwheel in a class of its own among aperiodic substitution tilings.

The same algebraic framework ($g = 2-i$ in $\mathbb{Z}[i]$, five maps contracting by $1/\sqrt{5}$) also supports two fractal single-tile variants with a fractal boundary, discovered by Bandt and Mekhontsev via computer search (2019).

Classical Pinwheel (Radin, 1994)

The classical tile is the solid 1-2-$\sqrt{5}$ right triangle. The five maps partition the triangle exactly — their images cover the whole tile with pairwise intersections of measure zero. The IFS attractor is therefore a solid triangular region.

The tiling is non-periodic: the group of symmetries of the tiling contains no translations or rotations. Yet Radin proved that the frequency of tiles in each orientation is uniform over the full circle — every angle appears equally often. This “statistical” rotational symmetry is the pinwheel’s most remarkable property.

Conway and Radin extended the construction to three dimensions: the quaquaversal tiling uses a single tetrahedral tile such that every 3D orientation appears equally.

Fractal Single-Tile Pinwheel (Bandt & Mekhontsev)

An exhaustive computer search (Mekhontsev, 2019) over all five-map IFS with expansion $g = 2-i$ and maps drawn from the group generated by $s$ and $r$ revealed that exactly two substitutions produce a non-periodic tiling with dense tile orientations using a single prototile with a fractal boundary. Both are shown below.

Like the classical tile, each fractal variant is a substitution tile: the five images of the attractor $A$ under the maps $h_0,\ldots,h_4$ cover $A$ exactly, with pairwise intersections of measure zero. The attractor is a compact, connected, simply-connected set whose boundary has Hausdorff dimension strictly between 1 and 2 — in contrast to the straight edges of the classical triangle.

The existence of such a tile was not previously known; Frank and Whittaker (2011) had constructed a fractal version of the pinwheel tiling using 13 prototiles via a theoretical approach, but the one-tile version required a computer search.

Algebraic Structure

Both variants share the same algebraic object. In the Gaussian integers $\mathbb{Z}[i]$, the expansion is:

In matrix form (using $s = \bigl[\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\bigr]$ for $90°$ rotation and $r = \bigl[\begin{smallmatrix}-1&0\\0&1\end{smallmatrix}\bigr]$ for $y$-axis reflection):

The characteristic polynomial is $x^2 - 4x + 5 = 0$, with roots $2 \pm i$.

All three tiles satisfy $A = g^{-1}(h_0 \cup \cdots \cup h_4)(A)$ with the same $g$, $s$, $r$, but different translation vectors:

The maps $h_k$ are isometries from $\langle s, r\rangle$ composed with integer translations, with $g = 2+s^3$.

References

• Radin, C. (1994). The pinwheel tilings of the plane. Annals of Mathematics, 139(3), 661–702.
• Conway, J. H. & Radin, C. (1998). Quaquaversal tilings and rotations. Inventiones Mathematicae, 132(1), 179–188.
• Bandt, C., Mekhontsev, D. & Tetenov, A. (2018). A single fractal pinwheel tile. Proceedings of the American Mathematical Society, 146(3), 1271–1285.
• Frank, N. P. & Whittaker, M. F. (2011). A fractal version of the pinwheel tiling. The Mathematical Intelligencer, 33(2), 7–17.
• Pinwheel tiling — Wikipedia
• Pinwheel — Tilings Encyclopedia
• Fractal Single-Tile Pinwheel — Tilings Encyclopedia
• Mekhontsev, D. (2019). An algebraic framework for finding and analyzing self-affine tiles and fractals.

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