A single fractal pinwheel tile

TL;DR

This paper introduces a novel single-tile fractal tiling that retains the symmetry and spectral properties of the classic pinwheel tiling, simplifying the complex multi-tile constructions.

Contribution

It presents a new single-tile fractal tiling with self-similarity and circular symmetry, reducing complexity compared to previous multi-tile examples.

Findings

The new tile exhibits the same symmetry as the pinwheel tiling.

It has a fractal boundary with a simple geometric structure.

The spectral properties are preserved in the new tiling.

Abstract

The pinwheel triangle of Conway and Radin is a standard example for tilings with self-similarity and statistical circular symmetry. Many modifications were constructed, all based on partitions of triangles or rectangles. The fractal example of Frank and Whittaker requires 13 different types of tiles. We present an example of a single tile with fractal boundary and very simple geometric structure which has the same symmetry and spectral properties as the pinwheel triangle.