A fractal version of the pinwheel tiling

TL;DR

This paper presents a novel fractal tiling based on the pinwheel pattern, featuring tiles with fractal boundaries and unique rotational properties, expanding the understanding of aperiodic tilings.

Contribution

It introduces a new fractal version of the pinwheel tiling with thirteen prototiles, enriching the class of aperiodic tilings with fractal boundaries and detailed symmetry analysis.

Findings

Tiles have fractal boundaries.

Tiling space is mutually locally derivable from the original pinwheel tiling.

Distinct rotational properties and symmetries are identified.

Abstract

We introduce a fractal version of the pinwheel substitution tiling. There are thirteen basic prototiles, all of which have fractal boundaries. These tiles, along with their reflections and rotations, create a tiling space which is mutually locally derivable from the pinwheel tiling space. Interesting rotational properties, symmetries, and relative tile frequency are discussed for the tiling space associated with the fractal pinwheel tiling.