An Algebraic Realization of the Taylor-Socolar Aperiodic Monotilings

TL;DR

This paper provides an algebraic framework for understanding the Taylor-Socolar aperiodic monotiling, simplifying its description by removing decorations and deriving the pattern from a single algebraic equation.

Contribution

It introduces decoration-free algebraic descriptions of the Taylor-Socolar monotiling and shows how its aperiodic pattern can be generated from one algebraic equation.

Findings

Decoration-free algebraic descriptions of the monotiling

Aperiodic pattern derived from a single algebraic equation

Unified algebraic framework for related monotilings

Abstract

The first aperiodic monotiling, introduced by Taylor, was based on a trapezoidal prototile equipped with 14 distinct decorations. A presentation of the closely related Taylor-Socolar aperiodic monotiling is based on a hexagonal prototile equipped with 7 decorations. This paper gives decoration-free algebraic descriptions equivalent to each of these presentations. It also shows how the monotilings and Taylor triangles pattern that characterizes the aperiodicity can be obtained from just one algebraic equation.