Cyclotomic Aperiodic Substitution Tilings

TL;DR

This paper introduces Cyclotomic Aperiodic Substitution Tilings (CAST), a new class of aperiodic tilings supported on cyclotomic fields, with applications in identifying symmetric tilings with dihedral symmetry.

Contribution

It defines CASTs, explores their substitution matrices and inflation multipliers, and demonstrates their use in identifying tilings with specific dihedral symmetries.

Findings

CASTs encompass many known aperiodic tilings.

They can be used to generate tilings with infinite patches of specific symmetries.

Practical methods for symmetry identification in tilings are provided.

Abstract

The class of Cyclotomic Aperiodic Substitution Tilings (CAST) is introduced. Its vertices are supported on the 2n-th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmetry Dn or D2n, i.e. the tiling contains an infinite number of patches of any size with dihedral symmetry Dn or D2n only by iteration of substitution rules on a single tile.