Cyclotomic Aperiodic Substitution Tilings

Stefan Pautze

doi:10.3390/sym9020019

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.

Highlights

Tilings have been the subject of wide research
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
Cyclotomic Aperiodic Substitution Tilings (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

Summary

Introduction

Tilings have been the subject of wide research. Many of their properties are investigated and discussed in view of their application in physics and chemistry, e.g., in detail in the research of crystals and quasicrystals, such as D. It seems its first consequent application to tile the whole Euclidean plane aperiodically appeared with the Penrose tiling [2,3,27,28,29] Among those methods, substitution rules may be the easiest approach to construct aperiodic tilings. Substitution rules may be the easiest approach to construct aperiodic tilings They have some other advantages: The inflation multipliers of tilings obtained by the cut-and-project scheme are limited to PV-numbers. 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. CASTs with minimal or at least small inflation multiplier are presented in Sections 3 and 4, which includes a generalization of the Lançon-Billard tiling.

Properties of Cyclotomic Aperiodic Substitution Tilings

The Odd n Case

The Even n Case

Rhombic CASTs with Symmetric Edges and Substitution Rules

Gaps to Prototiles Algorithm

Summary and Outlook