Crystallographic Tilings

TL;DR

This paper explores the classification of crystallographic tilings in Euclidean space by their automorphism groups, extending existing frameworks to include more general isometries and constructing specific tilings with given symmetry groups.

Contribution

It introduces an extended equivalence relation and metric on tiling spaces, enabling classification by automorphism groups and provides a method to construct tilings with specified crystallographic groups.

Findings

Extended the concept of mutual local derivability to include general isometries.

Developed new metrics on tiling spaces for better classification.

Constructed explicit examples of tilings with prescribed automorphism groups.

Abstract

Crystallographic tilings of the Euclidean space $\mathbb{E}^n$ are defined as simple tilings whose group of isometric automorphisms is crystallographic. To classify crystallographic tilings by their automorphism groups it is necessary to extend the standard equivalence relation of mutual local derivability to a version taking more general isometries than translations into account. This also requires the extension of the standard metrics on tiling spaces. Finally, a tiling with a given crystallographic group as automorphism group is constructed.