Tiling the Euclidean and Hyperbolic planes with ribbons

TL;DR

This paper introduces a topological method for classifying Euclidean and hyperbolic plane tilings with ribbons, extending combinatorial tiling theory and considering symmetries and orbifold decorations.

Contribution

It develops a purely topological classification approach for crystallographic tilings, generalizing Delaney-Dress theory and applicable to non-closed disk topologies.

Findings

Classification of tilings up to equivariant equivalence

Enumeration techniques for such tilings

Extension of tiling theory to hyperbolic and Euclidean planes

Abstract

We describe a method to classify crystallographic tilings of the Euclidean and hyperbolic planes by tiles whose stabiliser group contains translation isometries or whose topology is not that of a closed disk. We tackle this problem from two different viewpoints, one with constructive techniques to enumerate such tilings and the other from a viewpoint of classification. The methods are purely topological and generalise Delaney-Dress combinatorial tiling theory. The classification is up to equivariant equivalence and is achieved by viewing tilings as decorations of orbifolds.