Semi-regular tilings of the hyperbolic plane

TL;DR

This paper investigates semi-regular tilings of the hyperbolic plane, establishing criteria for their existence and uniqueness based on vertex-type configurations, and discusses open questions in the field.

Contribution

It provides combinatorial criteria for the existence and uniqueness of semi-regular hyperbolic tilings with specified vertex-types.

Findings

Criteria for existence of tilings based on vertex-type

Conditions for uniqueness of tilings

Open questions on classification and properties

Abstract

A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.