Combinatorial Tilings of the Sphere by Pentagons

TL;DR

This paper investigates the combinatorial tilings of the sphere using pentagons, aiming to classify all such tilings and exploring constraints on vertex degrees and tiling configurations.

Contribution

It provides new classifications of sphere tilings by pentagons, including the construction of earth map tilings under specific vertex degree conditions.

Findings

Tilings cannot have only one vertex of degree >3.

Construction of earth map tilings with vertices of degree >3 at least 4 apart.

Classification results under specific vertex degree constraints.

Abstract

A combinatorial tiling of the sphere is naturally given by an embedded graph. We study the case that each tile has exactly five edges, with the ultimate goal of classifying combinatorial tilings of the sphere by geometrically congruent pentagons. We show that the tiling cannot have only one vertex of degree >3. Moreover, we construct earth map tilings, which give classifications under the condition that vertices of degree >3 are at least of distance 4 apart, or under the condition that there are exactly two vertices of degree >3.