Spherical Tiling by 12 Congruent Pentagons

TL;DR

This paper classifies all edge-to-edge tilings of the sphere by 12 congruent pentagons, revealing a major class with two parameters and four isolated examples, expanding understanding beyond triangle tilings.

Contribution

It provides the first complete classification of spherical tilings by 12 congruent pentagons, including parametric and isolated cases.

Findings

One major class with two continuous parameters

Four isolated tiling examples

Complete classification of such pentagonal tilings

Abstract

The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this paper, we classify the simplest case, which is the edge-to-edge tilings of the 2-dimensional sphere by 12 congruent pentagons. We find one major class allowing two independent continuous parameters and four classes of isolated examples. The classification is done by first separately classifying the combinatorial, edge length, and angle aspects, and then combining the respective classifications together.