Tilings of the Sphere by Congruent Pentagons III: Edge Combination $a^5$

TL;DR

This paper classifies all possible edge-to-edge tilings of the sphere using congruent equilateral pentagons, identifying eight unique tilings across different categories and tile counts.

Contribution

It provides a complete enumeration of eight specific sphere tilings by congruent equilateral pentagons, expanding the understanding of pentagonal tilings on spherical surfaces.

Findings

Eight distinct tilings identified

Tilings include subdivision and earth map types

Specific tile counts for each tiling

Abstract

There are exactly eight edge-to-edge tilings of the sphere by congruent equilateral pentagons: three pentagonal subdivision tilings with 12, 24, 60 tiles; four earth map tilings with 16, 20, 24, 24 tiles; and one flip modification of the earth map tiling with 20 tiles.