Tilings of the Sphere by Congruent Pentagons I: Edge Combinations $a^2b^2c$ and $a^3bc$

Abstract

We develop the basic tools for classifying edge-to-edge tilings of the sphere by congruent pentagons. Then we prove that, for the edge combination $a^2b^2c$, such tilings are three two-parameter families of pentagonal subdivisions of the Platonic solids, with $12$, $24$ and $60$ tiles. We also prove that, for the edge combination $a^3bc$, such tilings are two unique double pentagonal subdivisions of the Platonic solids, with $48$ and $120$ tiles.