Tilings of the sphere by congruent pentagons I: Edge combinations a2b2c and a3bc

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 a2b2c, 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 a3bc, such tilings are two unique double pentagonal subdivisions of the Platonic solids, with 48 and 120 tiles.