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

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 $a^2{b}^{2}c$, 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 $a^{3}bc$, such tilings are two unique double pentagonal subdivisions of the Platonic solids, with $48$ and $120$ tiles.

42 pages, 27 figures

References in corpus (3)

• Exhaustive search of convex pentagons which tile the plane
• On Deformed Dodecahedron Tiling
• Tilings of the Sphere by Congruent Pentagons II: Edge Combination $a^3{b}^2$