Tiling the Sphere with Regular Polygons

Abstract

We give a complete classification of edge-to-edge tilings of the sphere by regular polygons under a unified framework. Without assuming convexity of the tiles or polyhedrality of the underlying graph, our proof is independent of the Johnson-Zalgaller classification of solids with regular faces (1967), which took over 200 pages. We apply a blend of trigonometric, algebraic and combinatorial tools of independent interest.