Tiling the Sphere with Regular Polygons

TL;DR

This paper provides a comprehensive classification of how regular polygons can tile a sphere edge-to-edge, using a novel, unified approach that does not rely on previous complex classifications.

Contribution

It introduces a new, unified framework for classifying spherical tilings by regular polygons without assuming convexity or polyhedrality.

Findings

Complete classification of spherical tilings by regular polygons

New combinatorial and algebraic methods for tiling analysis

Independent proof not relying on Johnson-Zalgaller classification

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.