Tiling the Sphere with Regular Polygons

TL;DR

This work delivers a complete classification of spherical tilings by regular polygons under a unified framework that does not rely on the Johnson–Zalgaller solids classification or on convexity/polyhedrality assumptions. It combines trigonometric, algebraic, and combinatorial methods to enumerate all edge-to-edge tilings, revealing correspondences with Platonic and Archimedean families, circumscribable Johnson solids, and infinite prism/antiprism/hosohedron/dihedra families. The authors also introduce and leverage subdivision and shrinking operations to generate and relate tilings, and provide interactive models and data to illuminate the structure. The results advance understanding of spherical tilings and have potential applications in materials science and chemistry where such geometric configurations model molecular frameworks.

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.

Figures (21)

• Figure 1: Platonic
• Figure 2: Archimedean
• Figure 3: Circumscribable Johnson solids
• Figure 4: Prisms and antiprisms
• Figure 5: Hosohedra and dihedra

Theorems & Definitions (49)

• Theorem
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5