Tiling the Sphere with Regular Polygons

2025/12/05 by Hoi Ping Luk, Luk, Hoi Ping, Roman Nedela +3
Materials Science · Mathematics · #05B45 #51M10 #51M20 #52B10 #52C20 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Metric Geometry (math.MG) #Quasicrystal Structures and Properties #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2512.05577

openalex publication_date 2025/12/05 · openalex created_date 2025/12/09 · openalex updated_date 2026/07/28

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.

Citations

• Tilings of the Sphere by Congruent Quadrilaterals or Triangles
• Areas of spherical polyhedral surfaces with regular faces

Related

• The Rest of the Tilings of the Sphere by Regular Polygons
• Tilings of the Sphere by Congruent Quadrilaterals or Triangles
• Dihedral Tilings of the Sphere by Regular Polygons and Quadrilaterals I: Squares and Rhombi
• Dihedral Tilings of the Sphere by Regular Polygons and Quadrilaterals II: Regular Polygons with High Gonality and Rhombi
• Dihedral Tilings of the Sphere by Regular Polygons and Quadrilaterals I: Quadrilaterals with Equal Opposite Edges