Edge-to-edge Tilings of the Sphere by Angle Congruent Pentagons

TL;DR

This work analyzes tilings of the sphere by angle-congruent pentagons through the anglewise vertex combination ($AVC$) framework, showing that angle data alone can determine many geometrically congruent tilings, including pentagonal subdivisions of Platonic solids and earth-map tilings. It introduces systematic reductions that collapse angle distinctions and connects those to 2D tilings and non-pentagonal subdivisions, revealing a rich combinatorial structure behind tilings. The paper provides complete classifications for several AVC families with two, three, and four distinct angles, and demonstrates both rigid and highly intricate tiling families arising from angle reductions, exemplifying a unifying combinatorial view of angle-congruent sphere tilings. These results illuminate how angle information governs tiling topology, enabling exhaustive enumeration and linking to classical polyhedral subdivisions and earth-map tilings, with implications for purely combinatorial interpretations of tiling theory.

Abstract

Congruent polygons are congruent in angles as well as in edge lengths. We concentrate on the angle aspect, and investigate how tilings of the sphere by congruent pentagons can be determined by the angle information only. We also investigate how the features of tilings are changed under reductions, i.e., by ignoring the difference among the angles.

Figures (34)

• Figure 1: Angle arrangements for $\alpha\beta\gamma\delta\epsilon$, and AADs for $|\delta|\delta|$.
• Figure 2: $N(\delta^3)$, and tilings for AVC(5A24) and AVC(5A60).
• Figure 3: No tiling for AVC(5A36).
• Figure 4: Pentagonal subdivision $PP_6$ of the cube.
• Figure 5: Time zone, and half earth map tiling.

Theorems & Definitions (26)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof