Tilings of the Sphere by Congruent Pentagons I

TL;DR

This paper proves that the only edge-to-edge tiling of the sphere by congruent pentagons is the dodecahedron, under specific vertex and edge length conditions, establishing a classification result in spherical tilings.

Contribution

It demonstrates the non-existence of additional tilings beyond the dodecahedron for certain pentagon configurations on the sphere.

Findings

No other edge-to-edge tilings exist beyond the dodecahedron under given conditions.

The specific vertex degree and edge length constraints are critical for the classification.

The result narrows the possibilities for spherical pentagonal tilings with congruence constraints.

Abstract

We show that there are no edge-to-edge tilings of the sphere by congruent pentagons beyond the minimal dodecahedron tiling, such that there is a tile with all vertices having degree 3 and the edge length combinations are three of the five possibilities.