Tilings of the Sphere by Congruent Pentagons IV: Edge Combination $a^4b$

TL;DR

This paper completes the classification of all edge-to-edge tilings of the sphere by congruent polygons, focusing on pentagons with four equal edges, expanding the understanding of spherical tilings.

Contribution

It provides a complete classification of sphere tilings by congruent pentagons with four equal edges, filling gaps from previous classifications of other polygon types.

Findings

Classified tilings with edge combination a^4b

Extended previous classifications to include this pentagon type

Contributed to the complete taxonomy of spherical tilings by congruent polygons

Abstract

We classify edge-to-edge tilings of the sphere by congruent almost equilateral pentagons, in which four edges have the same length. Together with our earlier classifications of edge-to-edge tilings of the sphere by congruent equilateral pentagons of other types, and our classification of edge-to-edge tilings of the sphere by congruent quadrilaterals or triangles, we complete the classification of edge-to-edge tilings of the sphere by congruent polygons.