Tilings of the sphere by congruent pentagons V: Edge combination $a^{4}b$ with rational angles

TL;DR

This paper classifies all edge-to-edge tilings of the sphere by congruent pentagons with a specific edge pattern and rational angles, revealing a family of symmetric subdivisions, unique tilings, and their modifications.

Contribution

It provides a complete classification of sphere tilings by congruent pentagons with edge combination a^4b and rational angles, including symmetric and non-symmetric cases.

Findings

A one-parameter family of symmetric subdivisions of the tetrahedron with 12 tiles.

Existence of symmetric 3-layer earth map tilings for all m ≥ 4.

Identification of a unique non-symmetric degenerate pentagon tiling with 20 tiles.

Abstract

We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with rational angles in degree: they are a one-parameter family of symmetric $a^4b$-pentagonal subdivisions of the tetrahedron with $12$ tiles; a sequence of unique symmetric $a^4b$-pentagons admitting a symmetric $3$-layer earth map tiling by $4m$ tiles for any $m\ge4$, among which each odd $m$ case admits two standard flip modifications; and a unique non-symmetric and degenerate $a^4b$-pentagon admitting a non-symmetric $3$-layer earth map tiling and its standard flip modification with $20$ tiles. The full classification from this series and all induced non-edge-to-edge quadrilateral tilings from degenerate pentagons are summarized with their 3D pictures.