Tilings of the sphere by congruent pentagons IV: Edge combination $a^4b$ with general angles

TL;DR

This paper classifies all edge-to-edge tilings of the sphere by congruent pentagons with edge pattern $a^4b$, revealing three families related to Platonic solids and earth map tilings, with detailed geometric data and counts.

Contribution

It provides a comprehensive classification of sphere tilings by congruent pentagons with edge combination $a^4b$, including parameter families and geometric data, extending previous work.

Findings

Three 1-parameter families of tilings related to Platonic solids

Sequences of non-symmetric earth map tilings with various rearrangements

Explicit counts of tilings for fixed pentagons and new quadrilateral tilings

Abstract

We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with any irrational angle in degree: they are three $1$-parameter families of pentagonal subdivisions of the Platonic solids, with $12, 24$ and $60$ tiles; and a sequence of $1$-parameter families of pentagons admitting non-symmetric $3$-layer earth map tilings together with their various rearrangements under extra conditions. Their parameter moduli and geometric data are all computed in both exact and numerical form. The total numbers of different tilings for any fixed such pentagon are counted explicitly. As a byproduct, the degenerate pentagons produce naturally many new non-edge-to-edge quadrilateral tilings. A sequel of this paper will handle $a^4b$-pentagons with all angles being rational in degree by solving some trigonometric Diophantine equations, to complete our full classification of edge-to-edge tilings of the sphere by congruent pentagons.