Tilings of the sphere by congruent quadrilaterals III: edge combination $a^3b$ with general angles

TL;DR

This paper classifies edge-to-edge tilings of the sphere using congruent quadrilaterals of type $a^3b$ with some irrational angles, identifying continuous families and sporadic cases of such tilings.

Contribution

It completes the classification of sphere tilings by congruent quadrilaterals, focusing on the $a^3b$ case with irrational angles, including new families and sporadic solutions.

Findings

Identified 1-parameter families of tilings with earth map structure

Discovered 5 sporadic quadrilaterals with unique tilings

Provided a comprehensive summary of the full classification

Abstract

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This last one classifies the case of $a^3b$-quadrilaterals with some irrational angle: there are a sequence of $1$-parameter families of quadrilaterals admitting $2$-layer earth map tilings together with their basic flip modifications under extra condition, and $5$ sporadic quadrilaterals each admitting a special tiling. A summary of the full classification is presented in the end.