Rational Angles and Tilings of the Sphere by Congruent Quadrilaterals

TL;DR

This paper uses Diophantine analysis to systematically classify sphere tilings by congruent almost equilateral quadrilaterals, providing detailed geometric data and extending previous classifications.

Contribution

It introduces a more systematic method for classifying spherical tilings by congruent quadrilaterals, applicable to related problems, building on prior work.

Findings

Complete classification of tilings by congruent almost equilateral quadrilaterals

Detailed geometric data for each tiling

Method applicable to other tiling classification problems

Abstract

We apply Diophantine analysis to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination a3b). Parallel to a complete classification by Cheung, Luk and Yan, the method implemented here is more systematic and applicable to other related tiling problems. We also provide detailed geometric data for the tilings.