Tilings of the Sphere by Congruent Quadrilaterals or Triangles

TL;DR

This paper provides a complete classification of edge-to-edge tilings of the sphere using congruent quadrilaterals and revisits the classification for triangles, contributing to the broader understanding of spherical tilings by congruent polygons.

Contribution

It offers a comprehensive classification of spherical tilings by congruent quadrilaterals and updates the classification for triangles, completing the series on congruent polygon tilings.

Findings

Complete classification of quadrilateral tilings on the sphere.

Modernized classification of triangular tilings.

Part of a series completing all congruent polygon tilings.

Abstract

We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles. Together with our series of papers that classifies edge-to-edge tilings of the sphere by congruent pentagons, we complete the classification of edge-to-edge tilings of the sphere by congruent polygons.