Side-to-side Tiling of the Sphere by Congruent Curvilinear Triangles

TL;DR

This paper classifies the curvilinear triangular tilings of the sphere, extending the known classification of straight-edged tilings by identifying their curvilinear counterparts as modifications of the straight tilings.

Contribution

It provides the first classification of sphere tilings by congruent curvilinear triangles, expanding the understanding of spherical tilings beyond straight-edged polygons.

Findings

Curvilinear triangular tilings are modifications of straight-edged tilings.

All such tilings can be classified based on known straight tilings.

The classification completes the understanding of edge-to-edge spherical tilings by congruent polygons.

Abstract

The edge-to-edge tilings of the sphere by congruent polygons, where all edges are straight, have been completely classified. We classify the curvilinear version of the similar triangular tilings, where the edges may not be straight, and find that these are the modifications of the straight triangular tilings.