Tilings of the Sphere by Congruent Quadrilaterals with Exactly Two Equal Edges

TL;DR

This paper classifies spherical tilings by congruent quadrilaterals with two equal edges, detailing their geometric parameters and symmetry groups, including earth map tilings and subdivisions of polyhedra.

Contribution

It provides a complete classification of such tilings, including their geometric constraints and symmetry groups, expanding understanding of spherical tiling configurations.

Findings

Classification of all such tilings including earth map and subdivisions

Determination of geometric ranges for edges and angles

Identification of symmetry groups of the tilings

Abstract

In this paper we give a classification of tilings of the sphere by congruent quadrilaterals with exactly two equal edges. The tilings are the earth map tilings, $(p,q)$-earth map tilings and their flip modifications, and quadrilateral subdivisions of the cube and the triangular prism. We described the ranges of values of the edges and angles for the tile to be geometrically realisable. The symmetry groups of the tilings are also determined.