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

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.