S^2 coverings by isosceles and scalene triangles – adjacency case I

Abstract

The aim of this paper is the study and classification of spherical f-tilings by scalene triangles T and isosceles triangles T′. Due to the complexity of this wide class of tilings, we consider a subclass performed by the adjacency of the shortest side of T and the longest side of T′. It consists of seven families of f-tilings (four families with one discrete parameter and one continuous parameter, two families with one discrete parameter and one sporadic f-tiling). We also analyze the combinatorial structure of all these families of f-tilings, as well as the group of symmetries of each tiling and the transitivity classes of isohedrality and isogonality.