Spherical f-tilings by two non congruent classes of isosceles triangles - I

The theory of f-tilings is related to the theory of isometric foldings, initiated by S. Robertson [8] in 1977. The study of dihedral f-tilings of the Euclidean sphere $S^2$ by triangles and r-sided regular polygons was initiated in 2004, where the case r=4$was considered [4]. In a subsequent paper [1], the study of all spherical f-tilings by triangles and r-sided regular polygons, for any$rge 5$, was described. Recently, in [2] and [3] a classification of all triangular dihedral spherical f-tilings for which one of the prototiles is an equilateral triangle is given. In this paper, we extend these results considering the dihedral case of two non congruent isosceles triangles in a particular way of adjacency ending up to a class of$f$-tilings composed by three parametrised families, denoted by$mathcal{F}_{k, alpha}$,$mathcal{E}_{alpha}$and$mathcal{L}_{k}, ; kgeq 3, ; alpha>frac{pi}{2}$, respectively, and one isolated tiling, denoted by$mathcal{G}$. The combinatorial structure including the symmetry group of each tiling is also given. Dawson and Doyle in [6], [7] have also been working on spherical tilings, relaxing the edge to edge condition