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The study of dihedral f-tilings of the sphere $S^2$ by spherical triangles and equiangular spherical quadrangles (which includes the case of 4-sided regular polygons) was presented by Breda and Santos [ Beitrage zur Algebra und Geometrie ,  45  (2004), 447–461].  Also, in a subsequent paper, the study of dihedral f-tilings of $S^2$ whose prototiles are an equilateral triangle (a 3-sided regular polygon) and an isosceles triangle was described (we believe that the analysis considering scalene triangles as the prototiles will lead to a wide family of f-tilings).  In this paper we extend these results, presenting the study of dihedral f-tilings by spherical triangles and $r$-sided regular polygons, for any $r \ge 5$. The combinatorial structure, including the symmetry group of each tiling, is given.

Spherical F-Tilings by Triangles and $r$-Sided Regular Polygons, $r \ge 5$