Dihedral \(f\)-tilings of the sphere by equilateral and scalene triangles. II (Q1010818)

Summary: 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 [\textit{A. M. d'Azevedo Breda} and \textit{A. F. Santos}, Beitr. Alg. Geom. 45, No. 2, 447--461 (2004; Zbl 1084.52011)]. In a subsequent paper [\textit{C. P. Avelino} and \textit{A. F. Santos}, Electron. J. Comb. 15, No. 1, 21p. (2008; Zbl 1177.52010)], the study of all spherical \(f\)-tilings by triangles and \(r\)-sided regular polygons, for any \(r\geq 5\), was described. Later on, in [the authors, Ew. J. Comb. 30, No. 1, 119--132 (2009; Zbl 1161.52014)], the classification of all \(f\)-tilings of \(S^2\) whose prototiles are an equilateral triangle and an isosceles triangle is obtained. The algebraic and combinatorial description of spherical \(f\)-tilings by equilateral triangles and scalene triangles of angles \(\beta, \gamma\) and \(\delta(\beta>\gamma>\delta)\) whose edge adjacency is performed by the side opposite to \(\beta\) was done in [part I of this paper, the authors, same title, submitted for publication]. In this paper we extend these results considering the edge adjacency performed by the side opposite to \(\delta\).