Spherical f-tilings by scalene triangles and isosceles trapezoids. III (Q2380242)

Summary: The study of the dihedral f-tilings of the sphere \(S^2\) whose prototiles are a scalene triangle and an isosceles trapezoid was initiated in [\textit{C. P. Avelino} and \textit{A. F. Santos}, Eur. J. Comb. 30, No. 5, 1221--1244 (2009; Zbl 1203.52015); Acta Math. Sin., Engl. Ser. 28, No. 5, 1013--1032 (2012; Zbl 1273.52025)]. In this paper we complete this classification presenting the study of all dihedral spherical f-tilings by scalene triangles and isosceles trapezoids in the remaining case of adjacency. A list containing all the f-tilings obtained in this paper is presented in Table 1. It is composed by isolated tilings as well as discrete and continuous families of tilings. The combinatorial structure is also achieved.