Abstract
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 parts I and II. 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. It is composed by isolated tilings as well as discrete and continuous families of tilings. The combinatorial structure is also achieved.
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