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Geometric and combinatorial structure of a class of spherical folding tessellations. I. (English) Zbl 06819562
Summary: A classification of dihedral folding tessellations of the sphere whose prototiles are a kite and an equilateral or isosceles triangle was obtained in recent four papers by C. P. Avelino and A. F. Santos [Australas. J. Comb. 53, 109–125 (2012; Zbl 1255.05043); “Spherical folding tesselations by kites and isosceles triangles II”, Int. J. Pure Appl. Math. 85, No. 1, 45–67 (2013); Math. Commun. 19, No. 1, 1–28 (2014; Zbl 1298.52025); Ars Math. Contemp. 11, No. 1, 59–78 (2016; Zbl 1354.52022)]. In this paper we extend this classification, presenting all dihedral folding tessellations of the sphere by kites and scalene triangles in which the shorter side of the kite is equal to the longest side of the triangle. Within two possible cases of adjacency, only one will be addressed. The combinatorial structure of each tiling is also analysed.

##### MSC:
 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry) 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 20B35 Subgroups of symmetric groups
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