Rendiconti del Seminario Matematico della Università di Padova
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Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles-IA.M. d'Azevedo Breda, Patrícia S. Ribeiro and Altino F. Santos (1) Departamento de Matemática, Universidade de Aveiro, 3810-193, Aveiro, Portugal
(2) Departamento de Matemática, E.S.T. Setúbal, 2910-761, Setúbal, Portugal
(3) Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, Quinta de Prados, Apartado 1013, 5001-801, Vila Real, Portugal
The study of dihedral f-tilings of the Euclidean sphere S2 by triangles and r-sided regular polygons was initiated in 2004 where the case r = 4 was considered . In a subsequent paper , the study of all spherical f-tilings by triangles and r-sided regular polygons, for any r ≥ 5, was described. Later on, in  the classfication of all f-tilings of S2 whose prototiles are an equilateral triangle (a 3-sided regular polygon) and an isosceles triangle are obtained. In this paper we extend these results presenting the study of all spherical f-tilings by equilateral triangles and scalene triangles in a particular way of adjacency. For this particular case of adjacency, we get two continuous families of f-tilings denoted by F1δ and F2δ. The combinatorial structure including the transitive properties under the action of their symmetry groups is given in Table 1.
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d'Azevedo Breda A.M., Ribeiro Patrícia, Santos Altino: Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles-I. Rend. Sem. Mat. Univ. Padova 124 (2010), 1-23. doi: 10.4171/RSMUP/124-1