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Báo cáo toán học: "Spherical F-Tilings by Triangles and r-Sided Regular Polygons, r ≥ 5"
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Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Spherical F-Tilings by Triangles and r-Sided Regular Polygons, r ≥ 5. | Spherical F-Tilings by Triangles and r-Sided Regular Polygons r 5 Catarina P. Avelino Altino F. Santos y Department of Mathematics UTAD 5001 - 801 Vila Real Portugal Submitted Dec 19 2007 Accepted Jan 24 2008 Published Feb 4 2008 Mathematics Subject Classification 52C20 52B05 20B35 Abstract The study of dihedral f-tilings of the sphere S2 by spherical triangles and equiangular spherical quadrangles which includes the case of 4-sided regular polygons was presented in 3 . Also in 6 the study of dihedral f-tilings of S2 whose prototiles are an equilateral triangle a 3-sided regular polygon and an isosceles triangle was described we believe that the analysis considering scalene triangles as the prototiles will lead to a wide family of f-tilings . In this paper we extend these results presenting the study of dihedral f-tilings by spherical triangles and r-sided regular polygons for any r 5. The combinatorial structure including the symmetry group of each tiling is given in Table 1. Keywords dihedral f-tilings isometric foldings spherical trigonometry 1 Introduction Let S2 be the Euclidean sphere of radius 1. By a dihedral folding tiling f-tiling for short of the sphere S2 whose prototiles are a spherical r-sided regular polygon Pr and a spherical triangle T we mean a polygonal subdivision T of S2 such that each cell tile of T is congruent to Pr or T and the vertices of T satisfy the angle-folding relation i.e. each vertex of T is of even valency 2n n 2 and the sums of alternate angles are equal that is n n T . 2i-i i 1 i 1 cavelino@utad.pt Research Unit CM-UTAD of University of Trás-os-Montes e Alto Douro. y afolgado@utad.pt Supported partially by the Research Unit Mathematics and Applications through the Foundation for Science and Technology FCT . THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R22 1 where the angles ữị around any vertex of T are ordered cyclically. In this paper we shall discuss dihedral f-tilings by spherical triangles and spherical r-sided regular .