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Báo cáo toán hoc:" Spherical f-Tilings by Scalene Triangles and Isosceles Trapezoids III "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Spherical f-Tilings by Scalene Triangles and Isosceles Trapezoids III. | Spherical f-Tilings by Scalene Triangles and Isosceles Trapezoids III Catarina P. Avelino Altino F. Santos t Department of Mathematics UTAD 5001 - 801 Vila Real Portugal Submitted Jun 23 2009 Accepted Jul 13 2009 Published Jul 24 2009 Mathematics Subject Classification 52C20 52B05 20B35 Abstract The study of the dihedral f-tilings of the sphere S2 whose prototiles are a scalene triangle and an isosceles trapezoid was initiated in 7 8 . 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. Keywords dihedral f-tilings combinatorial properties 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 isosceles trapezoid Q and a spherical triangle T we mean a polygonal subdivision T of S2 such that each cell tile of T is congruent to Q 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 @2i 2Í-1 n i 1 i 1 where the angles di around any vertex of T are ordered cyclically. cavelino@ut ad.pt 1 afolgado@ut ad.pt Supported partially by the Research Unit CM-UTAD of University of Tras-os-Montes e Alto Douro through the Foundation for Science and Technology FCT . THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R87 1 Folding tilings are intrinsically related to the theory of isometric foldings on Rie-mannian manifolds. In fact the set of singularities of any spherical isometric folding corresponds to a folding tiling of the sphere see 9 for the foundations of this subject. The study of dihedral .