Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles - II. | Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles - II A. M. d Azevedo Breda Department of Mathematics University of Aveiro 3810-193 Aveiro Portugal ambreda@ Patricia S. Ribeiro Department of Mathematics . Setúbal 2910-761 Setubal Portugal pribeiro@ Altino F. Santos Department of Mathematics . 5001-801 Vila Real Portugal afolgado@ Submitted Jun 25 2008 Accepted Jul 2 2008 Published Jul 14 2008 Mathematics Subject Classifications 52C20 52B05 20B35 Abstract 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 5 . In a subsequent paper 1 the study of all spherical f-tilings by triangles and r-sided regular polygons for any r 5 was described. Later on in 3 the classification of all f-tilings of S2 whose prototiles are an equilateral triangle and an isosceles triangle is obtained. The algebraic and combinatorial description of spherical f-tilings by equilateral triangles and scalene triangles of angles p 7 and Ỗ p 7 Ỗ whose edge adjacency is performed by the side opposite to p was done in 4 . In this paper we extend these results considering the edge adjacency performed by the side opposite to Ỗ. Keywords dihedral f-tilings combinatorial properties symmetry groups Supported partially by the Research Unit Mathematics and Applications of University of Aveiro through the Foundation for Science and Technology FCT . y Research Unit CM-UTAD of University of Trás-os-Montes e Alto Douro. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R91 1 1 Introduction Spherical folding tilings or f-tilings for short are edge-to-edge decompositions of the sphere by geodesic polygons such that all vertices are of even valency and the sum of alternate angles around each vertex is X. A f-tiling T is said to be monohedral if it is composed by congruent cells and dihedral if every tile of T is congruent to one of two fixed sets X and Y .
