H. L. Davies, Packings of spherical triangles and tetrahedra, in Proceedings of the Colloquium on Convexity (ed. W. Fenchel), 42--51, Ko/benhavns Univ. Mat. Inst., Copenhagen, 1967.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....developed in [8] for the case of triangles. 1. Introduction This paper is a continuation of our previous paper [8] In the paper [8] we gave a new classification of tilings of the 2 dimensional sphere consisting of congruent triangles, and clarified some obscure points in Davies classification [1]. As our next problem, we consider monohedral tilings by quadrangles and pentagons. Especially its classification as we carried out for the case of triangles is an interesting and important problem. We can easily show that if the sphere is tiled by n gons, then we have n = 3, 4 or 5. See 3 ....

H. L. Davies, Packings of spherical triangles and tetrahedra, in Proceedings of the Colloquium on Convexity (ed. W. Fenchel), 42--51, Ko/benhavns Univ. Mat. Inst., Copenhagen, 1967.

....a restricted case, i.e. under the condition of regularity , meaning that the corners at each vertex have the same angle. 2000 Mathematics Subject Classification. Primary 52C20; Secondary 05B45, 51M20. Key words and phrases. Tiling, spherical triangle, regular polyhedron. Later, Davies [4] gave a classification without the assumption of regularity . But Davies only gave a rough outline of the proof of the classification in [4] and detailed examinations are left to the readers. It seems to the authors that to fill this blank space and reconstruct the complete proof is by no means ....

....Subject Classification. Primary 52C20; Secondary 05B45, 51M20. Key words and phrases. Tiling, spherical triangle, regular polyhedron. Later, Davies [4] gave a classification without the assumption of regularity . But Davies only gave a rough outline of the proof of the classification in [4], and detailed examinations are left to the readers. It seems to the authors that to fill this blank space and reconstruct the complete proof is by no means an easy (rather a quite hard) problem. In addition, Davies final classification contains some duplicates (though his list covers all tilings ....

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H. L. Davies, Packings of spherical triangles and tetrahedra, in Proceedings of the Colloquium on Convexity (ed. W. Fenchel), 42--51, Ko/benhavns Univ. Mat. Inst., Copenhagen, 1967.

....hierarchically subdivide the sphere into more than 120 equivalent domains. The spherical icosidodecahedron (120 equivalent triangles) is obtained by dividing each face of a spherical icosahedron (twenty triangles) into six congruent scalene spherical triangles by bisecting each angle [8, p. 90] [9], 40] Therefore, the construction of a hierarchical global data structure is in part an exercise in compromise. The cells at a given level will differ in size or shape or both. Perfect adjacency preservation is also unobtainable in the ideal. In practice, design decisions and compromises will ....

H. L. Davies. Packings of spherical triangles and tetrahedra. In W. Fenchel, editor, Proc. of the Colloquium on Convexity,

.... However, the ve regular polyedra octahedron, cube, tetrahedron, dodecahedron, icosahedron , which yield regular meshes by straightforward projection, have been known for a long time (see for example [2, 3] In the same way, sphere triangulations by congruent triangles have long been studied [5, 6, 8]. Some of edge to edge triangulations involving isosceles triangles can be used to yield partitions by congruent (hence equal area) quadrilaterals. We cannot use these partitions because, either the number of cells is limited (less than a few hundreds) or some vertices have an enormous degree ....

H.L. Davies. Packings of spherical triangles and tetrahedra. In Proc. Colloquium on Convexity, pages 42-51, Copenhagen, 1965.

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