D. M. Y. Sommerville, Division of space by congruent triangles and tetrahedra, Proc. Royal Soc. Edinburgh 43 (1922--3), 85--116.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....d must be situated adjacently in one quadrangle, which is a contradiction. Therefore, we conclude that there does not exist a monohedral tiling of the sphere by quadrangles of type (v) q.e.d. Note that monohedral tilings by quadrangles of type (i) rhombus) are already classified in Sommerville [6] and Ueno Agaoka [8] A classification of tilings by quadrangles of type (iii) can be also obtained by our previous result [8] because by drawing one diagonal line in each quadrangle, we obtain a monohedral tiling of the sphere by triangles. a b a # # # # # # # # # # # # # # # # # # # # # # b ....

D. M. Y. Sommerville, Division of space by congruent triangles and tetrahedra, Proc. Royal Soc. Edinburgh 43 (1922--3), 85--116.

....Introduction In this paper, we give a complete classification of tilings of the 2 dimensional sphere consisting of one congruent triangle. We consider this problem as a purely combinatorial problem, not assuming a transitive group action on the set of tiles. Concerning this problem, Sommerville [9] gave a partial classification, particularly he classified tilings by isosceles triangles. But for the scalene case, he only treated 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. ....

....The rest sections are devoted to the proof of Theorem 1. After treating a preliminary case (equilateral triangles and the case where the number of faces takes the smallest value four) in 3, we give a classification by isosceles triangles in 4. This result was already proved by Sommerville [9]. But we give here a complete proof because Sommerville [9; p.90] stated only a brief outline of the proof. The remaining scalene case is the most Table V E #, #, # type of vertices [number] F 4 4 6 # # # = 2, #, #, # 1 # # # [4] 3 3# [4] 6# [4] 3# [2] 2# 2# [2] ....

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D. M. Y. Sommerville, Division of space by congruent triangles and tetrahedra, Proc. Royal Soc. Edinburgh 43 (1922--3), 85--116.

.... 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 ....

D.M.Y. Sommerville. Division of space by congruent triangles and tetrahedra. In Proc. Roy. Soc. Edinburgh, pages 85-116, 1923. 231 Session C6.2 12th Canadian Conference on Computational Geometry

....around an axis of direction (1; 1; 1) without changing the linear components of the symmetries, thus the convex realizations are the same as for TT1 0 . In [Sen81] the derivation of space filling convex tetrahedra by splitting certain classes of triangular prisms, as worked out by Sommerville [Som23a, Som23b] is discussed. The author states that all known such tetrahedra can be obtained from Sommerville s four families of prisms. These, as is easy to see, lead to just two families of space filling tetrahedra, which in fact correspond to our types TT1 and TT1 0 . Moreover, it turns out ....

D.M.Y. Sommerville. Division of space by congruent triangles and tetrahedra. Proc. Royal Society of Edinburgh, 43:85--116, 1923.

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