@MISC{Ueno01bycongruent, author = {Yukako Ueno and Yoshio Agaoka}, title = {by congruent triangles}, year = {2001} }

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Abstract

Abstract. We give a new classification of tilings of the 2-dimensional sphere by congruent triangles accompanied with a complete proof. This accomplishes the old classification by Davies, who only gave an outline of the proof, regrettably with some redundant tilings. We clarify Davies ’ obscure points, give a complete list, and show that there exist ten sporadic and also ten series of such tilings, including some unfamiliar twisted ones. We also give their figures, development maps in a way easy to understand their mutual relations. In Appendix, we give curious examples of tilings on noncompact spaces of constant positive curvature with boundary possessing a special 5valent vertex that never appear in the tiling of the usual sphere. In this paper, we give a complete classification of tilings of the 2dimensional 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.