BibTeX
@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.
Keyphrases
congruent triangle usual sphere ten series redundant tiling constant positive curvature complete list transitive group action davy obscure point unfamiliar twisted one curious example new classification old classification development map complete classification complete proof combinatorial problem 2-dimensional sphere noncompact space mutual relation
