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In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere, or the hyperbolic plane by congruent triangles, a fundamental domain for the action, called a Möbius triangle. ==Definition== Let ''l'', ''m'', ''n'' be integers greater than or equal to 2. A triangle group Δ(''l'',''m'',''n'') is a group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/''l'', π/''m'' and π/''n'' (measured in radians). The product of the reflections in two adjacent sides is a rotation by the angle which is twice the angle between those sides, 2π/''l'', 2π/''m'' and 2π/''n'' Therefore, if the generating reflections are labeled ''a'', ''b'', ''c'' and the angles between them in the cyclic order are as given above, then the following relations hold: # # It is a theorem that all other relations between ''a, b, c'' are consequences of these relations and that Δ(''l,m,n'') is a discrete group of motions of the corresponding space. Thus a triangle group is a reflection group that admits a group presentation : An abstract group with this presentation is a Coxeter group with three generators. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「triangle group」の詳細全文を読む スポンサード リンク
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