|
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges. They were classified in . These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group. A Schwarz triangle is represented by three rational numbers (''p'' ''q'' ''r'') each representing the angle at a vertex. The value ''n/d'' means the vertex angle is ''d''/''n'' of the half-circle. "2" means a right triangle. When these are whole numbers, the triangle is called a Möbius triangle, and corresponds to a ''non''-overlapping tiling, and the symmetry group is called a triangle group. In the sphere there are 3 Möbius triangles plus one one-parameter family; in the plane there are three Möbius triangles, while in hyperbolic space there is a three-parameter family of Möbius triangles, and no exceptional objects. == Solution space == A fundamental domain triangle, (''p'' ''q'' ''r''), can exist in different spaces depending on the value of the sum of the reciprocals of these integers: : This is simply a way of saying that in Euclidean space the interior angles of a triangle sum to π, while on a sphere they sum to an angle greater than π, and on hyperbolic space they sum to less. 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「Schwarz triangle」の詳細全文を読む スポンサード リンク
|