![]() The triangles do not overlap except at the edges, half of them have their orientation reversed and they fit together to tile a neighborhood of the point. ![]() At each of the three vertices the triangle can be successively reflected through edges emanating from the vertices to produce 2 m copies of the triangle where the angle at the vertex is π/ m. The original triangle Δ gives a convex polygon P 1 with 3 vertices. The construction of a tessellation will first be carried out for the case when a, b and c are greater than 2. In the sphere there are three 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.Ī fundamental domain triangle ( p q r), with vertex angles π⁄ p, π⁄ q, and π⁄ r, can exist in different spaces depending on the value of the sum of the reciprocals of these integers:ġ p + 1 q + 1 r > 1 : Sphere 1 p + 1 q + 1 r = 1 : Euclidean plane 1 p + 1 q + 1 r 1 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. The value n⁄ d means the vertex angle is d⁄ n of the half-circle. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group.Ī Schwarz triangle is represented by three rational numbers ( p q r), each representing the angle at a vertex. ![]() These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere ( spherical tiling), possibly overlapping, through reflections in its edges. Spherical triangle that can be used to tile a sphere ![]()
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