Möbius triangle

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 (Schwarz 1873).

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 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.

A 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:

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.

A Schwarz triangle is represented graphically by a triangular graph. Each node represents an edge (mirror) of the Schwarz triangle. Each edge is labeled by a rational value corresponding to the reflection order, being π/vertex angle.

Order-2 edges represent perpendicular mirrors that can be ignored in this diagram. The Coxeter-Dynkin diagram represents this triangular graph with order-2 edges hidden.

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