Schwarz 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 . 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 /"p", /"q", and /"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 , while on a sphere they sum to an angle greater than , 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. A Coxeter group can be used for a simpler notation, as ("p" "q" "r") for cyclic graphs, and ("p" "q" 2) = ["p","q"] for (right triangles), and ("p" 2 2) = ["p"]×[]. Schwarz triangles with whole numbers, also called Möbius triangles, include one 1-parameter family and three exceptional cases:   The Schwarz triangles ("p" "q" "r"), grouped by density:  Density 1:  Density 2:   Density ∞:   Density 1:   Density 2:   Density 3:   Density 4:   Density 6:   Density 10:   The (2 3 7) Schwarz triangle is the smallest hyperbolic Schwarz triangle, and as such is of particular interest.