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A Coxeter decomposition of a polygon is a decomposition into a finite number of polygons in which any two sharing a side are reflections of each other along that side. Hyperbolic polygons are the analogues of Euclidean polygons in hyperbolic geometry. A hyperbolic n-gon is an area bounded by n segments, rays, or entire straight lines. The standard model for this geometry is the Poincaré disk model. A major difference between Euclidean and hyperbolic polygons is that the sum of internal angles of a hyperbolic polygon is not the same as Euclidean polygons. In particular, the sum of the angles of a hyperbolic triangle is less than 180 degrees.
Coxeter decompositions are named after Harold Scott MacDonald Coxeter, an accomplished 20th century geometer. He introduced the Coxeter group, an abstract group generated by reflections. These groups have many uses, including producing the rotations of Platonic solids and tessellating the plane.
Given a polygon P, a group G can be generated by reflecting P around its sides. If the angles of P are π/k for natural numbers k, then G will be discrete. A Coxeter decomposition of a polygon is a decomposition into a finite number of polygons in which any two sharing a side are reflections of each other along that side.
The goal of a Coxeter decomposition is to break up a polygon into a composition of congruent triangles reflected on its sides.
If triangle ABC can undergo Coxeter decomposition and has angles , where is the number of times the th angle is broken up, the triangle ABC can be written as . Several properties of these fundamental polygons are known for hyperbolic triangles.
Quadrilaterals may also have Coxeter decompositions.
This article needs additional citations for
verification. (June 2017) |
A Coxeter decomposition of a polygon is a decomposition into a finite number of polygons in which any two sharing a side are reflections of each other along that side. Hyperbolic polygons are the analogues of Euclidean polygons in hyperbolic geometry. A hyperbolic n-gon is an area bounded by n segments, rays, or entire straight lines. The standard model for this geometry is the Poincaré disk model. A major difference between Euclidean and hyperbolic polygons is that the sum of internal angles of a hyperbolic polygon is not the same as Euclidean polygons. In particular, the sum of the angles of a hyperbolic triangle is less than 180 degrees.
Coxeter decompositions are named after Harold Scott MacDonald Coxeter, an accomplished 20th century geometer. He introduced the Coxeter group, an abstract group generated by reflections. These groups have many uses, including producing the rotations of Platonic solids and tessellating the plane.
Given a polygon P, a group G can be generated by reflecting P around its sides. If the angles of P are π/k for natural numbers k, then G will be discrete. A Coxeter decomposition of a polygon is a decomposition into a finite number of polygons in which any two sharing a side are reflections of each other along that side.
The goal of a Coxeter decomposition is to break up a polygon into a composition of congruent triangles reflected on its sides.
If triangle ABC can undergo Coxeter decomposition and has angles , where is the number of times the th angle is broken up, the triangle ABC can be written as . Several properties of these fundamental polygons are known for hyperbolic triangles.
Quadrilaterals may also have Coxeter decompositions.