Coxeter decompositions of hyperbolic polygons 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 /"k" for natural numbers "k", then "G" will be discrete. A Coxeter decomposition of a polygon is a decomposition into a finite number
Hyperbolic triangle relations between the sides "s" and the angles "A" of an equilateral triangle (a triangle where all sides have the same length and all angles are equal). The relations are: Whether "C" is a right angle or not, the following relationships hold: The hyperbolic law of cosines is as follows: Its dual theorem is There is also a "law of sines": and a four-parts formula: For hyperbolic trigonometry: Hyperbolic triangle In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called "sides" or "edges" and three points called "angles" or "vertices".