Equilateral triangle is the only triangle with integer sides and three rational angles as measured in degrees. The equilateral triangle is the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes) (the heptagonal triangle being the only obtuse one). Equilateral triangles are found in many other geometric constructs. The intersection of circles whose centers are a radius width apart is a pair of equilateral arches, each of which can be inscribed with an equilateral triangle. They form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral
Schwarz triangle function Δ has angles /"a", /"b" and /"c" with "a", "b", "c" integers greater than 1. The hyperbolic area of Δ equals – /"a" – /"b" – /"c", so that The construction of a tessellation will first be carried out for the case when "a", "b" and "c" are greater than 2. The original triangle Δ gives a convex polygon "P" with 3 vertices. 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 /"m". The triangles do not