Elliptic geometry geometry is that the sum of the interior angles of a triangle is greater than 180 degrees. In the spherical model, for example, a triangle can be constructed with vertices at the locations where the three positive Cartesian coordinate axes intersect the sphere, and all three of its internal angles are 90 degrees, summing to 270 degrees. For sufficiently small triangles, the excess over 180 degrees can be made arbitrarily small. The Pythagorean theorem fails in elliptic geometry. In the 90°–90°–90° triangle described above, all three sides have the same length, and consequently do not satisfy formula_1. The Pythagorean result
Triangle group is Euclidean if the angle sum is exactly π, spherical if it exceeds π and hyperbolic if it is strictly smaller than π. Moreover, any two triangles with the given angles are congruent. Each triangle group determines a tiling, which is conventionally colored in two colors, so that any two adjacent tiles have opposite colors. In terms of the numbers "l", "m", "n" > 1 there are the following possibilities. formula_4 The triangle group is the infinite symmetry group of a certain tessellation (or tiling) of the Euclidean plane by triangles whose angles add up to π (or 180°). Up