Ultimate fate of the universe cosmological constant, which would correspond to a negative energy density and positive pressure, would cause even an open universe to re-collapse to a big crunch. This option has been ruled out by observations. If the average density of the universe exactly equals the critical density so that formula_8, then the geometry of the universe is flat: as in Euclidean geometry, the sum of the angles of a triangle is 180 degrees and parallel lines continuously maintain the same distance. Measurements from Wilkinson Microwave Anisotropy Probe have confirmed the universe is flat with only a 0.4% margin of error. In absence
Schwarz triangle function classification of arithmetic triangle groups (there are only finitely many) and determined when two of them are commensurable. The particular example is related to Bring's curve and the arithmetic theory implies that the triangle group for Δ contains the triangle group for the triangle Δ with angles /4, /4 and /5 as a non-normal subgroup of index 6. Doubling the triangles Δ and Δ, this implies that there should be a relation between 6 triangles Δ with angles /2, /5 and /5 and hyperbolic area /10 and a triangle Δ with angles /5, /5 and /10 and hyperbolic area 3/5.