Triangle see that the Earth's surface is locally flat: If we draw an arbitrarily small triangle in the neighborhood of one point on the Earth's surface, the fraction "f" of the Earth's surface which is enclosed by the triangle will be arbitrarily close to zero. In this case the angle sum formula simplifies to 180°, which we know is what Euclidean geometry tells us for triangles on a flat surface. Rectangles have been the most popular and common geometric form for buildings since the shape is easy to stack and organize; as a standard, it is easy to design furniture and
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.