Special right triangle are in the ratio 1 : 2 : 3 and respectively measure 30° (), 60° (), and 90° (). The sides are in the ratio 1 : : 2. The proof of this fact is clear using trigonometry. The geometric proof is: The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if "α", , are the angles in the progression then the sum of the angles = 180°. After dividing by 3, the angle must be 60°. The right angle
Schwarz triangle of the reciprocals of these integers: This is simply a way of saying that in Euclidean space the interior angles of a triangle sum to , while on a sphere they sum to an angle greater than , and on hyperbolic space they sum to less. A Schwarz triangle is represented graphically by a triangular graph. Each node represents an edge (mirror) of the Schwarz triangle. Each edge is labeled by a rational value corresponding to the reflection order, being π/vertex angle. Order-2 edges represent perpendicular mirrors that can be ignored in this diagram. The Coxeter-Dynkin diagram represents this triangular