The stereographic projection of the Stern–Brocot tree
I sketch how the stereographic projection of the Stern–Brocot tree forms an ordered binary tree of Pythagorean triples, which can be used to compute best approximations of turn angles of points on the circle and finally trigonometric functions.
With each row the number of columns increases by one. Each cell shows a right triangle with side lengths that are natural numbers. If we number the rows m , the columns n , then the legs a,b and the hypotenuse c are defined by
The triplet [a\;b\;c] is called Pythagorean triple and this method produces every possible Pythagorean triple [X].
makes it clear, that all Pythagorean triples can be mapped onto rational points on the circle. Conversely, it can also be shown, that all all rational points on the circle can be mapped to primitive triples — those where a and b are coprime. So, primitive Pythagorean triples and rationals points on the circle are isomorphic.
The following table shows again the triangles with their classical m,n coordinates and the actual triples at the bottom. Triangles of non-primitive triples are hatched.
