The Modular Tree of Pythagorus

The Pythagorean triples of integers satisfying x + y = z have been studied and enumerated since Babylonian times. Since Diophantus, it has been known that this set of triples is related to the standard rational parameterization of the unit circle, ( t 2−1 t2+1 , 2t t2+1 ). The Pythagorean triple solutions, which are relatively prime, have the elementary and beautiful characterization as integers x = m − n, y = 2mn, z = m + n for relatively prime integers m,n. One can also realize that the Pythagorean triples are related to the Gaussian integers, Z[i], the lattice in the complex numbers with integer coordinates, u = x+iy, where x, y, are integers. The Pythagorean equation N(u) = uu = (x + iy)(x − iy) = z is an equation among Gaussians. It is perhaps, not surprising, that the Pythagorean triples are just the squares of the set of Gaussian integers, that is v = (m+ni) = (m2−n2) + (2mn)i gives the endpoint of the hypotenuse of a right triangle with integer length sides. This Gaussian square v obviously has a norm, N(v), which is a square. Both of these expressions secretly involve the double angle formulae for sine and cosine since the stereographic projection formula uses the central and chordal angles. By considering the action of the modular group, Γ = PSL2(Z), by conjugation on the set of all integer matrices M2(Z), we can blend together these two perspectives. We shall show (Corollary 3.2) that the Pythagorean triples can be identified with an orbit of a subgroup (of index 6), Γ(2), which is generated by the images of U = ( 1 2 0 1 ) and L = ( 1 0 2 1 ) . Moreover Γ(2) is the free product of the subgroups