The Modular Tree of Pythagoras

The Pythagorean triples that are relatively prime (called the primitive triples) have the elementary and beautiful characterization as integers x = m2 − n2, y = 2mn, z = m2 + n2 (when y is even) for relatively prime integers m and n of opposite parity. One can think of this as replacing the parameter t for the circle with the fraction m/n and then scaling. Our motivation for understanding the triples stems from the realization that one can enumerate the rational numbers on the line by using the modular group, in a sense reversing the Euclidean algorithm [2]. Now the line can be transformed by a linear fractional transformation to the circle. This transformation changes fractions to rational points on the circle, and after scaling this process gives rise to Pythagorean triples. Roughly speaking, we can establish a correspondence of a Pythagorean triple [m2 − n2, 2mn,m2 + n2] in which m and n are relatively prime with a matrix belonging to SL2(Z) (the group of two-by-two integral matrices of determinant one) whose entries depend on m and n. Since the modular group = PSL2(Z) = SL2(Z)/{±I } is essentially a free group, it follows that there is an underlying tree structure to Pythagorean triples. Making this tree structure and its connection to the modular group explicit is a bit delicate, but the payoff is worth the effort. Our main results can be summarized as follows: