Structure of Primitive Pythagorean Triples in Generating Trees

A Pythagorean triple is a triple of positive integers (a, b, c) such that a²+b²=c². If a, b are coprime, then it is called a primitive Pythagorean triple. It is known that every primitive Pythagorean triple can be generated from the triple (3, 4, 5) using multiplication by unique number and order of three specific 33 matrices, which yields a ternary tree of triplets. Two such trees were described by Berggren and Price, respectively. A different approach is to view the primitive Pythagorean triples as points in the three-dimensional Euclidean space. In this paper, we prove that the triple of descendants of any primitive Pythagorean triple in Berggren's or Price's tree forms a triangle (and therefore defines a plane), and we present our results related to these triangles (and these planes).