Metric properties in Berggren tree of primitive Pythagorean triples

TL;DR

This paper explores metric properties of primitive Pythagorean triples within Berggren's ternary tree, analyzing their geometric and spatial characteristics when viewed as side lengths and coordinate points.

Contribution

It introduces new metric analyses of triples in Berggren's tree, considering their geometric and spatial properties in novel ways.

Findings

Identifies metric properties of triples as side lengths.

Analyzes triples as points in 3D space.

Provides insights into the structure of the Berggren tree.

Abstract

A Pythagorean triple is a triple of positive integers $(x,y,z)$ such that $x^2+y^2=z^2$. If $x,y$ are coprime and $x$ is odd, then it is called a primitive Pythagorean triple. Berggren showed that every primitive Pythagorean triple can be generated from triple $(3,4,5)$ using multiplication by uniquely number and order of three $3\times3$ matrices, which yields a ternary tree of triplets. In this paper, we present some metric properties of triples in Berggren tree. Firstly, we consider primitive Pythagorean triple as lengths of sides of right triangles and secondly, we consider them as coordinates of points in three-dimensional space.