Clifford Algebras and Euclid's Parameterization of Pythagorean Triples

TL;DR

This paper reveals that Euclid's parameters for Pythagorean triples form a symplectic space linked to Clifford algebras, presenting a novel geometric and algebraic perspective with extensions to higher-dimensional Pythagorean tuples.

Contribution

It introduces a Clifford algebra and spinor framework for Euclid's parameterization, connecting classical number theory with modern algebraic structures and symmetries.

Findings

Euclid's parameters form a symplectic space with Clifford algebra structure

Explicit formulas for generating Pythagorean quadruples, hexads, and decuples

Geometric interpretation of Hall matrices as symmetries of an Apollonian gasket

Abstract

We show that the space of Euclid's parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra $\mathbb{R}_{2,1}$, whose minimal version may be conceptualized as a 4-dimensional real algebra of "kwaternions." We observe that this makes Euclid's parameterization the earliest appearance of the concept of spinors. We present an analogue of the "magic correspondence" for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geometric meaning as certain symmetries of an Apollonian gasket. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided.