Article

Advances in Applied Clifford Algebras

, Volume 17, Issue 1, pp 71-93

First online:

Clifford Algebras and Euclid’s Parametrization of Pythagorean Triples

  • Jerzy KocikAffiliated withDepartment of Mathematics, Southern Illinois University Email author 

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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 R21, whose minimal version may be conceptualized as a 4-dimensional real algebra of “kwaternions.” We observe that this makes Euclid’s parametrization 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.

Keywords.

Pythagorean triples Euclid’s parametrization spinors Clifford algebra Minkowski space pseudo-quaternions unimodular group Hall matrices Apolloniam gasket Lorentz group Pythagorean quadruples n-tuples