Clifford Algebras and Euclid's Parameterization of Pythagorean Triples

Details Clifford Algebras and Euclid's Parameterization of Pythagorean Triples Clifford Algebras and Euclid's Parameterization of Pythagorean Triples

2012-01-21

arxiv.org/abs/1201.4418v1

Advances in Applied Clifford Algebras 17 (2007), 71-93. Online version:December 13, 2006

Mathematics

Number Theory

22 pages, 7 figures. The sign convention is fixed, one comment on terminology is added

Scientific paper

10.1007/s00006-006-0019-2

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.

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