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Clifford algebras and Euclid’s parametrization of Pythagorean triples. (English) Zbl 1116.15024
The author proves that the space of Euclid’s parameters for Pythagorean triples is endowed with a natural symplectic structure. This space can be considered as a spinor space for the Clifford algebra 12 build over a 3-dimensional Minkowski space whose integer light-like vectors represent the Pythagorean triples. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples and hexads are provided.
MSC:
15A66Clifford algebras, spinors
20H05Unimodular groups, congruence subgroups (matrix groups)
22E43Structure and representation of the Lorentz group