  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.
