  We introduce the Pythagorean dimension: a natural number (or infinity) for
all representations of the Cuntz algebra and certain unitary representations of
the Richard Thompson groups called Pythagorean. For each natural number d we
completely classify (in a functorial manner) all such representations using
finite dimensional linear algebra. Their irreducible classes form a real
manifold of dimension $2d^2+1$ playing the role of a moduli space. Apart from a
finite disjoint union of circles, each point of the manifold corresponds to an
irreducible unitary representation of Thompson's group F (which extends to the
other Thompson groups and the Cuntz algebra) that is not monomial. The
remaining circles provide monomial representations which we previously fully
described and classified. We translate in our language a large number of
previous results in the literature. We explain how our techniques extend them.
