A note on Penrose's Spin-Geometry Theorem and the geometry of `empirical quantum angles'

In the traditional formalism of quantum mechanics, a simple direct proof of (a version of) the Spin Geometry Theorem of Penrose is given; and the structure of a model of the `space of the quantum directions', defined in terms of elementary $SU(2)$-invariant observables of the quantum mechanical systems, is sketched.