Abstract
The spin geometry theorem of Penrose is extended from SU(2) to E(3) (Euclidean) invariant elementary quantum mechanical systems. Using the natural decomposition of the total angular momentum into its spin and orbital parts, the \emph{distance} between the centre-of-mass lines of the elementary subsystems of a classical composite system can be recovered from their \emph{relative orbital angular momenta} by E(3)-invariant classical observables. Motivated by this observation, an expression for the `empirical distance' between the elementary subsystems of a \emph{composite quantum mechanical system}, given in terms of E(3)-invariant quantum observables, is suggested. It is shown that, in the classical limit, this expression reproduces the \emph{a priori} Euclidean distance between the subsystems, though at the quantum level it has a discrete character. `Empirical' angles and 3-volume elements are also considered.