  An arbitrarily dense discretisation of the Bloch sphere of complex Hilbert
states is constructed, where points correspond to bit strings of fixed finite
length. Number-theoretic properties of trigonometric functions (not part of the
quantum-theoretic canon) are used to show that this constructive discretised
representation incorporates many of the defining characteristics of quantum
systems: completementarity, uncertainty relationships and (with a simple
Cartesian product of discretised spheres) entanglement. Unlike Meyer's earlier
discretisation of the Bloch Sphere, there are no orthonormal triples, hence the
Kocken-Specker theorem is not nullified. A physical interpretation of points on
the discretised Bloch sphere is given in terms of ensembles of trajectories on
a dynamically invariant fractal set in state space, where states of physical
reality correspond to points on the invariant set. This deterministic
construction provides a new way to understand the violation of the Bell
inequality without violating statistical independence or factorisation, where
these conditions are defined solely from states on the invariant set. In this
finite representation there is an upper limit to the number of qubits that can
be entangled, a property with potential experimental consequences.
