  The search for a quantum theory of gravity is one of the major challenges
facing theoretical physics today. While no complete theory exists, a promising
avenue of research is the loop quantum gravity approach. In this approach,
quantum states are represented by spin networks, essentially graphs with
weighted edges. Since general relativity predicts the structure of space, any
quantum theory of gravity must do so as well; thus, "spatial observables" such
as area, volume, and angle are given by the eigenvalues of Hermitian operators
on the spin network states. We present results obtained in our investigations
of the angle and volume operators, two operators which act on the vertices of
spin networks. We find that the minimum observable angle is inversely
proportional to the square root of the total spin of the vertex, a fairly slow
decrease to zero. We also present numerical results indicating that the angle
operator can reproduce the classical angle distribution. The volume operator is
significantly harder to investigate analytically; however, we present
analytical and numerical results indicating that the volume of a region scales
as the 3/2 power of its bounding surface, which corresponds to the classical
model of space.
