  Results that illuminate the physical interpretation of states of
nonperturbative quantum gravity are obtained using the recently introduced loop
variables. It is shown that: i) While local operators such as the metric at a
point may not be well-defined, there do exist {\it non-local} operators, such
as the area of a given 2-surface, which can be regulated diffeomorphism
invariantly and which are finite {\it without} renormalization; ii)there exist
quantum states which approximate a given flat geometry at large scales, but
such states exhibit a discrete structure at the Planck scale; iii) these
results are tied together by the fact that the spectra of the operators that
measure the areas of surfaces are quantized in integral units of the Planck
area.
