  Starting from the working hypothesis that both physics and the corresponding
mathematics and in particular geometry have to be described by means of
discrete concepts on the Planck-scale, one of the many problems one has to face
in this enterprise is to find the discrete protoforms of the building blocks of
our ordinary continuum physics and mathematics living on a smooth background,
and perhaps more importantly find a way how this continuum limit emerges from
the mentioned discrete structure. We model this underlying substratum as a
structurally dynamic cellular network (basically a generalisation of a cellular
automaton). We regard these continuum concepts and continuum spacetime in
particular as being emergent, coarse-grained and derived relative to this
underlying erratic and disordered microscopic substratum, which we would like
to call quantum geometry and which is expected to play by quite different
rules, namely generalized cellular automaton rules. A central role in our
analysis is played by a geometric renormalization group which creates (among
other things) a kind of sparse translocal network of correlations between the
points in classical continuous space-time and underlies, in our view, such
mysterious phenomena as holography and the black hole entropy-area law. The
same point of view holds for quantum theory which we also regard as a
low-energy, coarse-grained continuum theory, being emergent from something more
fundamental. In this paper we review our approach and compare it to the quantum
graphity framework.
