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Starting from the working hypothesis that both physics and the correspondingmathematics and in particular geometry have to be described by means ofdiscrete concepts on the Planck-scale, one of the many problems one has to facein this enterprise is to find the discrete protoforms of the building blocks ofour ordinary continuum physics and mathematics living on a smooth background,and perhaps more importantly find a way how this continuum limit emerges fromthe mentioned discrete structure. We model this underlying substratum as astructurally dynamic cellular network (basically a generalisation of a cellularautomaton). We regard these continuum concepts and continuum spacetime inparticular as being emergent, coarse-grained and derived relative to thisunderlying erratic and disordered microscopic substratum, which we would liketo call quantum geometry and which is expected to play by quite differentrules, namely generalized cellular automaton rules. A central role in ouranalysis is played by a geometric renormalization group which creates (amongother things) a kind of sparse translocal network of correlations between thepoints in classical continuous space-time and underlies, in our view, suchmysterious phenomena as holography and the black hole entropy-area law. Thesame point of view holds for quantum theory which we also regard as alow-energy, coarse-grained continuum theory, being emergent from something morefundamental. In this paper we review our approach and compare it to the quantumgraphity framework.

The Structurally Dynamic Cellular Network and Quantum Graphity Approaches to Quantum Gravity and Quantum Geometry - A Review and Comparison