Planck-Scale exemplified by means of a Class of ’Cellular Network Modells’ and their Dynamics

Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a class of ’cellular networks’ consisting of cells (nodes) interacting with each other via bonds according to a certain ’local law’ which governs their evolution. Both the internal states of the cells and the strength/orientation of the bonds are assumed to be dynamical variables. We introduce a couple of candidates of such local laws which, we think, are capable of catalyzing the unfolding of the network towards increasing complexity and pattern formation. In section 3 the basis is laid for a version of ’discrete analysis’ and ’discrete topology/geometry’ which, starting from different, perhaps more physically oriented principles, manages to make contact with the much more abstract machinery of Connes et al. and may complement the latter approach. In section 4 a, as far as we can see, promising concept of ’topological dimension’ in form of a ’degree of connectivity’ for graphs, networks and the like is developed. It is then indicated how this ’dimension’, which for continuous structures or lattices being embedded in a continuous background agrees with the ”usual” notion of dimension (i.e. the respective embedding dimension) , may vary dynamically as a result of a ’phase transition like’ change of the ’connectivity’ in the network. A certain (highly) speculative argument, along the lines of statistical mechanics, is supplied in favor of the naturalness of dimension 4 of ordinary (classical) space-time.