Discrete Mathematics and Physics on the 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 'cellu- lar 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 ca..