Velocity polytopes of periodic graphs and a no-go theorem for digital physics

A periodic graph in dimension d is a directed graph with a free action of Z with only finitely many orbits. It can conveniently be represented in terms of an associated finite graph with weights in Z. Here we use the weight sums along cycles in this associated graph to construct a certain polytope in R, which we regard as a geometrical invariant associated to the periodic graph. It is the unit ball of a norm on R describing the large-scale geometry of the graph. It has a physical interpretation as the set of attainable velocities of a particle on the graph which can hop along one edge per timestep. Since a polytope necessarily has distinguished directions, there is no periodic graph for which this velocity set is isotropic. In the context of classical physics, this can be viewed as a no-go theorem for the emergence of an isotropic space from a discrete structure.