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 Zd with only finitely many orbits. It can conveniently be represented in terms of an associated finite graph with weights in Zd, corresponding to a Zd-bundle with connection. Here we use the weight sums along cycles in this associated graph to construct a certain polytope in Rd, which we regard as a geometrical invariant associated to the periodic graph. It is the unit ball of a norm on Rd 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.