  A periodic graph in dimension $d$ is a directed graph with a free action of
$\Z^d$ with only finitely many orbits. It can conveniently be represented in
terms of an associated finite graph with weights in $\Z^d$, corresponding to a
$\Z^d$-bundle with connection. Here we use the weight sums along cycles in this
associated graph to construct a certain polytope in $\R^d$, which we regard as
a geometrical invariant associated to the periodic graph. It is the unit ball
of a norm on $\R^d$ 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.
