For each integer
, we obtain a characterization of all graphs in which the ball of radius
around each vertex is isomorphic to the ball of radius 3 in
, the graph of the
-dimensional integer lattice. The finite, connected graphs with this property have a highly rigid, ‘global’ algebraic structure; they can be viewed as quotient lattices of
in various compact
-dimensional orbifolds which arise from crystallographic groups. We give examples showing that ‘radius 3’ cannot be replaced by ‘radius 2’, and that ‘orbifold’ cannot be replaced by ‘manifold’. In the
case, our methods yield new proofs of structure theorems of Thomassen [‘Tilings of the Torus and Klein bottle and vertex-transitive graphs on a fixed surface’, Trans. Amer. Math. Soc.
323 (1991), 605–635] and of Márquez et al. [‘Locally grid graphs: classification and Tutte uniqueness’, Discrete Math.
266 (2003), 327–352], and also yield short, ‘algebraic’ restatements of these theorems. Our proofs use a mixture of techniques and results from combinatorics, geometry and group theory.