  In this article, we extend the definition of Christoffel words to directed
subgraphs of the hypercubic lattice in arbitrary dimension that we call
Christoffel graphs. Christoffel graphs when $d=2$ correspond to well-known
Christoffel words. Due to periodicity, the $d$-dimensional Christoffel graph
can be embedded in a $(d-1)$-torus (a parallelogram when $d=3$). We show that
Christoffel graphs have similar properties to those of Christoffel words:
symmetry of their central part and conjugation with their reversal. Our main
result extends Pirillo's theorem (characterization of Christoffel words which
asserts that a word $amb$ is a Christoffel word if and only if it is conjugate
to $bma$) in arbitrary dimension. In the generalization, the map $amb\mapsto
bma$ is seen as a flip operation on graphs embedded in $\mathbb{Z}^d$ and the
conjugation is a translation. We show that a fully periodic subgraph of the
hypercubic lattice is a translate of its flip if and only if it is a
Christoffel graph.
