A d-dimensional extension of Christoffel words

Abstract: 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.