# A \(d\)-dimensional Extension of Christoffel Words

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## Abstract

In this article, we extend the definition of Christoffel words to directed subgraphs of the hypercubic lattice in an arbitrary dimension that we call Christoffel graphs. Christoffel graphs, when \(d=2\), correspond to the 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 \(bma\)) to an 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 translation of its flip if and only if it is a Christoffel graph.

## Keywords

Christoffel words Christoffel graphs Digital hyperplane Flip Hypercubic lattice Pirillo’s theorem## Mathematics Subject Classification

05C75 52C35 68R15## Notes

### Acknowledgments

We wish to thank the anonymous referee for his many valuable comments and for having noticed that Lemma 13 was missing. Both authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). The first author acknowledges support from the ANR project Dyna3S (ANR-13-BS02-0003). Sage open source software was used to generate tikz code to create the artwork.

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