A $$d$$ 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 → bma is seen as a flip operation on graphs embedded in 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.

Generation of Digital Planes Using Generalized Continued-Fractions Algorithms

“…The result is a set of points called pattern. By analogy with the terminology used in [17], we say that a pattern is made of a body, noted B n , and legs, noted L n . Euclid's algorithm ensures that there exists N such that v N = (1, 1).…”

“…In the field of combinatorics on words, the recursive structure of digital lines has been studied, in particular, via Christoffel words. Recent work by Labbé and Reutenauer [17] extends Christoffel words as subgraphs of the hypercubic lattice in arbitrary dimensions. The motivation for the present paper is to provide a better understanding of the self-similarities of what is called a Christoffel parallelogram in [17].…”

“…Recent work by Labbé and Reutenauer [17] extends Christoffel words as subgraphs of the hypercubic lattice in arbitrary dimensions. The motivation for the present paper is to provide a better understanding of the self-similarities of what is called a Christoffel parallelogram in [17]. This is, roughly speaking, the smallest pattern with parallel sides that tiles the digital plane by translation.…”

Approximation of Digital Surfaces by a Hierarchical Set of Planar Patches

“…There has been much effort done in order to find similar results in three dimensions despite the lack of a canonical algorithm and the diversity of existing generalizations of the Euclidean algorithm. Some combinatorial results, involving symmetries, piece exchanges and flips, have been stated thanks to an appropriate representation of digital planes [16]. However, most of other related works depend on a multi-dimensional generalization of the Euclidean algorithm such as Brun, e.g., [4], Jacobi-Perron, e.g., [6], Fully substractive, e.g.…”

Palindromic Complexity of Trees

“…For example, the notion of palindrome appears in the study of multidimensional geometric structures, thus introducing a new characterization. Some known classes of words are often redefined as digital planes [3,16], and the adjacency graph of structures obtained by symmetries appeared more recently [9]. In the latter article, authors show that the obtained graph is a tree and its palindromes have been described by Domenjoud, Provençal and Vuillon [8].…”