Sébastien Labbé

Christoffel and Fibonacci tiles

Among the polyominoes that tile the plane by translation, the so-called squares have been conjectured to tile the plane in at most two distinct ways (these are called double squares). In this paper, we study two families of tiles : one is directly linked to Christoffel words while the other stems from the Fibonacci sequence. We show that these polyominoes are double squares, revealing strong connections between discrete geometry and other areas by means of combinatorics on words.

Palindromic complexity of codings of rotations

We study the palindromic complexity of infinite words obtained by coding rotations on partitions of the unit circle by inspecting the return words. The main result is that every coding of rotations on two intervals is full, that is, it realizes the maximal palindromic complexity. As a byproduct, a slight improvement about return words in codings of rotations is obtained: every factor of a coding of rotations on two intervals has at most 4 complete return words, where the bound is realized only for a finite number of factors. We also provide a combinatorial proof for the special case of complementary-symmetric Rote sequences by considering both palindromes and antipalindromes occurring in it.

Reconstructing words from a fixed palindromic length sequence

To every word ω is associated a sequence Gω built by computing at each position i the length of its longest palindromic suffix. This sequence is then used to compute the palindromic defect of a finite word Ω defined by D(Ω) = |Ω|+1−|Pal(Ω)| where Pal(Ω) is the set of its palindromic factors. In this paper we exhibit some properties of this sequence and introduce the problem of reconstructing a word from GΩ. In particular we show that up to a relabelling the solution is unique for 2‐letter alphabets.

An arithmetic and combinatorial approach to three-dimensional discrete lines

The aim of this paper is to discuss from an arithmetic and combinatorial viewpoint a simple algorithmic method of generation of discrete segments in the three-dimensional space. We consider discrete segments that connect the origin to a given point (u1, u2, u3) with coprime nonnegative integer coordinates. This generation method is based on generalized three-dimensional Euclids algorithms acting on the triple (u1, u2, u3). We associate with the steps of the algorithm substitutions, that is, rules that replace letters by words, which allow us to generate the Freeman coding of a discrete segment. We introduce a dual viewpoint on these objects in order to measure the quality of approximation of these discrete segments with respect to the corresponding Euclidean segment. This viewpoint allows us to relate our discrete segments to finite patches that generate arithmetic discrete planes in a periodic way.

Two infinite families of polyominoes that tile the plane by translation in two distinct ways

It has been proved that, among the polyominoes that tile the plane by translation, the so-called squares tile the plane in at most two distinct ways. In this paper, we focus on double squares, that is, the polyominoes that tile the plane in exactly two distinct ways. Our approach is based on solving equations on words, which allows us to exhibit properties about their shape. Moreover, we describe two infinite families of double squares. The first one is directly linked to Christoffel words and may be interpreted as segments of thick straight lines. The second one stems from the Fibonacci sequence and reveals some fractal features.

Equations on palindromes and circular words

In this paper we consider several types of equations on words, motivated by the attempt of characterizing the class of polyominoes that tile the plane by translation in two distinct ways. Words coding the boundary of these polyominoes satisfy an equation whose solutions are in bijection with a subset of the solutions of equations of the form ABA@?B@?=XYX@?Y@?. It turns out that the solutions are strongly related to local periodicity involving palindromes and conjugate words.

Combinatorial properties of double square tiles

We study the combinatorial properties and the problem of generating exhaustively double square tiles, i.e. polyominoes yielding two distinct periodic tilings by translated copies such that every polyomino in the tiling is surrounded by exactly four copies. We show in particular that every prime double square tile may be obtained from the unit square by applying successively some invertible operators on double squares. As a consequence, we prove a conjecture of Provencal and Vuillon (2008) [17] stating that these polyominoes are invariant under rotation by angle @p.

Complexity of the Fibonacci snowflake

The object under study is a particular closed and simple curve on the square lattice ℤ2 related with the Fibonacci sequence Fn. It belongs to a class of curves whose length is 4F3n+1, and whose interiors tile the plane by translation. The limit object, when conveniently normalized, is a fractal line for which we compute first the fractal dimension, and then give a complexity measure.

A parallelogram tile fills the plane by translation in at most two distinct ways

We consider the tilings by translation of a single polyomino or tile on the square grid Z^2. It is well-known that there are two regular tilings of the plane, namely, parallelogram and hexagonal tilings. Although there exist tiles admitting an arbitrary number of distinct hexagon tilings, it has been conjectured that no polyomino admits more than two distinct parallelogram tilings. In this paper, we prove this conjecture.

Uniformly balanced words with linear complexity and prescribed letter frequencies

We consider the following problem. Let us fix a finite alphabet A; for any given d-uple of letter frequencies, how to construct an infinite word u over the alphabet A satisfying the following conditions: u has linear complexity function, u is uniformly balanced, the letter frequencies in u are given by the given d-uple. This paper investigates a construction method for such words based on the use of mixed multidimensional continued fraction algorithms.