Table des matières

Journées sur les pavages en mathématiques, informatique et musique

pavages-cercle-tonnetz.jpg Montpellier
26-27 septembre 2014

LIRMM (le 26 septembre) 161 rue Ada
F-34095 Montpellier Cedex 5

La Chapelle-Gély (le 27 septembre) 170 Rue Joachim du Bellay
F-34070 Montpellier


Vendredi 26 septembre (10h-18h, LIRMM)
Entrée libre sur réservation

Samedi 27 septembre (14h-18h, La Chapelle-Gély)
Séance grand public – entrée libre.

Résumés des interventions (26/09, LIRMM)

Tiling by translation: combinatorics, number theory, Fourier analysis and algorithms

Mihalis Kolountzakis, Professor and Department Chair, Department of Mathematics and Applied Mathematics, University of Crete, Greece.

I will talk about tiling by translation, meaning that we work in an additive group G and we have two sets A and B (we will be talking mostly about finite G) such that A+B=G is a direct sum, i.e. every g in G can be written uniquely as a+b, with a in A and b in B. In other words the translates A+b, with b in B, of A are non-overlapping and cover G. There are many very natural, very interesting and very difficult questions that concern such tilings. Some of these questions have to do with the structure that these sets can have (e.g. must B or A be a periodic set?) and some other questions have to do with the decidability or computational complexity of deciding "is A a tile of G?". There are many interconnections between these problems and I hope to give the flavor of the tools being used (the most prominent of which is Fourier analysis, given the translation invariance of the problem) and point out some surprising open problems.

Tilings, domino problem, and an aperiodic tile set from Kari and Culik built on a 1D dynamical system

Guilhem Gamard, Doctorant au LIRMM

A Wang tile is a unit square with colored edges. Given a finite tileset, can we cover the plane with copies of the tiles, in such manner that adjacent borders have matching colors? This puzzle is known as the Domino Problem and was proven undecidable. Quite surprisingly, this result implies that some tilesets can cover the plane, but never in a periodic way. We call them aperiodic tilesets.

In this talk, I will introduce the Wang tiles formalism, then focus on a particular aperiodic tileset named the "Kari-Culik tiles". I will show in details how these tiles work and prove their aperiodicity. Then I will give a sketch of proof of the undecidability of the Domino problem based on the Kari-Culik tiles.

From large to small rhythmic canons. Reducing canons and conjectures

Emmanuel Amiot, Professeur en CPGE, Perpignan

Since Dan Tudor Vuza invented the rhythmic canons that now bear his name (1991) it became clear that those canons, which are in a way what the human mind perceives while hearing a musical canon (= a rhythmic mosaic with translates of one motif tile), may generate all rhythmic canons by concatenation and duality; as a corollary, several hard conjectures about tilings need only be studied for these specific canons. These transformational techniques which can transfer features and conjectures from general to Vuza canons can also be, and are, used to generate new compositional material, interesting on its own. The underlying concept is the set of zeroes of the Fourier transform, which induces a finer classification than the absolute value of DFT, aka homometry. Future prospects may include even finer decomposition techniques for Vuza canons themselves, or conversely the study and elucidation of those Vuza canons which cannot be be further decomposed in any way.

An Elementary Construction of a Self-Similar Aperiodic Tiling

Victor Poupet, MdC, Université Montpellier II, LIRMM

Wang tiles were introduced by Hao Wang in the 1960s to represent simple logic problems as a tiling problem. They are described as unit squares with colored edges, and the associated problem (called "domino problem”) is to decide whether a given finite tile set can tile the plane, which is to say that one can fill the whole plane with copies of these tiles in such a way that colors on adjoining edges match.

While studying the decidability of this problem, it appears that it would be decidable if any tile set that can tile the plane can do so in a periodic way. The problem was later proved undecidable by Robert Berger and in doing so Berger described the first example of an aperiodic tile set : a set of tiles that can tile the plane, but such that all valid tilings are aperiodic. Since then, many other aperiodic tile sets have been discovered (amongst the most famous are Raphael Robinson’s and Jarkko Kari’s). In this talk, we will present the construction of yet another aperiodic tile set, whose structure is similar to Robinson’s tiling but for which the aperiodicity can be fully proved in a (hopefully) simpler and more elementary way.

On Modulus p canons

Hélianthe Caure, Doctorante à l’IRCAM/UPMC, Equipe Représentations Musicales

The concept of rhythmic canons, as it has been introduced by mathematician Dan Vuza in the 1990s, is the art of filling the time axis with some finite rhythmic patterns and their translations, without onsets superposition. Rhythmic tiling canons are equivalent to products of two polynomials with coefficients in {0, 1}. But since the property of being a polynomial with coefficients in {0, 1} is not closed under product, the idea of working in the polynomial field F_2 [X] came to mind, and by extension, the complete concept of modulus p canons.

It appeared that this idea involved huge improvements of the notion, both mathematically and musically. Modulus p tiling enriches classical rhythmic tiling canons with harmony, allowing notes superposition. Thus, it breaks the monotony of a strict tiling with some controlled covering. It is also a powerful mathematical tool, because thanks to some of its properties, it is extremely easy to compute a modulus p tiling by translating any given finite pattern.

Fixed point tilings

Alexander Shen, DR CNRS, LIRMM

Geometrical example of self-similar tilings (and aperiodic tiling in general) are based on some nice construction; we can check all the required properties but still this construction is miraculous, it is not clear that it should exist until we see it. There are several constructions, and each of them is a piece of art. On the other hand, there is an "industrial" technology of producing aperiodic tilings based on "fixed point theorem" (Kleene recursion theorem, self-referencial programs), going back to Kleene, von Neumann, Gacs and others. In this way we do not get any nice examples, and the resulting tile sets are large so nobody bothers to construct them explicitly, but we somehow understand why their existence is avoidable, and also are able to construct aperiodic tile sets with additional properties.

Résumés des interventions (Samedi 27/09, Chapelle-Gély)

Comment remplir l'espace ?

Nicolas Saby (mathématicien, IREM de Montpellier, université Montpellier 2)

La nature et l'homme ont plusieurs façons de remplir l'espace. On s'interrogera à partir de manipulations comment on peut aborder cette question. La question de l'espace à remplir et de la procédure de remplissage, nous fera explorer les frises, les pavages, et d'autres empilements.Ces manipulations nous montreront comment on trouve des invariants dans ces ornementations et comment être sur de les connaitre toutes.

La géométrie, art ou mesure ?

Hervé Lehning (mathématicien, membre du comité de rédaction de la revue Tangente)

La géométrie est née sous l’Antiquité pour mesurer la Terre. Dès cette époque, elle s’est également occupée d’étudier les formes, et donc d’art. Nous aborderons ce domaine très riche d’un point de vue artistique et pratique, en évitant les trop grandes envolées abstraites, même si cette étude mène à la théorie des groupes. Nous parlerons donc de l’art de paver, comme nous pouvons l’admirer à l’Alhambra de Grenade, de la notion de perspective et de sa subversion dont Escher, le graveur, s’était fait une spécialité, et de façon plus contemporaine, du morphing, qui a permis de vieillir en quelques secondes l’actrice principale du film Titanic, et qui est utile dans l’art du dessin animé.

Les pavages en musique : du Tonnetz aux canons mosaïques

Emmanuel Amiot (mathématicien), Moreno Andreatta (chercheur CNRS à l'Ircam), Hélianthe Caure (doctorante UPMC/Ircam)

Cette présentation proposera un tour d'horizon des diverses applications de la notion de pavage en musique, aussi bien dans l'organisation de l'espace des hauteurs (autour - en particulier - du Tonnetz et des ses variantes) ainsi que dans la constructions des canons rythmiques mosaïques.

Organisation des journées

Bruno Durand
Contact : Bruno.Durand[at]

Hélianthe Caure et Moreno Andreatta
Contact :