Combinatorics of Bricard’s octahedra

Matteo Gallet; Georg Grasegger; Jan Legerský; Josef Schicho

doi:10.5802/crmath.132

Géométrie

Combinatorics of Bricard’s octahedra
[Combinatoire des octaèdres de Bricard]

Matteo Gallet ¹ ; Georg Grasegger ² ; Jan Legerský ^3,⁴ ; Josef Schicho ⁴

¹ International School for Advanced Studies/Scuola Internazionale Superiore di Studi Avanzati (ISAS/SISSA), Trieste, Italy
² Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria
³ Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic
⁴ Johannes Kepler University Linz, Research Institute for Symbolic Computation (RISC), Linz, Austria

Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 7-38.

Résumé
Abstract

Dans cet article, on donne une preuve alternative de la classification des mouvements d’un octaèdre, originalement obtenue par Bricard au début du XX $^{e}$ siècle. On utilise une construction combinatoire avec un certain nombre de règles essentielles. Ces règles reposent sur une machinerie bien connue dans la géométrie algébrique moderne : l’espace de modules des courbes rationnelles stables avec des points marqués, utilisé pour codifier les configurations de graphes sur la sphère. On introduit un certain nombre d’objets et de règles : une fois que l’on les assume, la classification des mouvements d’un octaèdre telle que l’on expose devient élémentaire (bien que pas triviale) et peut être appréciée par le lecteur sans besoin de connaissances préalables très approfondies sur le sujet. We thank Celeste Damiani for helping us with the translation into French.

We re-prove the classification of motions of an octahedron — obtained by Bricard at the beginning of the XX century — by means of combinatorial objects satisfying some elementary rules. The explanations of these rules rely on the use of a well-known creation of modern algebraic geometry, the moduli space of stable rational curves with marked points, for the description of configurations of graphs on the sphere. Once one accepts the objects and the rules, the classification becomes elementary (though not trivial) and can be enjoyed without the need of a very deep background on the topic.

Reçu le : 2020-06-15
Révisé le : 2020-08-27
Accepté le : 2020-10-17
Publié le : 2021-03-01

DOI : 10.5802/crmath.132

Classification : 52C25

Affiliations des auteurs :

Matteo Gallet ¹ ; Georg Grasegger ² ; Jan Legerský ^{3, 4} ; Josef Schicho ⁴

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Matteo Gallet and Georg Grasegger and Jan Legersk\'y and Josef Schicho},
     title = {Combinatorics of {Bricard{\textquoteright}s} octahedra},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {7--38},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {1},
     year = {2021},
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     language = {en},
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Matteo Gallet; Georg Grasegger; Jan Legerský; Josef Schicho. Combinatorics of Bricard’s octahedra. Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 7-38. doi : 10.5802/crmath.132. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.132/

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