Geometry
Combinatorics of Bricard’s octahedra
Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 7-38.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.132
Classification: 52C25
Gallet, Matteo 1; Grasegger, Georg 2; Legerský, Jan 3, 4; Schicho, Josef 3

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 Johannes Kepler University Linz, Research Institute for Symbolic Computation (RISC), Linz, Austria
4 Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic
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Gallet, Matteo; Grasegger, Georg; Legerský, Jan; Schicho, Josef. Combinatorics of Bricard’s octahedra. Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 7-38. doi : 10.5802/crmath.132. http://www.numdam.org/articles/10.5802/crmath.132/

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