[Combinatoire des octaèdres de Bricard]
Dans cet article, on donne une preuve alternative de la classification des mouvements d’un octaèdre, originalement obtenue par Bricard au début du XX 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.
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@article{CRMATH_2021__359_1_7_0,
author = {Gallet, Matteo and Grasegger, Georg and Legersk\'y, Jan and Schicho, Josef},
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},
doi = {10.5802/crmath.132},
language = {en},
url = {http://www.numdam.org/articles/10.5802/crmath.132/}
}
TY - JOUR AU - Gallet, Matteo AU - Grasegger, Georg AU - Legerský, Jan AU - Schicho, Josef TI - Combinatorics of Bricard’s octahedra JO - Comptes Rendus. Mathématique PY - 2021 SP - 7 EP - 38 VL - 359 IS - 1 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.132/ DO - 10.5802/crmath.132 LA - en ID - CRMATH_2021__359_1_7_0 ER -
%0 Journal Article %A Gallet, Matteo %A Grasegger, Georg %A Legerský, Jan %A Schicho, Josef %T Combinatorics of Bricard’s octahedra %J Comptes Rendus. Mathématique %D 2021 %P 7-38 %V 359 %N 1 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.132/ %R 10.5802/crmath.132 %G en %F CRMATH_2021__359_1_7_0
Gallet, Matteo; Grasegger, Georg; Legerský, Jan; Schicho, Josef. Combinatorics of Bricard’s octahedra. Comptes Rendus. Mathématique, Tome 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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