Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Point configurations that are asymmetric yet balanced

Authors: Henry Cohn, Noam D. Elkies, Abhinav Kumar and Achill Schürmann
Journal: Proc. Amer. Math. Soc. 138 (2010), 2863-2872
MSC (2010): Primary 52B15; Secondary 05B40, 52C17, 82B05
Published electronically: March 23, 2010
MathSciNet review: 2644899
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Abstract: A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in $ \mathbb{R}^3$, and his classification is equivalent to the converse for  $ \mathbb{R}^3$. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.

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Additional Information

Henry Cohn
Affiliation: Microsoft Research New England, One Memorial Drive, Cambridge, Massachusetts 02142

Noam D. Elkies
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Abhinav Kumar
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Achill Schürmann
Affiliation: Institute of Applied Mathematics, TU Delft, Mekelweg 4, 2628 CD Delft, The Netherlands

Received by editor(s): March 30, 2009
Published electronically: March 23, 2010
Additional Notes: The first, second, and third authors thank the Aspen Center for Physics for its hospitality and support. The first, third, and fourth authors thank the Hausdorff Research Institute for Mathematics. The third and fourth authors thank Microsoft Research. The third author was supported in part by a Clay Liftoff Fellowship and by National Science Foundation grant DMS-0757765. The second author was supported in part by NSF grant DMS-0501029, and the fourth author was supported in part by Deutsche Forschungsgemeinschaft grant SCHU 1503/4-2.
Communicated by: Jim Haglund
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.