# Point Configurations That Are Asymmetric Yet Balanced

 Title: Point Configurations That Are Asymmetric Yet Balanced Author: Cohn, Henry; Kumar, Abhinav; Elkies, Noam David; Schürmann, Achill Note: Order does not necessarily reflect citation order of authors. Citation: Cohn, Henry, Elkies, Noam, Kumar, Abhinav, and Achill Schuermann. 2010. Point configurations that are asymmetric yet balanced. Proceedings of the American Mathematical Society 138 (2010): 2863-2872. Full Text & Related Files: 0812.2579v2.pdf (166.5Kb; PDF) 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 $$R^3$$, and his classification is equivalent to the converse for $$R^3$$. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group. Published Version: doi://10.1090/S0002-9939-10-10284-6 Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:5351575

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Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University