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

DOI:
https://doi.org/10.1090/S0002-9939-10-10284-6

Published electronically:
March 23, 2010

MathSciNet review:
2644899

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Abstract | References | Similar Articles | Additional Information

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 , and his classification is equivalent to the converse for . In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.

**[BV]**C. Bachoc and B. Venkov,*Modular forms, lattices and spherical designs*, Réseaux euclidiens, designs sphériques et formes modulaires, 87-111, Monogr. Enseign. Math.**37**, Enseignement Math., Geneva, 2001. MR**1878746 (2003d:11096)****[BBCGKS]**B. Ballinger, G. Blekherman, H. Cohn, N. Giansiracusa, E. Kelly, and A. Schürmann,*Experimental study of energy-minimizing point configurations on spheres*, Experiment. Math.**18**(2009), 257-283.**[B1]**A. E. Brouwer,*Parameters of strongly regular graphs*, tables published electronically at`http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html`.**[B2]**A. E. Brouwer,*Paulus graphs*, tables published electronically at`http://www.win.tue.nl/~aeb/graphs/Paulus.html`.**[CGS]**P. J. Cameron, J. M. Goethals, and J. J. Seidel,*Strongly regular graphs having strongly regular subconstituents*, J. Algebra**55**(1978), 257-280. MR**523457 (81d:05034)****[CK]**H. Cohn and A. Kumar,*Universally optimal distribution of points on spheres*, J. Amer. Math. Soc.**20**(2007), 99-148. MR**2257398 (2007h:52009)****[CSl]**J. Conway and N. J. A. Sloane,*Sphere Packings, Lattices and Groups*, third edition, Grundlehren der Mathematischen Wissenschaften, 290, Springer-Verlag, New York, 1999. MR**1662447 (2000b:11077)****[CSm]**J. H. Conway and D. A. Smith,*On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry*. A K Peters, Ltd., Natick, MA, 2003. MR**1957212 (2004a:17002)****[DGS]**P. Delsarte, J. Goethals, and J. Seidel,*Spherical codes and designs*, Geometriae Dedicata**6**(1977), 363-388. MR**0485471 (58:5302)****[KV]**H. Koch and B. Venkov,*Über gerade unimodulare Gitter der Dimension . III*, Math. Nachr.**152**(1991), 191-213. MR**1121234 (92j:11064)****[L]**J. Leech,*Equilibrium of sets of particles on a sphere*, Math. Gaz.**41**(1957), 81-90. MR**0086325 (19:165b)****[NS]**G. Nebe and N. Sloane,*A catalogue of lattices*, tables published electronically at`http://www.research.att.com/~njas/lattices/`.**[P]**A. J. L. Paulus,*Conference matrices and graphs of order*, Technische Hogeschool Eindhoven, report WSK 73/06, Eindhoven, 1973, 89 pp.**[T]**J. J. Thomson,*On the structure of the atom: An investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure*, Phil. Mag. (6)**7**(1904), 237-265.

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Additional Information

**Henry Cohn**

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

Email:
cohn@microsoft.com

**Noam D. Elkies**

Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Email:
elkies@math.harvard.edu

**Abhinav Kumar**

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

Email:
abhinav@math.mit.edu

**Achill Schürmann**

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

Email:
a.schurmann@tudelft.nl

DOI:
https://doi.org/10.1090/S0002-9939-10-10284-6

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.