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), 28632872
MSC (2010):
Primary 52B15; Secondary 05B40, 52C17, 82B05
Published electronically:
March 23, 2010
MathSciNet review:
2644899
Fulltext PDF Free Access
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]
Christine
Bachoc and Boris
Venkov, Modular forms, lattices and spherical designs,
Réseaux euclidiens, designs sphériques et formes modulaires,
Monogr. Enseign. Math., vol. 37, Enseignement Math., Geneva, 2001,
pp. 87–111. MR 1878746
(2003d:11096)
 [BBCGKS]
B. Ballinger, G. Blekherman, H. Cohn, N. Giansiracusa, E. Kelly, and A. Schürmann, Experimental study of energyminimizing point configurations on spheres, Experiment. Math. 18 (2009), 257283.
 [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), no. 2,
257–280. MR
523457 (81d:05034), 10.1016/00218693(78)90220X
 [CK]
Henry
Cohn and Abhinav
Kumar, Universally optimal distribution of
points on spheres, J. Amer. Math. Soc.
20 (2007), no. 1,
99–148. MR
2257398 (2007h:52009), 10.1090/S0894034706005467
 [CSl]
J.
H. Conway and N.
J. A. Sloane, Sphere packings, lattices and groups, 3rd ed.,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Sciences], vol. 290, SpringerVerlag, New York, 1999.
With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S.
P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447
(2000b:11077)
 [CSm]
John
H. Conway and Derek
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.
M. Goethals, and J.
J. Seidel, Spherical codes and designs, Geometriae Dedicata
6 (1977), no. 3, 363–388. MR 0485471
(58 #5302)
 [KV]
Helmut
Koch and Boris
B. Venkov, Über gerade unimodulare Gitter der Dimension 32.
III, Math. Nachr. 152 (1991), 191–213 (German).
MR
1121234 (92j:11064), 10.1002/mana.19911520117
 [L]
John
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), 237265.
 [BV]
 C. Bachoc and B. Venkov, Modular forms, lattices and spherical designs, Réseaux euclidiens, designs sphériques et formes modulaires, 87111, 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 energyminimizing point configurations on spheres, Experiment. Math. 18 (2009), 257283.
 [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), 257280. MR 523457 (81d:05034)
 [CK]
 H. Cohn and A. Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), 99148. MR 2257398 (2007h:52009)
 [CSl]
 J. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, third edition, Grundlehren der Mathematischen Wissenschaften, 290, SpringerVerlag, 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), 363388. MR 0485471 (58:5302)
 [KV]
 H. Koch and B. Venkov, Über gerade unimodulare Gitter der Dimension . III, Math. Nachr. 152 (1991), 191213. MR 1121234 (92j:11064)
 [L]
 J. Leech, Equilibrium of sets of particles on a sphere, Math. Gaz. 41 (1957), 8190. 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), 237265.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
52B15,
05B40,
52C17,
82B05
Retrieve articles in all journals
with MSC (2010):
52B15,
05B40,
52C17,
82B05
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:
http://dx.doi.org/10.1090/S0002993910102846
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 DMS0757765. The second author was supported in part by NSF grant DMS0501029, and the fourth author was supported in part by Deutsche Forschungsgemeinschaft grant SCHU 1503/42.
Communicated by:
Jim Haglund
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
