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Point configurations that are asymmetric yet balanced. (English) Zbl 1201.52017

Summary: 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.

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
52B15 Symmetry properties of polytopes
05B40 Combinatorial aspects of packing and covering
82B05 Classical equilibrium statistical mechanics (general)
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[1] 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. · Zbl 1061.11035
[2] 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. · Zbl 1185.68771
[3] A. E. Brouwer, Parameters of strongly regular graphs, tables published electronically at http://www.win.tue.nl/ aeb/graphs/srg/srgtab.html. · Zbl 0538.05024
[4] A. E. Brouwer, Paulus graphs, tables published electronically at http://www.win.tue.nl/ aeb/graphs/Paulus.html.
[5] 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. · Zbl 0444.05045 · doi:10.1016/0021-8693(78)90220-X
[6] Henry Cohn and Abhinav Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), no. 1, 99 – 148. · Zbl 1198.52009
[7] 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, Springer-Verlag, 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. · Zbl 0915.52003
[8] John H. Conway and Derek A. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Ltd., Natick, MA, 2003. · Zbl 1098.17001
[9] P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), no. 3, 363 – 388. · Zbl 0376.05015
[10] Helmut Koch and Boris B. Venkov, Über gerade unimodulare Gitter der Dimension 32. III, Math. Nachr. 152 (1991), 191 – 213 (German). · Zbl 0736.11022 · doi:10.1002/mana.19911520117
[11] John Leech, Equilibrium of sets of particles on a sphere, Math. Gaz. 41 (1957), 81 – 90. · Zbl 0080.14106 · doi:10.2307/3610579
[12] G. Nebe and N. Sloane, A catalogue of lattices, tables published electronically at http://www.research.att.com/ njas/lattices/.
[13] A. J. L. Paulus, Conference matrices and graphs of order \( 26\), Technische Hogeschool Eindhoven, report WSK 73/06, Eindhoven, 1973, 89 pp. · Zbl 0275.05123
[14] 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. · JFM 35.0790.01
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