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A365709
Number of rigid configurations of n non-overlapping congruent spheres in three-dimensional Euclidean space.
0
1, 1, 1, 1, 1, 2, 5, 13, 52
OFFSET
1,6
COMMENTS
A configuration of n non-overlapping congruent spheres in three-dimensional Euclidean space is called ``rigid'' if the Galilean ``rigid-body motions'' of that configuration are the only motions of this system of n spheres that do not change the number of contact points of the configuration. A system consisting of a single sphere is thus rigid. Rigid as well as non-rigid configurations exist when n>1: for example a configuration of two congruent spheres in contact is rigid, while two congruent spheres not in contact with each other are not in a rigid configuration; and so on. The sequence of numbers listed above consecutively from n=1 to n=9 has mainly been computer-generated, originally by Arkus et al. (reference given below) with the help of computer algebra, and more recently by Miranda Holmes-Cerfon (references given below) using a dynamical algorithm, and agrees with the pertinent computer-generated results found in a reference by other authors (also given below). The papers by Holmes-Cerfon start listing putatively complete number counts at n=5, respectively n=3, and give these number counts consecutively not only up to n=9, but also for n = 10, ..., 14, namely: 263 for n=10, 1659 for n=11, 11980 for n=12, 98529 for n=13, and 895478 for n=14; however, already for n=10 there are differences compared to the number counts by other authors. Holmes-Cerfon points out that her algorithm may miss some difficult configurations.
REFERENCES
N. Arkus, V. N. Manoharan, and M. P. Brenner, ``Deriving finite sphere packings,'' SIAM J. Discrete Math., vol. 25 (2011), pp. 1860-1901.
R. Hoy, J. Harwayne-Gidansky, and C. O'Hern, ``Structure of finite sphere packing via exact enumeration: Implications for colloidal crystal nucleation,'' Phys. Rev. E, vol. 85 (2012), art.051403.
M. C. Holmes-Cerfon, ``Enumerating Rigid Sphere Packings,'' SIAM Review, vol.58 (2016), no.2, pp. 229-244.
M. Holmes-Cerfon, ``Sticky-Sphere Clusters,'' Annual Review of Cond. Matter Phys. vol.8 (2017), pp. 77-98.
LINKS
Miranda Holmes-Cerfon, Sphere packings, singularities, and statistical mechanics, Experimental Mathematics Seminar, Rutgers University, Nov. 30, 2023.
FORMULA
No generating formula seems to be known.
CROSSREFS
Sequence in context: A082938 A303792 A059103 * A260709 A112836 A353719
KEYWORD
nonn,more
AUTHOR
Michael Kiessling, Sep 16 2023
STATUS
approved