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
KEYWORD
nonn,more
AUTHOR
Michael Kiessling, Sep 16 2023
STATUS
approved