Minimal-Energy Clusters of Hard Spheres 

N. J. A. Sloane(*), R. H. Hardin, T. D. S. Duff 
Mathematical Sciences Research Center 
AT&T Bell Laboratories 
Murray Hill, New Jersey 07974 

and 

J. H. Conway 
Mathematics Department 
Princeton University 
Princeton, New Jersey 08544 

(*) Present address:
Information Sciences Research
AT&T Shannon Lab
Florham Park, NJ 07932-0971 USA
Email: njas@research.att.com


ABSTRACT

What is the tightest packing of N equal nonoverlapping spheres,
in the sense of having minimal energy, i.e. smallest second
moment about the centroid?
The putatively optimal arrangements are described for N <= 32.
A number of new and interesting polyhedra arise.


Abstract of a related talk:

The Tightest Packings (or Clusters) of N Balls
N. J. A. Sloane

What is the tightest packing of N unit balls, in the sense
of minimal second moment? In 2 dimensions (Graham and Sloane,
Penny-Packing ..., Discrete Comp. Geom., 1990) the answers are
subsets of the best lattice packing, but this is definitely not
so in 3 dimensions.  The putatively optimal configurations
are in fact surprising: 7 = vertices of pentagonal bipyramid,
8 = two meshing C-curves (the French kiss), 9 = tetrakis triangular
prism, 10 = tetrakis square antiprism, 11 and 12 are subsets of
13 = center of a sphere plus 12 points on boundary in rings of
1+5+1+4+1 around N pole, etc.  19 is not the obvious arrangement
formed from 3 shells of the f.c.c. lattice.


This paper was published (in a somewhat different form) in
Discrete Computational Geom., 14 (1995), 237-259.


For the full version see
http://NeilSloane.com/doc/duff.pdf (pdf) or
http://NeilSloane.com/doc/duff.ps (ps)

[note:
<a href="http://NeilSloane.com/doc/dufffig11.ps"> Fig. 11 </a>,
<a href="http://NeilSloane.com/doc/dufffig12.ps"> Fig. 12 </a>
and "photos" of the putatively optimal clusters of
<a href="http://NeilSloane.com/doc/clust4_10.gif"> 4 to 10</a> and
<a href="http://NeilSloane.com/doc/clust13_20.gif"> 13 to 20</a>
spheres are in separate files].