The Science of Sticky Spheres
On the strange attraction of spheres that like to stick together
At Sixes and Sevens
Up to this point, each value of n has had a unique cluster that maximizes Cn. Furthermore, in each case the best-connected cluster with n+1 spheres can be assembled incrementally by sticking a new sphere somewhere on the surface of the max(Cn) cluster. These properties come to an end at n=6. With six spheres, two cluster shapes both yield the same maximum contact number, C6=12. (Note that a hypothetical six-sphere clique would have 15 contacts.) One of the max(C6) clusters is built incrementally from the five-sphere triangular dipyramid. But the other max(C6) cluster is a “new seed”—a structure that cannot be created simply by gluing a sphere to the surface of a smaller optimum cluster. The new seed is the octahedron (which might also be described as a square dipyramid).
Beyond n=6, the problem of finding all the maximum-contact clusters becomes more daunting. For n=7, the incremental approach of adding another sphere to the surface of an n=6 cluster yields four solutions that have 15 contact points. Three of these C7=15 clusters consist of four tetrahedra glued together face-to-face in various ways. The remaining product of incremental construction consists of an octahedron with a tetrahedron erected on one face. (One of the seven-sphere solutions has both left-handed and right-handed forms, but the convention is to count these “chiral” pairs as variants of a single cluster, not as separate structures.)
Finding this particular set of structures is not especially difficult. If you spend some time playing with Geomags or some other three-dimensional modeling device, you are likely to stumble upon them. But having identified these four clusters with C7=15, how do you know there aren’t more? And how do you prove that no seven-sphere cluster has 16 or more contacts?
As it turns out, 15 is indeed the maximum contact number for seven spheres, but there is another C7=15 cluster. It is a new seed, called a pentagonal dipyramid. With its fivefold symmetry, it has no structural motifs in common with any of the smaller clusters. The novelty of this object again raises the question: How can we ever be sure there aren’t still more arrangements waiting to be discovered?
A successful program for answering such questions was initiated about five years ago by Natalie Arkus, who was then a graduate student at Harvard University. (She is now at Rockefeller University.) In a series of papers written with her Harvard colleagues Michael P. Brenner and Vinothan N. Manoharan, she enumerated all the max(Cn) configurations for n=7 through n=10. The results were later extended to n=11 by Robert S. Hoy, Jared Harwayne-Gidansky and Corey S. O’Hern of Yale University. (Hoy is now on the faculty of the University of South Florida.) All of the results I describe here come from the work of these two groups.