The Science of Sticky Spheres
On the strange attraction of spheres that like to stick together
Eight, Nine, Ten
The results of the survey of sticky-sphere clusters are summarized in the table at right. Arkus and her colleagues determined max(Cn)—and identified all clusters that exhibit these highest contact numbers—for all n≤10. Along the way, they discovered quite a few clusters with interesting quirks and personalities.
At n=8 the maximum contact number is 18, and there are 13 distinct ways of achieving this bound. All but one of the clusters can be built up incrementally by attaching a new sphere to the surface of one of the n=7 clusters.
Clusters of nine spheres have up to 21 contacts; there are 52 varieties, including four new seeds. In this crowd of sphere packings, one stands out from all the rest. It has a property not seen in any other max(Cn) cluster up to this point: flexibility. The structure can be twisted around one axis without breaking any bonds between spheres. This ability to wiggle may seem surprising, given that the adjacency matrices were designed with the explicit aim of ensuring “minimal rigidity.” But there’s a reason this form of rigidity is called minimal. The requirement that every sphere make contact with at least three others implies that no individual sphere can move relative to the rest of the cluster without breaking at least one bond. But other modes of motion, in which larger groups of spheres flex or rotate, are not ruled out. In the flexible n=9 cluster, two square faces joined by an edge can twist and deform slightly. The animation at right shows the flexibility.
The n=10 clusters cross another threshold: For the first time the number of contacts exceeds 3n–6 (which is 24 in this case). Some 259 clusters of 10 spheres have exactly 24 contacts, but another three clusters have 25 each. Again, it’s mildly surprising that these objects find their way onto the list. The search algorithm begins with a list of matrices that specify exactly 3n–6 adjacencies, so how can the search uncover a cluster with even more contacts? The explanation is that even if two spheres are not required to touch, they are not forbidden to do so. When you solve a system of equations that specifies 24 adjacencies, it can happen, as if by coincidence, that a 25th pair of spheres is also at a distance of exactly 1.
One of these happy accidents is shown in the animation at right. The structure is derived from the flexible n=9 cluster by adding an octahedral cap to one of the square faces. (The addition eliminates the flexibility.)