How many ways are there to arrange N spheres in space, so they form a rigid cluster?

A cluster is rigid if it cannot be continuously deformed by any small but finite amount, and maintain all pairs in contact. Equivalently, it is an isolated solution to the system of equations listing the pairs of spheres that are exactly touching, after fixing the translational and rotational degrees of freedom. This is a nonlinear notion that is a natural set of clusters to consider, because these clusters are metastable when the spheres interact with a short-range potential.

Unfortunately there is no efficient way to test for rigidity computationally; this is in the class of co-MP hard problems. However, there are methods to test for stronger versions. All clusters listed below are prestress stable (to numerical tolerance), a notion which can be tested with semidefinite programming. This is a slightly stronger flavour of rigidity than second-order rigidity, but not as restrictive as first-order rigidity, which is a linear notion of rigidity. For more information on testing rigidity, see

  • M. Holmes-Cerfon, L. Theran, S. Gortler. "Almost-rigidity of frameworks." Communications on Pure and Applied Mathematics, 74(10), pp. 2185-2247, (2020).

Data

The following data was generated by an algorithm that follows all possible one-dimensional transition paths between rigid clusters, as in this publication:

  • M. Holmes-Cerfon. Enumerating rigid sphere packings. SIAM Review, 58(2), pp. 229-244 (2016).

The files (once unzipped) are txt files, with one cluster listed per line.

All coordinates

Ground states (clusters with the maximum number of contacts)

All clusters for N<10 are ground states.

Coordinates for N<=13 were sharpened afterward using Bertini, so the coordinates of all but the hypostatic clusters (those with fewer than 3N-6 contacts) are accurate to floating-point precision. It is possible to sharpen the coordinates for N=14-19 if needed. The hypostatic and singular clusters should be accurate to roughly 3e-8, and the regular clusters for N=14-19 to roughly 9e-16.

For N=15—19, only clusters with a certain minimum number of contacts are listed. The goal here was to find the ground states: those with the maximum number of contacts.

This data appears on the Online Encyclopedia of Integer Sequences (OEID): https://oeis.org/A365709.

A talk about this data can be found here: https://sites.math.rutgers.edu/~zeilberg/expmath/archive23.html