Intrinsic Geometry of Hard Disk Clusters

Abstract

Put \(n\) identical coins on a table with no two overlapping. Which arrangement makes the perimeter of the convex hull of the cluster as small as possible? Despite its elementary statement, the solution of this problem is known only up to four disks. We produce a calculus for hard disk clusters of arbitrary finite size, providing class criticality conditions, first order descent tests, and second order spectral criteria for perimeter minimisation. A central difficulty is that the perimeter formula changes with the hull combinatorics, while the admissible first order geometry changes with the realised contacts. Our approach is guided by the principle that the realised geometry intrinsically determines both the local form of the functional and the admissible motions. As an application, this article takes the first step beyond four disks by providing a solution for the five disk case. The minimum perimeter is \(10+2π\), attained in exactly three realised classes. Two admit perimeter preserving flexes, of dimensions one and two, while the third is rigid modulo rigid motions. The first order theory provides pruning criteria, and the reduced admissible space together with its intrinsic Hessian distinguish rigidity, second order instability, and perimeter flat degeneracy.