Authors: Flikkema, Edwin and Cox, Simon

DOI:

Abstract:

Candidates to the least perimeter partition of various polygonal shapes into N planar connected equal-area regions are calculated for N ≤ 42, compared to partitions of the disc, and discussed in the context of the energetic groundstate of a two-dimensional monodisperse foam. The total perimeter and the number of peripheral regions are presented, and the patterns classified according to the number and position of the topological defects, that is non-hexagonal regions (bubbles). The optimal partitions of an equilateral triangle are found to follow a pattern based on the position of no more than one defect pair, and this pattern is repeated for many of the candidate partitions of a hexagon. Partitions of a square and a pentagon show greater disorder. Candidates to the least perimeter partition of the surface of the sphere into N connected equal-area regions are also calculated. For small N these can be related to simple polyhedra and for N ≥ 14 they consist of 12 pentagons and N −12 hexagons.

The surface energy of a two-dimensional foam is simply its perimeter multiplied by surface tension (Weaire and Hutzler, 1999). A foam attains a local minimum of this perimeter, subject to the constraint of fixed bubble volumes. Here, we seek the arrangement of bubbles that gives the global minimum. The local structure of perimeter-minimizing bubble clusters is well defined: perimeter minimization implies Plateau’s rules (Plateau, 1873; Taylor, 1976): three and only three edges meet at a point at 120◦. The Laplace Law relating pressure difference and curvatures gives the further condition that each edge is a circular arc.

For bubbles tiling the plane, the hexagonal honeycomb gives the least perimeter (Hales, 2001). Here, we tackle the problemof tiling the sphere, for which, because of the curvature, non-hexagonal bubblesmust be introduced. We assume that each bubble is connected (Morgan, 2000) and seek the least perimeter partition of the sphere into N cells of equal area, equivalent to the energetic groundstate for N monodisperse bubbles or the optimal packing of equal-area objects. We examine values of N up to 32 and record the least perimeter and the configuration that realizes it.

Link to Paper