### The Minimal Perimeter for $N$ Confined Deformable Bubbles of Equal Area

#### Abstract

Candidates to the least perimeter partition of various polygonal shapes into $N$ planar connected equal-area regions are calculated for $N\le 42$, 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 $N \ge 14$ they consist of 12 pentagons and $N-12$ hexagons.