The Space of Planar Soap Bubble Clusters

TL;DR

This paper investigates the mathematical structure of planar soap bubble clusters, showing that those with positive second variation form an n-dimensional manifold, and explores their geometric properties and realizations.

Contribution

It characterizes the space of planar soap bubble clusters, proving it forms an n-dimensional manifold under certain conditions and analyzing its geometric structure.

Findings

Clusters with positive second variation form an n-dimensional manifold.

The larger space of clusters can have singularities.

Connections to generalized Voronoi partitions are discussed.

Abstract

Soap bubbles and foams have been extensively studied by scientists, engineers, and mathematicians as models for organisms and materials, with applications ranging from extinguishing fires to mining to baking bread. Here we provide some basic results on the space of planar clusters of n bubbles of fixed topology. We show for example that such a space of clusters with positive second variation is an n-dimensional manifold, although the larger space without the positive second variation assumption can have singularities. Earlier work of Moukarzel showed how to realize a cluster as a generalized Voronoi partition, though not canonically.