Geometrical consequences of foam equilibrium

TL;DR

This paper explores the geometric constraints of two-dimensional foam equilibrium, revealing that such foams correspond to sectional multiplicative Voronoi partitions generated by three-dimensional point sources.

Contribution

It establishes that equilibrated foams must be sectional multiplicative Voronoi partitions, linking foam geometry to three-dimensional point sources and reciprocal figures.

Findings

Vertices are aligned due to equilibrium conditions.

Equilibrated foams admit reciprocal figures.

Foams correspond to sectional multiplicative Voronoi partitions.

Abstract

The equilibrium conditions impose nontrivial geometrical constraints on the configurations that a two-dimensional foam can attain. In the first place, the three centers of the films that converge to a vertex have to be on a line, i.e. all vertices are aligned. Moreover an equilibrated foam must admit a reciprocal figure. This means that it must be possible to find a set of points P_i on the plane, one per bubble, such that the segments (P_i P_j) are normal to the corresponding foam films. It is furthermore shown that these constraints are equivalent to the requirement that the foam be a Sectional Multiplicative Voronoi Partition (SMVP). A SMVP is a cut with a two-dimensional plane, of a three-dimensional Multiplicative Voronoi Partition. Thus given an arbitrary equilibrated foam, we can always find point-like sources (one per bubble) in three dimensions that reproduce this foam as a generalized Voronoi partition. These sources are the only degrees of freedom that we need in oder to fully describe the foam.