Star-triangle equivalence in soap froths

TL;DR

This paper proves a theorem linking triangular bubbles and 3-fold stars in 2D foams at equilibrium, and explores related properties of gas diffusion and bubble configurations.

Contribution

It introduces a novel theorem connecting triangles and stars in soap froths using Moukarzel's duality, and investigates gas flow effects on bubble transformations.

Findings

Triangular bubbles can be exchanged with 3-fold stars at equilibrium.

T2 processes are continuous for triangles but not for other bubbles.

Gas flow influences bubble configurations differently depending on bubble type.

Abstract

In two dimensional foams at equilibrium, triangular bubbles can be freely exchanged with 3-fold stars --three edges ending at a central vertex. This theorem is deduced here from Moukarzel's duality. Moreover, to probe the method, a few related properties are established: under slow gas diffusion, T2 processes are continuous for triangles but not for other types of bubbles. In general, the gas flow results in different configurations in the presence of a triangle than in the presence of a star.