Decorated vertices with 3-edged cells in 2D foams: exact solutions and properties

TL;DR

This paper provides exact analytical solutions for decorated vertices in 2D foams, exploring their shapes, energies, and stability thresholds using a parametric Moebius transformation framework.

Contribution

It introduces a unified analytical approach to describe decorated vertices with various cell types in 2D foams, including stability analysis.

Findings

Exact formulas for energy, area, and excess energy of decorated vertices

A parametric model capturing different decoration types

Insights into the stability threshold in the flower problem

Abstract

The energy, area and excess energy of a decorated vertex in a 2D foam are calculated. The general shape of the vertex and its decoration are described analytically by a reference pattern mapped by a parametric Moebius transformation. A single parameter of control allows to describe, in a common framework, different types of decorations, by liquid triangles or 3-sided bubbles, and other non-conventional cells. A solution is proposed to explain the stability threshold in the flower problem.