Stable soap bubble clusters with multiple torus bubbles
Fabrice Delbary
TL;DR
This work addresses Almgren’s question by numerically constructing stable soap-bubble clusters that contain torus bubbles of genus $g=3$, $g=5$, and $g=11$. Using Surface Evolver with an immiscible-fluid template, the authors assemble clusters around tetrahedra, cubes, and dodecahedra to enforce Plateau’s laws and achieve energy-minimizing configurations. The results show stable clusters across multiple discretizations, evidenced by positive Hessian eigenvalues and robustness to perturbations, and they provide data and tools for further exploration of complex foam geometries. This numerical evidence supports the existence of stable torus-containing bubble clusters and offers a pathway to investigate higher-genus structures and their mathematical properties.
Abstract
In the last two centuries and more particularly in the last decades, the geometry of foams has become an important research domain, in mathematics, physics, material sciences and biology. Most of the simplest geometrical observations of bubble clusters have long resisted rigorous mathematical proofs. Geometries can even get more complicated if immiscible fluids are considered. Although they have to fulfill Plateau's laws like soap bubble clusters if the surface tensions are close to unity, this is not the case in general. In 1996, Frederick J. Almgren asked whether there is "any stable cluster of bubbles in $\mathbb{R}^3$ with some bubble being topologically a torus". We propose to answer the latter numerically with simple numerical examples. We build stable soap bubble clusters with a triple torus bubble, a fivefold torus bubble or an elevenfold torus bubble. The construction uses the geometry of a simple immiscible fluids cluster with a torus bubble.