Stable soap bubble clusters with multiple torus bubbles: getting a bit more exotic

TL;DR

This work expands the construction of stable soap-bubble clusters with multiple torus bubbles from Platonic solids to semiregular prisms and Archimedean solids, using a Surface Evolver–driven optimization framework. For prisms with $n$ sides, the approach yields clusters with $3n+8$ bubbles and a torus bubble of genus $n+1$, with concrete realizations for triangular, pentagonal, and hexagonal prisms (genuses $4$, $6$, and $7$ respectively) and extensive stability data. The study further catalogs architectures from truncated and highly symmetric Archimedean solids, producing clusters with up to $188$ bubbles and genus as high as $61$, accompanied by starting geometries, areas, and Hessian-eigenvalue analyses across discretizations. All simulations and starting configurations are openly available, enabling replication and exploration of even more exotic linkers of inner double bubbles, which could pave the way for designing increasingly complex stable bubble networks.

Abstract

Recently, numerical examples of stable soap bubble clusters with multiple torus bubbles have been presented. The geometry of these clusters is based on the Platonic solids whose vertices have valence $3$ (in order to fulfill Plateau's laws): the tetrahedron, the cube, the dodecahedron. The clusters respectively contain a bubble of genus $3, 5, 11$. The construction is quite generic and can be used with any convex polyhedron. If stable, the cluster obtained using a polyhedron with $n$ faces has $3n+2$ bubbles and one of these bubbles has genus $n-1$. We propose here to show that is it possible to get stable soap bubble clusters with multiple torus bubbles using a geometry based on prisms and Archimedean solids as well.

Figures (23)

• Figure 1: $2$D scheme of the starting configuration.
• Figure 2: $2$D schematic view of the stable cluster.
• Figure 3: Starting geometry. $s_{\mathrm{i}}=0.3\;,\;s_{\mathrm{m}}=0.37\;,\;s_{\mathrm{e}}=0.5\;,\;s_{\mathrm{f}}=0.85$.
• Figure 4: Stable cluster. Total area: $4.96$.
• Figure 5: Starting geometry. $s_{\mathrm{i}}=0.35\;,\;s_{\mathrm{m}}=0.45\;,\;s_{\mathrm{e}}=0.6\;,\;s_{\mathrm{f}}=0.8$.