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

Technical Deep Dive: Stable soap bubble clusters with multiple torus bubbles

Overview

This work explores the construction of stable soap bubble clusters with multiple torus (donut-shaped) bubbles. The key findings are:

• Stable soap bubble clusters containing multiple torus bubbles can be created using geometries based on prisms and Archimedean solids, in addition to the previously demonstrated Platonic solids.
• The number of bubbles in the clusters and the genus (number of holes) of the torus bubbles depends on the specific polyhedron used as the starting geometry.
• The stability of the clusters is evaluated by computing the condition number of the Hessian matrix, which measures the sensitivity of the system to perturbations.

Methodology

The authors use the same general approach as their previous work on Platonic solids:

1. Define a starting geometry using a polyhedron (prism or Archimedean solid) with vertices of valence 3 to satisfy Plateau's laws.
2. Create three scaled versions of the polyhedron (inner, middle, and outer) to define the different bubble regions.
3. Use right prisms with the faces of the inner polyhedron as bases to define the inner double bubbles.
4. Minimize the surface area of the system under volume constraints using the Surface Evolver software.

The authors tested a variety of polyhedra, including triangular, pentagonal, and hexagonal prisms, as well as several Archimedean solids (truncated tetrahedron, truncated cube, truncated octahedron, great rhombicuboctahedron, truncated dodecahedron, truncated icosahedron, and great rhombicosidodecahedron).

Results

The key results for each polyhedron tested are:

• Triangular prism: 17 bubbles, torus bubble genus 4
• Pentagonal prism: 23 bubbles, torus bubble genus 6
• Hexagonal prism: 26 bubbles, torus bubble genus 7
• Truncated tetrahedron: 26 bubbles, torus bubble genus 7
• Truncated cube: 44 bubbles, torus bubble genus 13
• Truncated octahedron: 44 bubbles, torus bubble genus 13
• Great rhombicuboctahedron: 80 bubbles, torus bubble genus 25
• Truncated dodecahedron: 98 bubbles, torus bubble genus 31
• Truncated icosahedron: 98 bubbles, torus bubble genus 31
• Great rhombicosidodecahedron: 188 bubbles, torus bubble genus 61

The authors also computed the condition number of the Hessian matrix for each configuration, which provides a measure of the stability. Lower condition numbers indicate more stable systems.

Limitations & Uncertainties

• The authors note that for prisms with more than 6 sides, the bubbles associated with the n-gon faces become much larger than the side bubbles. They suggest exploring non-regular prisms or using different scaling factors for the faces to address this.
• The paper does not provide a comprehensive exploration of all possible polyhedra or discuss the theoretical limits on the complexity of achievable bubble cluster configurations.
• The stability analysis focused on the Hessian matrix, but other factors like kinetic stability are not considered.

Future Work

The authors suggest several directions for future work:

• Investigate ways to further tune the bubble volumes and sizes, such as using irregular polyhedra or varying the scaling factors for different faces.
• Explore whether the concept of "inner double bubbles as linkers" can be generalized to build even more complex bubble cluster configurations.
• Analyze the kinetic stability of the bubble clusters in addition to the static stability considered in this work.