A new counter-example to Kelvin's conjecture on minimal surfaces

Abstract

A new counter-example to Kelvin's conjecture on minimal surfaces has been found. The conjecture stated that the minimal surface area partition of space into cells of equal volume was a tiling by truncated octahedra with slightly curved faces (K). Weaire and Phelan found a counter-example whose periodic unit includes two different tiles, a dodecahedron and a polyhedron with 14 faces (WP). Successively, Sullivan showed the existence of an infinite number of partitions by polyhedra having only pentagonal and hexagonal faces that included WP, the so-called tetrahedrally close packed structures (TCP). A part of this domain contains structures with lower surface area than K. Here, we present a new partition with lower surface area than K, the first periodic foam containing in the same structure quadrilateral, pentagonal and hexagonal faces, in ratios that are very close to those experimentally found in real foams by Matzke. This and other new partitions have been generated via topological modifications of the Voronoi diagram of spatially periodic sets of points obtained as local maxima of the stationary solution of the 3D Swift–Hohenberg partial differential equation in a triply periodic boundary, with pseudorandom initial conditions. The motivation for this work is to show the efficacy of the adopted method in producing new counter-examples to Kelvin's conjecture, and ultimately its potential in discovering a periodic partition with lower surface area than the Weaire–Phelan foam. The method seems tailored for the problem examined, especially when compared to methods that imply the minimization of a potential between points, where a criterion for neighboring points needs to be defined. The existence of partitions having a lower surface area than K and an average number of faces greater than the maximum value allowed by the TCP domain of 13.5 suggests the presence of other partitions in this range.

Keywords:

• tetrahedra packing
• bubbles
• crystal geometry
• foams

Acknowledgements

The author wishes to thank Michael Cross. His java demo 35 of the Swift–Hohenberg equation provided the idea for this work. Thanks are also due to Ken Brakke for the support received on the Surface Evolver, John Sullivan for the scripts used to evaluate foam properties, David Lloyd for the Matlab solver of the two-dimensional case of the Swift–Hohenberg equation and Davide Proserpio and Keri Collins for the many suggestions received on the first draft of this manuscript. Research has been supported by the University of Bath and by a Flexible Award within the EPSRC Bridging the Gaps Project between the Department of Mathematical Sciences and the Department of Mechanical Engineering at the University of Bath.