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# Weaire-Phelan structure

The Weaire-Phelan structure is a complex 3-dimensional structure. In 1993, Denis Weaire and Robert Phelan, two physicists based at Trinity College Dublin found that in computer simulations of foam, this structure was a better solution of the "Kelvin problem" than the previous best-known solution, the Kelvin structure. [Weaire D and Phelan R, "A counterexample to Kelvin's conjecture on minimal surfaces", Phil. Mag. Lett. 69, 107-110 (1994)]

Kelvin structure

In 1887, Lord Kelvin asked how space could be partitioned into cells of equal volume with the least area of surface between them, i.e., what was the most efficient soap bubble foam? [ [http://zapatopi.net/kelvin/papers/on_the_division_of_space.html "On the Division of Space with Minimum Partitional Area", By Lord Kelvin (Sir William Thomson),Philosophical Magazine, Vol. 24, No. 151, p. 503 (1887)] ] This problem has since been referred to as the Kelvin problem.

He proposed the foam of bitruncated cubic honeycomb, which is called the Kelvin structure. This is the convex uniform honeycomb formed by the truncated octahedron, which is a 14-sided space-filling polyhedron (a tetrakaidecahedron), with 6 square sides and 8 hexagonal sides. To conform to Plateau's laws governing the structures of foams, the hexagonal faces are slightly curved.

The Kelvin conjecture is that the Kelvin structure solves the Kelvin problem: that the foam of the bitruncated cubic honeycomb is the most efficient foam. The Kelvin conjecture was believed and no counter-examples were known for more than 100 years, until it was disproved by the discovery of the Weaire-Phelan structure.

Compare with the Kepler conjecture (on the densest packing of spheres), which is generally considered to have been proven in 1998.

Description of Weaire-Phelan structure

The Weaire-Phelan structure uses two kinds of cells of equal volume; an irregular pentagonal dodecahedron and a tetrakaidecahedron with 2 hexagons and 12 pentagons, again with slightly curved faces. The surface area is 0.3% less than the Kelvin structure, quite a large difference in this context. It has not been proved that the Weaire-Phelan structure is optimal, but it is generally believed to be likely: the Kelvin problem is still open, but the Weaire-Phelan structure is conjectured to be the solution.

Clathrate structure

The honeycomb associated to the Weaire-Phelan structure (obtained by flattening the faces and straightening the edges) is also referred to loosely as the Weaire-Phelan structure, and it was known well before the Weaire-Phelan structure was discovered, but the application to the Kelvin problem was overlooked. [A diagram can be found incite book
last=Pauling
first=Linus
title=The Nature of the Chemical Bond
year=1960
publisher=Cornell University Press
edition=3rd
page=471
, as shown on [http://www.susqu.edu/brakke/kelvin/Pauling.htm Ken Brakke's page] .
]

It is found as a crystal structure in chemistry where it is usually known as the 'Type I clathrate structure'. Gas hydrates formed by methane, propane and carbon dioxide at low temperatures have a structure in which water molecules lie at the nodes of the Weaire-Phelan structure and are hydrogen bonded together, and the larger gas molecules are trapped in the polyhedral cages.

Some alkali metal silicides and germanides also form this structure (Si/Ge at nodes, alkali metals in cages), as does the silica mineral melanophlogite (silicon at nodes, linked by oxygen along edges). Melanophlogite is a metastable form of SiO2 that is stabilized in this structure because of gas molecules trapped in the cages. The International Zeolite Association uses the symbol MEP to indicate the framework topology of melanophlogite.

Applications

The Weaire-Phelan structure is the inspiration for the design of the aquatic centre for the 2008 Olympics in Beijing in China. [Henry Fountain: [http://www.nytimes.com/2008/08/05/sports/olympics/05swim.html A Problem of Bubbles Frames an Olympic Design] , "New York Times", August 5, 2008; retrieved 2008 October 9.]

* Kepler conjecture
* Minimal surface
* Soap bubble
* Beijing National Aquatics Centre

References

* [http://www.vongirsewald.com/foam/ Weaire-Phelan structure unfolded dodecahedron and tetrakaidecahedron in .pdf / .dxf formats]
* [http://www.mathaware.org/mam/03/images/sullivan/phw00000.jpgAn image of the Weaire-Phelan structure]
* [http://www.steelpillow.com/polyhedra/wp/wp.htm Weaire-Phelan Bubbles] page with illustrations and freely downloadable 'nets' for printing and making models.

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