Weaire–Phelan structure
In geometry, the Weaire–Phelan structure is a complex 3-dimensional structure representing an idealised foam of equal-sized bubbles. In 1993, Trinity College Dublin physicist Denis Weaire and his student Robert Phelan 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.
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 bubble foam? This problem has since been referred to as the Kelvin problem.
He proposed a foam, based on the bitruncated cubic honeycomb, which is called the Kelvin structure. This is the convex uniform honeycomb formed by the truncated octahedron, which is a 14-faced space-filling polyhedron (a tetradecahedron), with 6 square faces and 8 hexagonal faces. To conform to Plateau's laws governing the structures of foams, the hexagonal faces of Kelvin's variant are slightly curved.
The Kelvin conjecture is that this structure solves the Kelvin problem: that the foam of the bitruncated cubic honeycomb is the most efficient foam. The Kelvin conjecture was widely believed, and no counter-example was known for more than 100 years, until it was disproved by the discovery of the Weaire–Phelan structure.
The Weaire–Phelan structure differs from Kelvin's in that it uses two kinds of cells, although they have equal volume.
One is the pyritohedron, an irregular dodecahedron with pentagonal faces, possessing tetrahedral symmetry (Th).
The second is the truncated hexagonal trapezohedron, a species of tetrakaidecahedron with two hexagonal and twelve pentagonal faces, possessing antiprismatic symmetry (D2d).
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