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Concept
Weaire–Phelan structure
Summary
In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution of the Kelvin problem of tiling space by equal volume cells of minimum surface area than the previous best-known solution, the Kelvin structure.
In two dimensions, the subdivision of the plane into cells of equal area with minimum average perimeter is given by the hexagonal tiling, but although the first record of this honeycomb conjecture goes back to the ancient Roman scholar Marcus Terentius Varro, it was not proven until the work of Thomas C. Hales in 1999.
In 1887, Lord Kelvin asked the corresponding question for three-dimensional space: how can space be partitioned into cells of equal volume with the least area of surface between them? Or, in short, what was the most efficient soap bubble foam?
This problem has since been referred to as the Kelvin problem.
Kelvin proposed a foam called the Kelvin structure. His foam is based on the bitruncated cubic honeycomb, a convex uniform honeycomb formed by the truncated octahedron, a space-filling convex polyhedron with 6 square faces and 8 hexagonal faces. However, this honeycomb does not satisfy Plateau's laws, formulated by Joseph Plateau in the 19th century, according to which minimal foam surfaces meet at angles at their edges, with these edges meeting each other in sets of four with angles of . The angles of the polyhedral structure are different; for instance, its edges meet at angles of on square faces, or on hexagonal faces. Therefore, Kelvin's proposed structure uses curvilinear edges and slightly warped minimal surfaces for its faces, obeying Plateau's laws and reducing the area of the structure by 0.2% compared with the corresponding polyhedral structure.
Although Kelvin did not state it explicitly as a conjecture, the idea that the foam of the bitruncated cubic honeycomb is the most efficient foam, and solves Kelvin's problem, became known as the Kelvin conjecture.
Official source
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