Weaire–Phelan structure

Weaire–Phelan structureSpace groupFibrifold notationCoxeter notationPmn (223)2o[[4,3,4]+]
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.

The Weaire–Phelan structure differs from Kelvin's in that it uses two kinds of cells, although they have equal volume. Like the cells in Kelvin's structure, these cells are combinatorially equivalent to convex polyhedra. One is a pyritohedron, an irregular dodecahedron with pentagonal faces, possessing tetrahedral symmetry (Th). The second is a form of truncated hexagonal trapezohedron, a species of tetrakaidecahedron with two hexagonal and twelve pentagonal faces, in this case only possessing two mirror planes and a rotoreflection symmetry. Like the hexagons in the Kelvin structure, the pentagons in both types of cells are slightly curved. The surface area of the Weaire–Phelan structure is 0.3% less than that of the Kelvin structure. The tetrakaidecahedron cells, linked up in face to face chains of cells along their hexagonal faces, form chains in three perpendicular directions. A combinatorially equivalent structure to the Weaire–Phelan structure can be made as a tiling of space by unit cubes, lined up face to face into infinite square prisms in the same way to form a structure of interlocking prisms called tetrastix. These prisms surround cubical voids which form one fourth of the cells of the cubical tiling; the remaining three fourths of the cells fill the prisms, offset by half a unit from the integer grid aligned with the prism walls. Similarly, in the Weaire–Phelan structure itself, which has the same symmetries as the tetrastix structure, 1/4 of the cells are dodecahedra and 3/4 are tetrakaidecahedra. The polyhedral honeycomb associated with the Weaire–Phelan structure (obtained by flattening the faces and straightening the edges) is also referred to loosely as the Weaire–Phelan structure. It was known well before the Weaire–Phelan structure was discovered, but the application to the Kelvin problem was overlooked.

Experiments have shown that, with favorable boundary conditions, equal volume bubbles spontaneously self assemble into the Weaire–Phelan structure. The associated polyhedral honeycomb is found in two related geometries of crystal structure in chemistry. Where the components of the crystal lie at the centres of the polyhedra it forms one of the Frank–Kasper phases, the A15 phase. Where the components of the crystal lie at the corners of the polyhedra, it is 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 hydrides silicides and germanides also form this structure, with silicon or germanium at nodes, and alkali metals in cages.

The Weaire–Phelan structure is the inspiration for the design by Tristram Carfrae of the Beijing National Aquatics Centre, the 'Water Cube', for the 2008 Summer Olympics.