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  "body": "\n## Example\n\nThe *Weaire–Phelan structure* is a periodic partition of $\\R^3$ into cells of equal volume, of two kinds.\n\n1. Irregular dodecahedra, with $12$ pentagonal faces.\n2. Tetrakaidecahedra, with $14$ faces: $2$ hexagons and $12$ pentagons.\n\nA cubic repeating unit contains $8$ cells: $2$ dodecahedra and $6$ tetrakaidecahedra.\nThe faces are slightly curved, so that the cell walls meet in threes at $120^\\circ$ and the edges meet in fours, as [[Plateau's laws|381RB634WS]] require.\n\nThe check for the [[Kelvin problem|CVTKSCP55A]] is a computation that Weaire and Phelan made in 1993.\nFor the same cell volume, the total area of the walls per cell is about $0.3\\%$ less than for Kelvin's partition into truncated octahedra.\nSo this structure shows that Kelvin's proposed answer is not optimal.\n\n## Remarks\n\nWithout curving the faces, the cells form a [[honeycomb|HTJ03RFKFH]] of flat-faced [[polyhedra|0325SGSJAR]] related to the crystal structure of some clathrate hydrates.\nThe structure was the model for the walls of the Beijing National Aquatics Centre built for the 2008 Olympic Games.\nIt is not known whether the Weaire–Phelan structure solves the Kelvin problem.\n\n## Explanations\n\n### L1 human\n\nImagine a foam where every bubble has the same volume.\nKelvin thought the best such foam used one shape of bubble with fourteen faces.\nWeaire and Phelan found, with computer help, a foam mixing two shapes of bubble that uses slightly less soap film.\nIt is still the best known answer.\n",
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