Math 2.0

Convex and discrete geometry Example

Weaire–Phelan structure

Builds on HoneycombPolyhedronPlateau's laws

Example

The Weaire–Phelan structure is a periodic partition of ℝ3\R^3 into cells of equal volume, of two kinds.

  1. Irregular dodecahedra, with 1212 pentagonal faces.
  2. Tetrakaidecahedra, with 1414 faces: 22 hexagons and 1212 pentagons.

A cubic repeating unit contains 88 cells: 22 dodecahedra and 66 tetrakaidecahedra. The faces are slightly curved, so that the cell walls meet in threes at 120∘120^\circ and the edges meet in fours, as Plateau's laws require.

The check for the Kelvin problem is a computation that Weaire and Phelan made in 1993. For the same cell volume, the total area of the walls per cell is about 0.3%0.3\% less than for Kelvin's partition into truncated octahedra. So this structure shows that Kelvin's proposed answer is not optimal.

Remarks

Without curving the faces, the cells form a honeycomb of flat-faced polyhedra related to the crystal structure of some clathrate hydrates. The structure was the model for the walls of the Beijing National Aquatics Centre built for the 2008 Olympic Games. It is not known whether the Weaire–Phelan structure solves the Kelvin problem.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Honeycomb, concept, no evidence levelHoneycombconceptMean curvature, concept, no evidence levelMean curvatureconceptMinimal surface, concept, no evidence levelMinimal surfaceconceptPolyhedron, concept, no evidence levelPolyhedronconceptPlateau's laws, statement, no evidence levelPlateau's lawsstatementWeaire–Phelan structure, example, no evidence levelWeaire–Phelan structu…example
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Honeycomb concept
  2. Mean curvature concept
  3. Minimal surface concept
  4. Polyhedron concept
  5. Plateau's laws statement
  6. Weaire–Phelan structure example

Further reading

  1. Web https://en.wikipedia.org/wiki/Weaire%E2%80%93Phelan_structure
About this page
TagVH7MSMN03Y
URIm2://VH7MSMN03Y
Kindexample
WritingW0 stub
MSC52C22, 49Q10, 52B10
IllustratesKelvin problem
UsesHoneycomb, Polyhedron, Plateau's laws
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