---
tag: VH7MSMN03Y
kind: example
name: Weaire–Phelan structure
msc: [52C22, 49Q10, 52B10]
illustrates: [CVTKSCP55A]
uses: [HTJ03RFKFH, 0325SGSJAR, 381RB634WS]
sources:
  - url: https://en.wikipedia.org/wiki/Weaire%E2%80%93Phelan_structure
same_as:
  wikidata: Q869539
  wikipedia: Weaire–Phelan structure
---

## Example

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

1. Irregular dodecahedra, with $12$ pentagonal faces.
2. Tetrakaidecahedra, with $14$ faces: $2$ hexagons and $12$ pentagons.

A cubic repeating unit contains $8$ cells: $2$ dodecahedra and $6$ tetrakaidecahedra.
The 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.

The check for the [[Kelvin problem|CVTKSCP55A]] 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 $0.3\%$ 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|HTJ03RFKFH]] of flat-faced [[polyhedra|0325SGSJAR]] 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.

## Explanations

### L1 human

Imagine a foam where every bubble has the same volume.
Kelvin thought the best such foam used one shape of bubble with fourteen faces.
Weaire and Phelan found, with computer help, a foam mixing two shapes of bubble that uses slightly less soap film.
It is still the best known answer.