```To:             geometry-research@forum.swarthmore.edu
Date:           Mon, 13 Dec 1993 16:02:05 GMT
From:           Evelyn Sander <sander@geom.umn.edu>
Organization:   The Geometry Center, University of Minnesota
Subject:        Kelvin Conjecture Overthrown
```

```I received the following email from Ken Brakke:

I've got some news from the wonderful world of soap bubbles
that I'd like to publicize on the Geometry Forum.
Here's the news:

A long-standing conjecture about soap bubbles has been overthrown.
In 1887, Lord Kelvin pondered how to partition space into cells
of equal volume with the least area of surface between them, i.e.
the most efficient soap bubble froth.  He came up with a 14-sided
space-filling polyhedron with 6 square sides and 8 hexagonal sides.
The faces have to curve a bit where they meet to form the proper
soap film angles.  Now Denis Weaire (dweaire@vax1.tcd.ie) and
Robert Phelan (rphelan@alice.phy.tcd.ie) of Trinity College, Dublin,
have beaten Kelvin.  The Weaire-Phelan structure uses
two kinds of cells, a dodecahedron and a tetrakaidecahedron
with 2 hexagons and 12 pentagons.  The surface area is 0.3% less
than the Kelvin structure, which is a whopping big amount to all
of us who have tried to beat Kelvin over the years.

The Weaire-Phelan structure has cubic symmetry, and the fundamental
region is a 2x2x2 cube.  The 8 cells start as Voronoi cells on centers

0   0   0
1   1   1
0.5 0   1
1.5 0   1
0   1   0.5
0   1   1.5
1   0.5 0
1   1.5 0

There are two dodecahedra (centered at (0,0,0) and (1,1,1)) and six
tetrakaidecahedra. The cells must be adjusted a bit to get exactly equal
volumes, and the faces must curve a bit to get the proper soap film angles.
The tetrakaidecahedra stack on their hexagonal faces to form three sets
of perpendicular interlocking columns, with the interstices filled
by the dodecahedra.

Ken Brakke
brakke@geom.umn.edu
```