Weaire-Phelan Structure
The best known solution to the Kelvin problem: how to partition three-dimensional space into cells of equal volume with the least total surface area between them. In 1887, Lord Kelvin proposed the truncated octahedron. In 1993, Denis Weaire and Robert Phelan found a structure with 0.3% less surface area — and nobody has beaten it since.
What’s happening
In 1887, Lord Kelvin asked: what arrangement of equal-volume cells fills all of 3D space while minimizing the total surface area between cells? He proposed a solution using a single cell shape — the truncated octahedron (a 14-faced polyhedron with 8 hexagonal faces and 6 square faces), with slightly curved faces to satisfy Plateau’s laws.
For over a century, this was the best known answer. Then in 1993, Denis Weaire and Robert Phelan, physicists at Trinity College Dublin, discovered a better solution using two types of cells:
- Irregular dodecahedra (12 pentagonal faces) — 2 per unit cell, shown in gold
- Tetrakaidecahedra (12 pentagons + 2 hexagons) — 6 per unit cell, shown in blue-silver
The Weaire-Phelan structure has 0.3% less surface area per unit volume than Kelvin’s solution. This might seem tiny, but it solved a 106-year-old conjecture and demonstrated that the optimal foam need not use identical cells.
The structure was famously used as inspiration for the Beijing National Aquatics Center (“Water Cube”) at the 2008 Olympics, whose facade is a slice through the Weaire-Phelan foam.
Whether Weaire-Phelan is truly optimal remains an open problem. Nobody has proved it, and nobody has found anything better.