Kelvin's Conjecture

Kelvin's conjecture concerns the space-filling arrangement of cells of equal volume with minimal surface area. This question arises naturally in the theory of foams when the liquid content is small. Kelvin (Thomson 1887) proposed that the solution was a 14-sided truncated octahedron having a very slight curvature of the hexagonal faces.

The isoperimetric quotient of the uncurved truncated octahedron is given by

			(1)
			(2)
			(3)

while Kelvin's slightly curved variant has a slightly larger isoperimetric quotient of 0.757.

Despite one hundred years of failed attempts and Weyl's (1952) opinion that the curved truncated octahedron could not be improved upon, Weaire and Phelan (1994) discovered a space-filling unit cell consisting of six 14-sided polyhedra and two 12-sided polyhedra with irregular faces and only hexagonal faces remaining planar. This structure has an isoperimetric quotient of 0.765, or approximately 1.0% more than Kelvin's cell.

The building for water events at the 2008 Beijing Olympics had a structure based on Weaire-Phelan foam (Rehmeyer 2008, Beijing Organizing Committee of the Olympiad 2008).

The truncated octahedral conjecture restricts the minimization to three-dimensional parallelohedra, which are convex polyhedra tiling by translations. Hales and Song (2026) and independently Cesaroni and Novaga (2026) proved that, within this class, the Archimedean truncated octahedron uniquely minimizes surface area at fixed volume, up to similarity. This does not establish optimality for general foams.