Kelvin's Conjecture

What space-filling arrangement of similar cells of equal volume has minimal surface area? This questions 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 the uncurved truncated octahedron is given by

			(1)
			(2)
			(3)

while Kelvin's slightly curved variant has a slightly less optimal 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).