Wikipedia geometry item 71: The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the…

Kelvin posed the problem in 1887: divide three-dimensional space into equal-volume cells with the least total boundary area. Weaire and Phelan found a better arrangement in 1993, but its global optimality remains unknown.

Known results

• Kelvin, 1887: proposed a slightly curved truncated-octahedral foam.
• Weaire and Phelan, 1993: found a structure with about $0.3\%$ less area than Kelvin’s, disproving Kelvin’s conjecture.
• Kusner and Sullivan: showed that the polyhedral Weaire–Phelan foam beats every foam with Kelvin’s topology.
• Cesaroni and Novaga: established results for restricted periodic or lattice classes, not the unrestricted problem.

Claimed constrained theorem

A recent arXiv paper claims that the Archimedean truncated octahedron uniquely minimizes area among fixed-volume three-dimensional parallelohedra. This is a narrower class than all partitions and does not prove Weaire–Phelan optimality; the claim is unverified here.

Current status (as of September 2026): The unrestricted minimum and global optimality of the Weaire–Phelan structure remain open; a claimed result for parallelohedra does not settle them.