The Kelvin problem
Some time ago my Great Uncle Juan got interested in an unsolved math problem known as the Kelvin problem. This is the three dimensional version of the two dimensional honeycomb conjecture, which considers the following question: What shape tessellates a plane into equal area subdivisions, with the least perimeter? The answer happens to be the regular hexagon, proven in 1999. The first shape that might have come to your mind is the circle, if you knew that the circle has the least perimeter for a given area. Circles however cannot tessellate the plane. Hexagons result from packing and squeezing circles as tightly as possible to fill the empty space. We say that regular hexagons pack the two dimensional plane the most efficiently.
We can define a measure of efficiency, called the isoperimetric quotient Q = 4π A/P^2, where A is the area and P is the perimeter of the shape tessellating the plane. For a given shape, Q is always less than or equal 1, and only equal to 1 for a circle. The regular hexagon has a Q = π /2√3 = 0.907.
In three dimensions, the analogous question to ask is: What shape fills/tessellates 3d space into equal volume subdivisions, with the least surface area? To the surprise of both myself and my Uncle, this problem is still unsolved!
This problem was first considered in 1887 by Lord Kelvin, who conjectured that the shape which most efficiently packs 3d space is the truncated octahedron, which is obtained by taking an octahedron and chopping off its pointy vertices. We may define a 3d isoperimetric quotient Q = 36π V^2/A^3, which is always less than or equal to 1, and only equal to 1 for the sphere. The truncated octahedron has a Q = 0.753. This shape certainly appears to be the natural generalization of the regular hexagon. One can imagine packing spheres tightly and squeezing them together, producing truncated octahedra.
Is this the most efficient shape?
A lattice of truncated octahedra
It is not. More than 100 years later in 1993, a counterexample to this conjecture was found! It is known as the Weaire-Phelan structure, named after the discoverers. This shape is found naturally in a cubic clathrate lattice structure of Na8Si46. It involves two types of polyhedra, the pyritohedron, and an irregular dodecahedron of equal volume. This shape has a Q = 0.761, about 1% more efficient than the truncated octahedron. As of 2024, this is still the most efficient way to pack 3d space into equal volume subdivisions.
The fundamental unit of the Weaire-Phelan structure
Let's discuss some of the physics behind these shapes.