The binary sphere packing problem
What is the densest way to fill space with spheres of the same size? Is there a way to do it that is denser than the obvious regular way to stack cannonballs on a square base, as shown in the picture above? According to the Kepler conjecture, posed by the mathematician and astronomer Johannes Kepler in 1611, the answer is “no”. The Kepler conjecture was confirmed to be correct in 1998 by Thomas Hales, using a computer assisted proof that was formally checked in 2014.
The cannonball stack is known in crystallography as a face-centred cubic (fcc) lattice, and stacking the balls using a triangular base instead of a square base achieves an equivalent result. Another regular way to stack spheres is the hexagonal close-packed (hcp) lattice, which arises if the spheres are stacked on a hexagonal base, like the snowballs shown above.
The diagram below illustrates a structural difference between the two arrangements: the layers in the fcc arrangement follow an ABCABC… repeating pattern, and the layers in the hcp arrangement follow an ABABAB… repeating pattern. Both these packings have an average density of π/√18, which is around 0.74048. This means that a large number of spheres packed in a fcc or hcp arrangement will occupy just over 74% of the total space available.
Carl Friedrich Gauss (1777–1855) proved that no lattice packing of spheres (meaning a regular, repeating pattern) can achieve a higher average density than π/√18. It is possible to change the order of the ABC layers in the fcc packing to create a disordered arrangement that also achieves this highest possible average density. However, the proof of the Kepler conjecture shows that it is not possible for any disordered arrangement of spheres to achieve an average density that is strictly greater than π/√18.
The binary packing problem is the problem of filling as much as possible of three dimensional space by using two types of spheres of different radii. The picture above shows part of a salt crystal, with the smaller sodium ions shown in purple, and the larger chloride ions shown in green. Each sodium ion sits in the middle of an octahedron formed by the nearest six chloride ions, and vice versa. Although ions are not literally rigid spheres, this arrangement suggests that if we use the small spheres to fill as much as possible of the octahedral holes between the big spheres, then this is likely to result in a dense packing.
If purple spheres in the sodium chloride packing were smaller, they would allow the green spheres to touch each other. More precisely, if the ratio of the big radius to the small radius is 1 to √2–1 (about 1 to 0.4142), then the small spheres fill out as much space as possible in the octahedral holes between the big spheres, as shown above. This arrangement of spheres has an average density of (5/3–√2)π, which is about 0.793. In other words, using these two sizes of spheres, it is possible to fill at least 79.3% of three dimensional space. The binary packing problem remains unsolved, but it is possible that this arrangement represents the optimal density.
The picture above illustrates the octahedral holes in a packing of hcp type, by showing what happens when a new layer of the packing is filled in. Each new sphere creates a tetrahedron (shown in orange, on the right) with three of the spheres from the previous layer, but doing this only uses half of the available holes. Each of the other holes becomes the centre of an octahedron consisting of three spheres in the old layer and three spheres in the new layer.
The contact graph of a packing is formed by taking the centres of the spheres as vertices, and joining a pair of vertices by an edge if the spheres touch each other. In many cases of sphere packings where the maximum density has been proven, the contact graph turns out to be a network of simplices, or n-dimensional tetrahedra. For example, the contact graph of the densest packing of circles in the plane, shown above, forms a network of triangles. In 2019, Thomas Fernique proved that in order for the contact graph of a binary sphere packing to be a network of tetrahedra, it is necessary for the two radii to have a 1 to √2–1 ratio, which is further evidence that binary packings with a 1 to √2–1 radio are especially interesting.
I found out about binary sphere packing from the recent paper Bounding the density of binary sphere packing by Thomas Fernique and Daria Pchelina. The authors use a computer-assisted proof to derive the best known upper bound for the average density of a binary sphere packing with radii in the ratio 1 to √2–1. Their bound is approximately 0.8125, which implies that there is certainly no way to fill more than 81.3% of three dimensional space by using spheres with radii in this ratio.
Picture credits and relevant links
The cannonball photograph is by Funkdooby and appears on the Wikipedia page on the cannonball problem.
The snowball photograph is by Yvette Cendes and appears on the Wikipedia page on close packing of equal spheres.
The green, blue, and purple pictures of sphere packings are by Cdang and Muskid and appear on the Wikipedia page on the Kepler conjecture.
The diagrams of sodium chloride crystals are by Goran tek-en and appear on Wikipedia.
The other three pictures come from the paper by Thomas Fernique and Daria Pchelina.
Substack management by Buzz & Hum.