Sphere packings in 8 and 24 dimensions
How densely can spheres be packed? In three dimensions Hales proved it in 1998, in eight and 24 dimensions Maryna Viazovska in 2016. There lie the E8 lattice and the Leech lattice, and the number 24.
How densely can equal spheres be packed? In the plane it is six circles around one, in space Kepler’s packing with twelve spheres around one (see Kissing numbers and 24). That nothing denser is possible was conjectured by Kepler in 1611 and only proved by Thomas Hales in 1998, with the help of computers (see From point to solid).
The question can be asked for any number of dimensions, and it is no game: packing spheres in high dimensions is exactly the problem of finding error-correcting codes with which data is sent over noisy lines. What is surprising is how little is known. Apart from dimensions 1, 2 and 3, the densest packing is proved in only two dimensions: 8 and 24.
Density of the best known sphere packing by dimension, on a logarithmic scale. Filled points are proved densest. In 8 dimensions the spheres still fill a quarter of space, in 24 dimensions barely two thousandths, and yet it is precisely there that the packing is perfect.
Viazovska’s proof
In March 2016 the Ukrainian mathematician Maryna Viazovska published a proof of a little over twenty pages that the E8 lattice is the densest packing in eight-dimensional space. A week later, with Henry Cohn, Abhinav Kumar, Stephen Miller and Danylo Radchenko, she proved the same for the Leech lattice in 24 dimensions. The key was a “magic function” from the world of modular forms, the same mathematics that lies behind deep statements about primes. In 2022 Viazovska received the Fields Medal for it, only the second woman ever.
In these two dimensions the spheres sit unusually tightly: in the E8 lattice each sphere touches 240 neighbours, in the Leech lattice 196,560. Both numbers are the largest possible, the so-called kissing numbers of these dimensions.
E8
The 240 neighbours of a sphere in the E8 lattice are at the same time the root system of the Lie group E8, the largest of the “exceptional groups” of mathematics. Projected onto the right plane, they arrange themselves in eight rings of 30 points each (see Quantum physics, magic and the unconscious).
The 240 roots of E8 projected into the plane: eight rings of 30 points each. Each point stands for one of the 240 neighbouring spheres in the E8 lattice.
In 2010 an experiment showed that E8 does not live only in mathematics. A group led by Radu Coldea cooled crystals of cobalt niobate almost to absolute zero and exposed them to a strong magnetic field. The magnetic chains in the crystal showed excitations with particular energies, and the first two stood in the ratio 1.618, the golden ratio. This is exactly what the physicist Alexander Zamolodchikov had predicted from the symmetry E8 in 1989. The golden ratio, the number of the pentagram, appears here as the ratio of two measured energies in a solid.
The 24
The Leech lattice stands at the centre of a web of astonishing connections. It leads to the largest of the “sporadic” symmetry groups, the so-called Monster, and via the connection known as “monstrous moonshine” to the modular forms, for which Richard Borcherds received the Fields Medal in 1998. The number 24 also plays a part in string theory: the simplest version, bosonic string theory, needs 26 dimensions, 24 of them transverse to the motion of the string.
That 24 of all numbers is the dimension in which spheres find a perfect order fits with what this site collects about the number 24: 1 × 2 × 3 × 4 = 24, the limit of the complete twin primes and the kissing number in four dimensions (see The number 24). And 8 is the number of the cube’s corners, whose solid angles together make exactly one full sphere (see The Platonic solids).