Suppose somebody hands you a bunch of oranges and asks you to stack them on a shelf, I’ll bet I can guess how you would do it. You’d build a pyramid by laying down a base layer and then fill in the upper levels by placing oranges in each of the divots provided by the layer below. If you’ve done this before, you might have noticed that the oranges in the base layer create a repeating hexagon pattern. In case you don’t have a crate of oranges next to you right now to try this out, check out the photo below.
A hexagonal packing of oranges creates the base layer on the left, and then the pyramid continues to rise using the hexagonal close packing. Image via Jeremy Jenum Flickr Creative Commons.
This is called a hexagonal circle packing, and it’s the densest way to pack a bunch of circles together. By densest, I mean that any other way you pack together circles is going to have much more empty space left over. When you place the subsequent layers on top by filling in the divots, what you’re doing is creating a well-studied arrangement called the hexagonal close packing of spheres. Just like the hexagonal packing in 2-dimensions, the hexagonal close packing is the densest way you can pack 3-dimensional spheres together. This was a result proved by Thomas Hales in 1998.
These both belong to the broader family of n-dimensional sphere packings, and it’s been a long standing open problem to find the densest sphere packings in each dimension. While we have the nice orange stacking analogy to help us visualize dimensions 2 and 3, in higher dimensions we can’t visualize things in the same way. But here is the essence of the problem. In any dimensions, a sphere is just a set of points that are equidistant from some center point, and a dense sphere packing is just an arrangement of non-overlapping spheres that fills up as much ambient space as possible.
Maryna Viazovska unlocked the solution in 8 dimensions, paving the way for the 24-dimensional solution. Photo Courtesy of Oberwolfach photo archives.
A few weeks ago Maryna Viazovska, currently a post-doc at the Berlin Mathematical School and the Humboldt University of Berlin, solved the sphere packing problem in 8-dimensions. Erica Klarreich, a math journalist for Quanta Magazine gives details on how Viazovska arrived at her solution, and some stories about the people she met along the way.
And then not a week went by before she and her coauthors, Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko generalized her methods to solve the sphere packing problem in a 24-dimensions. On his blog, mathematician Gil Kalai gives some historical background for the 8- and 24-dimensional sphere packing problems.
In a video posted by the Institute for Advanced Study, Stephen Miller gets into the details of the proof, he says “there’s something very special about 8 and 24, we can’t expect every week to keep proceeding like this.” Although the dimensions 8 and 24 might seem totally random, the reason these solutions came so close on each other’s heels is that these sphere packings — unlike those in other dimensions, as discussed on the n-category cafe — are related to two special lattices, E8 and the Leech lattice. Having this connection to lattices, which, full disclosure, I’m obsessed with, means that there is a world of machinery in the realm of modular forms for dealing with the packings. In a very broad sense, solving the packing problem came down to finding some suitable modular function that satisfied an appropriate list of properties that are derived from methods in harmonic analysis.
Sphere packing problems, of course, have many interesting applications, but the one that has always fascinated me is the link between dense sphere packings and error correcting codes. Trying to pack n-dimensional spheres as close to each other as possible is like trying to find points (namely, the center point of the sphere) that are as close to each other as possible, while maintaining some prescribed amount of distance between them (namely, the buffer created by the sphere around each center point). This acts just like an error correcting code, in the sense that we want to find code words that are similar enough that we can build a language out of them, but far enough apart that they can be transmitted over noisy channels and not be totally degraded by interference.
Like all good problems, sphere packings touch on many branches of mathematics: number theory, geometry, analysis. The fact that this problem has so many approaches and that its solutions are simultaneously so diverse in flavor, John Baez points out so perfectly in his blog post, “hints at the unity of mathematics.”