Packing Spheres in a Thousand Dimensions

Head-in-the-clouds math has surprisingly useful results.

- Evelyn Lamb

In March 2016, Ukrainian mathematician Maryna Viazovska solved the problem of how best to pack spheres in eight-dimensional space. She and four other researchers used her breakthrough to solve the 24-dimensional question just a week later. Woo hoo!

Now what does that mean?

First there’s the word “pack.” It makes sense to want to know how to pack things efficiently. Tennis ball manufacturers and fruit stand owners want to fit as much of their product into as little space as possible to cut down on shipping and storage costs. You want to be able to get your suitcase shut when you pack for a trip. But tennis balls, fruit, and clothes are three-dimensional objects, and we live in a three-dimensional world. Viazovska was packing in eight dimensions.

Math We See, Math We Can’t See

Three measurements identify three-dimensional objects: length, width, and height. These objects sit nicely in our three-dimensional space, where we can see them with our three-dimensional eyes and touch them with our three-dimensional hands. But mathematicians aren’t constrained by the three dimensions around us. Instead of considering three measurements, or three coordinates, mathematicians can look at spaces with any number of coordinates, including eight. It’s impossible to visualize eight-dimensional space, but mathematicians can still manipulate formulas and equations using eight coordinates the same way you do if you graph a two-dimensional line using a formula like y=3x+2.