Packing For Higher Dimensions
You may have heard of the sphere-packing problem. If you haven’t, let me brief you. It’s a problem about how to pack a bunch of spheres. Particularly, it’s about how to place spheres, all the same size, so there’s as little wasted space as possible.
It’s not an easy problem. Johannes Kepler, whom you remember as the astronomer with the gold nose because you’ve mixed him up with Tycho Brahe, studied it. He conjectured, in 1611, that the best packing you could do was the “close packing”. You know this pattern because it’s what a stack of oranges ends up being. We believe he was right. A computer-assisted proof was published in 2005.
But if we’re comfortable with mathematics we know a sphere, or a ball, doesn’t have to be something as boring as the balls we have in the real world. We could consider a circle to be a two-dimensional sphere. We could make something four-dimensional that looks a lot like a sphere. Or five-dimensional. Or 800-dimensional, if we have some reason to do this. (We do!) And optimization problems can be strange things. How many dimensions of space something has can affect how easy or hard a problem is. But just having more dimensions doesn’t mean the problem is harder. Sometimes having a vaster space means the problem becomes easier.
There’s recently been a breakthrough in the eight dimension. A paper by Maryna S Viazovska, with the Berlin Mathematical School and the Humboldt University of Berlin, seems to have worked out the densest possible packing for eight-dimensional spheres. And better, it ties into this beautiful pattern known as the E8 lattice. The MathsByAGirl blog recently posted an essay about that, and I’d like to recommend folks over there.
And, because I’m like this, I’d like to point folks over to one of my old essays. I’d got to wondering what the least efficient sphere packings were. The answers might surprise you.