So my post, from early in the quarantine, on 4-dimensional hypercubes has shown up in my site statistics. Which got me thinking… always a dangerous proposition. But mostly just searching for additional material.
Maryna Viazovska, a Ukrainian mathematician and a postdoctoral researcher at the Berlin Mathematical School and the Humboldt University of Berlin, discovered something in 2016 about n-dimensional hyperspheres. (Well, those are the postdoctoral positions she held in 2016.) Hyperspheres are also fascinating creatures, with unexpected properties.
I’m putting the video first, because the math goes off the deep end fairly quickly. And it is an entertaining 8 minutes. And I tripped over the video first myself. So grab a beverage of your choice, probably not coffee at this time of day, and enjoy. You might learn something, and I promise, it will only hurt for a minute.
The article, which isn’t the mathematical proof, is from Quanta. Sphere Packing Solved in Higher Dimensions. Not in arbitrary dimensions but in dimensions 8 and 24.
Mathematicians have been studying sphere packings since at least 1611, when Johannes Kepler conjectured that the densest way to pack together equal-sized spheres in space is the familiar pyramidal piling of oranges seen in grocery stores. Despite the problem’s seeming simplicity, it was not settled until 1998, when Thomas Hales, now of the University of Pittsburgh, finally proved Kepler’s conjecture in 250 pages of mathematical arguments combined with mammoth computer calculations.
And that is in 3 dimensions. As always, higher dimensions are just weird.
Oh, and while packing the most amount of oranges in the least space is interesting, there was a time when packing the most number of cannonballs in the least space was of interest to every navy in the world. And as is mentioned in the video, packing problems do have applications in communications.
You can find the original post on 4-dimentional hypercubes, and other objects at the following link. Fun With Four Dimensions.