The most underappreciated formula
Exploring spheres in higher dimensions
This talk centers around deriving the general formula for the volumes of spheres in higher dimensions. More broadly, it makes an attempt to take various initially counterintuitive facts about higher dimensions and make them (hopefully) a bit more intuitive.
Thanks to UC Santa Cruz for letting me film there, and to all its students for being such a lovely audience. Actually, the same thanks are owed to the International House at Berkeley, to the University of Wisconsin, and to Stanford. Not entirely intentionally, I ended up with a small de facto tour, giving this a couple of times.
Great talk!
I think an appeal to the unit square in higher dimensions being spiky isn’t necessary, and is suspicious anyway. It’s easy to understand why the central ball pokes through the unit square, in high dimensions: the unit balls at the corners have vanishingly small volumes, so the central ball has plenty of room to grow.
Also, I wonder if the fact that most of the volume of a high-dimensional ball is near its boundary is a possible explanation for the holographic principle in string theory. If the universe is indeed 10-dimensional, then most of its volume would be squeezed to its boundary so most everything that happens, happens at the boundary.
I was always better at stories than equations, so I come at math the way a tourist comes at a foreign language: enthusiastic, largely incompetent, occasionally getting the gist. But the thing I keep finding is that the "underappreciated" formulas almost always turn out to be the ones where the mathematician was solving something that seemed purely abstract and then it turned out to describe the physical world almost embarrassingly well. It's the part of science history that feels most like a practical joke. The universe keeps doing the homework.
I'll be reading this one twice. Possibly three times. Still won't fully understand it, but that's sort of the point.