In my last post on higher dimensions, I alluded to the fact that I don’t agree completely with certain notions about higher dimensions. Specifically, I disagree with the idea that the intuition that you take for granted in low dimensions is necessarily ill-equipped to serve you in higher dimensions. Low-dimensional intuition is ill-equipped for many problems, and like most other topics in math, it’s usually most sensible to do the calculations anyway.
Hyperspheres often get brought up with the subject of weirdness in higher dimensions, mostly because they’re easy to understand, and it’s easy to demonstrate the weirdness very quickly. But are they completely weird? Are the examples really fair, or are hyperspheres getting a bad rap?
First, let’s get some notation out of the way. We often like to call a hypersphere an n-sphere, because it’s an n-dimensional manifold. Technically, one of these can exist in any metric space with more than n dimensions (because I’m talking about intuition, I’m assuming it’s Euclidean space). For simplicity, though, we’ll say that it lives in n+1 space, so that we can define it easily:
That’s not all that I want to talk about, though. I also want to talk about the volume of the n-sphere, and in that case, we often talk about a ball, which is just the interior of a sphere. The interior of an n-sphere is an (n+1)-ball, because if the sphere is an n-dimensional manifold, its interior is an (n+1)-dimensional manifold:
Or more simply:
The volume of this object has a somewhat simple formula:
Where Γ(x) represents the Gamma function (which is a tad more complicated).
So where’s the counter-intuition? Say that we took the unit ball for all n > 0 and graphed its volume:
Volume vs. dimension of unit n-ball
This does seem a little odd. The volume goes up, hits a peak at 5, and then drops, and eventually bottoms out. In fact, with high enough dimension, you won’t see an n-ball have any volume at all. The limit of the volume of any n-ball as n goes to infinity is 0. That is
weird. That’s not necessarily something that you’d expect. It also seems weird that the volume starts dropping after a while.
But is all this really that strange? What if we fixed the radius at, say, 1/sqrt(π)? The volume vs. dimension is then just a decreasing function, even at low dimensions. Not surprising when you consider that radii less than 1 should make the volume diminish rapidly. So what about radii greater than 1? What if we fix r at say, 3? The volume peaks out at n = 56, and the volume is about 143 billion … somethings. After that, the volume diminishes back to zero again. All that we’re really saying here is that the geometry of the sphere dominates rn, but rn has enough power to dominate at low dimensions until the geometry cuts over.
What’s so special about rn though? Why is this the gold standard by which we judge the hypersphere? It’s just the hypercube with sides of length r. In fact, the unit sphere is inscribed in a cube with sides of length 2r. What if we considered a hypercube of circumradius r instead of inradius r? That means that a sphere of radius r contains it. If that’s the case, then it has volume strictly less than the sphere’s volume. In fact, its volume is:
which diminishes even faster than the sphere’s volume. So it can’t be the geometry of a cube that makes it keep its volumetric power.
So what’s my point? This is all sounding very counterintuitive. My point is that when you talk about counter-intuition in higher dimensions, it’s helpful to talk about what’s actually going on, instead of maligning poor innocent constructs like the hypersphere. What’s actually going on? More about that later.
But for now, consider this: no matter how many dimensions a sphere has, it’s always perfectly round, and perfectly isotropic. That’s intuition that isn’t lost in higher dimensions.
[Someone posted this to Reddit! Thanks!]