The Set Tour, Part 10: Lots of Spheres
The next exhibit on the Set Tour here builds on a couple of the previous ones. First is the set Sn, that is, the surface of a hypersphere in n+1 dimensions. Second is Bn, the ball — the interior — of a hypersphere in n dimensions. Yeah, it bugs me too that Sn isn’t the surface of Bn. But it’d be too much work to change things now. The third has lurked implicitly since all the way back to Rn, a set of n real numbers for which the ordering of the numbers matters. (That is, that the set of numbers 2, 3 probably means something different than the set 3, 2.) And fourth is a bit of writing we picked up with matrices. The selection is also dubiously relevant to my own thesis from back in the day.
Sn x m and Bn x m
Here ‘n’ and ‘m’ are whole numbers, and I’m not saying which ones because I don’t need to tie myself down. Just as with Rn and with matrices this is a whole family of sets. Each different pair of n and m gives us a different set Sn x m or Bn x m, but they’ll all look quite similar.
The multiplication symbol here is a kind of multiplication, just as it was in matrices. That kind is called a “direct product”. What we mean by Sn x m is that we have a collection of items. We have the number m of them. Each one of those items is in Sn. That’s the surface of the hypersphere in n+1 dimensions. And we want to keep track of the order of things; we can’t swap items around and suppose they mean the same thing.
So suppose I write S2 x 7. This is an ordered collection of seven items, every one of which is on the surface of a three-dimensional sphere. That is, it’s the location of seven spots on the surface of the Earth. S2 x 8 offers similar prospects for talking about the location of eight spots.
With that written out, you should have a guess what Bn x m means. Your guess is correct. It’s a collection of m things, each of them within the interior of the n-dimensional ball.
Now the dubious relevance to my thesis. My problem was modeling a specific layer of planetary atmospheres. The model used for this was to pretend the atmosphere was made up of some large number of vortices, of whirlpools. Just like you see in the water when you slide your hand through the water and watch the little whirlpools behind you. The winds could be worked out as the sum of the winds produced by all these little vortices.
In the model, each of these vortices was confined to a single distance from the center of the planet. That’s close enough to true for planetary atmospheres. A layer in the atmosphere is not thick at all, compared to the planet. So every one of these vortices could be represented as a point in S2, the surface of a three-dimensional sphere. There would be some large number of these points. Most of my work used a nice round 256 points. So my model of a planetary atmosphere represented the system as a point in the domain S2 x 256. I was particularly interested in the energy of this set of 256 vortices. That was a function which had, as its domain, S2 x 256, and as range, the real numbers R.
But the connection to my actual work is dubious. I was doing numerical work, for the most part. I don’t think my advisor or I ever wrote S2 x 256 or anything like that when working out what I ought to do, much less what I actually did. Had I done a more analytic thesis I’d surely have needed to name this set. But I didn’t. It was lurking there behind my work nevertheless.
The energy of this system of vortices looked a lot like the potential energy for a bunch of planets attracting each other gravitationally, or like point charges repelling each other electrically. We work it out by looking at each pair of vortices. Work out the potential energy of those two vortices being that strong and that far apart. We call that a pairwise interaction. Then add up all the pairwise interactions. That’s it.  The pairwise interaction is stronger as each vortex is stronger; it gets weaker as the vortices get farther apart.
In gravity or electricity problems the strength falls off as the reciprocal of the distance between points. In vortices, the strength falls off as minus one times the logarithm of the distance between points. That’s a difference, and it meant that a lot of analytical results known for electric charges didn’t apply to my problem exactly. That was all right. I didn’t need many. But it does mean that I was fibbing up above, when I said I was working with S2 x 256. Pause a moment. Do you see what the fib was?
I’ll put what would otherwise be a footnote here so folks have a harder time reading right through to the answer.
 Physics majors may be saying something like: “wait, I see how this would be the potential energy of these 256 vortices, but where’s the kinetic energy?” The answer is, there is none. It’s all potential energy. The dynamics of point vortices are weird. I didn’t have enough grounding in mechanics when I went into them.
That’s all to the footnote.
Here’s where the fib comes in. If I’m really picking sets of vortices from all of the set S2 x 256, then, can two of them be in the exact same place? Sure they can. Why couldn’t they? For precedent, consider R3. In the three-dimensional vectors I can have the first and third numbers “overlap” and have the same value: (1, 2, 1) is a perfectly good vector. Why would that be different for an ordered set of points on the surface of the sphere? Why can’t vortex 1 and vortex 3 happen to have the same value in S2?
The problem is if two vortices were in the exact same position then the energy would be infinitely large. That’s not unique to vortices. It would be true for masses and gravity, or electric charges, if they were brought perfectly on top of each other. Infinitely large energies are a problem. We really don’t want to deal with them.
We could deal with this by pretending it doesn’t happen. Imagine if you dropped 256 poker chips across the whole surface of the Earth. Would you expect any two to be on top of each other? Would you expect two to be exactly and perfectly on top of each other, neither one even slightly overhanging the other? That’s so unlikely you could safely ignore it, for the same reason you could ignore the chance you’ll toss a coin and have it come up tails 56 times in a row.
And if you were interested in modeling the vortices moving it would be incredibly unlikely to have one vortex collide with another. They’d circle around each other, very fast, almost certainly. So ignoring the problem is defensible in this case.
Or we could be proper and responsible and say, “no overlaps” and “no collisions”. We would define some set that represents “all the possible overlaps and arrangements that give us a collision”. Then we’d say we’re looking at S2 x 256 except for those. I don’t think there’s a standard convention for “all the possible overlaps and collisions”, but Ω is a reasonable choice. Then our domain would be S2 x 256 \ Ω. The backslash means “except for the stuff after this”. This might seem unsatisfying. We don’t explicitly say what combinations we’re excluding. But go ahead and try listing all the combinations that would produce trouble. Try something simple, like S2 x 4. This is why we hide all the complicated stuff under a couple ordinary sentences.
It’s not hard to describe “no overlaps” mathematically. (You would say something like “vortex number j and vortex number k are not at the same position”, with maybe a rider of “unless j and k are the same number”. Or you’d put it in symbols that mean the same thing.) “No collisions” is harder. For gravity or electric charge problems we can describe at least some of them. And I realize now I’m not sure if there is an easy way to describe vortices that collide. I have difficulty imagining how they might, since vortices that are close to one another are pushing each other sideways quite intently. I don’t think that I can say they can’t, though. Not without more thought.