Just 24 light-hours away it’s still Sunday, right? Oh well. On to the math!
If you ask a mathematician to define a circle, you’ll probably hear something along the lines of “A circle is the set of points in a plane equidistant from a given center point. The name of the distance from the center is the radius.” A mathematician will state it more precisely, but that’s the gist. As a technical matter – and mathematicians love technical matters – this is actually somewhat different from what kids learn about circles in school. Kids would probably say a frisbee is a circle, but really it’s more of a disc. A disc is the area contained within a circle, which is the same as saying a circle is the boundary of a disc.
Having made that note to satisfy the geometry-savvy readers, I’ll promptly go back to the grade-school convention and say “circle” when formally I mean disc. Ditto for “sphere”, which mathematically is properly the surface of a ball in 3-d space, but here I’ll take spheres to have a volume and circles to have an area.
So after all that rigmarole, we want to think more generally. Spheres have volume and circles have area, concepts which are more or less specialized to three dimensions and two dimensions respectively. But there’s nothing stopping us from thinking about them more generally, as a type measure of the space. The name for that measure varies depending on the number of dimensions, but in both cases it’s perfectly possible to think of area and volume as just measuring the amount of space in a region. Thinking along those general lines we might as well scrap the word “area” entirely and just rename the 2-d concept of area to “2-volume”, while what we previously called volume can be renamed “3-volume”.
Heck, while we’re at it why not just call circles “2-spheres” and spheres “3-spheres”?
For a 2-sphere (formerly called a circle) of radius r, the 2-volume (or what we used to call area) is:
For a 3-sphere (formerly just called a sphere) of radius r, the 3-volume (or what we used to just call volume) is:
We’ve seen those formulas before, in middle school or earlier. Aside from our wonky renaming nothing has changed. Try another case for yourself. What’s the 1-volume of a 1-sphere? Think about it for a minute, and then go on to the next paragraph.
In one-dimensional space, we just have a line. Pick a point and call it the center of our 1-sphere. By direct copying of our definitions for spheres in 2 and 3 dimensions, you pick a radius, call it r, and mark all the points which are a distance r from the center. But we’re on a line, and so there’s only two such points – one on each side of the center. The distance from the center to one of those points is r. Thus the total 1-volume (you may have realized 1-volume is our general way of saying “length”) of a 1-sphere is just the distance from one side of the 1-sphere to the other, passing through the center. That is:
Whew. That seems like an absurd amount of effort to go through to say a few basic geometrical facts that we already knew. Is all this renaming really worth anything?
No, it’s all garbage.
Ok, ok. I’m just kidding. Turns out this stuff is in fact both very important and very useful. Not so much for the old 1, 2, and 3 dimensional cases we’re used to, but for higher dimensions. Now as far as we know there aren’t literal higher spatial dimensions, protests of string theorists notwithstanding. But there are times when we might have a huge set of coordinates whose values happen to be geometrically connected in a way corresponding to our definition of the n-sphere.
As an example, take a box full of gas molecules. Each of them are going to be described by three numbers representing the momentum in the x, y, and z directions. The total energy of those particles is going to be described as proportional to the sum of the squares of those momentum coordinates. So if you have 10 particles you’ll have 30 momentum variables, each of which is squared and added together. That’s in a sense like an abstract 30-sphere. For many calculations (most especially the entropy) we want to know how many combinations of those numbers correspond to a certain energy – and the energy is the radius of the 30-sphere. To do that, we need the 30-volume of a 30-sphere.
That paragraph probably lost a few people. I had to stare cross-eyed at the statistical mechanics book for a little while while I was working through it too. In short, we’ll just say that there’s numerous physical reasons you might need to know the “volumes” of some preposterously high dimensional “spheres”. And sure enough, there’s a function that can tell us what those volumes are. It’s our Sunday Function. The n-volume of an n-sphere of radius r is:
Where that thing in the denominator of the fraction is itself the gamma function. That may be a more bizarre expression than you or I might have hoped but I’m sorry to say that it’s true nonetheless.
I know of two ways to prove it*. You can do the straightforward method and construct a generalization of spherical coordinates in n dimensions and integrate over them. Here “straightforward” means that it’s monstrously horrible and I have no doubt that I couldn’t do it without a reference in front of me. There’s also a trick you can do involving Gaussian integrals. It’s still rather head-spinning but nonetheless more elegant. I might save that for another time.
I’m not sure I’d describe any of this as useful outside of some fairly deep and abstract physics. On the other hand that deep and abstract physics is intimately entangled with our most basic understanding of the the statistical behavior of… well, pretty much everything. It’s a straight-line path from this stuff to engineering as mundane as the way your car engine works. I tip my hat to whoever first figured out what volume meant in higher dimensions. It was a pretty awesome feat.
*I’m doing neither of them in this post, though I’ll probably do at least the second later on. If you just can’t wait they’re both in the appendices of Pathria’s Statistical Mechanics.