The Summer 2017 Mathematics A To Z: N-Sphere/N-Ball


Today’s glossary entry is a request from Elke Stangl, author of the Elkemental Force blog, which among other things has made me realize how much there is interesting to say about heat pumps. Well, you never know what’s interesting before you give it serious thought.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

N-Sphere/N-Ball.

I’ll start with space. Mathematics uses a lot of spaces. They’re inspired by geometry, by the thing that fills up our room. Sometimes we make them different by simplifying them, by thinking of the surface of a table, or what geometry looks like along a thread. Sometimes we make them bigger, imagining a space with more directions than we have. Sometimes we make them very abstract. We realize that we can think of polynomials, or functions, or shapes as if they were points in space. We can describe things that work like distance and direction and angle that work for these more abstract things.

What are useful things we know about space? Many things. Whole books full of things. Let me pick one of them. Start with a point. Suppose we have a sense of distance, of how far one thing is from one another. Then we can have an idea of the neighborhood. We can talk about some chunk of space that’s near our starting point.

So let’s agree on a space, and on some point in that space. You give me a distance. I give back to you — well, two obvious choices. One of them is all the points in that space that are exactly that distance from our agreed-on point. We know what this is, at least in the two kinds of space we grow up comfortable with. In three-dimensional space, this is a sphere. A shell, at least, centered around whatever that first point was. In two-dimensional space, on our desktop, it’s a circle. We know it can look a little weird: if we started out in a one-dimensional space, there’d be only two points, one on either side of the original center point. But it won’t look too weird. Imagine a four-dimensional space. Then we can speak of a hypersphere. And we can imagine that as being somehow a ball that’s extremely spherical. Maybe it pokes out of the rendering we try making of it, like a cartoon character falling out of the movie screen. We can imagine a five-dimensional space, or a ten-dimensional one, or something with even more dimensions. And we can conclude there’s a sphere for even that much space. Well, let it.

What are spheres good for? Well, they’re nice familiar shapes. Even if they’re in a weird number of dimensions. They’re useful, too. A lot of what we do in calculus, and in analysis, is about dealing with difficult points. Points where a function is discontinuous. Points where the function doesn’t have a value. One of calculus’s reliable tricks, though, is that we can swap information about the edge of things for information about the interior. We can replace a point with a sphere and find our work is easier.

The other thing I could give you. It’s a ball. That’s all the points that aren’t more than your distance away from our point. It’s the inside, the whole planet rather than just the surface of the Earth.

And here’s an ambiguity. Is the surface a part of the ball? Should we include the edge, or do we just want the inside? And that depends on what we want to do. Either might be right. If we don’t need the edge, then we have an open set (stick around for Friday). This gives us the open ball. If we do need the edge, then we have a closed set, and so, the closed ball.

Balls are so useful. Take a chunk of space that you find interesting for whatever reason. We can represent that space as the joining together (the “union”) of a bunch of balls. Probably not all the same size, but that’s all right. We might need infinitely many of these balls to get the chunk precisely right, or as close to right as can be. But that’s all right. We can still do it. Most anything we want to analyze is easier to prove on any one of these balls. And since we can describe the complicated shape as this combination of balls, then we can know things about the whole complicated shape. It’s much the way we can know things about polygons by breaking them into triangles, and showing things are true about triangles.

Sphere or ball, whatever you like. We can describe how many dimensions of space the thing occupies with the prefix. The 3-ball is everything close enough to a point that’s in a three-dimensional space. The 2-ball is everything close enough in a two-dimensional space. The 10-ball is everything close enough to a point in a ten-dimensional space. The 3-sphere is … oh, all right. Here we have a little squabble. People doing geometry prefer this to be the sphere in three dimensions. People doing topology prefer this to be the sphere whose surface has three dimensions, that is, the sphere in four dimensions. Usually which you mean will be clear from context: are you reading a geometry or a topology paper? If you’re not sure, oh, look for anything hinting at the number of spatial dimensions. If nothing gives you a hint maybe it doesn’t matter.

Either way, we do want to talk about the family of shapes without committing ourselves to any particular number of dimensions. And so that’s why we fall back on ‘N’. ‘N’ is a good name for “the number of dimensions we’re working in”, and so we use it. Then we have the N-sphere and the N-ball, a sphere-like shape, or a ball-like shape, that’s in however much space we need for the problem.

I mentioned something early on that I bet you paid no attention to. That was that we need a space, and a point inside the space, and some idea of distance. One of the surprising things mathematics teaches us about distance is … there’s a lot of ideas of distance out there. We have what I’ll call an instinctive idea of distance. It’s the one that matches what holding a ruler up to stuff tells us. But we don’t have to have that.

I sense the grumbling already. Yes, sure, we can define distance by some screwball idea, but do we ever need it? To which the mathematician answers, well, what if you’re trying to figure out how far away something in midtown Manhattan is? Where you can only walk along streets or avenues and we pretend Broadway doesn’t exist? Huh? How about that? Oh, fine, the skeptic might answer. Grant that there can be weird cases where the straight-line ruler distance is less enlightening than some other scheme is.

Well, there are. There exists a whole universe of different ideas of distance. There’s a handful of useful ones. The ordinary straight-line ruler one, the Euclidean distance, you get in a method so familiar it’s worth saying what you do. You find the coordinates of your two given points. Take the pairs of corresponding coordinates: the x-coordinates of the two points, the y-coordinates of the two points, the z-coordinates, and so on. Find the differences between corresponding coordinates. Take the absolute value of those differences. Square all those absolute-value differences. Add up all those squares. Take the square root of that. Fine enough.

There’s a lot of novelty acts. For example, do that same thing, only instead of raising the differences to the second power, raise them to the 26th power. When you get the sum, instead of the square root, take the 26th root. There. That’s a legitimate distance. No, you will never need this, but your analysis professor might give you it as a homework problem sometime.

Some are useful, though. Raising to the first power, and then eventually taking the first root, gives us something useful. Yes, raising to a first power and taking a first root isn’t doing anything. We just say we’re doing that for the sake of consistency. Raising to an infinitely large power, and then taking an infinitely great root, inspires angry glares. But we can make that idea rigorous. When we do it gives us something useful.

And here’s a new, amazing thing. We can still make “spheres” for these other distances. On a two-dimensional space, the “sphere” with this first-power-based distance will look like a diamond. The “sphere” with this infinite-power-based distance will look like a square. On a three-dimensional space the “sphere” with the first-power-based distance looks like a … well, more complicated, three-dimensional diamond. The “sphere” with the infinite-power-based distance looks like a box. The “balls” in all these cases look like what you expect from knowing the spheres.

As with the ordinary ideas of spheres and balls these shapes let us understand space. Spheres offer a natural path to understanding difficult points. Balls offer a natural path to understanding complicated shapes. The different ideas of distance change how we represent these, and how complicated they are, but not the fact that we can do it. And it allows us to start thinking of what spheres and balls for more abstract spaces, universes made of polynomials or formed of trig functions, might be. They’re difficult to visualize. But we have the grammar that lets us speak about them now.

And for a postscript: I also wrote about spheres and balls as part of my Set Tour a couple years ago. Here’s the essay about the N-sphere, although I didn’t exactly call it that. And here’s the essay about the N-ball, again not quite called that.

The Set Tour, Part 9: Balls, Only The Insides


Last week in the tour of often-used domains I talked about Sn, the surfaces of spheres. These correspond naturally to stuff like the surfaces of planets, or the edges of surfaces. They are also natural fits if you have a quantity that’s made up of a couple of components, and some total amount of the quantity is fixed. More physical systems do that than you might have guessed.

But this is all the surfaces. The great interior of a planet is by definition left out of Sn. This gives away the heart of what this week’s entry in the set tour is.

Bn

Bn is the domain that’s the interior of a sphere. That is, B3 would be all the points in a three-dimensional space that are less than a particular radius from the origin, from the center of space. If we don’t say what the particular radius is, then we mean “1”. That’s just as with the Sn we meant the radius to be “1” unless someone specifically says otherwise. In practice, I don’t remember anyone ever saying otherwise when I was in grad school. I suppose they might if we were doing a numerical simulation of something like the interior of a planet. You know, something where it could make a difference what the radius is.

It may have struck you that B3 is just the points that are inside S2. Alternatively, it might have struck you that S2 is the points that are on the edge of B3. Either way is right. Bn and Sn-1, for any positive whole number n, are tied together, one the edge and the other the interior.

Bn we tend to call the “ball” or the “n-ball”. Probably we hope that suggests bouncing balls and baseballs and other objects that are solid throughout. Sn we tend to call the “sphere” or the “n-sphere”, though I admit that doesn’t make a strong case for ruling out the inside of the sphere. Maybe we should think of it as the surface. We don’t even have to change the letter representing it.

As the “n” suggests, there are balls for as many dimensions of space as you like. B2 is a circle, filled in. B1 is just a line segment, stretching out from -1 to 1. B3 is what’s inside a planet or an orange or an amusement park’s glass light fixture. B4 is more work than I want to do today.

So here’s a natural question: does Bn include Sn-1? That is, when we talk about a ball in three dimensions, do we mean the surface and everything inside it? Or do we just mean the interior, stopping ever so short of the surface? This is a division very much like dividing the real numbers into negative and positive; do you include zero among other set?

Typically, I think, mathematicians don’t. If a mathematician speaks of B3 without saying otherwise, she probably means the interior of a three-dimensional ball. She’s not saying anything one way or the other about the surface. This we name the “open ball”, and if she wants to avoid any ambiguity she will say “the open ball Bn”.

“Open” here means the same thing it does when speaking of an “open set”. That may not communicate well to people who don’t remember their set theory. It means that the edges aren’t included. (Warning! Not actual set theory! Do not attempt to use that at your thesis defense. That description was only a reference to what’s important about this property in this particular context.)

If a mathematician wants to talk about the ball and the surface, she might say “the closed ball Bn”. This means to take the surface and the interior together. “Closed”, again, here means what it does in set theory. It pretty much means “include the edges”. (Warning! See above warning.)

Balls work well as domains for functions that have to describe the interiors of things. They also work if we want to talk about a constraint that’s made up of a couple of components, and that can be up to some size but not larger. For example, suppose you may put up to a certain budget cap into (say) six different projects, but you aren’t required to use the entire budget. We could model your budgeting as finding the point in B6 that gets the best result. How you measure the best is a problem for your operations research people. All I’m telling you is how we might represent the study of the thing you’re doing.