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.

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.

A Leap Day 2016 Mathematics A To Z: Riemann Sphere

To my surprise nobody requested any terms beginning with `R’ for this A To Z. So I take this free day to pick on a concept I’d imagine nobody saw coming.

Riemann Sphere.

We need to start with the complex plane. This is just, well, a plane. All the points on the plane correspond to a complex-valued number. That’s a real number plus a real number times i. And i is one of those numbers which, squared, equals -1. It’s like the real number line, only in two directions at once.

Take that plane. Now put a sphere on it. The sphere has radius one-half. And it sits on top of the plane. Its lowest point, the south pole, sits on the origin. That’s whatever point corresponds to the number 0 + 0i, or as humans know it, “zero”.

We’re going to do something amazing with this. We’re going to make a projection, something that maps every point on the sphere to every point on the plane, and vice-versa. In other words, we can match every complex-valued number to one point on the sphere. And every point on the sphere to one complex-valued number. Here’s how.

Imagine sitting at the north pole. And imagine that you can see through the sphere. Pick any point on the plane. Look directly at it. Shine a laser beam, if that helps you pick the point out. The laser beam is going to go into the sphere — you’re squatting down to better look through the sphere — and come out somewhere on the sphere, before going on to the point in the plane. The point where the laser beam emerges? That’s the mapping of the point on the plane to the sphere.

There’s one point with an obvious match. The south pole is going to match zero. They touch, after all. Other points … it’s less obvious. But some are easy enough to work out. The equator of the sphere, for instance, is going to match all the points a distance of 1 from the origin. So it’ll have the point matching the number 1 on it. It’ll also have the point matching the number -1, and the point matching i, and the point matching -i. And some other numbers.

All the numbers that are less than 1 from the origin, in fact, will have matches somewhere in the southern hemisphere. If you don’t see why that is, draw some sketches and think about it. You’ll convince yourself. If you write down what convinced you and sprinkle the word “continuity” in here and there, you’ll convince a mathematician. (WARNING! Don’t actually try getting through your Intro to Complex Analysis class doing this. But this is what you’ll be doing.)

What about the numbers more than 1 from the origin? … Well, they all match to points on the northern hemisphere. And tell me that doesn’t stagger you. It’s one thing to match the southern hemisphere to all the points in a circle of radius 1 away from the origin. But we can match everything outside that little circle to the northern hemisphere. And it all fits in!

Not amazed enough? How about this: draw a circle on the plane. Then look at the points on the Riemann sphere that match it. That set of points? It’s also a circle. A line on the plane? That’s also a line on the sphere. (Well, it’s a geodesic. It’s the thing that looks like a line, on spheres.)

How about this? Take a pair of intersecting lines or circles in the plane. Look at what they map to. That mapping, squashed as it might be to the northern hemisphere of the sphere? The projection of the lines or circles will intersect at the same angles as the original. As much as space gets stretched out (near the south pole) or squashed down (near the north pole), angles stay intact.

OK, but besides being stunning, what good is all this?

Well, one is that it’s a good thing to learn on. Geometry gets interested in things that look, at least in places, like planes, but aren’t necessarily. These spheres are, and the way a sphere matches a plane is obvious. We can learn the tools for geometry on the Möbius strip or the Klein bottle or other exotic creations by the tools we prove out on this.

And then physics comes in, being all weird. Much of quantum mechanics makes sense if you imagine it as things on the sphere. (I admit I don’t know exactly how. I went to grad school in mathematics, not in physics, and I didn’t get to the physics side of mathematics much at that time.) The strange ways distance can get mushed up or stretched out have echoes in relativity. They’ll continue having these echoes in other efforts to explain physics as geometry, the way that string theory will.

Also important is that the sphere has a top, the north pole. That point matches … well, what? It’s got to be something infinitely far away from the origin. And this make sense. We can use this projection to make a logically coherent, sensible description of things “approaching infinity”, the way we want to when we first learn about infinitely big things. Wrapping all the complex-valued numbers to this ball makes the vast manageable.

It’s also good numerical practice. Computer simulations have problems with infinitely large things, for the obvious reason. We have a couple of tools to handle this. One is to model a really big but not infinitely large space and hope we aren’t breaking anything. One is to create a “tiling”, making the space we are able to simulate repeat itself in a perfect grid forever and ever. But recasting the problem from the infinitely large plane onto the sphere can also work. This requires some ingenuity, to be sure we do the recasting correctly, but that’s all right. If we need to run a simulation over all of space, we can often get away with doing a simulation on a sphere. And isn’t that also grand?

The Riemann named here is Bernhard Riemann, yet another of those absurdly prolific 19th century mathematicians, especially considering how young he was when he died. His name is all over the fundamentals of analysis and geometry. When you take Introduction to Calculus you get introduced pretty quickly to the Riemann Sum, which is how we first learn how to calculate integrals. It’s that guy. General relativity, and much of modern physics, is based on advanced geometries that again fall back on principles Riemann noticed or set out or described so well that we still think of them as he discovered.

Packing For Higher Dimensions

You may have heard of the sphere-packing problem. If you haven’t, let me brief you. It’s a problem about how to pack a bunch of spheres. Particularly, it’s about how to place spheres, all the same size, so there’s as little wasted space as possible.

It’s not an easy problem. Johannes Kepler, whom you remember as the astronomer with the gold nose because you’ve mixed him up with Tycho Brahe, studied it. He conjectured, in 1611, that the best packing you could do was the “close packing”. You know this pattern because it’s what a stack of oranges ends up being. We believe he was right. A computer-assisted proof was published in 2005.

But if we’re comfortable with mathematics we know a sphere, or a ball, doesn’t have to be something as boring as the balls we have in the real world. We could consider a circle to be a two-dimensional sphere. We could make something four-dimensional that looks a lot like a sphere. Or five-dimensional. Or 800-dimensional, if we have some reason to do this. (We do!) And optimization problems can be strange things. How many dimensions of space something has can affect how easy or hard a problem is. But just having more dimensions doesn’t mean the problem is harder. Sometimes having a vaster space means the problem becomes easier.

There’s recently been a breakthrough in the eight dimension. A paper by Maryna S Viazovska, with the Berlin Mathematical School and the Humboldt University of Berlin, seems to have worked out the densest possible packing for eight-dimensional spheres. And better, it ties into this beautiful pattern known as the E8 lattice. The MathsByAGirl blog recently posted an essay about that, and I’d like to recommend folks over there.

And, because I’m like this, I’d like to point folks over to one of my old essays. I’d got to wondering what the least efficient sphere packings were. The answers might surprise you.

The Set Tour, Part 11: Doughnuts And Lots Of Them

I’ve been slow getting back to my tour of commonly-used domains for several reasons. It’s been a busy season. It’s so much easier to plan out writing something than it is to write something. The usual. But one of my excuses this time is that I’m not sure the set I want to talk about is that common. But I like it, and I imagine a lot of people will like it. So that’s enough.

T and Tⁿ

T stands for the torus. Or the toroid, if you prefer. It’s a fun name. You know the shape. It’s a doughnut. Take a cylindrical tube and curl it around back on itself. Don’t rip it or fold it. That’s hard to do with paper or a sheet of clay or other real-world stuff. But we can imagine it easily enough. I suppose we can make a computer animation of it, if by ‘we’ we mean ‘you’.

We don’t use the whole doughnut shape for T. And no, we don’t use the hole either. What we use is the surface of the doughnut, the part that could get glazed. We ignore the inside, just the same way we had S represent the surface of a sphere (or the edge of a circle, or the boundary of a hypersphere). If there is a common symbol for the torus including the interior I don’t know it. I’d be glad to hear if someone had.

What good is the surface of a torus, though? Well, it’s a neat shape. Slice it in one direction, the way you’d cut a bagel in half, and at the slice you get the shape of a washer, the kind you fit around a nut and bolt. (An annulus, to use the trade term.) Slice it perpendicular to that, the way you’d cut it if you’re one of those people who eats half doughnuts to the amazement of the rest of us, and at the slice you get two detached circles. If you start from any point on the torus shape you can go in one direction and make a circle that loops around the doughnut’s central hole. You can go the perpendicular direction and make a circle that brushes up against but doesn’t go around the central hole. There’s some neat topology in it.

There’s also video games in it. The topology of this is just like old-fashioned video games where if you go off the edge of the screen to the right you come back around on the left, and if you go off the top you come back from the bottom. (And if you go off to the left you come back around the right, and off the bottom you come back to the top.) To go from the flat screen to the surface of a doughnut requires imagining some stretching and scrunching up of the surface, but that’s all right. (OK, in an old video game it was a kind-of flat screen.) We can imagine a nice flexible screen that just behaves.

This is a common trick to deal with boundaries. (I first wrote “to avoid having to deal with boundaries”. But this is dealing with them, by a method that often makes sense.) You just make each boundary match up with a logical other boundary. It’s not just useful in video games. Often we’ll want to study some phenomenon where the current state of things depends on the immediate neighborhood, but it’s hard to say what a logical boundary ought to be. This particularly comes up if we want to model an infinitely large surface without dealing with infinitely large things. The trick will turn up a lot in numerical simulations for that reason. (In that case, we’re in truth working with a numerical approximation of T, but that’ll be close enough.)

Tⁿ, meanwhile, is a vector of things, each of which is a point on a torus. It’s akin to Rⁿ or S^{2 x n}. They’re ordered sets of things that are themselves things. There can be as many as you like. n, here, is whatever positive whole number you need.

You might wonder how big the doughnut is. When we talked about the surface of the sphere, S², or the surface and interior, B³, we figured on a sphere with radius of 1 unless we heard otherwise. Toruses would seem to have two parameters. There’s how big the outer diameter is and how big the inner diameter is. Which do we pick?

We don’t actually care. It’s much the way we can talk about a point on the surface of a planet by the latitude and longitude of the point, and never care about how big the planet is. We can describe a point on the surface of the torus without needing to refer to how big the whole shape is or how big the hole in the middle is. A popular scheme to describe points is one that looks a lot like latitude and longitude.

Imagine the torus sitting as flat as it gets on the table. Pick a point that you find interesting.

We use some reference point that’s as good as an equator and a prime meridian. One coordinate is the angle you make going horizontally, possibly around the hole in the middle, from the reference point to the point we’re interested in. The other coordinate is the angle you make vertically, going in a loop that doesn’t go around the hole in the middle, from the reference point to the point we’re interested in. The reference point has coordinates 0, 0, as it must. If this sounds confusing it’s because I’m not using a picture. I thought making some pictures would be too much work. I’m a fool. But if you think of real torus-shaped objects it’ll come to you.

In this scheme the coordinates are both angles. Normal people would measure that in degrees, from 0 to 360, or maybe from -180 to 180. Mathematicians would measure as radians, from 0 to 2π, or from -π to +π. Whatever it is, it’s the same as the coordinates of a point on the edge of the circle, what we called S¹ a few essays back. So it’s fair to say you can think of T as S¹ x S¹, an ordered set of points on circles.

I’ve written of these toruses as three-dimensional things. Well, two dimensional-surfaces wrapped up to suggest three-dimensional objects. You don’t have to stick with these dimensions if you don’t want or if your problem needs something else. You can make a torus that’s a three-dimensional shape in four dimensions. For me that’s easiest to imagine as a cube where the left edge and the right edge loop back and meet up, the lower and the upper edges meet up, and the front and the back edges meet up. This works well to model an infinitely large space with a nice and small block.

I like to think I can imagine a four-dimensional doughnut where every cross-section is a sphere. I may be kidding myself. There could also be a five-dimensional torus and you’re on your own working that out, or working out what to do with it.

I’m not sure there is a common standard notation for that, though. Probably the mathematician wanting to make clear she’s working with a torus in four dimensions just says so in text, and trusts that the context of her mathematics makes it clear this is no ordinary torus.

I’ve also written of these toruses as circular, as rounded shapes. That’s the most familiar torus. It’s a doughnut shape, or an O-ring shape, or an inner tube’s shape. It’s the shape you produce by taking a circle and looping it around an axis not on the ring. That’s common and that’s usually all we need.

But if you need some other torus, produced by rotating some other shape around an axis not inside it, go ahead. You’ll need to make clear what that original shape, the generator, is. You’ve seen examples of this in, for example, the washers that fit around nuts and bolts. They’re typically rectangles in cross-section. Or you might have seen that image of someone who fit together a couple dozen iMac boxes to make a giant wheel. I don’t know why you would need this, but it’s your problem, not mine. If these shapes are useful for your work, by all means, use them.

I’m not sure there is a standard notation for that sort of shape. My hunch is to say you’d define your generating shape and give it a name such as A or D. Then name the torus based on that as T(A) or T(D). But I would recommend spelling it out in text before you start using symbols like this.

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 S^n-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 S^n-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.

The Set Tour, Part 8: Balls, Only Made Harder

I haven’t forgotten or given up on the Set Tour, don’t worry or celebrate. I just expected there to be more mathematically-themed comic strips the last couple days. Really, three days in a row without anything at ComicsKingdom or GoComics to talk about? That’s unsettling stuff. Ah well.

Sⁿ

We are also starting to get into often-used domains that are a bit stranger. We are going to start seeing domains that strain the imagination more. But this isn’t strange quite yet. We’re looking at the surface of a sphere.

The surface of a sphere we call S². The “S” suggests a sphere. The “2” means that we have a two-dimensional surface, which matches what we see with the surface of the Earth, or a beach ball, or a soap bubble. All these are sphere enough for our needs. If we want to say where we are on the surface of the Earth, it’s most convenient to do this with two numbers. These are a latitude and a longitude. The latitude is the angle made between the point we’re interested in and the equator. The longitude is the angle made between the point we’re interested in and a reference prime longitude.

There are some variations. We can replace the latitude, for example, with the colatitude. That’s the angle between our point and the north pole. Or we might replace the latitude with the cosine of the colatitude. That has some nice analytic properties that you have to be well into grad school to care about. It doesn’t matter. The details may vary but it’s all the same. We put in a number for the east-west distance and another for the north-south distance.

It may seem pompous to use the same system to say where a point is on the surface of a beach ball. But can you think of a better one? Pointing to the ball and saying “there”, I suppose. But that requires we go around with the beach ball pointing out spots. Giving two numbers saves us having to go around pointing.

(Some weenie may wish to point out that if we were clever we could describe a point exactly using only a single number. This is true. Nobody does that unless they’re weenies trying to make a point. This essay is long enough without describing what mathematicians really mean by “dimension”. “How many numbers normal people use to identify a point in it” is good enough.)

S² is a common domain. If we talk about something that varies with your position on the surface of the earth, we’re probably using S² as the domain. If we talk about the temperature as it varies with position, or the height above sea level, or the population density, we have functions with a domain of S² and a range in R. If we talk about the wind speed and direction we have a function with domain of S² and a range in R³, because the wind might be moving in any direction.

Of course, I wrote down Sⁿ rather than just S². As with Rⁿ and with R^{m x n}, there is really a family of similar domains. They are common enough to share a basic symbol, and the superscript is enough to differentiate them.

What we mean by Sⁿ is “the collection of points in Rⁿ⁺¹ that are all the same distance from the origin”. Let me unpack that a little. The “origin” is some point in space that we pick to measure stuff from. On the number line we just call that “zero”. On your normal two-dimensional plot that’s where the x- and y-axes intersect. On your normal three-dimensional plot that’s where the x- and y- and z-axes intersect.

And by “the same distance” we mean some set, fixed distance. Usually we call that the radius. If we don’t specify some distance then we mean “1”. In fact, this is so regularly the radius I’m not sure how we would specify a different one. Maybe we would write Sⁿ_r for a radius of “r”. Anyway, Sⁿ, the surface of the sphere with radius 1, is commonly called the “unit sphere”. “Unit” gets used a fair bit for shapes. You’ll see references to a “unit cube” or “unit disc” or so on. A unit cube has sides length 1. A unit disc has radius 1. If you see “unit” in a mathematical setting it usually means “this thing measures out at 1”. (The other thing it may mean is “a unit of measure, but we’re not saying which one”. For example, “a unit of distance” doesn’t commit us to saying whether the distance is one inch, one meter, one million light-years, or one angstrom. We use that when we don’t care how big the unit is, and only wonder how many of them we have.)

S¹ is an exotic name for a familiar thing. It’s all the points in two-dimensional space that are a distance 1 from the origin. Real people call this a “circle”. So do mathematicians unless they’re comparing it to other spheres or hyperspheres.

This is a one-dimensional figure. We can identify a single point on it easily with just one number, the angle made with respect to some reference direction. The reference direction is almost always that of the positive x-axis. That’s the line that starts at the center of the circle and points off to the right.

S³ is the first hypersphere we encounter. It’s a surface that’s three-dimensional, and it takes a four-dimensional space to see it. You might be able to picture this in your head. When I try I imagine something that looks like the regular old surface of the sphere, only it has fancier shading and maybe some extra lines to suggest depth. That’s all right. We can describe the thing even if we can’t imagine it perfectly. S⁴, well, that’s something taking five dimensions of space to fit in. I don’t blame you if you don’t bother trying to imagine what that looks like exactly.

The need for S⁴ itself tends to be rare. If we want to prove something about a function on a hypersphere we usually make do with Sⁿ. This doesn’t tell us how many dimensions we’re working with. But we can imagine that as a regular old sphere only with a most fancy job of drawing lines on it.

If we want to talk about Sⁿ aloud, or if we just want some variation in our prose, we might call it an n-sphere instead. So the 2-sphere is the surface of the regular old sphere that’s good enough for everybody but mathematicians. The 1-sphere is the circle. The 3-sphere and so on are harder to imagine. Wikipedia asserts that 3-spheres and higher-dimension hyperspheres are sometimes called “glomes”. I have not heard this word before, and I would expect it to start a fight if I tried to play it in Scrabble. However, I do not do mathematics that often requires discussion of hyperspheres. I leave this space open to people who do and who can say whether “glome” is a thing.

Something that all these Sⁿ sets have in common are that they are the surfaces of spheres. They are just the boundary, and omit the interior. If we want a function that’s defined on the interior of the Earth we need to find a different domain.

Lines That Cross Infinitely Many Times

MathematicsHub is a new mathematics blog. Among its first entries was an effort to answer why two straight lines in space can intersect in at most one point. That is, if two lines intersect at more than one point, they have to intersect at all of them. And that’s indistinguishable from being the same line.

If we suppose we’re talking about ordinary old space and straight lines and intersections, then that’s true and there’s not much arguing it. But I got to wondering about non-ordinary spaces and lines and such. Could we work out something that looks like a pair of lines, and that intersect in more than one place and still aren’t the same line?

There’s an obvious answer. That comes in spherical geometry, the way shapes on the surface of a ball work. In this space, “lines” are instead “great circles”. Those the equator, lines of longitude, the paths that airplanes would travel if they didn’t have to deal with winds or restricted airspaces or radar paths. And two distinct great circles will intersect in two points, which you can convince yourself of by looking at the south and north poles of a globe.

Can we come up with something where lines intersect at three points? Or four? … Possibly. If we started with something ellipsoidal, perhaps. Or if we started with a sphere and pinched off a corner and twisted it around maybe we could make most “great circle” routes intersect several times. At least, I can imagine this. I admit I don’t have the background in non-Euclidean geometries to make a compelling case for it. It feels right to my instincts, is all, and I leave it as a homework problem for someone who wants homework problems.

It struck me I could think of something that’s very line-like and that offers infinitely many intersections, though. It’s again on a sphere, like the surface of the Earth. Take a path that has a constant compass heading. That is, something that’s always (say) 30 degrees counterclockwise of the south-north line. Following this path, always that same angle with respect to the local south-north line, creates a path called a “loxodrome”. It will look very much like a straight line, and on a Mercator-projection map of the world will be a straight line.

However, if you look at the sphere, this loxodrome traces out a path that spirals in towards the south and the north poles. Properly, it never reaches the south or north pole (unless your loxodrome was pointing directly south or directly north all the time). It just keeps looping around, infinitely many times.

And there’s my idea. Suppose we accept a loxodrome as being enough like a line. I admit you might not be willing to go along with that. If you’re not willing to make this supposition then you won’t accept my conclusion and that’s that. We’ll just have to disagree. (And I’d grant that in most cases I wouldn’t call a loxodrome a line, because it doesn’t behave enough like a line for most purposes.)

But if you will let me call loxodromes a kind of line, then I can give you this happy conclusion. If you start two loxodromes from the same point and going in different angles, then, they’re going to intersect. And not just once, nor twice, but many times as you go closer to the south or the north pole. And despite having infinitely many intersections, they’re not the same loxodromes. They point in different directions and touch many different points.

And isn’t that remarkable?

You may not be convinced. It depends whether you’ll accept a loxodrome as being enough like a line. But I like the idea of lines that intersect infinitely many times yet aren’t the same line. And let me know if you don’t buy this, and why; or if you have a better idea, please.

A Summer 2015 Mathematics A To Z: hypersphere

Hypersphere.

If you asked someone to say what mathematicians do, there are, I think, three answers you’d get. One would be “they write out lots of decimal places”. That’s fair enough; that’s what numerical mathematics is about. One would be “they write out complicated problems in calculus”. That’s also fair enough; say “analysis” instead of “calculus” and you’re not far off. The other answer I’d expect is “they draw really complicated shapes”. And that’s geometry. All fair enough; this is stuff real mathematicians do.

Geometry has always been with us. You may hear jokes about never using algebra or calculus or such in real life. You never hear that about geometry, though. The study of shapes and how they fill space is so obviously useful that you sound like a fool saying you never use it. That would be like claiming you never use floors.

There are different kinds of geometry, though. The geometry we learn in school first is usually plane geometry, that is, how shapes on a two-dimensional surface like a sheet of paper or a computer screen work. Here we see squares and triangles and trapezoids and theorems with names like “side-angle-side congruence”. The geometry we learn as infants, and perhaps again in high school, is solid geometry, how shapes in three-dimensional spaces work. Here we see spheres and cubes and cones and something called “ellipsoids”. And there’s spherical geometry, the way shapes on the surface of a sphere work. This gives us great circle routes and loxodromes and tales of land surveyors trying to work out what Vermont’s northern border should be.

Reading the Comics, March 10, 2015: Shapes Of Things Edition

If there’s a theme running through today’s collection of mathematics-themed comic strips it’s shapes: I have good reason to talk about a way of viewing circles and spheres and even squares and boxes; and then both Euclid and men’s ties get some attention.

Eric the Circle (March 5), this one by “regina342”, does a bit of shape-name-calling. I trust that it’s not controversial that a rectangle is also a parallelogram, but people might be a bit put off by describing a circle as a sphere, what with circles being two-dimensional figures and spheres three-dimensional ones. For ordinary purposes of geometry that’s a fair enough distinction. Let me now make this complicated.

What’s The Worst Way To Pack?

While reading that biography of Donald Coxeter that brought up that lovely triangle theorem, I ran across some mentions of the sphere-packing problem. That’s the treatment of a problem anyone who’s had a stack of oranges or golf balls has independently discovered: how can you arrange balls, all the same size (oranges are near enough), so as to have the least amount of wasted space between balls? It’s a mathematics problem with a lot of applications, both the obvious ones of arranging orange or golf-ball shipments, and less obvious ones such as sending error-free messages. You can recast the problem of sending a message so it’s understood even despite errors in coding, transmitting, receiving, or decoding, as one of packing equal-size balls around one another.

The “packing density” is the term used to say how much of a volume of space can be filled with balls of equal size using some pattern or other. Patterns called the cubic close packing or the hexagonal close packing are the best that can be done with periodic packings, ones that repeat some base pattern over and over; they fill a touch over 74 percent of the available space with balls. If you don’t want to follow the Mathworld links before, just get a tub of balls, or crate of oranges, or some foam Mystery Science Theater 3000 logo balls as packing materials when you order the new DVD set, and play around with a while and you’ll likely rediscover them. If you’re willing to give up that repetition you can get up to nearly 78 percent. Finding these efficient packings is known as the Kepler conjecture, and yes, it’s that Kepler, and it did take a couple centuries to show that these were the most efficient packings.

While thinking about that I wondered: what’s the least efficient way to pack balls? The obvious answer is to start with a container the size of the universe, and then put no balls in it, for a packing fraction of zero percent. This seems to fall outside the spirit of the question, though; it’s at least implicit in wondering the least efficient way to pack balls to suppose that there’s at least one ball that exists.