## My Little 2021 Mathematics A-to-Z: Torus

Mr Wu, a mathematics tutor in Singapore and author of the blog about that, offered this week’s topic. It’s about one of the iconic mathematics shapes.

# Torus

When one designs a board game, one has to decide what the edge of the board means. Some games make getting to the edge the goal, such as Candy Land or backgammon. Some games set their play so the edge is unreachable, such as Clue or Monopoly. Some make the edge an impassible limit, such as Go or Scrabble or Checkers. And sometimes the edge becomes something different.

Consider a strategy game like Risk or Civilization or their video game descendants like Europa Universalis. One has to be able to go east, or west, without limit. But there’s no making a cylindrical board. Or making a board infinite in extent, side to side. Instead, the game demands we connect borders. Moving east one space from just-at-the-Eastern-edge means we put the piece at just-at-the-Western-edge. As a video game this is seamless. As a tabletop game we just learn to remember those units in Alberta are not so far from Kamchatka as they look. We have the awkward point that the board doesn’t let us go over the poles. It doesn’t hurt game play: no one wants to invade Russia from the north. We can represent a boundless space on our table.

Sometimes we need more. Consider the arcade game Asteroid. The player’s spaceship hopes to survive by blasting into dust asteroids cluttered around them. The game ‘board’ is the arcade screen, a manageable slice of space. Asteroids move in any direction, often drifting off-screen. If they were out of the game, this would make victory so easy as to be unsatisfying. So the game takes a tip from the strategy games, and connects the right edge of the screen to the left. If we ask why an asteroid last seen moving to the right now appears on the left, well, there are answers. One is to say we’re in a very average segment of a huge asteroid field. There’s about as many asteroids that happen to be approaching from off-screen as recede from us. Why our local work destroying asteroids eliminates the off-screen asteroids is a mystery for the ages. Perhaps the rest of the fleet is also asteroid-clearing at about our pace. What matters is we still have to do something with the asteroids.

Almost. We’ve still got asteroids leaking away through the top and bottom. But we can use the same trick the right and left edges do. And now we have some wonderful things. One is a balanced game. Another is the space in which ship and asteroids move. It is no rectangle now, but a torus.

This is a neat space to explore. It’s unbounded, for example, just as the surface of the Earth is. Or (it appears) the actual universe is. Set your course right and your spaceship can go quite a long way without getting back to exactly where it started from, again much like the surface of the Earth or the universe. We can impersonate an unbounded space using a manageably small set of coordinates, a decent-size game board.

That’s a nice trick to have. Many mathematics problems are about how great blocks of things behave. And it’s usually easiest to model these things if there aren’t boundaries. We can, sure, but they’re hard, most of the time. So we analyze great, infinitely-extending stretches of things.

Analysis does great things. But we need sometimes to do simulations, too. Computers are, as ever, great tempting setups to this. Look at a spreadsheet with hundreds of rows and columns of cells. Each can represent a point in space, interacting with whatever’s nearby by whatever our rule is. And this can do very well … except these cells have to represent a finite territory. A million rows can’t span more than one million times the greatest distance between rows. We have to handle that.

There are tricks. One is to model the cells as being at ever-expanding distances, trusting that there are regions too dull to need much attention. Another is to give the boundary some values that, we figure, look as generic as possible. That “past here it carries on like that”. The trick that makes rhetorical sense to mention here is creating a torus, matching left edge to right, top edge to bottom. Front edge to back if it’s a three-dimensional model.

Making a torus works if a particular spot is mostly affected by its local neighborhood. This describes a lot of problems we find interesting. Many of them are in statistical mechanics, where we do a lot of problems about particules in grids that can do one of two things, depending on the locale. But many mechanics problems work like this too. If we’re interested in how a satellite orbits the Earth, we can ignore that Saturn exists, except maybe as something it might photograph.

And just making a grid into a torus doesn’t solve every problem. This is obvious if you imagine making a torus that’s two rows and two columns linked together. There won’t be much interesting behavior there. Even a reasonably large grid offers problems. There might be structures larger than the torus is across or wide, for example, worth study, and those will be missed. That we have a grid means that a shape is easier to represent if it’s horizontal or vertical. In a real continuous space there’s no directions to be partial to.

There are topology differences too. A famous result shows that four colors are enough to color any map on the plane. On the torus we need at least seven. Putting colors on things may seem like a trivial worry. But map colorings represent information about how stuff can be connected. And here’s a huge difference in these connections.

This all is about one aspect of a torus. Likely you came in wondering when I would get to talking about doughnut shapes, and the line about topology may have readied you to hear about coffee cups. The torus, like most any mathematical concept familiar enough ordinary people know the word, connects to many ideas. Some of them have more than one hole. Some have surfaces that intersect themselves. Some extend into four or more dimensions. Some are even constructs that appear in phase space, describing ways that complicated physical systems can behave. These are all reflections of this shape idea that we can learn from thinking about game boards.

## 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 S2 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.