## 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.

## Monopoly Chances

While the whole world surely heard about it before, I just today ran across a web page purporting to give the probabilities and expected incomes for the various squares on a Monopoly board. There are many similar versions of this table around — the Monopoly app for iPad even offers the probability that your opponents will land on any given square in the next turn, which is superlatively useful if you want to micromanage your building — and I wouldn’t be surprised if there are little variations and differences between tables.

What’s interesting to me is that the author, Truman Collins, works out the answers by two different models, and considers the results to probably be fairly close to correct because the different models of the game agree fairly well. There are some important programming differences between Collins’s two models (both of which are shown, in code written in C, so it won’t compile on your system without a lot of irritating extra work), but the one that’s most obvious is that in one model the effect of being tossed into jail after rolling three doubles in a row is modelled, while in the other it’s ignored.

Does this matter? Well, it matters a bit, since one is closer to the true game than the other, but at the cost of making a more complicated simulation, which is the normal sort of trade-off someone building a model has to make. Any simulation simplifies the thing being modelled, and a rule like the jail-on-three-doubles might be too much bother for the improvement in accuracy it offers.

Here’s another thing to decide in building the model: when you land in jail, you can either pay a \$50 fine and get out immediately, or can try to roll doubles. If there are a lot of properties bought by your opponents, sitting in jail (as the rolling-doubles method implies) can be better, as it reduces the chance you have to pay rent to someone else. That’s likely the state in the later part of the game. If there are a lot of unclaimed properties, you want to get out and buy stuff. Collins simulates this by supposing that in the early game one buys one’s way out, and in the late game one rolls for doubles. But even that’s a simplification: suppose you owned much of the sides of the board after jail. (You’re likely crushing me, in that case.) Why not get out and get closer to Go the sooner, as long as it’s not likely to cost you?

That Collins tries different models and gets similar results suggest that these estimates are tolerably close to right, and often, that’s the best one can really know about how well a model of a complicated thing represents the reality.