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To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and "identify opposite sides". We can imagine gluing together one pair of opposite sides to get a cylinder, and then gluing together opposite ends of the cylinder to get a torus. (Provided that our material is stretchy enough, which isn't an issue topologists concern themselves with.)

In the past, I've described this description of the torus by analogy with Pac-Man. In this video game, Pac-Man can leave the screen on one edge, and come back onto the screen from the same position on the other side.

But I want to abandon this analogy, and come up with something better, because:

  • If you haven't played Pac-Man, it's not very helpful - and how many people these days have? I think I've played Pac-Man on a TI calculator a total of once or twice in high school.
  • If you have played Pac-Man, it's not very helpful, because in a typical Pac-Man maze, the "tunnels" that allow this wrapping-around behavior only go one way: from left to right. So a Pac-Man level is more like a cylinder than a torus.

Are there better analogies?

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    $\begingroup$ “Asteroids” had a toroidal universe, but it’s even older than Pac-Man I think. Some students seem to know what it is and understand the toroidal topology, some students don’t play such videos at all. Not sure if there is an “excellent” example in video games. — Trigonometry, angles on a unit circle, sinusoidal signals, phase shifts give a one-dimensional analogy. Two angular or phase-shift variables give you coordinates on a torus. Note sure if that’s helpful in an intuitive, visualizable sort of way, though. $\endgroup$ – user615 Nov 7 '20 at 18:43
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    $\begingroup$ Would a world map do? Or do you need explicitly the square bit? $\endgroup$ – Thorbjørn Ravn Andersen Nov 8 '20 at 13:22
  • $\begingroup$ @ThorbjørnRavnAndersen It's not important for it to be square, but world maps that wrap around top to bottom as well as left to right don't happen often. $\endgroup$ – Misha Lavrov Nov 8 '20 at 15:39
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    $\begingroup$ @ThorbjørnRavnAndersen I think I want an analogy for the same reason everyone wants an analogy: to relate a new concept to something people have seen before. A bathing ring is an analogy for the torus, but not for the construction of the torus via identifying opposite sides of a square. $\endgroup$ – Misha Lavrov Nov 8 '20 at 17:13
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    $\begingroup$ @user1027: Asteroids is not meaningfully older than Pac Man. At least some of the Ultima series of games have world maps that wrap around in toroidal form; the layout of Ultima III's overworld is particularly interesting in that regard since it has one primary mass of land and one main ocean, but two detached coasts. $\endgroup$ – supercat Nov 10 '20 at 2:44
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A possibility is to show your students Google's implementation of the game Snake. If you enter the term snake into Google's search engine, the there is a box at the top showing

enter image description here

If your particular Google bubble doesn't show this result, I think that there is a direct link.

If you hit "Play", you can play a game of Snake, using the standard rules. Play a couple of short rounds, and demonstrate what happens if you run into a wall (you lose). Then, after losing a couple of rounds, hit the gear to open up a settings menu.

enter image description here

On the second line of settings, choose the infinity symbol.

enter image description here

This version of the game is played on a torus. If you eat enough apples, you can see how the wrapping around works. Alternatively, instead of selecting the infinity symbol, the peace symbol also allows you to play on a torus, but without the centering (i.e. the game will behave like Asteroids or Pac-Man, where you leave one side of the screen, and reappear on the other).

What is handy about this is that you get a very quick demonstration of a game being played on a torus, and this demonstration is available to any person who has a clever device (e.g. a desktop or laptop computer, a clever phone, etc). Moreover, it has a rather visceral, concrete quality to it which, to my mind, is more convincing than a bunch of hand drawn pictures (though maybe not as convincing as actually making a torus out of paper or cloth).

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    $\begingroup$ I like the idea of using this game as a prop! ☮️ mode is especially convenient for this demonstration, because it disables collisions between the snake's head and its body as well, which means I don't have to be good at the game :) $\endgroup$ – Misha Lavrov Nov 8 '20 at 17:19
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    $\begingroup$ In advance, I would also print a t-shirt where one snake is chasing another snake from left-to-right, about to bite its tail. But the snake wraps all the way around the shirt, so it is about to bite itself. I'd probably show the video game to illustrate the torus, meanwhile casually taking off my jacket to reveal the shirt (mindful to turn around at some point to show the back side). What about a clothing prop? $\endgroup$ – Nick C Nov 8 '20 at 22:23
  • $\begingroup$ @NickC Wouldn't that shirt still only imply a cylinder? Unless you had it crawl into your shirt, too... Not that it matters, as t-shirts have 3 holes, not 1 :) $\endgroup$ – Weckar E. Nov 9 '20 at 7:23
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    $\begingroup$ @NickC If you have the means, make it a button-down shirt instead, so you can demonstrate how it changes from a plane into a cylinder when you button it up. (Obviously wear another shirt underneath it.) $\endgroup$ – Darrel Hoffman Nov 9 '20 at 19:34
  • $\begingroup$ A shirt is not a torus, and it is not even a cylinder, as it has too many holes. It "unbuttons" to form a twice punctured square, with the buttons and holes giving an identification of one set of opposite sides. $\endgroup$ – Xander Henderson Nov 9 '20 at 19:43
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One thing that I like about the Pacman analogy is that, if you draw Pacman's eye (or Ms. Pacman's bow, which is my usual choice), then Pacman is not mirror symmetric. This means you can talk about nonorientable surfaces and draw how Pacman goes around a Mobius strip/Klein bottle/projective plane and comes back reversed.

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  • $\begingroup$ Wait. Does he reverse? I need to try this... $\endgroup$ – Weckar E. Nov 9 '20 at 7:25
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I don't think it's necessary or a good idea to invoke a specific game, because any given game will be one that only a small fraction of your students have played. Just say, "You know how in some video games, if you go off one side of the screen, you reappear on the other side? Say you have a game where this happens both left/right and top/bottom."

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    $\begingroup$ That's a fair objection, but on the other hand, this is strictly less helpful than "You know how in some video games, like Pac-Man, if you go off one side of the screen, you..." $\endgroup$ – Misha Lavrov Nov 7 '20 at 19:05
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    $\begingroup$ @MishaLavrov: I would claim that my answer is more helpful. Any answer invoking a specific video game will likely be unintelligible to the 90% of students who haven't played that game. Therefore it adds cognitive load and confusion for those students, and if you do this in class, there will be a lot of wasted time. $\endgroup$ – Ben Crowell Nov 8 '20 at 20:17
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    $\begingroup$ Some people don't play any video games, but we expect them to know what video games are for the sake of an analogy, and I think this is acceptable. Similarly, I don't think it's a huge ask to expect a person who hasn't played Pac-man to conceive of a game where such-and-such happens when an object moves off the screen. In essence, you are explaining a real-world application of a torus (at least its fundamental domain) and fully explaining the background is necessary. $\endgroup$ – Nick C Nov 8 '20 at 22:24
  • $\begingroup$ I feel like this is a less-than-helpful idea, because what kids play 2D games anymore? They're off playing games like Minecraft or Fortnite nowadays. $\endgroup$ – nick012000 Nov 9 '20 at 16:35
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    $\begingroup$ @nick012000 Those are certainly the games which garner the most press, but there is also a thriving indie scene, and a lot of retro-style games that come out of independent developers. The triple-A market is not the sum total of gaming (see, for example, a lo-fi game like Undertale). 2D games are alive and well, even if few of them are played on tori. $\endgroup$ – Xander Henderson Nov 9 '20 at 20:29
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When you're picking out a movie on Netflix, there's a grid of movies & TV available. If you keep scrolling to the right in any one of the rows, you eventually come back to the first movie in the row. And if you keep scrolling down through the categories, eventually you come back to the first category you started out with.

(At least, that's how the Netflix catalog interface works on my TV. Streaming UI being what it is, I wouldn't be surprised if it's different on other devices, or if it changes next week.)

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I have (once) used a square of felt with two strips of velcro glued to the felt. Then just "identify" one opposing pair of edges in an orientation preserving way via velcro. Demonstrate that one obtains a finite cylinder. Then "identify" the two circle edges in an orientation preserving way.

This also gave some room to talk about nonorientation preserving gluing, a Moebius strip, and the real projective plane (and its cross-cap). You can't actually perform the identifications for all of these (since you will obtain an embedding in $\Bbb{R}^3$ or $S^3$) which allows discussion of $2$-manifolds that don't embed in $3$-space.

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I'm not sure you need an analogy, I think you could explain it nicely just stating how to construct it from a square, but if you do, you could buy a couple of bicycle tubes (preferable for a children’s bike, which are smaller and wider). Then cut it open so you have a rectangle. You can then show the intact and the cut version side by side.

(It might not be a nice rectangle, but I think it will work anyway)

bicycle tube

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East to west on a world map

The world map is usually what I use to explain toroidal periodic boundary conditions. The analogy isn't perfect because the Earth isn't a torus, but it gets the important ideas across in a way everyone can understand. Unfortunately, it doesn't work north-to-south. I used to use Asteroids, but few students are familiar with the game. Everyone is familiar with the world map.

World map modified from Mercator projection by Wikipedia user Strebe ( https://en.wikipedia.org/wiki/World_map#/media/File:Mercator_projection_SW.jpg)

World map modified from Mercator projection by Wikipedia user Strebe.

If you take the map as is, then it looks like you have to fly east over Africa or Europe to get from California to China. However, using the minimum image convention, you see that a flight from California to China should go west over the periodic boundary to land in the left periodic image of China.

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  • $\begingroup$ If the map represented a torus, proceeding in a straight line north from the eastern edge of Canada should cause one to pass through Greenland, Antarctica, and South America in that order, without an intervening journey south through Asia. $\endgroup$ – supercat Nov 10 '20 at 17:17
  • $\begingroup$ @supercat Yes, that's why I say: "The analogy isn't perfect because the Earth isn't a torus, but it gets the important ideas across in a way everyone can understand. Unfortunately, it doesn't work north-to-south." $\endgroup$ – WaterMolecule Nov 10 '20 at 17:20
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    $\begingroup$ The north-to-south connection is what distinguishes a torus from a sphere. On a sphere, going from a point north of the equator to a point south of the equator would require crossing over the equator, but on a torus one could also wrap around from north to south without having to cross the equator anywhere. $\endgroup$ – supercat Nov 10 '20 at 17:28
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I have always found the best way to ingrain toroidals is by showing them in unexpected places. Drinking straws, tires, a hollow question mark, a mug... Figuring out how many holes real world objects have is in my opinion the best way to build intuition for their properties.

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  • $\begingroup$ I am very skeptical that you drink coffee out of a mug constructed by identifying opposite sides of a square. $\endgroup$ – Misha Lavrov Nov 9 '20 at 16:24
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    $\begingroup$ @MishaLavrov A coffee mug is topologically identical to a torus, because you can squish down the cup part until you're just left with the handle. $\endgroup$ – nick012000 Nov 9 '20 at 16:35
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    $\begingroup$ @nick012000 Yes, but my question is not about toruses in general. $\endgroup$ – Misha Lavrov Nov 9 '20 at 16:37
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    $\begingroup$ @nick012000 Pish! A coffee mug is topologically equivalent to a genus 1 handlebody, which, as an orientable 3-manifold with boundary, is completely different from a torus, which is an orientable 2-manifold without boundary. :P (removes tongue from cheek) $\endgroup$ – Xander Henderson Nov 9 '20 at 20:26

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