Exploring Spaces 1: Torus

elliot evans
Apr 22 2024


Have you ever heard that asteroids is played on a torus?

Objects wrap around the vertical and horizontal edges of space in the game.


sketch of a donut beside a yellow donut shape You may have seen a torus drawn or described as a (hollow) donut.

A hollow donut shape is an example of a torus.

More generally though (from topology), any space that wraps around like a torus, is a torus.

Let's investigate the relationship between the space in asteroids and a donut.

Taping a Rectangle into a Donut

Cube nets In school I learned how to take a flat "net", fold it up, and tape it into a cube. You can do a similar, although a bit more tricky, craft to turn a flat rectangle into a torus.
sketch of a donut beside a yellow donut shape sketch of a donut beside a yellow donut shape sketch of a donut beside a yellow donut shape
sketch of a donut beside a yellow donut shape I looked up how to convert a paper rectangle into a paper torus. I found this nice simple video tutorial and followed it.

If you use this method, you have to cut the paper a bit because paper is not very flexible. So you don't technically get a torus, but if you imagine that the gaps are not there then its a torus.

Here's how mathematicians usually give instructions on how to tape (they sometimes say glue) a rectangle to itself to get a torus.

The diagram says to tape the top edge to the bottom edge without twisting the paper, and also to tape the left edge to the right edge without twisting the paper.
In this section I tried to show that if you physically attach the parts of the rectangle that wrap around to eachother in asteroids, you get a hollow donut shape.

Thank you to Sharon for lending me these scissors lol

Input Spaces

Please try playing around with these inputs.

This torus input is connected in the same way as two circular inputs put beside eachother!

We can learn this from topology: a torus is the cartesian product of two circles.

A torus is 2D and circles are 1D. Two 1D circles make for a 2D torus.

Thank you to Red Blob Games (Amit Patel) for sharing your draggable input recipes! I used them to make these inputs.

Drawing on a Torus

Drawing on a torus represented as a rectangle that wraps around can give us a new perspective. Here are some ways to draw a circle on a torus.

Oh snap, a new way to draw a circle just dropped! Draw a circle as a line that wraps around a torus.

This isn't necessarily a circle in the usual geometric sense, but it does have constant curvature (like a usual circle) and is connected in the same way as a normal circle.
You can draw two circles that only intersect once on a torus! I don't think you can do that on a normal piece of paper.

I wanted to write this article because I want to write and draw about some other mathematical objects, and sometimes it is easier to draw things wrapping around the edge.

Further Notes

Let me know if you want to see illustrations or interactives of any of these notes!