(A prequel to the Mathematical Tapas course. We'll see be seeing tori in the algebraic topology and elliptic curves classes.)

Here's a little thought-origami for you. Take a square piece of paper. Label two opposite sides a and the other two b.



Now tape the a sides together (don't get all fancy; just glue them parallel). You should have a toilet paper roll.



The b sides are now circles. Go ahead and tape those together as well. This gives you something that looks like the outside of a donut.



(If you're playing along at home, you probably just learned that you can't actually tape these together without crushing the whole thing. What you just proved (well, plus or minus a few years of differential geometry and algebraic topology) is that you can't embed a flat torus into 3-space. Whatever "embed" means. And "flat torus." Well, 3-space makes sense anyway.)

The thing we just made is a called a torus. We needed to stretch some parts and shrink others in order to make it fit into space, but that's OK. In fact that's more than OK, it's called "algebraic topology." Technically, algebraic topology is the application of algebra to the understanding of shapes. Informally, algebraic topology is the branch of math that tells us about shapes, but only up to stretching and smooshing them around.

How can we distinguish a torus from a beachball? Well, I could wear a torus around my head, but not a beachball. Unless the beachball is deflated, of course. Then the beachball becomes a beret, which is, in some spheres (get it?), considered even more wearable than a donut.

We do have a pretty good idea about what "has a hole" actually means. A donut has a hole and a (functioning) beachball does not. What quantitative properties, though, encode this information? How do you prove mathematically that a beachball does not have a hole? How do we know that Earth does not have a hole through the middle?

Look at our picture of the torus again and look specifically at the green loop. Think of this green loop as a stretchable and shrinkable rope that always has to be touching the torus. How does the green loop tell us that the torus is not a sphere?

No matter what we do to the green rope, it's always going to be wrapped exactly once around the waist of the torus. Specifically we can't shrink it any smaller than it is initially. However, if you make a loop on the surface of the Earth, no matter how strange the loop is initially, you can always shrink it down to a point. (Think of putting a loop on the earth and try to find a trick that will shrink the loop to a point at the south pole. How does your trick fail if you bore a hole through the center of the earth?)

What about the red loop? It looks like you can just pop it off the torus and be done with it, but that breaks the rules! Since the red loop must always be touching the torus, we see that no matter how we move it around it always has to circle once around the donut's hole. Therefore the green loop keeps track of one flavor of circle that lives on the torus and the red loop keeps track of another.

So there's our proof! Loops on a sphere and loops on a torus behave differently. If you put a loop on the sphere, it can always shrink to nothing. However there are loops on the torus that can't shrink to nothing (they might get hung up on the hole or around the waist). These loops keep track of, or "encode" the shape of the torus.

Here are some fun questions:

Go back to the square. If we had glued the green sides together first and then the red sides, we would have gotten a different-looking torus. We probably would have drawn it horizontally instead of vertically, and the green loop would be the one that goes around the hole. However, as far as the torus is concerned we made the same figure: the horizontal edges are glued and the vertical edges are glued. How do you reconcile this? How are these tori the same?




Again on the square, draw a line from the bottom left corner to the top right. When you glue the red edges this gives a line traveling up the toilet paper roll, except the line has a single twist in it (a helix, if you're genetically inclined).



However the new line begins and ends where the green and red meet. That means that when we glue the green edges together we get another LOOP on the torus. What does it look like? Is it the same as either the red or the green loop? How many different loops can you put on the torus anyway?