How can a torus be turned to a cylinder if the circumference of the outer ring is larger?

Question

I’m not a math professional by any means, I’m just interested in math and this topic has been on my mind for a while, and I just couldn’t find an answer. Also, on the same vein, how can you make a cylinder out of a piece of paper? The inner circle would have a smaller circumference than the outer, but each side of the paper has the same length.

Answer 1

The math of topology is an abstraction that has no direct connection to manipulating physical objects. But in this case, we can tie things to physical intuition pretty well.

When we talk about turning a cylinder into a torus by gluing together the ends, there's two ways we can mean it:

• We can imagine it abstractly, where the ends of the cylinder are just declared to be the same. This is similar to video game behavior in games such as Pac-Man, where objects going off one side of the screen return on the other side as though those points were adjacent.
• We can perform a continuous deformation of the cylinder inside $\mathbb R^3$ until the ends are actually in the same location and we get a torus. Here, continuity is what matters, which means that points that start out close together stay close together (and there's a precise $\epsilon$-$\delta$ definition of this). The physical intuition is that you have an infinitely flexible and stretchable material that won't break: more like rubber than like paper.

A cylinder made out of paper should be thought of mathematically as a metric space, not a topological space, because paper can't be stretched: if you imagine an ant following a path on the surface of the paper, the length of the path won't change for the ant when you bend the paper. In this respect, a cylinder can't be turned into a torus - mathematically or physically.

We can actually prove this mathematically in terms of Gaussian curvature, which, informally, measures how much the way the surface bends affects distance. A cylinder has zero Gaussian curvature everywhere - it is actually no different in terms of small-scale distances from a flat piece of paper (and we can make a cylinder out of a sheet of paper by bending it). A torus has positive Gaussian curvature on the outside and negative on the inside. Gauss's theorema egregium says, therefore, that we can't turn the cylinder into anything like the torus by "local isometry", which translates almost exactly into our intuition of what's physically possible with a sheet of paper.