[rcthompson]

For people who don't want to read the full explanation: This is showing a distance-preserving embedding of a torus and other shapes in 3 dimensions. The simple method of embedding the torus as a doughnut shape requires stretching it, which means distances are not preserved. For example, the inner circumference is smaller than the outer one. But it turns out you can add corrugations of varying amplitude to counteract this effect. As you might expect, it requires an infinite sequence of ever-smaller layers of corrugations to exactly offset the stretching effect and achieve exact preservation of all distances in all directions, but just like constructing a fractal, we can generate the first few steps and stop once the corrugations of the next step would be too small to see. That's what the visualizations on this site are.