so i'm very happy to be here with greg leibon, a mathematician at dartmouth college. >> thank you, dan. >> our poor topologist, it turns out he was on a torus, but in fact he could have been on many other kinds of surfaces, right? >> that's right. when you go to classify or think about what are the possibilities of surfaces, there are two major kinds, especially in terms of what you need to get used to to think about them. >> so there's a first distinction: there are orientable surfaces and there are nonorientable surfaces. >> that's right. and it's nice to think -- maybe first the simplest example of something that's nonorientable. >> the most famous example, in fact. >> this is the most famous example, the mobius strip. and if we imagine that this has no thickness -- we have to keep those things in mind. and maybe one other thing, it has a boundary, which means it has some kind of edge to it. in fact, this is a fun thing to do right off the bat is you follow this edge around. it only has one edge. so it's important, we're thinking intrinsically here. we're living inside the surface.