we're here with greg leibon, research mathematician at dartmouth college, and, greg, can i see four dimensions without hurting myself? >> you're going to have to go through a little bit of pain to see into higher dimensions, but the pain comes in the form of analogies and metaphors. so we would probably start understanding four dimensions by thinking a little about what we know in lower dimensions. for example, we could start with a cube. >> okay, great, great. we've got our cube. >> we've got our cube here. and there's a few things worth observing first. that looks like a three-dimensional cube to us, but of course this is on a two-dimensional screen. this is some sort of projection. >> that's true, that's true. it's a total cheat. >> but our brains are, fortunately in this case, very hardwired to interpret this as a three-dimensional object. four dimensions is much trickier because, for example, here is a picture of a four-dimensional cube that we'll just look at for a second just to get a sense. it looks very cluttered. it's the same sort of projection where now we've gone from a cube in fo

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.