. >> so this is edwin abbott's flatland, a book first printed in 1884, and it's the story of beings that live in two dimensions -- on the plane, if you like, with no thickness -- and how they might experience three-dimensional life, which is analogous to our problem as beings living in a three-dimensional world trying to experience four-dimensional life. so the two-dimensional being is living on this sort of infinitesimally thin surface, right? so they can only move in two dimensions. >> and somebody enters their world from above, a sphere -- we can imagine if a sphere, it's this very symmetric object, and we're going to imagine it in the following sense: this sphere which actually lives in three dimensions is going to dip into this two-dimensional being's world, and the two-dimensional being is going to experience an aspect of this sphere, what we think of mathematically as slices of the sphere, and they're going to look at the geometry of those slices over time as the sphere moves through their world, and they're going to use that as a summary statement as to what three dimensions look