Rich asks: << Anyone care to ... come up with an accurate explanation of manifold, understandable at the level of a college graduate who majored in, say, chemistry?
Since manifolds are central to the kind of math I usually do, I've often been asked by a non-mathematician to explain them. I usually start by depicting a few curves and surfaces, and then point out that they don't need to "be anywhere" to make sense. Rather, they are the result of putting together pieces of 1- or 2-dimensional "modeling clay" according to certain instructions. This idea can be illustrated with the circle, the sphere and/or torus, and then the Klein bottle or projective plane, the last two of which can be described by sewing together certain edges of a square. I then point out that these two don't have any way of existing in ordinary space (except with flaws), much as a knot can't be flawlessly depicted in 2D. If someone follows this far, then the idea of modeling clay naturally can be elevated to of 3-manifolds (since the actual stuff is 3D, anyway). So it's then possible to explain the 3-sphere as two solid balls with corresponding points on the boundary "glued together" ("Rome to Rome, Paris to Paris, etc."), and/or the 3-torus as a cube with opposited faces identified. This approach seems to work with poets no worse than with chemists. (Note: Since someday I hope to put this kind of explanation in a book, I'll ask that if anyone wants to quote it then please attribute it to me -- thanks.) --Dan Asimov