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
