All this talk of Mobius strips makes me wonder ...
There are lots of 2D surfaces that can be embedded into 3 dimensions, for
example the infinite plane (R^2), sphere, torus etc can all be embedded in
R^3.
There are some 2D surfaces which cannot, and (I believe) require 4
dimensions such as the Klein bottle and cross-cap.
Is 4D sufficient for all unbounded surfaces? All surfaces? Can a surface
be
constructed which needs 5 dimensions?
And please, I can't provide a definition of a surface, other than it is a
2D
structure which can be represented by a reasonably behaved function, such
as
everywhere continuous (analytic even?).
|
