Re: sphereland

Paul Burchard (burchard@CS.Princeton.EDU)
Mon, 09 Dec 1996 13:13:05 -0500

Chuck Biehl wrote:
> I don't understand how a 4D bring could untie a 3D
> knot if a 3D being is holding both ends of the rope.
> I know it has to do with a twist in the fourth dimension,
> but I just can't picture it. There's no 2D analog (I think)
> because I think you have to be at least 3D in
> order to tie knots... any thoughts?

Here's one way to imagine it -- use color to denote the 4th dimension...
say ranging from blue for objects "close by" in the 4th direction, to
red for ones "far away". Then for two objects to be at the "same spot"
in 4D means they must appear to be at the same 3D point _and_ they must
be the same color.

So for the knot, as you bring two strands close to bumping into each
other in a frustrated attempt to undo it, you can make one strand bluer
(pulling it "closer" to you in the 4th dimension) and one strand redder
(pushing "farther" in the 4th dimension). Then, when the two strands
would appear to block each other in 3D, actually they are not colliding
at all in 4D, as the color difference vividly shows. You can pass the
strands right on past each other.

The simplest analog of this is the way you can't exchange the position
of a pair of points in 1D, but there's no problem doing it in 2D. (P.S.
Is this something where an applet might come in handy for the topology