P3D 4D space

[Nick Merz <merz@xxxxxxxxx>]

Nick Merz writes:
<< I would be particularly interested in 4D objects projected
  into 3 space.  For example, I know it's possible to tie a plane into
  a knot in 4D space without having it pierce itself, but I've never
  seen it in 3D. >>

Jim Norman asks:
<<Call me an old fuddy duddy, but what on earth (or any place else, for that
matter) is "4D space"?  And how do you "tie a plane into a knot"???? >>

Response:
I think 4D space is easy to conceptualize and really hard to 
visualize (I find it impossible).  The idea with 3D space is that we 
have 3 axis, X Y and Z, which are all orthogonal to one another.  For 
4D space, there is another axis which is orthogonal to all three of 
the others X Y and Z.  This can get abstracted onward and upward into 
5D spaces and N-dimensional spaces, then you wake up and realize you 
just slept through another linear algebra lecture.

As for knot tieing, consider that a line is a 1D object.  When you 
confine it to 1D space, you get a straight rail:

--------------------------------------

If you allow that line to swim around in 2D space (a plane), you can 
get any kind of open curve:

                                         __---
                                      /           \                    \
           \                       /              |                      |
             \                   /                \                  /
                ` -------- '                      `-------- '

but you can't tie a knot because when it swings around to close off, 
it bumps into itself.  However, if you move this 1D object into 3 
space, you can jump out of the plane and loop it into a knot and 
still have two loose ends.  Consider a plane is a 2D object.  In 2D 
space, it's a plane.  In 3D space, you can peak it and valley it and 
get all kinds of cool geography, but it would intersect itself if you 
tried to tie a knot.  Conceptually, you can tie it off as a knot in 
4D space and still have an open surface (the surface of the Earth is 
an example of a closed, self-intersecting 2D surface).

The cool part of this whole discussion is in considering ways of 
projecting this stuff into 3D, that is, into stereo.  Consider, for 
example, that a still photo is a 2D projection of 3D space (no 
surprise there).  Look at a sheet of paper edge on, and the line you 
see is a 1D projection of 2D space.  A computer can easily model 4D 
space (or a plane tied in a knot in 4D space).  Get the machine to 
generate 2 stereo views of that space, and you have a 3D projection 
of 4D space.  I imagine it would take a while to really learn to 
visualize these views, in much the way it takes some people a while 
to get a handle on orthographic projection drawings.

That's long-winded anough for one posting.

nick.