Visualizing a Klein Bottle

Date: 08/19/99 at 07:20:39
From: Sean
Subject: Klein bottle concept

I'm having a problem visualizing spaces of more than 3 dimensions. A 
good example is a Klein bottle. I read that it can't exist in 3 
dimensions, only 4 dimensions. What part of it can't be seen or 
represented in 3D? Is there some technique that can help me 
visualize higher dimensions?

Date: 08/19/99 at 12:37:21
From: Doctor Peterson
Subject: Re: Klein bottle concept

Hi, Sean.

The only thing that keeps a Klein bottle from existing in 3 dimensions 
is its self-intersection. A true Klein bottle is supposed to be a 
non-intersecting single-sided surface with no edges (that is, a closed 
non-orientable surface); when we make one in 3 dimensions, the 
"handle" has to poke through the "side" of the bottle to connect 
properly. If you have a fourth dimension available, you can just push 
the "handle" aside in that direction a little so that it doesn't 
touch.

You can often visualize concepts like this if you shift down one 
dimension. Let's think about a 2-dimensional equivalent: suppose we 
live on a plane, and want to draw a simple closed curve that looks 
like a figure eight:

          *********           **********
       ***         ***     ***          ***
      *               *   *                *
     *                 * *                  *
     *                  *                   *
     *                 * *                  *
      *               *   *                *
       ***         ***     ***          ***
          *********           **********

The trouble is that it intersects itself, so it's not a simple closed 
curve. But if we had a good enough imagination to picture a third 
dimension, we could lift one part of the curve a little in that 
direction, making a "bridge" so it didn't intersect! That's the sort 
of thing you have to do to make the Klein bottle work.

Here is more information:

Math 655 - Introduction to Topology - Zbigniew Fiedorowicz 
  http://www.math.ohio-state.edu/~fiedorow/math655/Klein2.html   

The Math of Non-Orientable Surfaces - Margaret Boittin, Erin Callahan, 
David Goldberg, and Jacob Remes, Yale University
  http://pantheon.yale.edu/~jar55/math/project/math.htm   

and, just for fun,

Acme Klein Bottles - Cliff Stoll
  http://www.kleinbottle.com/   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/