Klein Bottle Intersection

Date: 11/04/2002 at 09:23:53
From: Gilles Eveloy
Subject: Klein Bottle surface

I would like a detailed description of the intersection domain in a 
Klein bottle.

To be more specific, does the external surface contain a hole to let 
the handle cross it, or do we consider that if I consider a point A on 
the external surface and a point B on the handle surface, I can draw 
a continuous path that belongs to the Klein bottle surface ?

Thank you.
Gilles Eveloy

Date: 12/15/2002 at 17:31:23
From: Doctor Nitrogen
Subject: Re: Klein Bottle surface

Hi, Gilles:

There is no hole at all in the Klein bottle. There is only one surface 
for the Klein bottle, and it is not divided up into an "inside" and an 
"outside," although it seems that way from the view of our ordinary 
space.

Here is a description of the topology for a Klein bottle. 

Imagine a long, hollow circular cylinder, with circles for the ends. 
Imagine that the ends are open, and that the cylinder is made of some 
very pliable clear plastic. Suppose you could give the top opening a 
little clockwise "twist" and you could give the bottom opening a 
little counterclockwise "twist," and you could keep both ends twisted 
that way and then try to join the ends. If you want both circular ends 
to match, both clockwise or both counterclockwise, you obviously 
cannot just join them together, because each end is oriented a 
different way, one clockwise, the other counterclockwise. What a 
topologist does to join them so that the two orientations match, is 
"push" the top end directly "into" the cylinder's surface, but he does 
this in a higher-dimensional space, not in our ordinary three-
dimensional space. The result is that he makes both ends match in 
orientation (both clockwise) when he joins them in the higher-
dimensional space.

The result is the peculiar Klein bottle. The original cylinder had a 
hollow inside and an outside. But the Klein bottle does not divide our 
space into an inside and an outside as an ordinary bottle does. There 
is no "interior" and "exterior" surface; there is only one surface, so 
that if you were to connect your two points A and B, it might look as 
if point B is located at some mysterious "interior" of the bottle, but 
such is not the case....

You can also make a Klein bottle by "sewing" two Moebius strips 
together. You can read about this and some more about Klein bottles 
at:

Klein Bottle - Davide P. Cervone, The Geometry Center
http://www.math.union.edu/~dpvc/papers/RP2/Glossary/KleinBottle.html 

Another surface that is strange like the Klein bottle is the "twisted 
sphere," or the projective plane. This projective plane looks as if  
it's infinitely long, but if you start walking straight ahead from 
point A and keep going, you arrive at point A again, although the 
plane seems infinite in extent.

I hope this helped answer the questions you had.

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