Klein Bottles and Mobius Strips

Date: 08/09/99 at 12:37:41
From: Pam Evans
Subject: Klein Bottle/Mobius Strip

How is the Klein bottle related to the Mobius Strip? If I can 
construct a Mobius strip in 3-dimensional space, why can't I construct 
a Klein bottle without intersection?

Date: 08/09/99 at 14:34:54
From: Doctor Rob
Subject: Re: Klein Bottle/Mobius Strip

Thanks for writing to Ask Dr. Math.

A Moebius strip is constructed from a strip of paper by bending it 
around in a loop and fastening one end to the other, with a twist. You 
can think of it as being formed from the following rectangle

   D                                     C
    o-----------------------------------o
    |                                   |
    |                                   |
    o-----------------------------------o
   A                                     B

by joining side BC to side DA so that B and D coincide and A and C 
also coincide.

A Klein bottle is constructed from the same strip of paper by rolling 
it into a tube lengthwise, and joining side AB to side DC in the usual 
way, and then taking the two circles at each end and joining them 
together, not in the usual way (that would give you a torus), but so 
that one of them is reversed. Thus the circle formed by going from B 
to C upwards in the above diagram is stuck to the circle going from D 
to A downwards.

This means that you can take a Klein bottle and cut it along the line 
AB, and the result is a Moebius strip.

This can't be done in three-dimensional space, because it would form a 
bounded (2-dimensional) surface with no boundary that has only one 
side. Such cannot exist in 3-space. In 3-space every bounded surface 
with no boundary divides the space into two pieces, one of which is 
bounded, called the inside, and one which is unbounded, called the 
outside. The Klein bottle doesn't do that. It is a bounded surface 
with no boundary, so it cannot be realized in 3-space.

The proof of this fact is pretty difficult, I'm sorry to report, so I 
can't give it here. You may study this when you study topology as an 
advanced undergraduate or graduate student.

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