Klein Bottles and Mobius StripsDate: 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/ |
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