Klein Bottle

   
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  • A mathematician named Klein
  • Thought the Moebius Strip was divine.
    • He said, "If you glue
    • "the edges of two
  • "You get a weird bottle like mine!"
Take two disks. Deform each one just a little until they are bowls. Disks only have one edge, so each bowl has only one edge, so now glue the edge of one to the edge of the other, and what you get is a sphere (well, distorted somewhat). Since each disk has two sides the sphere also has two sides, only now it's an inside and an outside.

Got that?

OK, now consider the Moebius Strip. Just as a disk has only one edge, so the Moebius Strip has only one edge. We can imagine taking two Moebius Strips and gluing the edge of one to the edge of the other. It's a bit hard to do in reality because the twistiness gets in the way a bit, but we can imagine doing it (drawings will be provided here as soon as someone asks for them - click on the Edit This Page link at the bottom of this page and fill in the form)

images/big_klein.gif
images/klein.gif
The result is called a Klein Bottle, and it's pretty hard to visualise in all its detail. Here are a couple of pictures.

images/klein-medium.jpg
This one was drawn by Tom Banchoff and classes as instructive, artistic and inspired. I am somewhat in awe of this picture, having myself tried in the past to draw pretty Klein bottles and having failed miserably.
Visit Tom's page on the Klein bottle at: There you can also find animations of klein bottles which help enormously in understanding this particular projection into 3-space.


On the remainder of this page I want to talk about Klein bottles, their different representations and their relationships with Moebius Strips. In particular I want to talk about the three or four common embeddings of the Klein bottle into the usual 3-dimensional space, and the ways of viewing them in 4-dimensional space.

Then we want to talk about generalising processes. We've talked about gluing two disks together to get a sphere, and gluing two Moebius Strips together to get a Klein Bottle. What do you get when you glue the edge of a Moebius Strip to the edge of a disk? You get a Cross Cap or Projective Plane.

This page should also reference the pages on Mathematics and Topology. It's also worth noting that this Klein is the same Klein as appears in Colins Mathematical Ancestry.

See also http://www.google.com/search?q=%22Klein+Bottle%22


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