Visualizing a Klein BottleDate: 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/ |
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