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Monday, December 12, 2005

Embedding a Klein Bottle in 4-space

My wife is leaving for a 3 week trip to India and decided to give me my Christmas present early. What I got was a glass model of an immersed Klien bottle. On the left, you see some models and on the right, a drawing made by Mathematica software. She got the idea for the gift from Dus7's blog and went to Classy Glass to get it.
Ok, so what is a Klien bottle? And, why is it important, and why did I use the word "immersed"?
First, a Klien bottle is an example of what is known as a "2-manifold"; that is, a very nearsighted ant that lived on the surface of the bottle could not distinguish the area immediately around it from what it would see if it lived on an ordinary plane (the kind of plane you did geometry on in high school). Furthermore, for those who have had some calculus, there is a way of "parametrizing" the Klien bottle so we could do calculus on it.
To obtain the Klien bottle from a section of a plane, consider the following figure:

Take this rectangle and glue the upper edge to the lower edge as shown. One obtains a long open "tube". Now if one were to glue the remaining "circle" edges to each other in a normal way, would would obtain a surface that looks like the skin of a donut; this is called a "torus".

But if we glue the circles in the reverse way (as indicated by the arrows; this is similar to what one does when one makes a Mobius Strip) one obtains the Klien bottle. Go ahead and click on the link as it gives better drawings than I give here.

Now, back to the glass model and the drawing. Notice how the bottle seems to "intersect itself" in a circle (where the glass tube goes from the "outside" into the "inside"? This is unavoidable if one tries to put this bottle in 3-space.

But, if one goes up one dimension, one can avoid this self intersection.

Consider the following two figures.

Here, we pretend that we are in 4 dimensional space, with coordinates (x, y, z, w). In the first figure (the shorter tube) we assume that the two circle boundaries of the tube lie in the plane (x, y, 0,0). The rest of the tube lies in the hyperplane (x, y, 0,w). That is, for all points in the first figure, the z coordinate is "0". Label the first circle boundary (closest to you) by C3 and the second one by C1. Now the second tube lies in the hyperplane (x, y, z, 0); that is, all of the points in this tube have "w" coordinate "0". The three circle that I've shown lie in the plane (x, y, 0, 0). The circle closest to you is called C3, and the one just slightly to the right of that is called C1. Now the circles labeled C1 and C3 are the same circle as they have the same coordinates in the (x, y, 0, 0) plane. So, we can join these two tubes in 4 space to obtain a surface that has no "exposed" edges; we call such a thing a "closed" surface. Notice that these tubes do not intersect each other except for these two circles as all of their points differ in their z and w coordinates. And, when you join these two tubes up, you get a Klien bottle! This is because of the way the tubes connect at circle C1.


Blogger Zeno said...

Cool stuff, Ollie. One small correction: It's Klein bottle. One of my fellow grad students had a joke posted on his door back in the old days. It was based on the fact that "klein" means "small" in German. And "kein" means none. The cartoon panel was divided into three panels. In the first panel was a small bottle labeled "Klein bottle". In the second panel was an even smaller bottle labeled "Kleiner bottle". The third panel was blank, except for the label "Kein bottle".

Oh, how we laughed.

P.S.: dus7 tipped me off to your math blog.

12:00 AM  

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