Felix Klein and The Klein Bottle

Eggbot & Klein Bottle, posted by Lenore Edman from flickr.com, http://www.evilmadscientist.com/2013/eggbot-klein-bottle/

Felix Klein was born in Prussia, Germany on April 25, 1849. In Klein’s obituary records it states that Klein enjoyed taking note that his birth date consisted of three squares of primes: the 22-th month, on the 52-th day in the year 432. Klein attended the University of Bonn, where he studied mathematics and physics from 1856-1866. While attending the University of Bonn, Klein had the opportunity to be an assistant to Julius Plücker , who piqued Klein’s interest in geometry. Julius Plücker is a famous German mathematician, and physicist. He did extensive work with cathode rays, spectroscopy of gases, and also invented line geometry. After Plücker died, Klein was left with his mentor’s incomplete work on geometry. Klein is a legacy in the field of mathematics for his work such as Klein quartics, Klein quadrics, the Klein model, and Klein geometry. In their biography of Felix Klein, J.J. O’Connor and E.F. Robertson write, “It is a little hard to understand the significance of Klein’s contributions to geometry. This is not because it is strange to us today, quite the reverse, it has become so much a part of our present mathematical thinking that it is hard for us to realise the novelty of his results and also the fact that they were not universally accepted by all his contemporaries” (J. J. O’Connor, E. F. Robertson).

One of Felix Klein’s discoveries is the Klein bottle. The Klein bottle is a lot like a Möbius strip, so let’s start with that. A Möbius stip is a surface with one non-orientable side, and one boundary component.  It can be made by taking the two ends of a rectangle, twisting one end 180 degrees, and joining them together. To be non-orientable means that a manifold has “a path where the vector ends up as the mirror-image of the initial orientation” (mathforum.org). To visualize a one-sided surface, if you are “standing upright, you can walk along the surface and reach both sides of each point of the surface” (Polthier, Konrad).  From this interpretation of a Möbius stip, we can now describe the Klein bottle as the combination of two Möbius strips. The Klein bottle is a one-sided surface, that has no inside or outside, and no edges. The Klein bottle “can be constructed by gluing both pairs of opposite edges of a rectangle together giving one pair a half-twist, but can be physically realized only in four dimensions, since it must pass through itself without the presence of a hole” (Weinstein, Eric W.). To generalize, a Klein bottle is made up of two dimensional parts, but can only exist in the fourth dimension. One feature of the Klein bottle is that if you slide a symbol around on its surface, it will reappear in the same place but backwards.

Although you can’t obtain a true Klein bottle, because sadly we live in a three dimensional universe, you can buy a replica of a supposed Klein bottle that will exist in three dimensions. The reason a true Klein bottle cannot be obtained in our universe, is that there is no way it could pass through itself without leaving a hole. In a 4 dimensional universe the Klein bottle can contain itself, but in our 3 dimensional world it is not possible. The Klein bottles you can purchase though, are usually made out of glass, and follow the laws of physics in our universe, and 3-dimensional Euclidean geometry. There are many you things you can do with your Klein bottle, but I’m sure your math teacher would love to receive one of these to display on her desk.

References:

http://aperiodical.com/2012/04/klein-outside-the-bottle/

http://www.famous-mathematicians.com/felix-klein/

https://www.kleinbottle.com/whats_a_klein_bottle.htm

http://mathforum.org/mathimages/index.php/Topology_Glossary

Polthier, Konrad. “Imaging Maths – Inside the Klein Bottle.” Plus.maths.org. Plus Magazine, 23 Aug. 2003. Web. 18 Nov. 2014. <http://plus.maths.org/content/imaging-maths-inside-klein-bottle&gt;.

O’Connor, J. J., and E. F. Robertson. “Felix Christian Klein.” Klein Biography. Web. 18 Nov. 2014. <http://www-history.mcs.st-andrews.ac.uk/Biographies/Klein.html&gt;.

Weisstein, Eric W.  “Klein Bottle.” From MathWorld–A Wolfram Web Resource. http://wathworld.wolfram.com/KleinBottle.html

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