The Möbius Strip and the Klein Bottle

I have always been interested in figures like the Möbius Strip and the Klein Bottle. The Mobius strip is a is a one-sided nonorientable surface and the Klein Bottle is a closed nonorientable surface. Both figures have a Euler characteristic of 0. While the Möbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle can be embedded in R4. As an aspiring geometer, I always found it intriguing to visualize both of these figures. Since the Möbius strip can be embedded in three-dimensional Euclidean space R3, it was much simpler to visualize than the Klein Bottle. Still, it was figures like the Möbius Strip and the Klein Bottle that inspired me to pursue geometry beyond the foundations of Euclidean geometry, namely hyperbolic and elliptic geometry. Interestingly, the angles of a triangle in hyperbolic geometry add up to less than 180o.

 

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Möbius Strip

 KleinBottle-Figure8-01

 Klein Bottle

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