The Mobius Band and Other Surfaces

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Relevant examples from Escher's work:

Möbius Strip

The Möbius Strip is an interesting surface. It locally looks like any other surface. Close-up we see a 2-dimensional object. The surface becomes more interesting when we try to decide how many sides this surface has. If we take a flat piece of paperr, then we can easily convince ourselves that it has exactly 2 sides. We could for instance color one side of our paper blue and the other side red, and we would never run into any problem. This is not true for the Möbius Strip however. If we would take a marker and start coloring one side of the Möbius Strip we would realize after a while that after we had colored that side, we had colored the entire surface! In other words this is a one-sided surface.

It is actually very easy to create a Möbius Strip for one's self. Take a strip of paper and glue the ends together after giving the strip a half-twist. This proces is illustrated in the figure below:

Bande de moebius.png

To find out more about the Mobius Strip and related surfaces do the Mobius Strip Exploration

The Torus

The Klein Bottle

(Adapted from Wikipedia)

439px-Acme klein bottle.jpg

The Klein bottle is a certain non-orientable surface, i.e., a surface (a two-dimensional topological space) with no distinction between the "inside" and "outside" surfaces. The Klein bottle was first described in 1882 by the German mathematician Felix Klein. It was originally named the "Klein-Fläche" 'Klein surface'; however, this was incorrectly interpreted as "Klein-Flasche" 'Klein bottle', which ultimately, due to the dominance of the English language in science, led to the adoption of this term in the German language as well.

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External References

  1. [Mobius Strip (Wikipedia)]
  2. [The Torus (Wikipedia)]
  3. [The Klein Bottle (Wikipedia)]