The Mobius Band and Other Surfaces

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Relevant examples from Escher's work:


You can learn more about the Möbius Strip here:

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

Find out more about the Möbius Strip and related surfaces with the Möbius 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.

Below we see how we would create a klein bottle starting from a square. We first glue the left and right hand sides together to create a cylinder. Next we bend the cylinder and then glue together the ends so that the arrows agree.

We cannot realize this figure in three dimensions. We can pretend that we pass the end of the cylinder through itself before gluing the ends together, but mathematicians often prefer to think of this object as existing in 4-dimensional space.

Just like the Mobius stip, the Klein Bottle is a one-sided figure. Unlike the Mobius Strip, the Klein bottle does not have any boundary though. The Klein Bottle has no holes or punctures. We would say that the surface is closed.

Construction.png This section is unfinished.

External References

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