The Triple Klein Bottle

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The elementary building blocks of the Klein bottle may be slightly modified to produce several different geometric versions of the Klein bottles. Here we see a Klein bottle with a triple symmetry.

Every non-orientable tube is a Klein bottle. Here the tube has three inversions such that it is non-orientable. In fact, we can use increase the degree of symmetry to any odd number and always produce a topological Klein bottle. Even numbers yield an orientable torus.