Moebius Strip

Take a long strip of paper and glue one short end to the other short end. If you did this without a twist, you have a short cylinder. There are two edges, and two sides.

If you put a half-twist in it, you now have a MoebiusStrip. If you start colouring one side and keep going, you find you've coloured every part of the surface - it has only one side. If you put a felt tip pen on a part of an edge and run it around the edge, you'll find you've marked every part of the edge - it has only one edge.

Can you predict what you get if you cut it in half down the middle? What about cutting it in thirds?

If it has only one edge, can you glue the edge of one MoebiusStrip to the edge of another? You can, and you get a KleinBottle.

And by a neat analogy,

Take a long, flexible cylinder. Glue the two circular faces together. If you do this so that the faces have opposite orientations (i.e., that the outward vectors from each face point in opposite directions), you'll get a torus. If you do it so that they have the same orientation (you'll need to pass part of the cylinder through itself in threespace, but it works in 4d) you'll get a KleinBottle.


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