Moebius Strip

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Take a sheet of paper. Thinking of it as an idealised finite piece of a plane, without thickness, and with idealised lines as boundaries, it's got two sides (obviously). It also looks like it has four edges, but in Topology - also called Rubber Sheet Mathematics - if you can smoothly bend or distort something from one shape into another, the two shapes are thought of as being the same. If you look at one corner of your sheet of paper you can imagine the paper shrinking a bit away from the corner, making the corner rounded. Carry on further and you end up with a circular piece of paper, and that clearly has only one edge.

  • A mathematician confided
  • That a Möbius Strip is one-sided,
    • And you'll get quite a laugh,
    • If you cut one in half,
  • For it stays in one piece when divided.
So a disk of paper, or a sheet of paper, has one edge and two sides.

Punch a hole in it and throw away the spare bit.

Now you have two sides and two edges. You can see that it's two edges because if you take a felt-tip pen and start colouring at some place on an edge, and keep going, you eventually get back to where you started, but you never got to every edge bit. You've coloured one edge, but there's another edge not yet coloured. Use another coloured pen for that edge, and now you've got them all. Two colours, two edges.

Punch another hole and you have two sides and three edges. Punch another hole and you have two sides and four edges.

And so on. Every time you punch a hole you get another edge and need another pen to colour it.

Now take a piece of paper and glue the two long sides to each other, making a cylinder. How many sides? How many edges? Is this the same as one of the earlier examples?

Here's a picture drawn by Escher ...


Now take a long strip and glue the short ends together. How many sides? How many edges? Is this the same answer as one of the earlier examples?

Take another long strip and shade one side. Again glue the short edges together, but put in a half twist so that the shaded side meets the unshaded side.

Now how many sides and edges do you have ...

This is called a Moebius Strip and it's got some really interesting properties. For the remainder of this page I want to talk about Moebius strips, their relationship with the Klein Bottle and the Projective Plane or Cross Cap, and a little about topology in general. In particular there is more than one embedding of the Moebius strip, and looking at them gives insights into their relationships with other surfaces.

For a start, here are some facts:

For graphics jockeys who are buying a CAD package a good benchmark is
"Will it draw a moebius strip?"
Lots of graphics engines complain at this. Unigraphics definitely does it.

This page should also reference the page on Mathematics and Mathematics Talks.


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