Mobius Strips

Topology

Topology is the area of mathematics which studies how different points on an object (or in a space) are connected to each other. For example, it is often pointed out that to a topologist a donut and a tea cup are the same. If you ignore how far apart points are, and what shape the surface takes (both of which are studied by geometry, not topology) then the only significant feature about these two objects is the hole (in the middle of the donut and in the handle of the tea cup.) This may make topology sound a little bit silly; it is not!

Surprisingly, there are many unexpected and deep theorems in topology, and topology is very important in other areas of mathematics like geometry and even calculus! The easiest example of unexpected topological results is the "Mobius band", a simple twisted cylinder of paper with surprising topological properties.

Making a Mobius Band

Take a rectangular piece of paper which is at least twice as long as it is wide. Laying it on the table in front of you, color the top-left and bottom-right corners green and color the top-right and bottom-left corners red. Then roll the paper (with a twist) so that the two red corners meet each other and the two green corners meet each other. Fasten it in this shape with tape or glue. You have made a topological Mobius band (or "Mobius strip"). (It would be best if you now ignore the "seam" where you have fastened the two ends together. A perfect band would not have any such break but would be smooth all around...I just can't figure out how to tell you to make one of those!)

How Many Sides?

What to do:

Using a crayon, draw a circle on the band. Pinch the band between your index finger and thumb with your index finger inside the circle. Now, pull the band between your pinched fingers until the circle comes around again. Notice that this time your thumb is in the circle and your index finger is on the "other side".

What does it mean:

The Mobius band only has one side (even though the paper you made it out of had two). If you try to color "just one side" of the band with a crayon, you will find that you have colored the whole thing. It also only has one edge: draw a mark anywhere on the edge, start your finger at any other point on any edge (even opposite the mark) and run your finger along the edge. You will eventually touch the mark.

Cutting Tricks

There are two good tricks you can do with a Mobius band and some scissors. In each case, the outcome is surprising...unless you already know some topology in which case you can predict exactly what will happen!

Draw a line right down the middle of the band (in the long direction, that is) and cut along that line (all the way around until you get back to where you started cutting). What happens? How many Mobius bands do you have now?
Cut along a line which always stays exactly one quarter of the distance from the edge. (Note, you will have to cut twice as far this time to get back to where you started!) What happens? How many Mobius bands do you have now? Be careful: not every band is a Mobius band...some are just ordinary tubes. You can check whether a band is a Mobius band by checking how many sides it has.

Keep learning

If you find the Mobius band interesting, you should learn more about its properties (like "non-orientability") and about other topological objects (like the Klein bottle) by reading a book on topology or taking a course in topology.

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