Green paper Möbius strip

Möbius Strip summary

You have watched and played with Möbius strips. Let’s summarise what we’ve learnt.

Odd and even half-twisted paper strips

A strip of paper has two sides. Attaching its ends together produces a band. The band also has two sides, an inside and an outside. Cutting along the strip’s center line creates two new bands, each half the width of the original.

Making a half-twist before joining the ends of a strip generates the one-sided Möbius band. Other one-sided bands can be obtained by making any odd number of half-twists. Two-sided bands, like the trivial untwisted one, can be obtained by making an even number of half-twists in the strip before joining the ends. We will call the number of half-twists the order of the band. For examples, a 3-half-twisted band is one with three half-twists. Since all the bands we are discussing are half-twisted bands, we can omit the words half-twisted from the prefix and just call them 0-band (the regular non-half-twisted band), 1-band (the regular Möbius strip or band), 2-band (the two-half-twisted-Möbius-band), the 3-band and so on.

Cutting half-twisted paper strips

When you cut an n-band through its center line, the following happens according to the parity of n, n meaning the number of half-twists in the band:

  • If n is odd, there will be a knotted band with a total of 2n+2 half-twists.
  • If n is even, there will be two bands linked together n/2 times, each of which has n half-twists.

For example:

  • a 2-band, when cut through the middle, will create two 2-bands linked together once.
  • a 3-band when cut through the middle, will create a knotted 8-band.

Some more mathematical and topological properties

Mathematicians found that the outcome of bisecting a half-twisted band along its center line can be guessed even before cutting, just by observing the edge of the band. For example, if you follow along the edge of the Mōbius strip with your finger, you’ll find that your finger traces a closed band, half-twisted 4 times, which is what you get when cut the Mōbius band. Similarly, the boundary of the 2-band is what’s known as the Hopf link; the 3-band is the trefoil knot; the 4-band is Solomon’s knot; and so on, in one-to-one correspondence with the topology of the same bands cut through their centers.

Other cool stuff can be found by cutting the Mōbius strip one-third of the way from its edge, or making a Mōbius strip from two paper strips one on top of the other, or by glueing two (or more) Mōbius strips together as you have seen in the video and the quiz. If your interested in them, you can find many sites with more information, such as this. I also highly recommend Cliff Pickover’s book: The Mōbius Strip

Discussion

Any questions? If so, please post below so that we can answer!

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This article is from the free online course:

Flexagons and the Math Behind Twisted Paper

Weizmann Institute of Science