Skip to 0 minutes and 0 secondsUp until now, we’ve talked about paper objects with more than two sides, but now, I’m going to introduce you to something that has less than two sides. Take a strip of paper, Bring the ends together and tape them to make a band. This band has two sides, an inside and and outside, and two edges, left and right. When you cut the band in two, you get as expected two bands, each half as wide as the original band.

Skip to 0 minutes and 30 secondsThat’s not really exciting… but… if you take a strip of paper, bring the ends together like before, but just before you tape the ends, you make a half twist in the paper, and now tape together the ends, you get one of the most amazing paper constructions. A Möbius band or strip as it is usually called. Named after the famous 18th century German mathematician, August Ferdinand Möbius, it was actually discovered a few years before Möbius by another German mathematician, Johann Listing. The amazing thing about this band is that it has only one side, and one edge! I can prove this to you in the following way.

Skip to 1 minute and 12 secondsI’m going to take a marker and draw a line through the center of the band, until I return to the starting point, without taking the marker off the paper. If there were two sides to this band, the band would only be marked through the center on one of the sides, and you would see that half of the band is not marked at all. But that’s not what happens. You see, when I return to the beginning, the whole of the Möbius band is marked. So this indeed is a one-sided object. In the same manner, we could prove that it has only one edge.

Skip to 1 minute and 48 secondsIt may look as though there are two edges, but this edge here, is just a continuation of the first edge, due to the half-twist that we made in the strip of paper at the beginning. Different topologies have different properties and that’s where things get fun! When we cut the Möbius band through the middle, what do you think we get? Is it two regular bands? Two Möbius bands? Or perhaps, one regular and one Möbius? Tara! Actually, we get just one band, half-twisted four times! That’s the magic of the Möbius strip. There are many more tricks that exploit the Möbius strip topology, and you will find some links to them below.

Skip to 2 minutes and 30 secondsBut before I go let’s experiment and see what happens, if I make two half-twists before joining the ends. I get a 2-half-twisted Möbius strip, that has two sides and two edges. And when I cut this through the middle, I get, two Möbius bands, intertwined! If I make three half twists in the paper strip and once again join the ends to form a band, I again get a one-sided, one-edged band. In fact, it is easy to prove that an odd number of half-twists, results in a one-sided, one-edged Möbius band, which, when cut through the middle, gives one band, while an even number of half-twists, results in a two-sided, two-edged Möbius band, which, when cut through the middle, gives two bands.

Skip to 3 minutes and 18 secondsBut we can still distinguish between a one-half twisted Möbius band and a three-half twisted Möbius band, when cutting through the middle, because, although when we cut this three-half twisted Möbius band through the middle, we get just one band, it now has… … a knot in it! From 3 half-twists and above, the more half-twists there are, the more knots we’ll get in the bands that are created by cutting through the center of the original band. As a final magic trick, I’m going to leave you with this. Take two bands, one just a regular band, and the other a Möbius band, and assemble them one on top of the other, perpendicular to each other. Now I’m going to cut through both bands.

Skip to 4 minutes and 5 secondsWhat do you think we’ll get? Hmm? Tara! Now, isn’t that a nice picture…

The Möbius Strip and its variants

In this step, we’re going to take a look at some Möbius strips. I can’t emphasise enough, how important Möbius strips are to mathematics, so it is really important to learn how to make them.

First, take a strip of paper, about 70 centimetres long and then watch the video and follow along. It is imperative that you do exactly as is shown in the video, to get the full experience of the Möbius strip, so don’t be lazy… :-)

In the next step, we’ll make sure you did by testing you, so please keep the strips that you make (or make new ones).

The real reason we’re step-siding from flexagons to Möbius strips is that there is an important connection between the two, that we will see later on.

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This video is from the free online course:

Flexagons and the Math Behind Twisted Paper

Weizmann Institute of Science