Möbius strips, doughnuts, and Klein bottles.

mobiusantsLook carefully at the M. C. Escher picture to the left. If an ant keeps moving forward, he will end up on both sides of the strip he is walking on. This suggests that the strip has only one side (neglecting the edges of the strip) which is pretty cool. The funny walkway is a Möbius strip which does have only one side. Making one-sided objects is only a small part of what the math that makes Möbius strips can do. The same trick can also show you that 1980s video games were being played on the surface of a doughnut, give you a way to understand four-dimensional objects like Klein bottles, and even let us describe the shape of the observable universe.

Let’s start with the basic method of constructing a Möbius strip. Take a strip of paper, twist it half a turn, and tape or glue the ends together. That half turn matches opposite sides of the paper at the join making the paper strip Möbius. Brilliant has more detailed instructions and an animation of the process. Convince yourself that the resulting object has one side by tracing your finger along the middle of the strip.

mobiusexample

A Möbius strip activity!

Once you have constructed a Möbius strip, using a fairly wide strip of paper, try cutting the strip down the middle. The result is another Möbius strip, twice as long and half as wide. The new Möbius strip also has three half-twists of the strip of paper, rather than one. You can also cut the Möbius strip at one-third of the way across and something really interesting happens — because of the twist, cutting one-third of the way across also cuts two-thirds of the way across before the cuts join up. If your dexterity is not up to this, here is a good video on both cutting in half and in thirds.

This material is a good discovery activity for students. You start with making a Möbius strip, demonstrate the one-sided property, and then have the students cut the strip down the middle — the fact it stays one object is interesting and the fact that this preserves one-sidedness is cool. The discovery that the cutting adds two half twists (this will continue if you cut again) is a bonus observation for sharp students. The cutting into thirds is even stranger — and leaves a lot of room for students to discover things. There are many other Möbius strip activities, like knitting Möbius scarves. Occupy Math has one of these.

Working with Möbius strips and their cousins, mathematically

When you want to do math on a geometric object, like a Möbius strip, getting out paper and scissors is not really practical. Sometimes the geometries are solid, or even higher dimensional, so we need a technique. Consider the two diagrams of paper strips below, with arrows on the ends of the strips.

strips

The arrows tell you how to match up the ends of the strips. The upper diagram means to tape the ends together without a twist, which would give you a ring of paper — like you would use in making a paper chain for holiday decorations. The lower diagram, with the reversed arrow, specifies a Möbius strip. This specification — on a flat piece of paper — is much easier to work with when you are trying to tease out the mathematical properties of objects. Let’s take this a step farther.

torusID

There are other shapes we can specify with this method. The diagram above has two types of arrows. We match up the arrows that are the same type (single, double). Matching the single arrows makes the square into a tube; matching the double arrows then joins the ends of the tube to make a doughnut or torus. An example torus is shown below. This method of specifying a torus in abstract is called an identification diagram — we identify arrows that are the same. This leads to an odd fact. Old computer games, like asteroids had a screen that wrapped left-to-right and top-to-bottom. This means that you were playing these games on the surface of a doughnut, topologically speaking. In case this doesn’t make sense, Occupy Math tracked down an an animation of making a doughnut by identification.

torus

We got the Möbius strip by reversing one of the identification arrows at the ends of a strip of paper. The identification diagram for the torus has two sets of arrows, both forming aligned pairs. We can reverse either or both of them. If we reverse one of the pairs of arrows we get a Klein bottle. This is a one-sided object that cannot be constructed inside three dimensional space without hitting itself. Look at the picture below, of the Klein bottle. The neck of the bottle punches through the side of the bottle when, mathematically, those parts don’t touch. In fact, in four dimensions, a Klein bottle can be constructed, but in glass and in three dimensions, we get only a flattened image: flattened from the mathematical truth in four dimension to a “flat” three dimensional object. If we reverse both sets of arrows, the identification diagram specifies a projective plane which is also non-three-dimensional. Here are approximate pictures of these weird objects.

 

Occupy Math’s editorial staff noted that there are Klein bottle hats, follow the link for a pattern.

This goes on from here to some very weird places.

We were able to make a doughnut shape by identifying the left-and-right edges of a square as well as the top-and-bottom edges. If we identify the three pairs of opposite faces of a cube, we get a doughnut — but with an extra dimension. So we know how to diagram four-dimensional doughnuts! Data from the WMAP satellite suggested an example of an identification-based geometry for the whole universe. A dodecahedron is a solid with twelve faces that are regular pentagons. Analysis of cosmic background radiation suggests that the universe has the shape of a dodecahedron with the opposite faces twisted to match and identified. This means that, after rotating opposite faces to line up, we declare them to be the same, like the sides of a square we identified to make a doughnut. That turns the strangeness to eleven; this creates a finite universe with no boundary, because the opposite faces wrap to one another.

This post journeys from an interesting paper-and-tape exercise to the edges of cosmological theory. The idea of creating new shapes by identifying the boundaries of simple shapes is a pretty basic one in topology, but it creates some very interesting results. Occupy Math hopes you enjoyed this trip into the topological end of mathematics. If you find other activities that use Möbius strips, please comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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