Möbius strip sesquicentennial

The 19th-century mathematician August Möbius was born on this day in 1790. Möbius was responsible for a number of things in geometry, topology, and number theory, but he’s best known for the Möbius strip, a puzzling and endlessly fascinating entity that happens to be 150 years old this year.[1]

A Möbius strip is a two-dimensional object that, when viewed in three-dimensional space, has only one side. What’s that mean? Let’s make one, using paper to approximate a two-dimensional object (we’ll ignore the thickness of the paper).

Take a standard sheet of paper and cut a 3 cm (or 1 inch) strip off of a long edge. Of course, it has two sides. On one side, which we’ll call side A, write the letter “A” on each end, like this:

Turn it over to what we’ll call side B, and write the letter “B” on each end, similarly. We have a flat strip with two sides, and we’ve labelled the sides.

Pick up the strip and put the two short ends together so that the two letters “A” meet and the strip forms a ring, a band. Tape it together, as you see to the right.

We still have a two-dimensional strip, now formed into a circular band, with two sides. Let’s check that out. Get a pen and put it on one of the letters “A”. Keeping the pen on the paper, draw a line away from the other “A”, and keep drawing until you get to the other “A” and then return to your starting point (left).

The pen will have traced a line around side A of the band. Side B will be blank. That’s all as we’d expect.

Make another flat strip and mark it as before, with “A”s and “B”s. But this time, when you pick it up to form it into a band, give one end a half twist so that you join an “A” with a “B”, taping that together into a ring (right).

Take the pen and do as before: put it on an “A”, and, keeping it on the paper, draw a line away from the adjacent “B”. Keep drawing until the pen returns to where you started. Look at the result (left). You’ll see that there is no part of the band that’s blank, without the line traversing it. Unlike the un-twisted band, this one, a Möbius strip, has one side, not two. That is, we can’t define a side A that’s distinct from side B on this band.

That can be a little hard to get one’s head around, so think about it a while.

A fun thing to do with a Möbius strip is to cut it down the middle, a handy thing now that we have a line to cut along. Take first the “normal” band, the two-sided one that we made first. Get a pair of scissors and cut along the line you drew. The band will, of course, fall into two separate bands (right), each the same as the original, topologically. No surprise there.

Now do the same with the Möbius strip. What happens?

The Möbius strip does not separate into two bands, but instead winds up as a single, larger band — because it had only one edge, and the cut created a second. As a result, the cut band is not only twice as large, but is also no longer a Möbius strip (left). It’s a two-sided band with a full twist in it (two half-twists). It’d be very cool if you could keep cutting a Möbius strip and get a larger and larger one out of the cut... but that’s not how it works.

A reminder: the paper band is a model of a Möbius strip, using the paper to approximate a two-dimensional object. A true Möbius strip has no thickness, and exists only theoretically.

A similar concept is the Klein bottle: a two-dimensional object that can be thought of as a tube without distinct “inside” and “outside” surfaces — the inside and outside are one, much as side A and side B of the Möbius strip become one. A Klein bottle, too, exists only theoretically, and can’t even be modelled in three-dimensional space (modelling it requires four dimensions).

Topology is such fun!
 

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[1] Sesquicentennial, from sesqui, Latin for “one and a half”, and centennial, “100th anniversary”.