Paradromic Rings
There are a number of activities based on Möbius bands in recreational mathematics lore. Here we'll look at a variant with some topological and number sequence aspects.
Setup
For this activity, you'll need some paper strips, tape, and scissors. Colored markers, pencils, or crayons can be helpful for keeping track of and visualizing certain things, so you'll likely want those for more advanced work. I typically use printer paper 8.5" x 11" copy paper and cut it lengthwise into 0.5" x 11" or 1" x 11" strips. For kids with less fine motor control, thicker strips are the way to go. For pushing the activity in more advanced directions, thinner is better. (A substitute for thinner is longer, which can be accomplished by taping two 1" x 11" strips into a 1" x 22" strip, for example.) I'll be using construction paper here to make it more colorful, but a lot of the things my colors emphasize could be accomplished with markers.
To introduce this topic, walk students through building a Möbius band. I usually do it by modeling making a cylinder out of my paper strip but giving one side of it a half-twist at the last minute:
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Have the students tape the ends together. They'll want to use more than a minimal amount of tape, or otherwise they'll end up cutting the tape. I recommend at least taping across the entire width of the strip. Taping on both sides of the gap will give more reinforcement.
Prepare your own Möbius band and a cylinder (without any twists).
Puncture the cylinder and begin cutting it down the middle, pausing to ask students what they think will happen when you finish cutting all the way around. They will likely predict you will be left with two thinner cylinders:
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Now do the same with your Möbius band, having students cut along with theirs. They will likely be surprised to find that instead of two thinner bands they end up with a single longer, thinner band!
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Initial Questions
- Why is it that cylinder separated into two pieces when cut, while the Möbius band remained in one piece?
- The cut Möbius band seems to have some twists in it. Is it a longer Möbius band or is it something else?
- We've seen what happens with 0 half-twist and 1 half-twist. What happens with other numbers of half-twists?
Explorations
My main arc for this activity is broadly exploring the last question above: What are the different things we'll see when we add different numbers of half-twists and then cut the band? Below is a set of bands with 2, 3, 4, 5, 6, and 7 half-twists, from left to right, top to bottom:
As you can see, the more twists, the more bunched up it becomes, so 7 twists is already testing the limits of our ability to work with a 1" x 11" strip. If you want to investigate more twists, you'll likely want strips with a higher length to width ratio.
Here are some of results of cutting the bands:
Example: \(2\) half-twists
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Example: \(3\) half-twists
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Example: \(4\) half-twists
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Example: \(5\) half-twists
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Example: \(6\) half-twists
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Example: 7 half-twists
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Sorting through these, students should be able to articulate some descriptions of these configurations. The low-hanging fruit is that for odd numbers of half-twists, we end up with a single band after cutting while for even numbers of half-twists we end up with two bands. As the number of half-twists increases, these bands become more tangled, so you can miscount the bands if you're not careful!
For odd numbers of half-twists, examining the resulting knot is easier if you use a longer band. Here are the knots that result from 3, 5, and 7 half-twists:
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For even numbers of half-twists, examining the resulting link is easier if you use a band with two different color sides, like the one below. I've taped thin strips of construction paper together, but the same effect can be achieved with two colored markers. (This coloring system is also useful for explaining what is happening for even numbers of half-twists that is not happening for odd numbers of half-twists.)
Here are the links that result from 2, 4, and 6 half-twists:
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There is a pattern to discover. In the language of knot theory, the things the students are finding form the family of torus links \( T(2,n) \) where \( n \) is the number of half-twists. The top row below is for \( n \) odd while the bottom row is for \( n \) even:
After seeing a few, they might also be able to predict what comes next.
Another avenue to pursue is spurred by the question of whether the cut Möbius band is again a Möbius band or something else. Right now, the thing that sets a Möbius band apart from a cylinder is that the Möbius band has a single half-twist. If we are careful, we can disassemble the band at the gap and untwist it a half-twist at a time until we end with a cylinder. If we count these half-twists correctly, we'll find that it takes 4 to turn it back into a cylinder. This is a strange result and I've seen presenters confidently claim it has only 2, since that seems like the intuitive result if you haven't done it.
One way to think about it is that if you put on a belt, carefully making sure not to twist it as you thread it around your waist before buckling, you'll get a cylinder. On the other hand, if you have a really long belt, you might try feeding it around your waist twice, carefully keeping the top edge up and bottom edge down before buckling. If you were then to shimmy out of this belt, it would appear untwisted while coiled in this double loop, but if you were to pull it open into one big loop, you'd find it now has some twists in it! In fact, it has exactly two half-twists. When you cut the Möbius band, you can think about its two "halves" becoming two laps of a belt that maintain relative local orientation to one another, picking up an additional two half-twists when you uncoil it into a single loop.
The number of half-twists you find in each knot or link component is as follows:
| Half-twists before cut: | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) |
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| Components after cut | \(2\) | \(1\) | \(2\) | \(1\) | \(2\) | \(1\) | \(2\) | \(1\) |
| Half-twists after cut (per component): | \(0\) | \(4\) | \(2\) | \(8\) | \(4\) | \(12\) | \(6\) | \(16\) |
| Half-twists after cut (total): | \(0\) | \(4\) | \(4\) | \(8\) | \(8\) | \(12\) | \(12\) | \(16\) |
If the number \(n \) of half-twists is even, then each component ends up with \( n \) half twists. On the other hand, if \( n \) is odd, the number of half-twists in the band that results from cutting it is \( 2n + 2 \). In both cases, the number of half-twists roughly doubles, unsurprisingly, if we add up the number of half twists in all components. In the odd case, we pick up an extra 2 half-twists for the reasons discussed above.
Wrap-Up Questions
- If a strip is given 20 half-twists before taping and cutting, what can you say about the resulting object(s)? What about 21 half-twists?
- Can you come up with a general description for what happens when cutting these twisted bands?
- What difference do you think it would make if we twist clockwise versus counterclockwise, if any?
- Are there any other questions we could explore with these bands?
Extensions
There are many nice extensions to this activity based on taping different rings to each other. For some ideas, check out these Numberphile videos featuring Tadashi Tokieda: (One way to think about some of the objects discovered is by thinking of them as chunks of a torus or other 2D surface!)Some nice avenues of inquiry are:
- If we put \( m \) twists in one band and \( n \) in another and tape the bands together as shown in the linked videos, what are the different sorts of objects we could get after cutting? How does twisting clockwise versus counterclockwise change this in each case?
- If instead of cutting down the middle, effectively dividing the band into halves, what if we cut each band into thirds? Do we ever end up with just one object? Two? Three? More? How can you predict this?
- More generally, the name paradromic rings refers to what we get when we take a band with \( n \) twists and cut it into \( k \) equal-width lanes. What can we say about these, generally?
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