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Math improv: Fruit by the foot
So I saw these Fruit by the Foots, or Fruit by the Feet. Maybe Fruits by the Foot. Anyway, I figured they had mathematical potential. So I decided to just record myself playing with them. The first thing that comes to mind when there's a strip of paper, even paper covered in fruit flavored sugar, is to make a Mobius strip. So I did. Thing about Mobius strips is they have one side, while Fruits by the Foot are, by nature, two-sided. A normal loop would have a paper side and a sugar side. Putting a half twist in would make a sudden transition from one to the other. But you can also do this. Wrap it around twice so the sugary part sticks to itself and the entire outside is covered in paper. Here is our Mobius strip. After confirming that Sharpie flip chart markers won't bleed through the paper, I drew a line along a single side of the Mobius strip. But that's not telling us anything that we didn't already know. Just like one line can cover both sides, except really one side, the paper does, too. So unlike a normal loop of Fruit by the Foot covered in paper, which would need two pieces, one for each side, the Mobius strip can be unwrapped by pulling off a single strip of paper. I went and washed the Sharpie off my hands and then came back to see what else I could do. The strip of fruit-flavored gunk has these two lines going down it, perforations, dividing it into thirds. One of the go-to Mobius strip fun things is to cut it in half down the middle. But after that, cutting it into thirds is the next thing to do. And it's as if Fruit by the Foot is designed for this purpose. So I started separating it along this line. You might want to pause here and think about what you think would happen. So I continue around. And when I loop around once, now I'm on the other side. And it turns out to not have been two lines, but one line that goes around twice. Let's see what we've got. It's completely whoa. I mean, this is totally magical to me, that there's these two loops and they're linked together, and they're not even the same size. And they're made of flavored sugar gunk. But let's understand what's going on. You can do it with paper, too. I cut out a strip, twist, and tape. Now I'm coloring the edge, both to demonstrate that the Mobius strip only has one edge and so that we can keep track of it. Because when we cut a strip into thirds, you could also think of it like this. You're cutting the edge off. This leaves a thinner Mobius strip and a long looping edge. Because of the twist, the edge loops around the body of a Mobius strip so the two are linked. Really, though, you should just try it yourself. After making a Mobius strip and ripping it into thirds, I poked at the leftover bit of Fruit by the Foot, waiting to be inspired. I don't know. It's spirally? I opened up another package so I'd have more to play with. Bam. Whoa. It's got this pattern on it. I mean, I don't remember this at all about Fruit by the Foot. But then again, it's not like I've had one in the past 15 years. And last time I had a Fruit by the Foot, I wouldn't have recognized that this is a frieze pattern, a symmetric pattern that repeats in one dimension. I mean, repeating patterns are good because they have a rolling stamp that just kind of presses this pattern into it. But it's got other symmetry, too. The two halves of this pattern are exactly the same, but not the same as in mirror symmetry going the long way. But it does have mirror symmetry going the short way, plus this point of 180 degree rotational symmetry, which basically means if you turn the strip of Fruit Roll-Up around, the pattern is the same. But it's neat that it does this in two tessellating halves. So you can flip it around and fold it in half and everything. OK. So I open the last of my Fruits by the Foot, hoping for another frieze pattern. But it's the same as the last one. So I make a normal non Mobius loop. And then I make a couple more and decide to put them together in a Borromean configuration. The Borromean rings are this arrangement of three loops that you see sometimes, and they're linked in such a way that no two loops are actually linked with each other. I mean, if you look at just two of the three, they're not linked. If you undid any one loop, the other two would also be separated. Yet, the three together are stuck. I tighten the loops up because they form this nice wrappy cube. And then I decided to try taking out the paper, but it was tricky because everything sticky and linky, so it didn't turn out so well. So since I'm labeling things, I labeled the Mobius strip part. And that reminds me that the frieze patterns and Mobius strips are related. At least, this particular frieze pattern has, as a result of its other symmetries, glide reflection symmetry. And glide reflection symmetry means that a pattern can be Mobius stripped. See, when I loop it around into a Mobius strip, the pattern exactly matches up, even though it's upside down. Oh, glide reflections. The glide is the loop part and the reflect is the flip part. And what that means is that with just 1/2 of this nice, tessellating pattern, we can loop it with a twist and it matches up perfectly with itself to make a complete strip, at least, theoretically. In practice, it's a rather sticky process.