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##### Math Improv: Fruit by the Foot : Playing mathematically with fruits by the foot. Mobius Story, Wind and Mr. Ug: http://www.youtube.com/watch?v=4mdEsouIXGM Mobius Candy buttons: http://www.youtube.com/watch?v=OOLIB3cjFqw Mobius Music box: http://www.youtube.com/watch?v=3iMI_uOM_fY Doodling Snakes + Graphs (Useful for drawing Borromean rings): http://youtu.be/heKK95DAKms

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- So I saw these Fruit by the Foots...
- ...or Fruit by the Feet?
- Maybe Fruits by the Foots?
- Anyway I figured they had mathematical potential
- so I decided to 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.
- The 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 that the sugary part sticks to itself
- and the entire outside is covered in paper.
- Here's our Mobius strip.
- After confirming that Sharpie flip-chart markers
- won't bleed through the paper,
- I drew a line along the 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 like 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 the Mobius strip,
- so the two are linked.
- Really though, you should just try it yourself.
- After making the 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 like fifteen 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 this strip of Fruit
- Roll up around, the pattern is the same.
- But it's neat that it does this in two tesselating 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 seem 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 cube
- and then I decided to try taking off the paper, but
- it was tricky because everything's 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 pattern 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 one half 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.