Math Improv: Fruit by the Foot

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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.