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Studying for a test? Prepare with these 9 lessons on Kruseduller i Matematik og meget mere.

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# Matematik improvisering: Frugt per fod

Video transskription

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.