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>> Hello, this is
Julie Harland.
Yes, you could actually see me
this time and I
am YourMathGal.
So to see all my videos
organized by topic,
go to yourmathgal.com
and today we are going
to be talking
about the Moebius strip.
There are two spellings.
This is the German spelling.
It has this German letter we
don't have
in the English language an O
with two dots over it.
And you also see it
spelled M-O-E-B-I-U-S.
Okay. Before doing the Moebius
strip and that's another cool
thing, I want you
to get some paper
and some scissors
and some markers
that are not
permanent markers.
These are either highlighters
or cheap kid's markers.
And you can also just use
pencil if you don't have
anything like that or crayons.
And take--
and you want
to make some strips that are
about one and a half inches
wide et cetera
on the long side of the paper.
So go ahead and put the video
on pause and make I don't
know, maybe eight or so
of these strips of paper.
All right,
so here is how we're going
to start.
We're going to take one
of these pieces of paper
and close it up so
that we have this cylinder.
So we're going to take a piece
of tape and tape it.
And I get a long enough piece
of tape so I can go all the
way around like that, okay?
So we have this cylinder,
isn't that interesting.
What do we notice about this?
Well, it looks
like it has two sides
like it sort
of have this outside
and this inside.
And in fact,
we can define the inside
and outside.
This way, we could say, "Well,
if I start
on one side I could just color
it one color, right,
until I get back
to the beginning.
So you just have that sorry.
So I see that's the pink side
and then I see there's nothing
over on here
so there must be another side.
And you can color in here.
So I'm coloring this kind
of green.
Or you could just do one big
long color and kind
of do it all the way
through for a reason 'cause
we're going to cut it
in a minute until I get back.
And so I have this green side,
kind of the inside
and then I have the outside
which happens to be pink.
So this is something
that has two sides,
a pink side and a green side
and how about edges?
Well, like--
it looks like there're two
edges, the edge over here
and then edge over here.
And in fact,
if I take my finger
on one edge,
you want to make sure it never
really touches that.
Notice if I go around,
I end up back here.
It never touches this edge
over on this side.
So I have two edges as well,
all right?
So this cylinder has two sides
and two edges.
All right now, what happens
if we cut down the center?
So what you could do is kind
of fold it
and make a cut right here.
I hope you're doing this
with me, and then think
about what would happen
if you cut all the
way through.
So you can put the video
on pause if you want
to think about it.
I don't think it will take too
long to figure
out what's going
to happen here.
Cut all the way through
and what will happen?
Well, I'll just get two more
cylinders but notice it's half
as wide, okay?
So if this was 1 inch
at the beginning
that this would each be
1/2 inch.
I still have this green side
and a pink side
on each of them.
Okay, so that's what happens
if we take a cylinder.
You have to do more
interesting thing.
What about if we start again
with the strip and instead
of taping it like this,
we are going
to take the very top
and make a half twist
like that, okay?
So here is how it was
at the beginning,
that's a cylinder
but what happens
if I make a half twist
and then tape it.
What happens there?
Careful taping it all the way
around, okay?
The reason why I tape it all
the way around is 'cause we'll
be cutting it
and you don't want it
to fall apart.
This is a Moebius strip
so when you do
that little half twist,
you get what's called a
Moebius strip.
So put the video on pause
and make a couple more Moebius
strips on your own.
All right.
So now we're going to try
to figure out what are the
properties of a Moebius strip.
Well, let's start off
of at figuring how many sides
there are, you know.
It looks like it's got two
sides but, you know,
let's just do the same thing.
You start somewhere
and we just start shading
that side until we get back
to the beginning.
Or you just keep shading
and shading.
So this is the pink side,
right?
If you're doing this with me,
if you want, put the video
on pause and then come back
to it after you're done
shadings yours
or you're done shading yours.
And okay, I finally get back
to the beginning.
So I have the pink side.
Okay, so let's see.
There's the pink side,
let's do the other side.
Wait a minute,
the other side is pink also.
In fact, I have
shaded everything.
So this Moebius strip only has
one side.
Wow, isn't that cool?
Now, how did that happen?
How could you have a piece
of paper on one side?
Well, let's see how we
made it.
Let's say I think this has
two sides.
So I'm going to take,
you know, I'll think that's,
you know, it looks
like that would be different
in that side, right?
So in other words,
I'm on this side,
at the very ends of both
but I'm just going to change
so I could see this is kind
of the green side, right?
And this is my pink side,
right, before I tape it up,
but what happens?
So if I tape up like this,
that would be
like my original cylinder was
but what did I do?
I checked the top
and I did a little half twist
and I just merged the two
sides together.
So now the green side
and the pink side are all the
same sides
so you could see how
that became one side
by making a half twist.
Now, what about the edges,
okay.
So on this first
when we noticed we had,
you know, one edge over here,
right, and then this was
completely different.
But what if I said we'll get
another color here.
I am going
to say this is edge 1, okay,
I'm sorry,
and this is edge 2, right?
That looks like it's
on different sides.
So in other words,
I folded it together,
this is edge 1
and this is edge 2.
Okay. So originally
when I just had a cylinder,
I had it like that, right?
But when I twisted it,
see this one?
Emerges over here with edge 2.
So I think there's also going
to be one.
Actually, let's just try it.
It's kind of hard to do 'cause
on the edge,
you just have to--
you can't really see what's
going on so I'm just going
to kind of say, "Well,
this over here is an edge
right here" it's blue okay.
And it looks
like over here would be a
different edge, right,
the kind of green,
they're going
to count really--
I can't see it
if I just put it
on the real edge.
And if you go,
you take your finger right
here on the blue edge
and you go around.
Don't get a paper cut.
You keep going and going
and going, you'll notice
that it ends up over here
on the green.
So that green is all the same
edge, and I keep going
and going and going and I end
up with blue again.
So Moebius strip only has one
side and it also only has
one edge.
He's really ended up being
on the same edge, okay?
So that's kind of cool.
That's why I really
like the Moebius strip,
it's neat.
You know, in fact,
it means like if you're an ant
walking on this,
you're walking and walking,
you end up walking forever
and ever and ever and getting
on really on both sides.
If you've never heard
of MC Escher,
he has a famous drawing
like that.
So if you look it
up on the internet
or in the book about MC Escher
or on the video,
you might look
at the Moebius strip
with an ant on both sides,
but it looks like they're
in different size 'cause
they're on the same side,
okay?
Now, let's do something else
that's kind of interesting.
So when we did the other--
when we did the cylinder
and we made that cut
down in the center.
Let's figure
out what happens here.
I'm going to take the strip
and I'm going
to make a cut right here
to get it started.
And I want you to think
about what would happen
if I cut all the way
through to the center?
So put the video
on pause maybe and think
about that and try it
on your own.
I want you to cut it
through the center
and make a guess
or conjecture, postulate,
whatever you want to call it,
hypothesis, what will happen.
Okay, so I'm going to do it.
I'm going to just cut right
through the center here
until I get to this
where I started and I--
okay, what's going to happen?
Guess, guess.
Okay. I guessed the first time
I did this
that I would get two pieces
just like I did
when I cut the cylinder.
I will get two pieces.
But notice what happened here.
I did not get two pieces.
I got one long strip
and it looked like it's twice
as long but it's half this
wide so that's kind
of interesting what happens
when you cut a Moebius strip
right down the center.
Now, what you can do is find
out how many--
see if this is the same side
as the other side.
So in other words,
it's just one or two sides.
So if I say this is my blue
side and again I'm going
to do this kind
of fast this time.
I'm going to go through here.
I'm going to see
if this is a one
or two-sided object.
So I'm going, I'm going,
it's hard 'cause this is long.
I got back, you know what,
I didn't end up getting below
on that side.
So this ended
up being two-sided.
So it's not a Moebius strip,
right?
It ended up being two-sided
so once I cut through.
All right.
So that's one thing that's
kind of cool
about the Moebius strip.
You go down the center
and you get one long thing
that is two-sided.
So when you cut
down on one side
of thing right
down the center,
you get something twice
as long but it's only one
thing and it's two-sided.
Okay, let's do something else
with one of your other Moebius
strips that you have here.
And again,
so this is one side, one edge
and instead of cutting
through to the center,
let's kind of cut close
to one side.
Let's say a 3rd or a 4th
of the way in.
So make one little cut
to get started and I want you
to think what will happen
if I cut all the way along
until I get back.
All right,
so before you do it,
think about what
should happen.
Make a guess.
If you don't want
to see it before you're done,
put the video on pause
and I will just keep cutting
here, cutting and cutting.
A little tricky,
you can do it.
Keep cutting and cutting.
[ Noise ]
And cutting until I stop
and what's going to happen?
Isn't that cool?
What happened here?
That's the same Moebius strip
but you know what,
it's like I cut this much off.
I cut this much off, the edge,
of course there's only
one edge.
It's the tricky part
so I cut this off the edge.
We still have a Moebius strip.
So if I were to cut closer
to one end,
this would just be skinnier.
If I would have done a little
more toward the middle,
this part would be fatter.
This ends up being twice
as long and they are,
you know, linked together.
So that's another cool thing.
And what you can do,
I'm not going
to do it right now,
is you could figure
out whether this piece here is
a one-sided or two-sided
by again, taking something
and going all the way across.
And when you're done,
see if you also have blue
on what you would think would
be the other side.
So I'll let you think of that
out so you have something
to do on your own.
Okay, let's do one more thing.
Let's-- instead
of using a Moebius strip,
let's create something else.
So we started
out with just the cylinder.
The Moebius strip I did a half
twist, we're going
to do another half twist
which means now we've done
in it a whole revelation here.
So this is going
to be harder to cut.
But what happens here,
this is back
to being two-sided.
Remember each time you do a
half twist,
you've got a one-sided
and then goes back
to a two-sided, you know,
back to its own side.
In other words, look,
we take this green,
same side of thing.
I need blue
on the other side let's say.
Okay. So green side only,
now green and blue, right,
so this is green
and blue are together,
I've got a one-sided thing.
That's the Moebius strip
and then I twist it again.
I'm back to only green
on one side and blue
on the other side.
So that's what I've got,
green on one side,
blue on other side
and we will go ahead and--
yeah, here is the tricky part.
[ Noise ]
Not so simple.
And what will happen,
what do you think when you cut
down the center of that?
So try that,
that's a little
trickier cutting.
The trick is
to get the cut started
and then just kind
of keep moving the paper
out of the way
so you can keep cutting
down the center.
And what do you think
will happen?
[ Noise ]
This time I still have
inter-linked
but they're both the same
width, right?
So here are the different
things that we did.
We have-- this is my original
Moebius strip.
Okay, cut down the curve.
Here it is.
I think this is one
of the Moebius strips I cut
down the center.
That's when I cut
down the center
of a Moebius strip.
This is when I cut
down the center of a cylinder,
right, and this is what
happened if I have two twists.
I'm sorry,
this is what happened
if I cut off the edge
of the Moebius strip
and my last one,
this is what happens
when I did two twists and cut
down the center.
And then again, if you want,
you could figure
out what this is two sides
or one side.
All right.
So you could make your own
experiments up but basically,
the coolest part I think is--
what happens
with the Moebius strip
when you go down the center.
Anyway, and hopefully now you
know how to make a
Moebius strip.
I've seen rings
that are Moebius strips,
necklaces, lots of drawings.
It's [inaudible].
Okay, have fun.