All right here's a little follow-up to the Yellow Birds and Topology. We'll call it
"the Yellow Birds and Topology 2."
What I want to do is take this idea of the simple closed curves and just go up one
level of sophistication to what's called a Mobius strip,
but before I do that, I want to give you a little bit a personal history. I'm going to
put up a picture
of me, of my first year teaching at the Community College.
It took me about three years at community college to get to the point
where I was comfortable with all the curriculum, all the material in calculus,
statistics, things like that,
and at the end at three years, I decided, okay, I'm going to try to bring in something
extra every Friday into class. And so what you're going to see right now is the first thing
that I brought in many years ago--
one of the first things I brought in to try to kind of spice up my classes a
little bit--
and it's called a Mobius Strip. A Mobius Strip is a one-sided piece of paper.
So, I've got two strips right here, and so first of all,
I'm going to show you were two-sided piece of paper and we intuitively know what a side
a piece a paper is, and what an edge is. So I'm gonna make a loop out of this, so we don't
have to worry about this part right here
or the bottom part. So I'm just going to take and put these two
edges together to form this loop right here. And now this loop
is a loop of two-sided piece of paper. It's got this side,
the outside you might say, and the inside, and it's got this
edge, and this edge. But in mathematics, we like to pay attention to definitions and
really decide what we're talking about when we talk about the side and the edge
So, one of the things about a side of this
of this loop right here is that I can write all over this side right here
and I can't get to the other side without crossing the edge
similar to my simple closed curve, where I can't go from inside to outside
without crossing
that boundary. So, on this piece of paper, I'm going to show you that I can't go from
this side to this side without crossing
an edge. So, let me just write here--maybe I'll start
on the inside. I'm going to start right here, and I'm just going to write
on this loop.
And you see, I'm never crossing an edge. I'm simply going
through the whole thing until I end up right where I started.
I've drawn really all over this side--you can think of this as a paintbrush, maybe I'm just
drawing all over that side--
but I can't draw on the other side without crossing this edge right here.
Now, what do we mean by an edge? Well, this has two edges:
this edge, and this edge. One of the properties that edges have
if there's two of them, if I cut between them, I'm going to cut the piece of paper in half.
So let's do that.
I'm just gonna make a little hole right here,
and go in and cut
between the two edges, so that's what I'm doing right now.
I'll go all the way around this loop
and cut between the two edges until
I get back to where I started from, which is right here.
And sure enough, if I cut that last little piece, I end up with two pieces of paper.
So, if I have two sides, I can't get from one side to the other side without
crossing an edge,
and if I have two edges, I can cut between them and get two pieces of paper.
So that was a two-sided, two-edged piece of paper.
Now what I want to do is take another strip of paper,
and this time, instead of just connecting the edges like this,
I'm gonna put a half turn in it, and connect the edges like this.
And what I end up with is a one-sided piece a paper that has only one
edge. So, this side right here
this side right here are exactly the same side. From this point to this point,
is just further along
that side, and this edge and this edge are exactly the same thing,
this point to this edge is just farther along than this point on this edge
right here.
So let me start by showing you that I can draw all over this one side
without ever crossing
an edge. So, let's go to the board, and I'll start drawing
on this right here,
and so I'm just going to keep drawing,
I'm never crossing an edge, you can think of this as a paintbrush
and I'm just going across the whole thing,
coloring in this side as I go,
and sure enough, sooner or later,
I end up right back where I started from,
and I've colored all over the whole thing without ever crossing an
edge. So that's because this has only one side to it, and I've just colored
on that side. Now, the second thing is that this has only one edge, so if I think it has
two edges, if I cut
between those two edges, I'll end up with two pieces of paper.
So let's do that. So I'll make that cut, and I'll just start cutting,
if you think it has two edges, then you think I'm
cutting between the two edges. But it has only one edge,
so actually what I'm doing is just cutting next to that one edge.
So I'll just keep going here until I end up right back where I started from,
make that last cut, and so, if I cut between the two edges,
I get two separate pieces of paper. But I cut alongside the one edge, and so what I
end up with is just one piece of paper like this. I've cut next to that one
edge, so I haven't cut this thing in half.
So, there's a look at a one-sided piece of paper, we call it the Mobius Strip,
you might say, well what has this got to do with me? Well, a Mobius Strip kind of a nice design,
so you'll see them sometimes in designs on packaging
things like that. If you do a little search on the internet for Mobius Strip
and look at all the images that come up, you'll see a lot of really interesting
things.
Also, in the 1950s, the B.F. Goodrich
company patented a conveyor belt
that was a Mobius Strip because they thought by making that one
twist, and having one long side instead of
two shorter sides, they would cut down on the wear on their conveyor belts.
So in any case there's a follow-up to the Yellow Birds and Topology 2,
and one of my favorite little applications of mathematics, the Mobius Strip.