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PROFESSOR: All right!

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Welcome back to 6849.

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Today, we're going to
move into linkage folding.

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And before we get there, I
want to talk about an issue

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that sort of motivates
linkage folding.

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And it's about the
definition of folding.

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So we've talked
about origami a lot,

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and we've talked
about crease patterns,

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and we've talked about
the folded state-- which

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is the finished product
after you fold something.

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But there's another thing
we might care about.

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So whenever we wanted
to prove that we could

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fold a piece of paper
in a particular way,

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we described a folded state.

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We said what does the
paper look like in 3D,

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how are the layers stacked,
all these kinds of issues.

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But when you fold a
real piece of paper,

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you start from a
flat thing, and you

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want to continuously deform
it into some 3D shape.

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And that is another
kind of folding,

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which we call a folding
motion, which we haven't really

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worried about up until now.

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And it's going to be the central
concept in linkage folding.

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The way to define
a folding motion

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is just a continuum
of folded states.

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It's a movie.

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You start somewhere, usually
we start from the flat folded

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state, and then at all times
we specify a new folded state,

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a new way to fold
a piece of paper.

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As long as that's a
continuous process,

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then that is a folding motion.

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So we've done all
this work proving

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that folded states exist.

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The question is, what if
I want a folding motion?

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Because really, in
order to fold the paper,

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I've got to get there.

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And that's we're
going to worry about

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for the first few minutes.

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The good news is the reason we
haven't worried about this up

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'till now is there's an
equivalence between the two

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for paper folding.

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So if you have-- it's called
a simple polygonal piece

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of paper, and that
just means your piece

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of paper some polygon,
but it has no holes in it.

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It's sort of solid,
there's nothing like this.

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Then, it has a folding motion
into any desired folded state.

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So you say, I don't know.

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I have this folded state
where all the lines of a swan

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are aligned in that
second fold and cut.

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How do I actually fold
into that folded state?

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It's possible.

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Any folded state
we describe, you

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can reach by a folding motion.

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So we're going to prove this.

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When you look at it the right
way it's almost trivial,

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but maybe a little confusing.

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So I'll describe the proof to
see what you think about it.

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I'm going to start with the
case of one dimensional paper,

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because it's always
easier to think about.

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The idea is the following.

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You have a piece of paper, and
you have some folded state.

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I'm going to draw
a little letter

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m, what you want
to fold it into.

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They should have
the same length.

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We're given this map, we're
told how the piece of paper

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folds-- in this case
into two dimensions--

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and if there were
any touching layers

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here it would be described
which is on the top of which.

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We're told how to
instantaneously go

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from here to here.

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What we're not told is
the steps in between,

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and that's what we're
going to fill in over here.

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So the idea is rolling creases.

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I have my piece of
paper here, and I'm just

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going to curl in the
end a little bit.

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So I'm going to fold
this thing into--

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or I just roll up the end,
I can do this continuously.

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It's sort of like the
real piece of paper.

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If you start folding
and then pull,

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I continuously roll that crease.

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At all times, in fact, this
is supposed to stay flat.

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It's really hard to
get started in reality,

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but in theory, you
can get started

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with an infinitesimal
part of the crease.

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You just roll it in.

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So you make the piece
of paper smaller.

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It will be-- geometrically
these are on top of each other,

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so it'll be tiny.

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So it'll actually be a subset
of the original piece of paper.

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You keep doing this,
you end up, let's say,

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with a rolled up piece
of paper like that.

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A little hard to draw.

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Really, it lies along
the line segment,

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and you're just going around
and around in a line segment.

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But it's like a cinnamon bun.

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Why do we bother?

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Well, if we look at a tiny
piece of this folding--

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so in fact I should have
done this at the beginning,

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let me do that now--
away from the creases.

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If I look at a tiny portion,
it's nice and smooth.

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I'm not at a crease.

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I'm going to highlight which
portion of the paper here ends

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up-- this piece of paper's
roughly in quarters.

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This is like the third quarter--
that ends up over here.

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What I'd really like to
do is roll up everything

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to lie on top of that,
so I do some rolling.

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Maybe this is the-- and
I'm going to roll this up.

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So I end up with
the thick segment,

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and then some stuff
rolled on top of it.

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I don't really care too much
what it looks like, I just

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need that there's a
continuous rolling process

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to get everything to lie
on top of that segment.

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Now what I do going down--
this is all in one dimension.

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All of this lies
in one dimension.

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Now we're going to go
into the second dimension.

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Exciting.

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So we imagine that there's this
M, the folded shape that we

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want to build.

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And we're going to place it
where the thick part belongs,

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where that highlighted
part belongs.

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So it belongs
here, and of course

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everything comes
along for the ride.

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So you've got a little roll
here and a bigger roll here.

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Something like that.

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Just taking that-- or I guess
they were all on the same side,

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I'm not drawing
very consistently.

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OK, but it's all lying along
this segment of the M. Now

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I just need to unroll.

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This is where for
the mathematicians

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this will become obvious
in a little while.

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Maybe, I don't know.

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It's kind of a common trick.

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So we have this motion
that got us here.

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We can obviously play
that motion backwards.

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It's symmetric, I could unroll
just as easily as I rolled.

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But now what I want to do is
not unroll in one dimension--

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if I did that, I'd end
up lying along the line

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segment extending this old part.

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Actually, it's not
even quite straight,

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but let's pretend
it's kind of straight.

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But now I want to unroll along
the M. So unrolling here,

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the first thing I would
unroll is this stuff.

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It's going to go along the
long direction, the left side.

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So it's going to unroll
along, and but instead

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of going straight, I want
it to just move along

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the M. It's like
I'm an ant, and I

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don't realize that I live in
this two dimensional space.

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I'm going to pretend that I live
on this one dimensional space,

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even though it's not
physically there yet.

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That's why I drew it dashed.

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I just unroll.

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So I'll end up by the
end-- at this point

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I will have this half
of the M, the bold part,

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and then this half of the
M hasn't been made yet.

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And I'll have a little roll
that's enough to make that.

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I unroll more, and I
end up going there.

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So it's kind of crazy.

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Just roll up the
piece of paper so it's

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super tiny so you can put
it on the folded state

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without having any creases,
then undo everything

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you did, but living on the
surface instead of in the line.

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So you may or may not believe
that works, but it works.

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It's kind of cool.

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You can generalize that
to two dimensional paper.

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It's a little harder
to draw the diagram,

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so I bring the textbook.

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Here we have a polygon
of paper, but same idea.

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We're going to roll
up that piece of paper

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until it's super
tiny-- in this case,

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super tiny is a little triangle.

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We identify--
actually, ahead of time

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we identify a portion, a little
triangle on the surface here,

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that does not hit any creases.

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Here it is.

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And so we roll up the paper to
be contained in that triangle.

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Of course, there's
many layers above it,

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which you can't-- it's
hard to draw here.

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You just curl that
triangle a little bit.

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Now it lives on the crane.

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Here, we're supposing
we want to get

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a continuous motion
of the crane.

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It's multiple layers,
all sorts of fun things.

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And then you just
keep on rolling,

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just like you did in the
top thing but backwards.

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And instead of
living in the plane,

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you now live on
the crane surface.

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And the mapping between
them is given to you.

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You're given the folded
state that tells you

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how the piece of paper
lives on the crane,

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and so you just undo
everything on the crane,

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instead of in the plane.

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On the crane instead
of in the plane--

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should be a Doctor Seuss.

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Yeah, question.

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How do I pick the region?

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I just-- in general, we
have some crease pattern.

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I just take some region
interior to one face

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of the crease pattern.

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So just avoiding
creases is all I want.

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Yeah.

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The level of detail in
describing any region will do.

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In reality, we want
it small, so that's

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almost flat by smoothness.

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But that's sort of
a technical detail.

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Other questions?

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How many people does
this make sense to?

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Just curious.

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Yeah, almost everyone
kind of makes sense.

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Good.

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You have to see it a few times.

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I remember giving a talk
of this paper in 2001,

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and I sort of understood it, but
it was a very interactive talk.

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People kept asking questions.

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I think by the end
everyone understood it.

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Now I understand it.

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It's good stuff.

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There's some important
open questions, though.

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This crucially relies on--
this maybe not so-- well, yeah.

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If you had a hole in your piece
of paper, this would not work.

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Because you can't reduce--
you can't get rid of the hole

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by folding.

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And you can do all sorts
of rolling, folding corners

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over, and making
this thing smaller,

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but you'll never be able
to get rid of that hole.

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And so this approach
breaks down.

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Also, if your piece
of paper was a cube,

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the surface of the cube--
like, the end the last class

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we talked about
flattening, we had a cube,

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and you want to
collapse it flat.

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There we know that
there's a folded state,

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but we don't know about
the folding motions.

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It's kind of disconcerting.

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We presume there's
a folding motion,

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but again, this
approach doesn't work

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because you can't get a cube
down to a tiny triangle.

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You can't get rid of the--
it's kind of like having holes,

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but you surround a hole
in the case of the cube,

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and you can't get rid of that
topological feature by folding.

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So it's tricky.

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We avoid collision
throughout here.

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We also crease
basically every point.

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This is rather impractical
if you actually

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want to make something.

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It's comforting,
theoretically, but this is not

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how you actually fold
a crane, because you

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00:12:37,530 --> 00:12:39,630
have to do all this
continuous crease rolling.

247
00:12:39,630 --> 00:12:41,850
So over time, every
point of paper

248
00:12:41,850 --> 00:12:44,350
is at some point on a crease.

249
00:12:44,350 --> 00:12:46,120
And a big open problem
is what can you

250
00:12:46,120 --> 00:12:47,886
do with finitely many creases?

251
00:12:47,886 --> 00:12:48,760
I'll write that down.

252
00:12:52,320 --> 00:12:55,180
Let's suppose you have no
overlapping layers of paper,

253
00:12:55,180 --> 00:12:56,615
to make life a
little bit easier.

254
00:12:59,500 --> 00:13:03,720
So the paper doesn't
touch itself, so to speak.

255
00:13:03,720 --> 00:13:07,530
Can you do a finite
number of extra creases?

256
00:13:13,100 --> 00:13:15,040
I would conjecture
maybe yes, but this

257
00:13:15,040 --> 00:13:19,260
is a challenging problem.

258
00:13:19,260 --> 00:13:21,370
Here we have sort of
infinitely many creases.

259
00:13:21,370 --> 00:13:23,620
Everything becomes a crease.

260
00:13:23,620 --> 00:13:25,920
The extreme form
of this is what we

261
00:13:25,920 --> 00:13:32,285
call rigid origami, where
you have no extra creases.

262
00:13:35,010 --> 00:13:40,260
So you're supposed to
just use the creases that

263
00:13:40,260 --> 00:13:43,490
are given to you in
the crease pattern,

264
00:13:43,490 --> 00:13:50,240
and the faces of the crease
pattern are rigid polygons.

265
00:13:50,240 --> 00:13:54,060
They have to stay flat
throughout the folding motion.

266
00:13:54,060 --> 00:13:55,830
So rigid origami
is a thing about

267
00:13:55,830 --> 00:13:58,970
ridge-- about folding motions,
not about folded states.

268
00:13:58,970 --> 00:14:01,110
I have some folded
state I want to get to,

269
00:14:01,110 --> 00:14:03,730
can I get there
with rigid polygons?

270
00:14:03,730 --> 00:14:06,110
And the creases now
become hinges, basically.

271
00:14:06,110 --> 00:14:09,660
If you're making
origami out of metal

272
00:14:09,660 --> 00:14:13,020
with flat sheets,
or plastic polygons,

273
00:14:13,020 --> 00:14:15,100
whatever, hinging them together.

274
00:14:15,100 --> 00:14:17,450
Can use still fold this thing?

275
00:14:17,450 --> 00:14:20,120
Or did folding it
require this kind

276
00:14:20,120 --> 00:14:23,860
of bending the
paper in crazy ways,

277
00:14:23,860 --> 00:14:26,170
putting creases
all over the place?

278
00:14:26,170 --> 00:14:30,370
Sometimes you can do this,
most of the time you can't.

279
00:14:30,370 --> 00:14:33,110
And there's a lot we don't
know about rigid origami.

280
00:14:33,110 --> 00:14:35,460
I have one simple example.

281
00:14:35,460 --> 00:14:38,760
Something we know is bad,
which is kind of fun.

282
00:14:38,760 --> 00:14:40,980
Should have brought an
actual paper shopping bag.

283
00:14:40,980 --> 00:14:43,560
You take a paper shopping
bag, they're mass produced

284
00:14:43,560 --> 00:14:46,730
in this crease pattern,
so they're produced flat,

285
00:14:46,730 --> 00:14:48,360
I believe.

286
00:14:48,360 --> 00:14:54,860
And can't be folded,
doesn't fold rigidly.

287
00:14:54,860 --> 00:14:56,860
You just have these
creases-- the edges

288
00:14:56,860 --> 00:14:59,380
of the bag, this
kind of simple thing.

289
00:14:59,380 --> 00:15:03,390
It looks like straight skeleton
over here, and a perpendicular.

290
00:15:03,390 --> 00:15:06,330
Looks very nice from a
fold and cut standpoint.

291
00:15:06,330 --> 00:15:10,294
And it does have a folded
state, which is flat.

292
00:15:10,294 --> 00:15:11,460
There's no way to get there.

293
00:15:11,460 --> 00:15:14,980
If you look at all the
possible folded states

294
00:15:14,980 --> 00:15:19,050
of this pattern with rigid
panels, there are two.

295
00:15:19,050 --> 00:15:21,380
There's the 3D state
you should see here.

296
00:15:21,380 --> 00:15:23,109
There's the flat state.

297
00:15:23,109 --> 00:15:25,150
There's nothing that could
possibly connect them.

298
00:15:25,150 --> 00:15:27,740
There are no other states,
so there's no folding motion

299
00:15:27,740 --> 00:15:29,250
from one to the other.

300
00:15:29,250 --> 00:15:31,490
You could
instantaneously be flat,

301
00:15:31,490 --> 00:15:33,100
and you could
instantaneously be open,

302
00:15:33,100 --> 00:15:35,110
but you can't do
anything in between

303
00:15:35,110 --> 00:15:39,460
without curving the paper
or doing something invalid

304
00:15:39,460 --> 00:15:40,820
according to rigid origami.

305
00:15:40,820 --> 00:15:43,480
So obviously you don't want to
make a paper shopping bag out

306
00:15:43,480 --> 00:15:47,660
of metal, I guess is the
lesson, without adding

307
00:15:47,660 --> 00:15:48,715
extra creases at least.

308
00:15:51,190 --> 00:15:51,690
Yeah.

309
00:15:55,164 --> 00:15:57,580
I would love to find more open
questions on rigid origami,

310
00:15:57,580 --> 00:16:02,660
but so far we've just looked
at a few small things.

311
00:16:02,660 --> 00:16:09,420
But this turns us
over to linkages,

312
00:16:09,420 --> 00:16:12,310
which are all about finding
these motions without making

313
00:16:12,310 --> 00:16:12,985
extra creases.

314
00:16:16,840 --> 00:16:19,100
So I need to introduce a
little bit of terminology.

315
00:16:19,100 --> 00:16:22,010
This is like creases
and all those things

316
00:16:22,010 --> 00:16:25,050
from the origami world.

317
00:16:25,050 --> 00:16:28,120
But we care about various
levels of linkages.

318
00:16:28,120 --> 00:16:30,770
Let me remind you,
in general a linkage

319
00:16:30,770 --> 00:16:33,540
is going to be something
like these guys.

320
00:16:33,540 --> 00:16:36,000
This is the Peaucellier
linkage, these guys

321
00:16:36,000 --> 00:16:38,280
were pinned to the
projector screen,

322
00:16:38,280 --> 00:16:42,540
these guys were flexible,
these are rigid bars,

323
00:16:42,540 --> 00:16:45,870
and this guy ended up
tracing a straight line

324
00:16:45,870 --> 00:16:47,260
as this thing moved.

325
00:16:47,260 --> 00:16:49,694
So that's the thing we want
to map-- model mathematically.

326
00:16:49,694 --> 00:16:51,360
We're going to have
zero thickness, just

327
00:16:51,360 --> 00:16:53,740
like with paper folding.

328
00:16:53,740 --> 00:16:57,760
We're going to have vertices,
where things come together.

329
00:16:57,760 --> 00:17:03,068
And we're going to have edges,
which are those rigid bars.

330
00:17:03,068 --> 00:17:05,109
And I'm going to try to
speak in two languages at

331
00:17:05,109 --> 00:17:07,340
once-- the fairly
intuitive one, and I'm

332
00:17:07,340 --> 00:17:09,089
going to use some
notation in parentheses.

333
00:17:09,089 --> 00:17:10,950
So if you follow
the math, great.

334
00:17:10,950 --> 00:17:12,420
If not, also fine.

335
00:17:16,869 --> 00:17:20,069
The graph is just
combinatorial structure.

336
00:17:20,069 --> 00:17:22,900
It just says, look, I've
got things joined together

337
00:17:22,900 --> 00:17:23,400
like this.

338
00:17:23,400 --> 00:17:25,329
There's no geometry,
there's just

339
00:17:25,329 --> 00:17:28,590
a sense of what edges are
joined together at vertices.

340
00:17:28,590 --> 00:17:30,830
It's an abstract thing, kind
of like the shadow trees

341
00:17:30,830 --> 00:17:31,840
we work with.

342
00:17:31,840 --> 00:17:33,570
They're not really
embedded anywhere,

343
00:17:33,570 --> 00:17:35,740
though here, you don't
even have lengths.

344
00:17:35,740 --> 00:17:40,480
Linkage, you assign lengths
to the edges, and that's it.

345
00:17:40,480 --> 00:17:47,940
So I take a graph and then
I add lengths to edges.

346
00:17:50,890 --> 00:17:53,730
Its lengths of edges.

347
00:17:53,730 --> 00:17:56,130
So in math, this
would be a function l

348
00:17:56,130 --> 00:18:00,470
that maps every edge to a
non-negative real number.

349
00:18:04,410 --> 00:18:07,060
And I might want
a little bit more.

350
00:18:07,060 --> 00:18:09,190
So this is sort of optional.

351
00:18:09,190 --> 00:18:10,720
You might have some
pinned vertices,

352
00:18:10,720 --> 00:18:14,280
things that are not
allowed to move.

353
00:18:14,280 --> 00:18:18,690
So you might add coordinates
for pinned vertices.

354
00:18:25,490 --> 00:18:28,230
And so in math, this
would be a function p

355
00:18:28,230 --> 00:18:30,250
that maps some subset
of the vertices E

356
00:18:30,250 --> 00:18:34,765
prime to wherever
you happen to live.

357
00:18:38,010 --> 00:18:40,420
So here, I'm supposing our
linkage lives in d dimensions,

358
00:18:40,420 --> 00:18:43,450
because we're going to think
about linkages in 2D today.

359
00:18:43,450 --> 00:18:46,230
We're going to think about
linkages in 3D, linkages in 4D,

360
00:18:46,230 --> 00:18:46,740
and higher.

361
00:18:46,740 --> 00:18:49,420
Everything is fair game.

362
00:18:49,420 --> 00:18:50,940
We're going to be
pretty general.

363
00:18:50,940 --> 00:18:53,230
And sometimes, I want to
think about the linkage

364
00:18:53,230 --> 00:18:54,675
as this abstract thing.

365
00:18:54,675 --> 00:18:56,450
It tells me how
things are connected,

366
00:18:56,450 --> 00:18:57,700
tells me what the lengths are.

367
00:18:57,700 --> 00:18:59,330
This is again like
a shadow tree.

368
00:18:59,330 --> 00:19:01,710
Maybe I'm also told
some vertices are pinned

369
00:19:01,710 --> 00:19:05,362
in particular places in the
plane or in 3D or whatever.

370
00:19:05,362 --> 00:19:07,570
But sometimes I want to
forget about that extra stuff

371
00:19:07,570 --> 00:19:08,920
and just think about the graph.

372
00:19:08,920 --> 00:19:10,770
We won't do that
'till next lecture,

373
00:19:10,770 --> 00:19:15,390
but I mention it now
because we will need it.

374
00:19:15,390 --> 00:19:18,075
And then where
the real action is

375
00:19:18,075 --> 00:19:21,150
is in the configuration
of a linkage.

376
00:19:21,150 --> 00:19:24,380
And that's the geometry,
that's how the thing is folded.

377
00:19:24,380 --> 00:19:26,885
This is basically a
folded state for linkages.

378
00:19:33,257 --> 00:19:35,340
Why is it different
terminology than folded state?

379
00:19:35,340 --> 00:19:41,530
Just because it's a
different sub-field, I guess.

380
00:19:41,530 --> 00:19:44,417
Avoid confusion with
the other notions.

381
00:19:44,417 --> 00:19:46,250
With folded states,
usually we allow creases

382
00:19:46,250 --> 00:19:47,030
all over the place.

383
00:19:47,030 --> 00:19:48,863
With the linkage, you're
really only allowed

384
00:19:48,863 --> 00:19:51,490
to bend at the vertices,
so it's a bit special.

385
00:19:54,390 --> 00:19:56,160
So a configuration
just gives you

386
00:19:56,160 --> 00:19:58,210
coordinates for every vertex.

387
00:20:01,190 --> 00:20:05,900
So we would say configuration
C maps every vertex

388
00:20:05,900 --> 00:20:10,216
to some point in d
dimensions, and you

389
00:20:10,216 --> 00:20:11,840
have to satisfy a
bunch of constraints.

390
00:20:24,630 --> 00:20:26,970
And the constraints are
given by the linkage,

391
00:20:26,970 --> 00:20:30,620
so you've got the lengths
and you've got the pinning.

392
00:20:30,620 --> 00:20:34,540
So you have to have that.

393
00:20:34,540 --> 00:20:38,270
If you look at two vertices
that are connected by an edge,

394
00:20:38,270 --> 00:20:40,390
that edge has a
length, the distance

395
00:20:40,390 --> 00:20:42,450
between those two
points should be

396
00:20:42,450 --> 00:20:48,280
equal to the length of that edge
as given by that function l.

397
00:20:48,280 --> 00:21:00,070
This is for phi w an edge, and
then for the pinned vertices

398
00:21:00,070 --> 00:21:02,100
the point you choose
better be exactly what's

399
00:21:02,100 --> 00:21:03,365
given by the pin function.

400
00:21:11,490 --> 00:21:15,100
Never mind the details,
it's pretty intuitive.

401
00:21:15,100 --> 00:21:16,830
But link-- whenever
I say linkage,

402
00:21:16,830 --> 00:21:18,150
I mean the abstract thing.

403
00:21:18,150 --> 00:21:22,260
Configuration, I mean actually
embedded in whatever space

404
00:21:22,260 --> 00:21:24,130
you're living in.

405
00:21:24,130 --> 00:21:25,603
So let's do a simple example.

406
00:21:37,020 --> 00:21:39,550
Basically, the simplest
linkage that you can think of

407
00:21:39,550 --> 00:21:45,670
is a square, and abstractly just
a bunch of vertices and edges.

408
00:21:45,670 --> 00:21:47,940
This is the graph.

409
00:21:47,940 --> 00:21:49,900
I haven't written down anything.

410
00:21:49,900 --> 00:21:56,110
When I add lengths-- let's say
they're all the same length,

411
00:21:56,110 --> 00:21:59,345
I want to represent a square--
that becomes a linkage.

412
00:22:02,100 --> 00:22:06,600
And maybe I also specify this
guy is at coordinates 0, 0;

413
00:22:06,600 --> 00:22:10,990
and this guy is at
coordinates 1, 0 that's

414
00:22:10,990 --> 00:22:12,677
one unit away from this point.

415
00:22:12,677 --> 00:22:14,260
So that would sort
of pin things down,

416
00:22:14,260 --> 00:22:16,620
and these guys are
still available to move.

417
00:22:16,620 --> 00:22:18,370
This is of course not
a valid drawing, not

418
00:22:18,370 --> 00:22:19,911
a valid configuration
of the linkage.

419
00:22:19,911 --> 00:22:21,710
The lengths aren't
all the same, this guy

420
00:22:21,710 --> 00:22:23,690
looks longer than this one.

421
00:22:23,690 --> 00:22:26,790
But then a valid configuration
would be an actual square.

422
00:22:30,160 --> 00:22:31,530
I should draw in the plane.

423
00:22:31,530 --> 00:22:37,170
Let's say-- so this is
at 0, 0; this is at 0, 1;

424
00:22:37,170 --> 00:22:39,720
this is 1, 1; 1, 0.

425
00:22:39,720 --> 00:22:42,379
So that's a configuration.

426
00:22:42,379 --> 00:22:44,420
There are a bunch of
configurations this linkage.

427
00:22:44,420 --> 00:22:51,090
For example, that would
be another configuration.

428
00:22:51,090 --> 00:22:55,221
And there's a whole
degree of freedom here.

429
00:22:55,221 --> 00:22:57,720
And we really care about all
those different configurations.

430
00:22:57,720 --> 00:22:59,490
We call that the
configuration space.

431
00:23:02,760 --> 00:23:06,230
Configuration space is the
set of all configurations.

432
00:23:09,260 --> 00:23:12,420
So it has one thing
for the square,

433
00:23:12,420 --> 00:23:16,060
there's one thing for this
particular parallelogram,

434
00:23:16,060 --> 00:23:18,140
and there's that whole picture.

435
00:23:18,140 --> 00:23:24,700
Why don't I show you
graphically what it looks like?

436
00:23:24,700 --> 00:23:30,210
So here we have a
square-- square, move.

437
00:23:30,210 --> 00:23:32,100
There we go.

438
00:23:32,100 --> 00:23:34,430
So yeah, it's flexible.

439
00:23:34,430 --> 00:23:36,270
That vertex C is
moving around a circle.

440
00:23:39,220 --> 00:23:41,770
We can go around, and
around, and around.

441
00:23:41,770 --> 00:23:43,290
All right, you get the idea.

442
00:23:43,290 --> 00:23:44,748
Are there any other
configurations?

443
00:23:47,660 --> 00:23:48,910
No or yes.

444
00:23:48,910 --> 00:23:49,730
Pick one.

445
00:23:49,730 --> 00:23:50,535
Yes is correct.

446
00:23:53,450 --> 00:23:55,450
Ah, yeah, there.

447
00:23:55,450 --> 00:23:57,630
So when I get to here
and everything's sort

448
00:23:57,630 --> 00:23:59,820
of overlapping--
little hard to see,

449
00:23:59,820 --> 00:24:01,840
but you see there's
the yellow and green.

450
00:24:01,840 --> 00:24:03,700
Then I can bend--
and this I can't

451
00:24:03,700 --> 00:24:05,880
do in the way this
program's set up-- I

452
00:24:05,880 --> 00:24:08,630
can now move d
without moving AB.

453
00:24:08,630 --> 00:24:11,360
I could move the segment
BD and spin that around.

454
00:24:14,480 --> 00:24:18,890
So in fact, if you think
about it for a while--

455
00:24:18,890 --> 00:24:21,360
I hope I got this
right, because I

456
00:24:21,360 --> 00:24:23,070
think every time I've
given this lecture

457
00:24:23,070 --> 00:24:25,610
I draw a different picture.

458
00:24:25,610 --> 00:24:28,540
The space, the
configuration space,

459
00:24:28,540 --> 00:24:35,775
it looks something like
three kissing circles.

460
00:24:38,570 --> 00:24:41,242
So it's not quite
drawn to scale.

461
00:24:41,242 --> 00:24:42,950
Each of these points
I want to correspond

462
00:24:42,950 --> 00:24:45,960
to one configuration.

463
00:24:45,960 --> 00:24:48,040
I should really draw this
with more board room.

464
00:24:59,170 --> 00:25:01,410
All right, so we
started with the square.

465
00:25:01,410 --> 00:25:05,840
And let's say these are the
pinned guys on the bottom.

466
00:25:05,840 --> 00:25:09,130
I had this one circle
of motion, which

467
00:25:09,130 --> 00:25:12,260
just moved this segment
around sort of in a circle.

468
00:25:12,260 --> 00:25:16,780
So at the opposite point would
be where these guys are pinned

469
00:25:16,780 --> 00:25:20,350
and the square is inside out.

470
00:25:20,350 --> 00:25:24,090
And halfway along
would be I don't know,

471
00:25:24,090 --> 00:25:28,120
let's say I'm going left
here, I've got this,

472
00:25:28,120 --> 00:25:29,670
and then my square is over here.

473
00:25:32,210 --> 00:25:35,040
Draw the right number
of points, yeah.

474
00:25:35,040 --> 00:25:39,060
And over on this side would
be these are the pinned guys,

475
00:25:39,060 --> 00:25:42,590
and my square is over there.

476
00:25:42,590 --> 00:25:44,620
From here, I can
do another circle

477
00:25:44,620 --> 00:25:47,910
of motion, which is I can
never move these points,

478
00:25:47,910 --> 00:25:50,960
but now I can turn
this guy around.

479
00:25:50,960 --> 00:25:53,610
And when I go halfway
around it will

480
00:25:53,610 --> 00:25:56,980
be sort of folded on itself.

481
00:25:56,980 --> 00:25:59,680
So these are the
pinned guys, and then

482
00:25:59,680 --> 00:26:02,950
the polygon's like that,
all on top of itself.

483
00:26:02,950 --> 00:26:05,340
We're only thinking about
where the vertices are,

484
00:26:05,340 --> 00:26:08,440
we're not thinking about
overlap order here, one

485
00:26:08,440 --> 00:26:11,360
of the differences
with linkages.

486
00:26:11,360 --> 00:26:13,840
And if I go around
from this guy,

487
00:26:13,840 --> 00:26:17,110
it turns out-- I'm pretty
sure these circles of things

488
00:26:17,110 --> 00:26:20,530
are different except right here.

489
00:26:20,530 --> 00:26:27,390
And so this thing is
equal to this thing.

490
00:26:27,390 --> 00:26:31,995
So hence, three circles
that, pairwise, kiss, I

491
00:26:31,995 --> 00:26:33,519
think is the topology
of this space.

492
00:26:33,519 --> 00:26:34,810
You can walk around this thing.

493
00:26:34,810 --> 00:26:37,360
It was a little hard to
draw, but in this very simple

494
00:26:37,360 --> 00:26:40,810
example, we can draw effectively
how the configuration

495
00:26:40,810 --> 00:26:44,100
space works, how you
can navigate operations.

496
00:26:44,100 --> 00:26:47,710
In general, I want points
in the configuration space

497
00:26:47,710 --> 00:26:50,070
to correspond to
configurations of the linkage.

498
00:26:54,810 --> 00:26:58,990
and I want paths in
the configuration space

499
00:26:58,990 --> 00:27:02,697
to corresponds to motions.

500
00:27:02,697 --> 00:27:05,280
This is actually the first time
we get a definition of motion.

501
00:27:05,280 --> 00:27:07,670
I haven't specified one
here, although motion is just

502
00:27:07,670 --> 00:27:09,480
going to be a continuum
of configurations,

503
00:27:09,480 --> 00:27:12,220
just like we had with paper.

504
00:27:12,220 --> 00:27:13,920
In the configuration
space, a motion

505
00:27:13,920 --> 00:27:16,100
is just some path
through the space.

506
00:27:16,100 --> 00:27:17,100
You start at some point.

507
00:27:17,100 --> 00:27:18,550
You end at some point.

508
00:27:18,550 --> 00:27:21,346
There may be multiple
ways to get there,

509
00:27:21,346 --> 00:27:22,970
but if your thing is
connected, there's

510
00:27:22,970 --> 00:27:25,342
a way to get from
anywhere to anywhere.

511
00:27:25,342 --> 00:27:26,800
We want to understand
the structure

512
00:27:26,800 --> 00:27:27,890
of paths in that space.

513
00:27:27,890 --> 00:27:29,150
That's motions.

514
00:27:29,150 --> 00:27:32,450
That's how you can get places.

515
00:27:32,450 --> 00:27:36,010
Mathematically, I guess
this is one big parenthesis,

516
00:27:36,010 --> 00:27:47,760
you can think of a conflagration
of an n-vertex linkage

517
00:27:47,760 --> 00:27:50,710
as a bunch of coordinates.

518
00:27:50,710 --> 00:27:56,030
You've got the coordinates
for the first vertex,

519
00:27:56,030 --> 00:27:58,500
coordinates for v1, so
if we're in d dimensions,

520
00:27:58,500 --> 00:28:00,560
there's going to be
d of these things.

521
00:28:00,560 --> 00:28:02,750
So you've got, x, y, whatever.

522
00:28:02,750 --> 00:28:06,200
Then you could write down
the coordinates for v2.

523
00:28:06,200 --> 00:28:07,700
Let's say we're in
three dimensions.

524
00:28:07,700 --> 00:28:11,050
So we've got three
here, three values here.

525
00:28:11,050 --> 00:28:12,912
I'm not going to write
down what they are,

526
00:28:12,912 --> 00:28:14,620
because I don't know
what they are-- then

527
00:28:14,620 --> 00:28:17,120
three values for
the third vertex,

528
00:28:17,120 --> 00:28:24,920
and so on for n vertices.

529
00:28:24,920 --> 00:28:27,430
Why is this useful?

530
00:28:27,430 --> 00:28:29,370
Because you can think
of a configuration,

531
00:28:29,370 --> 00:28:31,050
specify all these coordinates.

532
00:28:31,050 --> 00:28:33,155
How many coordinates in total?

533
00:28:33,155 --> 00:28:37,920
It's d times n numbers.

534
00:28:37,920 --> 00:28:41,130
You can think of a configuration
as just d times n numbers.

535
00:28:41,130 --> 00:28:43,110
So you can think
of a configuration

536
00:28:43,110 --> 00:28:52,990
as a point in d times
n dimensional space.

537
00:28:52,990 --> 00:28:55,080
So this is giant space.

538
00:28:55,080 --> 00:29:00,390
Like for our square, I drew it
as a two-dimensional diagram,

539
00:29:00,390 --> 00:29:03,770
but in reality
there's four vertices.

540
00:29:03,770 --> 00:29:06,230
Each has two dimensions.

541
00:29:06,230 --> 00:29:09,000
So it's an eight-dimensional
space-- a little hard

542
00:29:09,000 --> 00:29:11,590
to imagine.

543
00:29:11,590 --> 00:29:14,880
But it's so constrained--
you can't just

544
00:29:14,880 --> 00:29:16,280
take any set of numbers here.

545
00:29:16,280 --> 00:29:19,440
They have to satisfy the
edge length conditions,

546
00:29:19,440 --> 00:29:21,530
and they have to satisfy
the pinning constraints.

547
00:29:21,530 --> 00:29:24,150
And by the end, the space
is locally one-dimensional.

548
00:29:24,150 --> 00:29:27,480
It lives in eight
dimensions, but you can only

549
00:29:27,480 --> 00:29:32,070
move along
one-dimensional curves.

550
00:29:32,070 --> 00:29:34,410
So we say this
configuration space

551
00:29:34,410 --> 00:29:37,670
or that these configurations
have one degree of freedom.

552
00:29:40,462 --> 00:29:41,920
And "degree of
freedom" is probably

553
00:29:41,920 --> 00:29:43,380
a term you've heard before.

554
00:29:43,380 --> 00:29:45,380
It's used all over the place.

555
00:29:45,380 --> 00:29:47,130
And the formal notion
of degree of freedom

556
00:29:47,130 --> 00:29:51,390
here is just, locally, how
many dimensions do you have

557
00:29:51,390 --> 00:29:54,910
if you look at the paths
going out from where you are?

558
00:29:54,910 --> 00:29:57,380
If, for example, I have
just a free segment,

559
00:29:57,380 --> 00:30:01,251
it's got a bunch of dimensions--
a bunch of degrees of freedom.

560
00:30:01,251 --> 00:30:02,750
It can translate,
and it can rotate.

561
00:30:02,750 --> 00:30:04,242
I guess, three.

562
00:30:04,242 --> 00:30:05,950
If I pin one of the
vertices, now it only

563
00:30:05,950 --> 00:30:07,033
has one degree of freedom.

564
00:30:07,033 --> 00:30:08,620
It can spin around.

565
00:30:08,620 --> 00:30:12,050
Just locally, how many
different ways can you move?

566
00:30:12,050 --> 00:30:13,669
What is the dimension
of your space?

567
00:30:13,669 --> 00:30:15,460
And a lot of the time,
we care about things

568
00:30:15,460 --> 00:30:16,650
that have only one
degree of freedom

569
00:30:16,650 --> 00:30:18,060
because they're controlled.

570
00:30:18,060 --> 00:30:19,490
We get to say what they do.

571
00:30:23,010 --> 00:30:23,720
All right.

572
00:30:23,720 --> 00:30:28,336
Let's move on to something
interesting about all this,

573
00:30:28,336 --> 00:30:30,470
which is Kempe's
Universality Theorem.

574
00:30:59,800 --> 00:31:02,900
So this is the result that
you can sign your name, so

575
00:31:02,900 --> 00:31:05,120
to speak, but I'll
state it more formally.

576
00:31:09,840 --> 00:31:16,090
My input is an algebraic
planar curve, and all of this

577
00:31:16,090 --> 00:31:18,680
can be generalized to
beyond two dimensions,

578
00:31:18,680 --> 00:31:21,370
but I'll start with
the planar case.

579
00:31:37,340 --> 00:31:38,540
Let me get back to that.

580
00:31:41,470 --> 00:31:44,660
Think of that as
just this blind,

581
00:31:44,660 --> 00:31:46,990
some kind of polynomial curve.

582
00:31:51,740 --> 00:31:57,450
We're going to restrict that
curve to a bounded disk,

583
00:31:57,450 --> 00:32:02,020
then claim that is
exactly the trajectory

584
00:32:02,020 --> 00:32:03,890
of some vertex in a linkage.

585
00:32:22,724 --> 00:32:23,240
OK.

586
00:32:23,240 --> 00:32:27,410
So I have a lot of
things to define here.

587
00:32:27,410 --> 00:32:30,040
One of them is trajectory.

588
00:32:30,040 --> 00:32:31,660
So this is pretty intuitive.

589
00:32:31,660 --> 00:32:34,860
So for example, you take some
linkage like this square,

590
00:32:34,860 --> 00:32:39,130
and as it moves, you just
follow one of the vertices.

591
00:32:39,130 --> 00:32:43,210
So for example, if we look
at c, the trajectory of c

592
00:32:43,210 --> 00:32:47,840
is this white circle, at
least as we move to here.

593
00:32:47,840 --> 00:32:50,300
I think even here, when
we move d, c doesn't move.

594
00:32:50,300 --> 00:32:53,200
So the trajectory
of c is that circle.

595
00:32:53,200 --> 00:32:55,560
It's just-- you take
all the configurations,

596
00:32:55,560 --> 00:32:57,619
but then you just
focus on one vertex

597
00:32:57,619 --> 00:32:59,160
and see where it
can go in the plane.

598
00:32:59,160 --> 00:33:06,650
In general, it can be very
complicated, like here, maybe.

599
00:33:06,650 --> 00:33:07,830
Yeah.

600
00:33:07,830 --> 00:33:11,879
You may recall it seems
I do this every time.

601
00:33:11,879 --> 00:33:14,420
If you look at the trajectory,
this is not actually a vertex,

602
00:33:14,420 --> 00:33:17,850
and that is not a valid color
if you want to see something.

603
00:33:21,240 --> 00:33:23,080
This vertex, for
example, its trajectory

604
00:33:23,080 --> 00:33:26,140
is some kind of figure
eight curve like that.

605
00:33:26,140 --> 00:33:28,520
In general, it can
be some crazy curve,

606
00:33:28,520 --> 00:33:32,100
and we want that crazy curve
to match a given planar curve.

607
00:33:35,890 --> 00:33:38,200
What is this notation?

608
00:33:38,200 --> 00:33:43,080
All I mean, it's just
something like 3x cubed times

609
00:33:43,080 --> 00:33:52,630
y minus 7y to the
fourth power equals 0.

610
00:33:52,630 --> 00:33:55,180
You can also add 23 to it.

611
00:33:55,180 --> 00:33:56,212
Whatever.

612
00:33:56,212 --> 00:33:57,340
So you have some equation.

613
00:33:57,340 --> 00:33:58,830
This is an equation on points.

614
00:33:58,830 --> 00:34:01,140
In general, if you
think about it,

615
00:34:01,140 --> 00:34:03,860
you've got two degrees
of freedom for a point.

616
00:34:03,860 --> 00:34:06,140
It has an x-coordinate
and a y-coordinate.

617
00:34:06,140 --> 00:34:08,170
You add one equation,
one constraint,

618
00:34:08,170 --> 00:34:11,045
that effectively pins it
down to be one-dimensional.

619
00:34:11,045 --> 00:34:13,170
So it's going to define
some one-dimensional curve.

620
00:34:13,170 --> 00:34:14,250
It could be very general.

621
00:34:14,250 --> 00:34:16,060
It could even be disconnected.

622
00:34:16,060 --> 00:34:18,590
All sorts of weird stuff.

623
00:34:18,590 --> 00:34:21,360
But you can use this to write
down circles and ellipses.

624
00:34:21,360 --> 00:34:23,060
That's the quadratic
polynomials.

625
00:34:23,060 --> 00:34:27,070
But you could also write much
more complicated things here.

626
00:34:27,070 --> 00:34:30,510
So this defines some
crazy curve, locally

627
00:34:30,510 --> 00:34:32,360
one-dimensional thing.

628
00:34:32,360 --> 00:34:35,900
And it may also go off to
infinity if there's hyperbolas,

629
00:34:35,900 --> 00:34:36,944
or whatever, in there.

630
00:34:36,944 --> 00:34:39,110
We're not going to try to
capture the infinite curve

631
00:34:39,110 --> 00:34:40,568
because that's
actually impossible.

632
00:34:40,568 --> 00:34:43,630
If you have any pin
vertex, then you

633
00:34:43,630 --> 00:34:46,330
can only get so far away
from that pin vertex.

634
00:34:46,330 --> 00:34:47,880
If you have no pin
vertices, then you

635
00:34:47,880 --> 00:34:50,691
can show you can make
any point in the plane.

636
00:34:50,691 --> 00:34:52,440
So if I want to get a
one-dimensional set,

637
00:34:52,440 --> 00:34:55,290
it's got to be a bounded
set from a linkage.

638
00:34:55,290 --> 00:34:58,490
So I take some big
disk that captures

639
00:34:58,490 --> 00:35:02,110
the stuff I care about in
that one-dimensional curve,

640
00:35:02,110 --> 00:35:05,390
and then I am going to
trace out everything

641
00:35:05,390 --> 00:35:09,840
of that curve inside that disk
using one vertex of a linkage.

642
00:35:09,840 --> 00:35:12,110
That's my goal.

643
00:35:12,110 --> 00:35:14,720
So it's a pretty
powerful result.

644
00:35:14,720 --> 00:35:17,450
I like to call it Kempe's
Universality Theorem

645
00:35:17,450 --> 00:35:23,850
because Kempe wrote a paper
about it in 1891 or so-- 1876.

646
00:35:23,850 --> 00:35:26,550
Even earlier.

647
00:35:26,550 --> 00:35:28,610
He didn't really prove
the theorem, though.

648
00:35:28,610 --> 00:35:34,360
Kempe is quite famous for
two wrong proofs of theorems,

649
00:35:34,360 --> 00:35:35,890
but the theorems are true.

650
00:35:35,890 --> 00:35:38,460
The one he's most famous for
is the four color theorem,

651
00:35:38,460 --> 00:35:39,626
which you may have heard of.

652
00:35:39,626 --> 00:35:42,520
Any plane or map can be
colored with four colors

653
00:35:42,520 --> 00:35:45,250
such that no two
countries share boundary

654
00:35:45,250 --> 00:35:46,910
if they have the same color.

655
00:35:46,910 --> 00:35:49,960
He didn't prove that theorem,
but it was the late 1800s.

656
00:35:49,960 --> 00:35:52,860
It wasn't proved
until 1960 or so.

657
00:35:52,860 --> 00:35:59,540
But the technique he used is
called the discharging method,

658
00:35:59,540 --> 00:36:01,150
is the proof that
people use today.

659
00:36:01,150 --> 00:36:02,590
There are now two proofs
to the four color theorem.

660
00:36:02,590 --> 00:36:04,090
They both use discharging.

661
00:36:04,090 --> 00:36:08,060
So they both use Kempe's ideas.

662
00:36:08,060 --> 00:36:11,590
He was just ahead of his time.

663
00:36:11,590 --> 00:36:13,887
He also claimed to
prove this theorem.

664
00:36:13,887 --> 00:36:16,220
And he proved a slightly
weaker version of this theorem.

665
00:36:16,220 --> 00:36:18,400
I'm going to show you his
proof because it's nice,

666
00:36:18,400 --> 00:36:20,630
and it can be fixed,
actually, relatively easily--

667
00:36:20,630 --> 00:36:24,400
although that was done only in
the context of this class six

668
00:36:24,400 --> 00:36:25,930
years ago.

669
00:36:25,930 --> 00:36:28,560
So Kempe's proof has been
around for over 100 years.

670
00:36:28,560 --> 00:36:30,060
Other people had
proved this theorem

671
00:36:30,060 --> 00:36:31,811
but in more complicated ways.

672
00:36:31,811 --> 00:36:33,560
We can prove it in the
same way Kempe did,

673
00:36:33,560 --> 00:36:34,685
just a little bit of extra.

674
00:36:42,747 --> 00:36:45,330
So the rest of the lecture will
be about proving that theorem.

675
00:37:06,650 --> 00:37:07,780
All right.

676
00:37:07,780 --> 00:37:09,820
So we've got two things
we need to worry about.

677
00:37:09,820 --> 00:37:11,465
We have to worry
about this curve,

678
00:37:11,465 --> 00:37:13,790
and we have to worry about
being in a bounded disk.

679
00:37:13,790 --> 00:37:15,160
I like being in a bounded disk.

680
00:37:15,160 --> 00:37:16,320
That sounds really easy.

681
00:37:16,320 --> 00:37:17,810
Let's start with that.

682
00:37:17,810 --> 00:37:20,900
So this is this figure.

683
00:37:20,900 --> 00:37:24,090
I have, just for fun, a scan of
the original Kempe paper where

684
00:37:24,090 --> 00:37:25,639
he draws the
corresponding gadgets.

685
00:37:25,639 --> 00:37:26,930
This is, again, a gadget proof.

686
00:37:26,930 --> 00:37:30,450
We're going to have lots of
cool gadgets to do fun things.

687
00:37:30,450 --> 00:37:32,820
A rhombus looks like this.

688
00:37:32,820 --> 00:37:36,200
It has one pin vertex,
a, here, and you

689
00:37:36,200 --> 00:37:39,104
have a degree of freedom in
how you specify this thing.

690
00:37:39,104 --> 00:37:41,520
You specify the lengths of the
edges, but all of the edges

691
00:37:41,520 --> 00:37:44,370
have the same length.

692
00:37:44,370 --> 00:37:48,510
So it's like a square, except
I only pin one of the vertices.

693
00:37:48,510 --> 00:37:50,120
That's really the same thing.

694
00:37:50,120 --> 00:37:54,340
And you can see, it can only
go out to twice that distance,

695
00:37:54,340 --> 00:37:56,604
and then it fails
to exist out here.

696
00:37:56,604 --> 00:37:59,020
These figures are drawn using
a program called Cinderella,

697
00:37:59,020 --> 00:38:00,840
which is very, very cool.

698
00:38:00,840 --> 00:38:05,020
It's commercial, but
you can download a demo

699
00:38:05,020 --> 00:38:06,940
and make your own fun
constructions like this.

700
00:38:06,940 --> 00:38:10,990
These are all on the web,
the ones I drew here.

701
00:38:10,990 --> 00:38:13,110
So you can see,
basically, that point b

702
00:38:13,110 --> 00:38:15,320
is constrained to lie in a disk.

703
00:38:15,320 --> 00:38:16,240
That's it.

704
00:38:16,240 --> 00:38:17,980
Very simple.

705
00:38:17,980 --> 00:38:23,110
So that's the rhombus gadget.

706
00:38:29,690 --> 00:38:31,440
But I want to think
about it a little bit.

707
00:38:37,950 --> 00:38:40,524
So this vertex is pinned,
let's say, at 0,0.

708
00:38:40,524 --> 00:38:42,690
Well, I guess it's pinned
at the center of the disk.

709
00:38:42,690 --> 00:38:45,400
I don't know where that is.

710
00:38:45,400 --> 00:38:47,560
This vertex is going to be
the vertex I care about.

711
00:38:47,560 --> 00:38:50,330
It's the one that I want to
force to lie on the curve

712
00:38:50,330 --> 00:38:51,680
and trace out the curve.

713
00:38:51,680 --> 00:38:55,800
This is going to
be my magic vertex.

714
00:38:55,800 --> 00:38:57,770
Let's call it x,y.

715
00:38:57,770 --> 00:39:00,330
Say the point that
it lives on is x,y.

716
00:39:03,830 --> 00:39:08,510
I somehow have to evaluate this
multinomial, various powers

717
00:39:08,510 --> 00:39:10,550
of x and y, multiply
them by some constants,

718
00:39:10,550 --> 00:39:13,080
add them together, and
force that to equal 0

719
00:39:13,080 --> 00:39:15,134
by some constraint, by
adding-- bars, in general,

720
00:39:15,134 --> 00:39:15,800
are constraints.

721
00:39:15,800 --> 00:39:16,740
Edges are constraints.

722
00:39:16,740 --> 00:39:18,970
They give me
equality constraints.

723
00:39:18,970 --> 00:39:20,845
I say the distance
between these two vertices

724
00:39:20,845 --> 00:39:22,360
is equal to something.

725
00:39:22,360 --> 00:39:24,920
Somehow I have to set up the
distance between two vertices

726
00:39:24,920 --> 00:39:27,450
to be that crazy
function that I'm given.

727
00:39:27,450 --> 00:39:28,800
It could be anything.

728
00:39:28,800 --> 00:39:31,510
It could be a mess.

729
00:39:31,510 --> 00:39:35,350
So Kempe had this cool idea.

730
00:39:35,350 --> 00:39:39,410
He said, look, you
can write this point

731
00:39:39,410 --> 00:39:45,150
as a sequence of motions,
you can think of them.

732
00:39:45,150 --> 00:39:48,270
So you start at the
origin here, and then you

733
00:39:48,270 --> 00:39:50,020
move along this segment,
and then you move

734
00:39:50,020 --> 00:39:51,060
along this segment.

735
00:39:51,060 --> 00:39:52,935
So let's just write
down-- what's that angle?

736
00:39:52,935 --> 00:39:53,700
Call it alpha.

737
00:39:53,700 --> 00:39:56,170
Let me be consistent
with my notation.

738
00:39:56,170 --> 00:39:58,090
Yes, alpha.

739
00:39:58,090 --> 00:40:01,600
And these are horizontal lines.

740
00:40:01,600 --> 00:40:05,464
And you can think of this
angle-- I'll call it beta.

741
00:40:05,464 --> 00:40:06,630
So there's some length here.

742
00:40:06,630 --> 00:40:08,090
Did I give it a name?

743
00:40:08,090 --> 00:40:09,190
I do.

744
00:40:09,190 --> 00:40:13,710
I call it r/2, because
if these are both r/2

745
00:40:13,710 --> 00:40:17,440
and these are also r/2-- that's
the definition of a rhombus--

746
00:40:17,440 --> 00:40:20,180
then this point
can get to anywhere

747
00:40:20,180 --> 00:40:22,640
within a disk of radius
r centered at this point.

748
00:40:22,640 --> 00:40:25,870
So this would be the
center of the disk.

749
00:40:25,870 --> 00:40:28,350
So we're given r, we're
told what the disk is.

750
00:40:28,350 --> 00:40:29,200
We set this up.

751
00:40:29,200 --> 00:40:32,450
But now alpha and beta
are in some sense free.

752
00:40:32,450 --> 00:40:33,875
They're actually
related to-- no,

753
00:40:33,875 --> 00:40:35,500
they're not even
related to each other.

754
00:40:35,500 --> 00:40:38,460
This point still has
two degrees of freedom.

755
00:40:38,460 --> 00:40:40,527
Even though it can't
go out to infinity,

756
00:40:40,527 --> 00:40:42,110
it can move in x and
it can move in y.

757
00:40:42,110 --> 00:40:44,540
We saw because only
one point was pinned,

758
00:40:44,540 --> 00:40:48,060
it could float around in
a tw-dimensional space.

759
00:40:48,060 --> 00:40:53,030
So alpha and beta are actually
both free to some extent.

760
00:40:53,030 --> 00:40:55,210
So instead of
coordinatizing by x,y,

761
00:40:55,210 --> 00:40:57,270
you could coordinatize
by alpha and beta.

762
00:40:57,270 --> 00:41:01,280
And you can relate those two
coordinatizations by some trig.

763
00:41:03,765 --> 00:41:05,390
I'm going to cheat
here because I never

764
00:41:05,390 --> 00:41:07,990
remember which is sine or
cosine without thinking

765
00:41:07,990 --> 00:41:08,730
for 30 seconds.

766
00:41:12,500 --> 00:41:14,310
The way I remember
is always cosine

767
00:41:14,310 --> 00:41:17,000
is alphabetically
smaller than sine,

768
00:41:17,000 --> 00:41:18,880
and so cosine is
the x-coordinate

769
00:41:18,880 --> 00:41:22,450
and sine is the
y-coordinate, if you

770
00:41:22,450 --> 00:41:24,670
draw the triangle in the
conical orientation, which

771
00:41:24,670 --> 00:41:25,378
you might forget.

772
00:41:32,947 --> 00:41:34,530
If you were given
these angles and you

773
00:41:34,530 --> 00:41:35,946
want to construct
this point, it's

774
00:41:35,946 --> 00:41:38,290
like, well, I go
in this direction.

775
00:41:38,290 --> 00:41:41,210
That direction is cosine
alpha, sine alpha.

776
00:41:41,210 --> 00:41:43,970
And I go that direction
for r/2 distance.

777
00:41:43,970 --> 00:41:49,320
So I have r/2 times cosine alpha
in the x-coordinate, r/2 sine

778
00:41:49,320 --> 00:41:50,445
alpha in the y-coordinates.

779
00:41:50,445 --> 00:41:52,420
It's nice and alphabetical.

780
00:41:52,420 --> 00:41:55,610
Then from there, I
go in this direction,

781
00:41:55,610 --> 00:41:58,030
which is in the beta
direction-- the cosine beta,

782
00:41:58,030 --> 00:41:59,980
sine beta direction.

783
00:41:59,980 --> 00:42:02,860
I go that direction by
the same amount, r/2.

784
00:42:02,860 --> 00:42:03,677
So there you go.

785
00:42:03,677 --> 00:42:05,760
That's x and y written in
terms of alpha and beta.

786
00:42:08,340 --> 00:42:11,540
For the purpose of this exercise
I prefer cosines over sines.

787
00:42:11,540 --> 00:42:13,470
So I'm going to
rewrite these sines

788
00:42:13,470 --> 00:42:16,440
in terms of cosines
in the obvious way,

789
00:42:16,440 --> 00:42:27,880
which I will look up, which
is cosine alpha minus pi/2,

790
00:42:27,880 --> 00:42:32,360
Those r/2 cosine
beta minus pi/2.

791
00:42:35,170 --> 00:42:36,566
OK.

792
00:42:36,566 --> 00:42:39,160
Let's see if I can successfully
draw cosine and sine.

793
00:42:39,160 --> 00:42:42,220
Sine starts here and does that.

794
00:42:42,220 --> 00:42:44,560
Cosine starts here
and does that.

795
00:42:44,560 --> 00:42:46,340
They're just shifts
of each other, which

796
00:42:46,340 --> 00:42:49,367
I didn't draw so beautifully.

797
00:42:49,367 --> 00:42:50,200
Something like that.

798
00:42:53,650 --> 00:42:56,090
Cosine and sine,
just a shift by pi/2.

799
00:42:56,090 --> 00:42:58,830
So if I subtract
pi/2 from the angles,

800
00:42:58,830 --> 00:43:04,660
I can turn it into a
cosine instead of a sine.

801
00:43:04,660 --> 00:43:05,350
Great.

802
00:43:05,350 --> 00:43:07,320
So I have x and y
written as cosines

803
00:43:07,320 --> 00:43:09,710
of things involving
alpha and beta.

804
00:43:09,710 --> 00:43:11,470
So what?

805
00:43:11,470 --> 00:43:17,530
Well magically, when I think
about squaring or raising

806
00:43:17,530 --> 00:43:22,160
x to some power like 10, if
you look in angle space what

807
00:43:22,160 --> 00:43:23,740
happens to alpha
and beta, it's kind

808
00:43:23,740 --> 00:43:26,710
of like multiplying
the angles by 10.

809
00:43:26,710 --> 00:43:29,670
In some sense, going into
the trigonometric world

810
00:43:29,670 --> 00:43:31,924
is like taking logarithms
if you're into algebra.

811
00:43:31,924 --> 00:43:34,340
You have these exponents, and
they're hard to think about.

812
00:43:34,340 --> 00:43:36,006
You take logs, it's
just multiplication.

813
00:43:36,006 --> 00:43:37,510
No biggie.

814
00:43:37,510 --> 00:43:40,710
And for those of you who
know complex analysis,

815
00:43:40,710 --> 00:43:42,065
that should be obvious.

816
00:43:42,065 --> 00:43:45,010
For the rest, just
take it on faith,

817
00:43:45,010 --> 00:43:47,390
and we'll just find
out that it's true.

818
00:43:47,390 --> 00:43:49,820
Because you have
this good friend.

819
00:43:53,082 --> 00:43:54,730
You have the product
of two cosines.

820
00:44:03,470 --> 00:44:08,071
You can rewrite it in terms of
adding and subtracting angles.

821
00:44:08,071 --> 00:44:09,820
Now, you may not have
learned it this way.

822
00:44:09,820 --> 00:44:11,980
You probably learned
about rules for cosines

823
00:44:11,980 --> 00:44:15,270
of sums of things and cosines
of differences of things.

824
00:44:15,270 --> 00:44:19,050
If you take these two
expansions, add them together,

825
00:44:19,050 --> 00:44:21,090
lots of things
cancel, and you end up

826
00:44:21,090 --> 00:44:25,390
just being left with cosine a
times cosine b with a module

827
00:44:25,390 --> 00:44:27,520
a factor of 2.

828
00:44:27,520 --> 00:44:31,770
So in our situation, we
have this crazy thing.

829
00:44:31,770 --> 00:44:35,160
Let me write down another
one-- x to the seventh times y

830
00:44:35,160 --> 00:44:41,110
squared plus whatever,
maybe times 6 here.

831
00:44:41,110 --> 00:44:42,430
Well, I know what x and y are.

832
00:44:42,430 --> 00:44:43,500
I just expand them.

833
00:44:43,500 --> 00:44:45,730
I can just plug in
this thing involving

834
00:44:45,730 --> 00:44:48,930
cosine alpha and cosine beta
into x, and plug in this thing

835
00:44:48,930 --> 00:44:53,500
involving cosine of alpha
minus pi/2 and so on into y.

836
00:44:53,500 --> 00:44:54,820
Multiply all that stuff out.

837
00:44:54,820 --> 00:44:57,400
What you end up doing is
multiplying cosines times

838
00:44:57,400 --> 00:44:58,820
cosines.

839
00:44:58,820 --> 00:44:59,960
And here's how we do it.

840
00:44:59,960 --> 00:45:03,830
To multiply two cosines, I
absorb things and turn it

841
00:45:03,830 --> 00:45:09,080
into cosines of sums of angles
and differences of angles.

842
00:45:09,080 --> 00:45:10,890
And I brought an
example for you.

843
00:45:15,172 --> 00:45:17,190
I'm from Waterloo,
so I use Maple.

844
00:45:17,190 --> 00:45:22,060
You could use Mathematica,
whatever you want,

845
00:45:22,060 --> 00:45:24,440
because doing this
algebra is a pain.

846
00:45:24,440 --> 00:45:27,590
But at the top there, it
says substitute x equal

847
00:45:27,590 --> 00:45:30,000
that crazy thing and y
equal that crazy thing.

848
00:45:30,000 --> 00:45:33,720
Here I didn't bother
rewriting in terms of cosines

849
00:45:33,720 --> 00:45:37,960
because Maple's smarter than I
am, so I don't need to do that.

850
00:45:37,960 --> 00:45:40,110
And I have some crazy
equation at the top

851
00:45:40,110 --> 00:45:43,060
there, which is x cubed
times y minus 5 times x times

852
00:45:43,060 --> 00:45:46,900
y squared equals 0, but I'm
ignoring the equals 0 part.

853
00:45:46,900 --> 00:45:50,900
So I just plug that in, and the
first time, the first answer

854
00:45:50,900 --> 00:45:52,150
there, it doesn't do anything.

855
00:45:52,150 --> 00:45:53,740
It just plugs it in.

856
00:45:53,740 --> 00:45:58,170
Then I say expand that, and
it does all this crazy stuff.

857
00:45:58,170 --> 00:46:01,470
It multiplies all those things
out, uses binomial theorem

858
00:46:01,470 --> 00:46:01,970
or whatever.

859
00:46:01,970 --> 00:46:03,761
You get various new
coefficients out there,

860
00:46:03,761 --> 00:46:06,080
but you get various powers
of cosines and sines

861
00:46:06,080 --> 00:46:08,430
of alphas and betas.

862
00:46:08,430 --> 00:46:10,690
And then I use the
magic operation

863
00:46:10,690 --> 00:46:12,897
Combine Using Trig Formulas.

864
00:46:12,897 --> 00:46:15,230
And combine means when you
have a product of two things,

865
00:46:15,230 --> 00:46:17,750
try to make it one thing.

866
00:46:17,750 --> 00:46:21,260
And that's just a way of telling
Maple to apply this formula,

867
00:46:21,260 --> 00:46:24,800
but it does it for the sine case
also-- both cosines and sines.

868
00:46:24,800 --> 00:46:28,980
And then you get this pretty
equation, or pretty left-hand

869
00:46:28,980 --> 00:46:30,480
side I guess.

870
00:46:30,480 --> 00:46:32,080
You get sines.

871
00:46:32,080 --> 00:46:34,084
It's written it as
sines and cosines.

872
00:46:34,084 --> 00:46:35,500
In our situation,
we're only going

873
00:46:35,500 --> 00:46:39,240
to get cosines because I
was very reductionist here.

874
00:46:39,240 --> 00:46:42,100
And it'll always be cosines
of various integer multiples

875
00:46:42,100 --> 00:46:46,500
of alpha plus some
integer multiple of beta.

876
00:46:46,500 --> 00:46:49,155
And then there'll be some
power of r out there.

877
00:46:49,155 --> 00:46:49,780
It's no biggie.

878
00:46:49,780 --> 00:46:51,280
In fact, we're told what r is.

879
00:46:51,280 --> 00:46:52,830
We don't have to
think very hard.

880
00:46:52,830 --> 00:46:55,610
We can construct r
to the seventh power.

881
00:46:55,610 --> 00:46:57,344
It's not a big deal.

882
00:46:57,344 --> 00:46:58,760
In fact, all of
these coefficients

883
00:46:58,760 --> 00:46:59,640
are not a big deal.

884
00:46:59,640 --> 00:47:02,700
We can just say 15
over 32 times r cubed.

885
00:47:02,700 --> 00:47:05,160
We can just make an
edge that's that length.

886
00:47:05,160 --> 00:47:07,440
The hard part is making
these things because alpha,

887
00:47:07,440 --> 00:47:08,760
beta are variables.

888
00:47:08,760 --> 00:47:10,180
They're not a fixed thing.

889
00:47:10,180 --> 00:47:13,920
As this thing moves, we want to
be able to compute twice alpha

890
00:47:13,920 --> 00:47:18,010
plus beta, and then take
the cosine of that thing.

891
00:47:18,010 --> 00:47:19,960
Now taking the cosine
of a thing is easy.

892
00:47:19,960 --> 00:47:22,400
It's just the x-coordinate.

893
00:47:22,400 --> 00:47:28,900
If I have a segment that's
going in direction theta,

894
00:47:28,900 --> 00:47:30,390
then the x-coordinate
of this thing

895
00:47:30,390 --> 00:47:33,510
with respect to that
thing is cos theta.

896
00:47:33,510 --> 00:47:35,260
This is what I've been
using all the time.

897
00:47:35,260 --> 00:47:37,260
The y-coordinate is sine
theta, but x-coordinate

898
00:47:37,260 --> 00:47:38,480
is cosine theta.

899
00:47:38,480 --> 00:47:41,010
So if we can construct something
at an angle of twice alpha

900
00:47:41,010 --> 00:47:45,300
plus beta, then we just
project onto the x-axis.

901
00:47:45,300 --> 00:47:47,050
This length is cosine theta.

902
00:47:49,560 --> 00:47:50,280
Done.

903
00:47:50,280 --> 00:47:52,580
So all we need is to be
able to take an angle,

904
00:47:52,580 --> 00:47:55,940
multiply it by some
integer like 27,

905
00:47:55,940 --> 00:47:59,382
and we need to be able to
add two angles together.

906
00:47:59,382 --> 00:48:01,200
And that's what Kempe does.

907
00:48:03,750 --> 00:48:07,840
So he does it with
this crazy thing

908
00:48:07,840 --> 00:48:09,940
called a contraparallelogram.

909
00:48:12,460 --> 00:48:15,860
A parallelogram looks like this.

910
00:48:15,860 --> 00:48:17,136
You see the parallelogram.

911
00:48:17,136 --> 00:48:19,010
So it's a little more
general than a rhombus,

912
00:48:19,010 --> 00:48:19,810
but not much more.

913
00:48:19,810 --> 00:48:23,290
And then the contraparallelogram
is you take this diagonal,

914
00:48:23,290 --> 00:48:26,750
a,y I guess it's called here,
and you reflect one side down.

915
00:48:26,750 --> 00:48:29,350
And so the blue thing is
the contraparallelogram.

916
00:48:32,270 --> 00:48:33,830
How does it work?

917
00:48:33,830 --> 00:48:35,130
What does it do?

918
00:48:35,130 --> 00:48:37,340
Well, I'm assuming the
left point there is pinned.

919
00:48:37,340 --> 00:48:40,930
Actually, I'm assuming
both x and y are pinned.

920
00:48:40,930 --> 00:48:42,920
And it just moves around.

921
00:48:42,920 --> 00:48:43,890
It's kind of cool.

922
00:48:46,920 --> 00:48:51,185
It has this great
feature that the angle

923
00:48:51,185 --> 00:48:54,620
at a here between
the two blue segments

924
00:48:54,620 --> 00:48:57,150
is the same as the angle at y.

925
00:48:57,150 --> 00:49:01,165
And the angle at x is the
same as the angle at b.

926
00:49:01,165 --> 00:49:03,420
You can see that
throughout the motion.

927
00:49:03,420 --> 00:49:06,489
So it's basically
an angle copier.

928
00:49:06,489 --> 00:49:08,030
Angle copier sounds
good because if I

929
00:49:08,030 --> 00:49:10,550
want to multiply an angle by
2, I'd like two copies of it

930
00:49:10,550 --> 00:49:12,590
and then stick them together.

931
00:49:12,590 --> 00:49:15,504
So we're just going to take
this contraparallelogram

932
00:49:15,504 --> 00:49:20,190
and combine a bunch of them.

933
00:49:20,190 --> 00:49:24,280
So if we're going
to multiply by 2--

934
00:49:24,280 --> 00:49:26,850
it's a little harder
to see right on blue.

935
00:49:26,850 --> 00:49:30,030
So I have one
contraparallelogram here,

936
00:49:30,030 --> 00:49:34,300
and then I make a similar--
you just scale it up--

937
00:49:34,300 --> 00:49:38,300
contraparallelogram that
lies along that one.

938
00:49:38,300 --> 00:49:39,800
So there's two
contraparallelograms.

939
00:49:42,490 --> 00:49:45,420
And as I said
before, this angle is

940
00:49:45,420 --> 00:49:47,010
going to be equal to this angle.

941
00:49:47,010 --> 00:49:48,940
That's going to be
our input alpha.

942
00:49:48,940 --> 00:49:50,680
And we're trying to
compute twice alpha,

943
00:49:50,680 --> 00:49:52,840
which will be here.

944
00:49:52,840 --> 00:49:55,380
And because this
contraparallelogram

945
00:49:55,380 --> 00:49:58,670
is similar to this one,
the angles are the same.

946
00:49:58,670 --> 00:50:00,450
It's just a blowing up of it.

947
00:50:00,450 --> 00:50:03,470
So that means these angles,
which are equal to each other

948
00:50:03,470 --> 00:50:05,190
because it's
contraparallelogram,

949
00:50:05,190 --> 00:50:08,350
will be equal to these
angles because it's

950
00:50:08,350 --> 00:50:10,610
the same contraparallelogram
scaled up.

951
00:50:10,610 --> 00:50:13,762
Therefore, we have two copies
of the angle right there.

952
00:50:13,762 --> 00:50:16,340
I'll just show
you that it works.

953
00:50:16,340 --> 00:50:19,110
A little hard to see on a
digital screen with pixels.

954
00:50:19,110 --> 00:50:19,610
Here.

955
00:50:19,610 --> 00:50:22,026
When we get 90 degrees, we're
going to get the twice of it

956
00:50:22,026 --> 00:50:23,810
is exactly 180.

957
00:50:23,810 --> 00:50:26,400
Here's 45.

958
00:50:26,400 --> 00:50:29,930
Here's two 30s
make 60, and so on.

959
00:50:29,930 --> 00:50:33,010
It works basically all the time.

960
00:50:33,010 --> 00:50:35,330
Wow, cool.

961
00:50:35,330 --> 00:50:37,990
It's hard to see in some cases.

962
00:50:37,990 --> 00:50:43,760
I guess there, twice 180
is making 360, and so on.

963
00:50:43,760 --> 00:50:47,230
That's how you
multiply an angle by 2.

964
00:50:47,230 --> 00:50:50,620
But then you can extend that and
just repeat this construction.

965
00:50:50,620 --> 00:50:53,220
I take one contraparallelogram,
I attach it

966
00:50:53,220 --> 00:50:55,070
to another similar
contraparallelogram,

967
00:50:55,070 --> 00:50:57,577
I attach that to another
similar contraparallelogram,

968
00:50:57,577 --> 00:50:58,910
I get three copies of the angle.

969
00:51:03,810 --> 00:51:05,520
Amazing.

970
00:51:05,520 --> 00:51:08,090
Here's trisecting
180 degree angle.

971
00:51:08,090 --> 00:51:08,885
I get three 60s.

972
00:51:11,740 --> 00:51:13,420
Wow, that looks crazy.

973
00:51:16,730 --> 00:51:18,117
But it works.

974
00:51:18,117 --> 00:51:19,700
This is sort of a
proof that it works.

975
00:51:19,700 --> 00:51:21,170
It exists in all the states.

976
00:51:23,780 --> 00:51:24,280
Great.

977
00:51:24,280 --> 00:51:25,730
This is an angle
trisector, which

978
00:51:25,730 --> 00:51:27,870
was a big deal in
the late 1800s.

979
00:51:27,870 --> 00:51:30,260
You could build a linkage
where if-- instead

980
00:51:30,260 --> 00:51:32,070
of thinking of this
as being your input,

981
00:51:32,070 --> 00:51:34,653
or this, whatever-- any one of
these three could be the input,

982
00:51:34,653 --> 00:51:36,500
and the output is
three times that.

983
00:51:36,500 --> 00:51:37,750
That's what we want.

984
00:51:37,750 --> 00:51:40,760
You could turn it around and
say, if I'm given some angle,

985
00:51:40,760 --> 00:51:43,210
I line up these two bars
to be equal to that,

986
00:51:43,210 --> 00:51:46,700
and then I'll get three
trisections of the angle.

987
00:51:46,700 --> 00:51:48,417
You can use this to
quintisect an angle.

988
00:51:48,417 --> 00:51:50,000
You can divide into
any integer number

989
00:51:50,000 --> 00:51:53,780
parts using this crazy linkage.

990
00:51:53,780 --> 00:51:55,000
Fun stuff.

991
00:51:55,000 --> 00:51:58,750
And we can use it to multiply
an angle by some constant.

992
00:51:58,750 --> 00:52:03,690
I should write, we
do this, we expand,

993
00:52:03,690 --> 00:52:06,160
and we will get
a bunch of terms.

994
00:52:06,160 --> 00:52:10,990
We will get that our
function phi here

995
00:52:10,990 --> 00:52:13,390
is a sum of a
bunch of terms that

996
00:52:13,390 --> 00:52:16,300
look like some constant,
which we can compute ahead

997
00:52:16,300 --> 00:52:23,760
of time nothing's changing,
times cosine of some integer,

998
00:52:23,760 --> 00:52:32,380
what do I call it, it's
to match minutes here,

999
00:52:32,380 --> 00:52:40,680
r sub i times alpha,
plus s sub i times

1000
00:52:40,680 --> 00:52:44,150
beta, plus some
magic number delta,

1001
00:52:44,150 --> 00:52:49,479
and delta is 0, or plus
or minus pi over 2.

1002
00:52:49,479 --> 00:52:50,520
So that's not a big deal.

1003
00:52:50,520 --> 00:52:53,370
That's just saying whether
we're using sign or cosine.

1004
00:52:53,370 --> 00:52:55,050
What we need is to
construct, and these

1005
00:52:55,050 --> 00:52:58,801
are both integers could
be positive or negative,

1006
00:52:58,801 --> 00:53:00,300
negative's not such
a big deal, it's

1007
00:53:00,300 --> 00:53:03,640
just you look at the
angle the other way.

1008
00:53:03,640 --> 00:53:06,770
So we need some integer times
alpha, some integer times beta,

1009
00:53:06,770 --> 00:53:08,250
we can now do that.

1010
00:53:08,250 --> 00:53:10,680
Then we need to
add them together.

1011
00:53:10,680 --> 00:53:12,370
How do we do that?

1012
00:53:12,370 --> 00:53:13,980
We use the Kempe Additor.

1013
00:53:17,320 --> 00:53:19,010
There he is drawing it.

1014
00:53:19,010 --> 00:53:23,480
we drew a slightly more detailed
diagram, looks like this.

1015
00:53:23,480 --> 00:53:24,980
Now, this is a
little more confusing

1016
00:53:24,980 --> 00:53:26,690
so I had to label the thing.

1017
00:53:26,690 --> 00:53:30,320
We've got the x-axis,
I like that part.

1018
00:53:30,320 --> 00:53:32,610
The idea is we have two inputs.

1019
00:53:32,610 --> 00:53:35,960
Here's input one, defining
one angle to the x-axis.

1020
00:53:35,960 --> 00:53:38,580
Here's input two, another
angle to the x-axis,

1021
00:53:38,580 --> 00:53:39,991
here is about 90 degrees.

1022
00:53:39,991 --> 00:53:41,490
I want to add them
together and that

1023
00:53:41,490 --> 00:53:44,920
will be this output over here.

1024
00:53:44,920 --> 00:53:49,530
Our idea, I think I should
draw this separately,

1025
00:53:49,530 --> 00:53:52,700
not our idea this is
Kempe's idea, here's

1026
00:53:52,700 --> 00:53:56,540
0 here's input
one, and input two.

1027
00:53:56,540 --> 00:53:59,960
Both of these are measured
as angles to the x-axis.

1028
00:53:59,960 --> 00:54:04,260
All we need to do is sort of
rotate this angle to be here,

1029
00:54:04,260 --> 00:54:08,940
and then this guy
will be the sum.

1030
00:54:08,940 --> 00:54:12,010
How do we figure that out
when all we know how to do

1031
00:54:12,010 --> 00:54:16,770
is multiply angles by two or
divide them in half let's say?

1032
00:54:16,770 --> 00:54:18,190
I told you that.

1033
00:54:18,190 --> 00:54:20,330
You think about it
for a while, you

1034
00:54:20,330 --> 00:54:23,420
realize, oh well
let's instead think

1035
00:54:23,420 --> 00:54:26,279
of it as taking this
angle the smaller one,

1036
00:54:26,279 --> 00:54:27,820
doesn't really matter
but it's easier

1037
00:54:27,820 --> 00:54:30,570
to think about copying
that angle over here.

1038
00:54:30,570 --> 00:54:32,120
I want to copy that
angle over here,

1039
00:54:32,120 --> 00:54:36,040
and I'm an origamist
I think reflection.

1040
00:54:36,040 --> 00:54:40,350
I would really like to
reflect along the bisector.

1041
00:54:40,350 --> 00:54:42,610
Now what is it a bisector of?

1042
00:54:42,610 --> 00:54:46,330
It's a bisector of this angle,
which hasn't been marked here.

1043
00:54:46,330 --> 00:54:51,930
But if this a, this is b,
then this would be b minus a.

1044
00:54:51,930 --> 00:54:58,515
So I want to bisect that angle.

1045
00:54:58,515 --> 00:54:59,890
Really I want to
bisect the angle

1046
00:54:59,890 --> 00:55:03,566
between this edge and this edge.

1047
00:55:03,566 --> 00:55:04,590
How do I do that?

1048
00:55:04,590 --> 00:55:08,330
I use a Kempe
multiplicator, value of two,

1049
00:55:08,330 --> 00:55:13,000
apply it to this angle, then
I figure out that bisector.

1050
00:55:13,000 --> 00:55:16,250
How do I copy this
thing over to here?

1051
00:55:16,250 --> 00:55:19,310
Well I take this
angle and I double it,

1052
00:55:19,310 --> 00:55:23,160
I'll get that, actually that.

1053
00:55:23,160 --> 00:55:25,480
So I just used two
Kempe doublers,

1054
00:55:25,480 --> 00:55:29,480
one to have been one to double.

1055
00:55:29,480 --> 00:55:32,230
And I can we
possibly see it here?

1056
00:55:32,230 --> 00:55:33,980
I have this line marked middle.

1057
00:55:33,980 --> 00:55:36,530
That is the bisector between
the input two line and the input

1058
00:55:36,530 --> 00:55:37,580
one line.

1059
00:55:37,580 --> 00:55:38,950
How is it found?

1060
00:55:38,950 --> 00:55:42,750
Well it's here.

1061
00:55:42,750 --> 00:55:49,360
That's one part of the Kempe
halfer, doubler, whatever.

1062
00:55:49,360 --> 00:55:52,800
And it's attached to one
over there, similar one.

1063
00:55:52,800 --> 00:55:56,130
So it's attached on one side to
input two and on the other side

1064
00:55:56,130 --> 00:55:58,230
to input one.

1065
00:55:58,230 --> 00:56:00,960
And so therefore it
actually takes that angle

1066
00:56:00,960 --> 00:56:02,950
and it cuts it in half there.

1067
00:56:02,950 --> 00:56:04,220
That's what it does.

1068
00:56:04,220 --> 00:56:06,750
So now I have that
middle line and I

1069
00:56:06,750 --> 00:56:09,760
have another Kempe doubler
which is attached to the output

1070
00:56:09,760 --> 00:56:14,230
thing, and the middle, and
it's attached to the x-axis

1071
00:56:14,230 --> 00:56:15,080
to ground it.

1072
00:56:15,080 --> 00:56:18,380
So now we're taking this
angle and doubling it

1073
00:56:18,380 --> 00:56:20,670
and that gives us the output.

1074
00:56:20,670 --> 00:56:25,080
And you do it, and magically
thanks to Cinderella,

1075
00:56:25,080 --> 00:56:26,250
it does the right thing.

1076
00:56:26,250 --> 00:56:29,420
So as I move, here I'm
changing input one,

1077
00:56:29,420 --> 00:56:31,790
and I'm moving the output
and also this middle line,

1078
00:56:31,790 --> 00:56:35,680
it's staying a bisector there,
when they both get to 90

1079
00:56:35,680 --> 00:56:37,750
it'll be 180, and so on.

1080
00:56:37,750 --> 00:56:41,400
And I can change
and put two as well.

1081
00:56:41,400 --> 00:56:44,050
And once you build it it's kind
of clear that it will work.

1082
00:56:47,282 --> 00:56:48,740
There's a special
case of doubling,

1083
00:56:48,740 --> 00:56:50,670
but I can add any
two angles I want.

1084
00:56:50,670 --> 00:56:53,410
And what's fun is it also
works, I set it up for input one

1085
00:56:53,410 --> 00:56:56,140
being smaller input
two, but it works just

1086
00:56:56,140 --> 00:56:57,577
as fine the other way around.

1087
00:56:57,577 --> 00:56:59,410
I don't want to go too
big because I get out

1088
00:56:59,410 --> 00:57:02,880
of the screen.

1089
00:57:02,880 --> 00:57:06,350
There's 0 adding 0 to something.

1090
00:57:06,350 --> 00:57:07,880
Cool, huh?

1091
00:57:07,880 --> 00:57:08,900
Like magic.

1092
00:57:08,900 --> 00:57:10,120
So Kempe was really smart.

1093
00:57:10,120 --> 00:57:11,870
He knew about these linkages.

1094
00:57:11,870 --> 00:57:13,960
He wrote that book How
to Draw a Straight Line.

1095
00:57:13,960 --> 00:57:15,534
But he knew a lot
more about then

1096
00:57:15,534 --> 00:57:16,700
how to draw a straight line.

1097
00:57:16,700 --> 00:57:21,660
He could use, he could
add two angles, man.

1098
00:57:21,660 --> 00:57:24,280
Very cool.

1099
00:57:24,280 --> 00:57:26,710
All right, what's left.

1100
00:57:26,710 --> 00:57:29,490
Basically done,
for Kempe's proof.

1101
00:57:29,490 --> 00:57:31,690
I haven't told you what's
wrong with this proof yet.

1102
00:57:31,690 --> 00:57:35,285
Maybe you've been
thinking about it.

1103
00:57:35,285 --> 00:57:37,830
Then you just have to put
all these gadgets together.

1104
00:57:37,830 --> 00:57:41,800
We have a schematic
of it in the textbook.

1105
00:57:41,800 --> 00:57:46,575
So we started at the
top with our rhombus.

1106
00:57:46,575 --> 00:57:49,380
Not really drawn like a
rhombus there, but pretend it's

1107
00:57:49,380 --> 00:57:52,530
a rhombus, this guy.

1108
00:57:52,530 --> 00:57:57,302
I guess this is simplified
since we drew this figure.

1109
00:57:57,302 --> 00:57:58,760
So all the edge
links are the same.

1110
00:57:58,760 --> 00:58:00,742
We've got that
point p at the top,

1111
00:58:00,742 --> 00:58:01,950
that's the one we care about.

1112
00:58:01,950 --> 00:58:06,089
We've got the o in the bottom
left, it's pinned to the table.

1113
00:58:06,089 --> 00:58:07,880
And there's one gadget
I didn't talk about,

1114
00:58:07,880 --> 00:58:10,400
translators which is copies,
angles around, not too

1115
00:58:10,400 --> 00:58:12,810
big a deal.

1116
00:58:12,810 --> 00:58:15,150
Then we, in this case
say, we take alpha,

1117
00:58:15,150 --> 00:58:19,530
it's right there, we multiply
it by two, get two alpha,

1118
00:58:19,530 --> 00:58:22,920
take beta maybe add those two
together, we'll get twice alpha

1119
00:58:22,920 --> 00:58:25,060
plus beta.

1120
00:58:25,060 --> 00:58:26,970
And that is some angle
we want to measure

1121
00:58:26,970 --> 00:58:32,010
the x-coordinate of the
segment with that angle.

1122
00:58:32,010 --> 00:58:36,196
So we just make this thing
have the right length.

1123
00:58:36,196 --> 00:58:37,320
What length should it have?

1124
00:58:37,320 --> 00:58:40,940
Well we have this
cosine of whatever

1125
00:58:40,940 --> 00:58:45,680
of our angle in
our formula here.

1126
00:58:45,680 --> 00:58:47,780
And there's some
constant in front of it.

1127
00:58:47,780 --> 00:58:50,300
And for whatever reason
we have this constant 3a

1128
00:58:50,300 --> 00:58:53,610
squared times b, something
that we know how to compute.

1129
00:58:53,610 --> 00:58:57,167
So we just make this edge
have that length, that's

1130
00:58:57,167 --> 00:59:00,710
a scale marker, and
measure the projected

1131
00:59:00,710 --> 00:59:02,947
x length of that thing.

1132
00:59:02,947 --> 00:59:05,530
Well, in fact in general we have
a whole bunch of these terms.

1133
00:59:05,530 --> 00:59:07,820
We need to add them together.

1134
00:59:07,820 --> 00:59:11,360
We have a whole bunch of these
things you saw on the maple,

1135
00:59:11,360 --> 00:59:13,120
had I don't know, 20 of those.

1136
00:59:13,120 --> 00:59:16,030
So we just string them together.

1137
00:59:16,030 --> 00:59:19,040
That's our thing we start at
some point, which we call 0.

1138
00:59:19,040 --> 00:59:22,850
In fact, it will be the
same as that o, let's say.

1139
00:59:22,850 --> 00:59:24,276
It should be the same as that o.

1140
00:59:24,276 --> 00:59:26,650
We're going to measure relative
to the center of the disk

1141
00:59:26,650 --> 00:59:31,270
say, wherever 0 0 is.

1142
00:59:31,270 --> 00:59:34,632
And then we want to measure
how long this thing gets in x.

1143
00:59:34,632 --> 00:59:36,590
Some of these terms might
actually be negative,

1144
00:59:36,590 --> 00:59:39,080
some are positive.

1145
00:59:39,080 --> 00:59:42,040
But we're effectively adding
up all of these weighted sums

1146
00:59:42,040 --> 00:59:44,160
of cosines of our
angles which we

1147
00:59:44,160 --> 00:59:46,830
compute through
these crazy things.

1148
00:59:46,830 --> 00:59:50,500
At the end we want this to be 0.

1149
00:59:50,500 --> 00:59:53,850
So our goal is to make the
sum of all these things, which

1150
00:59:53,850 --> 00:59:56,770
is the sum of all these
things rewritten, equal to 0

1151
00:59:56,770 --> 00:59:59,640
That was the definition
of our curve.

1152
00:59:59,640 --> 01:00:00,840
How do we force it to be 0?

1153
01:00:00,840 --> 01:00:04,260
Well the actual value is
the x-coordinate here.

1154
01:00:04,260 --> 01:00:06,770
And I'm going to
force this to lie

1155
01:00:06,770 --> 01:00:09,970
on x equals 0 using a
peaucellier linkage,

1156
01:00:09,970 --> 01:00:12,529
because it forces
points to line a line.

1157
01:00:12,529 --> 01:00:14,070
In the way we drew
it, it would force

1158
01:00:14,070 --> 01:00:17,170
it to be on a vertical line.

1159
01:00:17,170 --> 01:00:19,700
So you have to imagine this
picture a little differently,

1160
01:00:19,700 --> 01:00:21,330
because in fact
this should be at x

1161
01:00:21,330 --> 01:00:24,070
equals 0, which is
normally right here.

1162
01:00:24,070 --> 01:00:25,980
So this thing actually
goes over to the right

1163
01:00:25,980 --> 01:00:28,440
and we'll get positive
and negative values

1164
01:00:28,440 --> 01:00:32,160
and it should lie along
the vertical line there.

1165
01:00:32,160 --> 01:00:36,510
And that's how you force, now
this point p prime has nothing

1166
01:00:36,510 --> 01:00:38,670
to do with them, is
not the same as p.

1167
01:00:38,670 --> 01:00:40,920
But we computed it based
on all these things

1168
01:00:40,920 --> 01:00:42,700
in terms of alpha and beta.

1169
01:00:42,700 --> 01:00:45,220
So this is a constraint
on p in the end,

1170
01:00:45,220 --> 01:00:47,120
because p determined
all these things.

1171
01:00:47,120 --> 01:00:50,280
And either this thing will
be on the line or not.

1172
01:00:50,280 --> 01:00:52,271
And we add this one
equality constraint

1173
01:00:52,271 --> 01:00:53,895
which would be exactly
this constraint.

1174
01:00:53,895 --> 01:00:58,200
It will constrain p
to lie on that curve,

1175
01:00:58,200 --> 01:01:01,375
and that's Kempe's construction.

1176
01:01:01,375 --> 01:01:01,875
Questions?

1177
01:01:06,700 --> 01:01:10,810
Three cool, just slightly wrong.

1178
01:01:10,810 --> 01:01:12,100
Any ideas what's wrong?

1179
01:01:23,130 --> 01:01:24,680
Has nothing do
with this picture.

1180
01:01:24,680 --> 01:01:26,565
It has everything to
do with the gadgets.

1181
01:01:31,960 --> 01:01:35,360
And actually it really has
to do with all this business

1182
01:01:35,360 --> 01:01:37,110
about the square.

1183
01:01:37,110 --> 01:01:39,340
I've lost my diagram of
the configuration space

1184
01:01:39,340 --> 01:01:40,140
to the square.

1185
01:01:40,140 --> 01:01:43,610
We had this thing that
when the square was flat,

1186
01:01:43,610 --> 01:01:45,560
it could fold in a new way.

1187
01:01:45,560 --> 01:01:47,782
And you saw I couldn't
even do that in Cinderella.

1188
01:01:47,782 --> 01:01:49,240
Cinderella is
designed to find sort

1189
01:01:49,240 --> 01:01:52,470
of one primary
arc of this thing.

1190
01:01:52,470 --> 01:01:57,150
But there are these
branch points, these guys.

1191
01:01:57,150 --> 01:01:59,520
And this construction
is perfect if you always

1192
01:01:59,520 --> 01:02:01,500
stay on your track.

1193
01:02:01,500 --> 01:02:04,140
But in a real linkage, you
can diverge at any point

1194
01:02:04,140 --> 01:02:06,922
and go to some other track, and
you can walk around this space.

1195
01:02:06,922 --> 01:02:08,880
When you have these branch
points where there's

1196
01:02:08,880 --> 01:02:12,840
more than two curves coming
together, you have a choice.

1197
01:02:12,840 --> 01:02:16,140
So you're not supposed
to have a choice here,

1198
01:02:16,140 --> 01:02:19,250
supposed to always
do this thing.

1199
01:02:19,250 --> 01:02:26,030
And sadly there's one
gadget, really two gadgets,

1200
01:02:26,030 --> 01:02:28,170
the parallelogram and
the contra-parallelogram.

1201
01:02:28,170 --> 01:02:30,540
The parallelogram
we're using here.

1202
01:02:30,540 --> 01:02:33,800
It's also used in the copy
gadget which I didn't show you.

1203
01:02:33,800 --> 01:02:40,230
What don't I tell you, a
copy gadget looks like this.

1204
01:02:40,230 --> 01:02:45,360
It's like a lamp, you
know like the Pixar lamp.

1205
01:02:45,360 --> 01:02:50,210
So if you have an angle here
it gets copied here, right.

1206
01:02:50,210 --> 01:02:52,809
These are just two
identical parallelograms.

1207
01:02:52,809 --> 01:02:54,600
So if you need multiple
copies of an angle,

1208
01:02:54,600 --> 01:02:56,430
you can make multiple
copies that way.

1209
01:02:56,430 --> 01:03:00,332
I don't think you really
need this, but Kempe used it.

1210
01:03:00,332 --> 01:03:03,460
It feels good to be
able to copy an angel.

1211
01:03:03,460 --> 01:03:05,290
And it's a parallelogram.

1212
01:03:05,290 --> 01:03:07,950
And we used a lot, the
contra-parallelogram.

1213
01:03:07,950 --> 01:03:10,550
The trouble is they
are the same linkage.

1214
01:03:10,550 --> 01:03:13,320
If you forget about
the configuration,

1215
01:03:13,320 --> 01:03:15,470
you just look at the linkage
which is who's pinned,

1216
01:03:15,470 --> 01:03:18,330
what are the edge links, what
is the graph connectivity,

1217
01:03:18,330 --> 01:03:22,932
it looks like this, just these
numbers are not all the same,

1218
01:03:22,932 --> 01:03:24,640
and only one of these
vertices is pinned.

1219
01:03:24,640 --> 01:03:26,580
Other than that it's
exactly this diagram.

1220
01:03:26,580 --> 01:03:28,870
You know, It's a square,
it's a quadrilateral,

1221
01:03:28,870 --> 01:03:31,730
has some lengths.

1222
01:03:31,730 --> 01:03:34,650
Opposite pairs are equal.

1223
01:03:34,650 --> 01:03:38,780
But you can't really
distinguish from the two

1224
01:03:38,780 --> 01:03:39,950
kinds of configurations.

1225
01:03:39,950 --> 01:03:41,990
There's one track which
is the parallelogram.

1226
01:03:41,990 --> 01:03:44,323
There's another track which
is the contra-parallelogram.

1227
01:03:46,060 --> 01:03:48,580
And you can do this.

1228
01:03:48,580 --> 01:03:53,410
Let me show you in animation,
give you some intuition.

1229
01:03:53,410 --> 01:03:56,200
So here I drew both, oh sorry,
not supposed to move that.

1230
01:03:56,200 --> 01:04:01,770
Here I drew both of them, got
the green contra-parallelogram.

1231
01:04:01,770 --> 01:04:06,130
And there's the, sorry green,
the blue contra-parallelogram.

1232
01:04:06,130 --> 01:04:09,150
And then we've got the
black and blue parallelogram

1233
01:04:09,150 --> 01:04:10,270
on the outside.

1234
01:04:10,270 --> 01:04:14,190
And ignore this diagonal, it
should be a different color.

1235
01:04:14,190 --> 01:04:18,360
And when we're flat,
that's the same thing.

1236
01:04:18,360 --> 01:04:20,970
And so we've got this one
cycle that goes around

1237
01:04:20,970 --> 01:04:22,880
and that's the blue picture.

1238
01:04:22,880 --> 01:04:25,950
Then at this point you can
switch into the black picture

1239
01:04:25,950 --> 01:04:30,420
and follow the parallelogram
around until it's flat.

1240
01:04:30,420 --> 01:04:33,880
And that point you could also
switch back to blue or not.

1241
01:04:33,880 --> 01:04:37,590
So I think this is two
circles joined to two points.

1242
01:04:37,590 --> 01:04:40,340
If I'm not mistaken,
the configuration space

1243
01:04:40,340 --> 01:04:42,420
would look like this.

1244
01:04:42,420 --> 01:04:45,874
So one of these tracks, say this
one, is a contra-parallelogram.

1245
01:04:45,874 --> 01:04:47,040
This one is a parallelogram.

1246
01:04:47,040 --> 01:04:51,230
They meet at the two
extremes, both flat cases.

1247
01:04:51,230 --> 01:04:55,250
And in Kempe's construction
he wants each of the things

1248
01:04:55,250 --> 01:04:57,360
that we call the
parallelogram like this one,

1249
01:04:57,360 --> 01:04:59,127
to stay on one, the
parallelogram track.

1250
01:04:59,127 --> 01:05:00,710
And he wants the
contra-parallelograms

1251
01:05:00,710 --> 01:05:03,950
to stay on the
contra-parallelograms track.

1252
01:05:03,950 --> 01:05:06,050
But how do you force that?

1253
01:05:06,050 --> 01:05:09,120
He didn't force it,
and so technically he

1254
01:05:09,120 --> 01:05:10,550
did not solve this problem.

1255
01:05:10,550 --> 01:05:13,492
He solved, that
solved this problem,

1256
01:05:13,492 --> 01:05:14,950
which is that you
would get exactly

1257
01:05:14,950 --> 01:05:16,750
the trajectory of a vertex.

1258
01:05:16,750 --> 01:05:20,050
He solved the slightly weaker
problem, which is his point

1259
01:05:20,050 --> 01:05:22,930
can travel along that curve.

1260
01:05:22,930 --> 01:05:25,660
It can also travel
on other curves.

1261
01:05:25,660 --> 01:05:27,160
But it's going to
be one dimensional

1262
01:05:27,160 --> 01:05:28,380
because it is constrained.

1263
01:05:28,380 --> 01:05:30,630
But there are these other
tracks you might conceivably

1264
01:05:30,630 --> 01:05:31,940
be able to go on.

1265
01:05:31,940 --> 01:05:33,679
In particular, you
can follow this track.

1266
01:05:33,679 --> 01:05:34,845
There might be other things.

1267
01:05:34,845 --> 01:05:37,050
And we'd like to
get rid of that.

1268
01:05:37,050 --> 01:05:39,750
It turns out you can get
rid of that pretty easily.

1269
01:05:43,290 --> 01:05:46,340
This problem was first noticed
by Kapovitch and Millson

1270
01:05:46,340 --> 01:05:52,650
in like 2002 I think.

1271
01:05:52,650 --> 01:05:53,970
Yeah.

1272
01:05:53,970 --> 01:05:55,720
Well that's when it
finally got published.

1273
01:05:55,720 --> 01:05:58,440
I think it was around for
a few years before that.

1274
01:05:58,440 --> 01:06:02,730
And they observed that
for parallelograms you

1275
01:06:02,730 --> 01:06:06,050
can fix it pretty easily
like in this picture.

1276
01:06:06,050 --> 01:06:09,230
You have the
parallelogram a-b-c-d.

1277
01:06:09,230 --> 01:06:12,040
You construct the
midpoints of a and b

1278
01:06:12,040 --> 01:06:13,940
and the midpoints of d and c.

1279
01:06:13,940 --> 01:06:15,910
How do you actually do this?

1280
01:06:15,910 --> 01:06:19,490
I want that point x to stay
on the middle of that bar.

1281
01:06:19,490 --> 01:06:21,020
You can do that
by making, if you

1282
01:06:21,020 --> 01:06:23,360
want to construct a point
in the middle of a bar,

1283
01:06:23,360 --> 01:06:25,630
you just add a tiny
triangle, where

1284
01:06:25,630 --> 01:06:30,380
it's forced to be that
way, where a b equals,

1285
01:06:30,380 --> 01:06:34,190
I mean the sum of a
and b equals a plus b.

1286
01:06:34,190 --> 01:06:37,977
And if you want,
do we need this?

1287
01:06:37,977 --> 01:06:40,060
I think you could just
construct those three bars.

1288
01:06:40,060 --> 01:06:42,300
This makes a little
bit more rigid,

1289
01:06:42,300 --> 01:06:47,050
but I don't think we
actually need that here.

1290
01:06:47,050 --> 01:06:49,500
So you can force a vertex
to be along the midpoint

1291
01:06:49,500 --> 01:06:54,120
there by making a tiny
triangle, a zero area triangle.

1292
01:06:54,120 --> 01:06:58,480
And when you do that, I don;t
actually have a picture of it,

1293
01:06:58,480 --> 01:07:02,070
we can look at the
countra-parallelogram I guess,

1294
01:07:02,070 --> 01:07:05,052
it forces the thing to work.

1295
01:07:05,052 --> 01:07:07,010
So for example, suppose
we go from the midpoint

1296
01:07:07,010 --> 01:07:09,420
here to the midpoint there.

1297
01:07:09,420 --> 01:07:12,690
That's going to stay parallel to
this segment and this segment.

1298
01:07:12,690 --> 01:07:15,040
It's like two parallelograms
joined together.

1299
01:07:15,040 --> 01:07:18,060
Okay, and if you went
here, and then you

1300
01:07:18,060 --> 01:07:22,889
tried to go off into the
contra-parallelogram state,

1301
01:07:22,889 --> 01:07:23,680
what would that do?

1302
01:07:23,680 --> 01:07:26,780
Well you'd have like from
the midpoint of this segment

1303
01:07:26,780 --> 01:07:29,350
to the midpoint of this segment.

1304
01:07:29,350 --> 01:07:30,320
Holy cow.

1305
01:07:30,320 --> 01:07:32,480
It's like they're right
on top of each other.

1306
01:07:32,480 --> 01:07:34,355
So that's not going to
have the right length.

1307
01:07:34,355 --> 01:07:37,210
Right now it's supposed to be
this length, same as the length

1308
01:07:37,210 --> 01:07:39,040
a x.

1309
01:07:39,040 --> 01:07:41,780
So if you try to jump
tracks you can't.

1310
01:07:41,780 --> 01:07:44,560
It's really easy to force
to stay on the parallelogram

1311
01:07:44,560 --> 01:07:45,060
track.

1312
01:07:48,120 --> 01:07:50,260
And then six years
ago in this class

1313
01:07:50,260 --> 01:07:52,070
I posed, hey can we
do the same thing

1314
01:07:52,070 --> 01:07:53,642
for the contra-parallelogram.

1315
01:07:53,642 --> 01:07:55,850
Kapovich and Millson gave
some other very complicated

1316
01:07:55,850 --> 01:07:56,980
solution.

1317
01:07:56,980 --> 01:08:00,340
Turns out there's also
pretty easy way to fix

1318
01:08:00,340 --> 01:08:05,320
the contra-parallelogram, which
is you add this pinpoint set up

1319
01:08:05,320 --> 01:08:07,680
to be--- So these midpoints
will all be aligned

1320
01:08:07,680 --> 01:08:09,950
because it's a
contra-parallelogram.

1321
01:08:09,950 --> 01:08:11,900
You take the middle
and you bisect,

1322
01:08:11,900 --> 01:08:16,060
you put x far away, it would
be very far down there.

1323
01:08:16,060 --> 01:08:19,540
And you just make these
lengths be what they should be.

1324
01:08:19,540 --> 01:08:21,399
And it's not so
obvious but you can

1325
01:08:21,399 --> 01:08:24,160
prove that as the
contra-parallelogram moves,

1326
01:08:24,160 --> 01:08:27,398
these edges will all
have the same length.

1327
01:08:27,398 --> 01:08:29,770
It works.

1328
01:08:29,770 --> 01:08:35,700
Whereas if you try to go
into the parallelogram state,

1329
01:08:35,700 --> 01:08:37,240
and you measure
these links, they're

1330
01:08:37,240 --> 01:08:38,439
going to be really tiny.

1331
01:08:38,439 --> 01:08:40,300
Here they're really large.

1332
01:08:40,300 --> 01:08:41,960
Am I doing that right?

1333
01:08:41,960 --> 01:08:43,050
What am I measuring?

1334
01:08:43,050 --> 01:08:45,500
No, the point is it's
not about those links.

1335
01:08:45,500 --> 01:08:48,979
These guys have to be co-linear
in the contra-parallelogram

1336
01:08:48,979 --> 01:08:51,800
case, but they are not collinear
in the parallelogram case,

1337
01:08:51,800 --> 01:08:54,640
and this is essentially
forcing that co-linearity.

1338
01:08:54,640 --> 01:08:57,140
So I don't want to go into the
details of how that's proved,

1339
01:08:57,140 --> 01:08:58,630
it's in the textbook.

1340
01:08:58,630 --> 01:09:02,060
But this bracing will force it
to say contra-parallelogram,

1341
01:09:02,060 --> 01:09:04,580
that bracing will force
it to be a parallelogram,

1342
01:09:04,580 --> 01:09:06,529
and then we actually
have proofed

1343
01:09:06,529 --> 01:09:09,030
Kempe's universality theorem,
with just a slight tweak

1344
01:09:09,030 --> 01:09:09,920
to his proof.

1345
01:09:13,410 --> 01:09:15,770
So that's the bug and the fix.

1346
01:09:15,770 --> 01:09:18,430
Let me tell you some
generalizations,

1347
01:09:18,430 --> 01:09:20,210
some open problems,
some applications.

1348
01:09:28,819 --> 01:09:32,069
Maybe start with
the obvious stuff

1349
01:09:32,069 --> 01:09:37,990
which is something called
Weierstrass approximation

1350
01:09:37,990 --> 01:09:57,750
theorem, which is that if you
have any continuous function,

1351
01:09:57,750 --> 01:10:04,090
I should say let's
think of it as a curve,

1352
01:10:04,090 --> 01:10:10,150
has an epsilon approximation
as a polynomial.

1353
01:10:15,950 --> 01:10:17,530
OK, so in particular
what this tells

1354
01:10:17,530 --> 01:10:19,790
us is if we have some
crazy curve we want

1355
01:10:19,790 --> 01:10:24,830
to make, like signature,
then you can approximate it

1356
01:10:24,830 --> 01:10:29,670
by a polynomial of the form that
we analyzed, this [INAUDIBLE]

1357
01:10:29,670 --> 01:10:34,240
of x y equals 0.

1358
01:10:34,240 --> 01:10:36,340
And the polynomial will
actually look like this,

1359
01:10:36,340 --> 01:10:38,040
It's a little bit
ugly, if you've ever

1360
01:10:38,040 --> 01:10:39,980
seen these constructions.

1361
01:10:39,980 --> 01:10:44,930
But it will be always within
an epsilon thickness of Eric,

1362
01:10:44,930 --> 01:10:48,770
always Eric, never
any other curve.

1363
01:10:48,770 --> 01:10:51,310
All right so you can apply
this to make your signature

1364
01:10:51,310 --> 01:10:53,962
or make whatever crazy
curve you have in mind.

1365
01:10:53,962 --> 01:10:54,795
That's kind of nice.

1366
01:10:57,320 --> 01:10:59,765
So this is why you can sign
your name using polynomials.

1367
01:11:02,390 --> 01:11:03,899
So that's an easy application.

1368
01:11:03,899 --> 01:11:04,773
Some generalizations.

1369
01:11:13,276 --> 01:11:14,260
All right.

1370
01:11:14,260 --> 01:11:20,340
You can do curves
in d dimensions.

1371
01:11:20,340 --> 01:11:22,520
We talked about two dimensions.

1372
01:11:22,520 --> 01:11:24,130
You can generalize that.

1373
01:11:24,130 --> 01:11:28,530
You can also do surfaces
in d dimensions.

1374
01:11:28,530 --> 01:11:31,440
Basically if you want to specify
a curve in three dimensions,

1375
01:11:31,440 --> 01:11:35,820
you need two equations
to constrain it down

1376
01:11:35,820 --> 01:11:37,449
to a one dimensional thing.

1377
01:11:37,449 --> 01:11:38,990
Generally you start
with d dimensions

1378
01:11:38,990 --> 01:11:41,156
and you can remove however
many dimensions you want.

1379
01:11:41,156 --> 01:11:42,710
You'll get some
subsurface in there.

1380
01:11:42,710 --> 01:11:44,860
Just add a bunch of
polynomial constraints.

1381
01:11:44,860 --> 01:11:47,350
You can do that.

1382
01:11:47,350 --> 01:11:49,520
Other fun facts.

1383
01:11:49,520 --> 01:11:54,852
The number of edges you
need is about n to the d.

1384
01:11:54,852 --> 01:11:58,960
This is being careful
about reusing structure.

1385
01:11:58,960 --> 01:12:01,950
In the plane you can deal
with n squared edges.

1386
01:12:01,950 --> 01:12:03,050
And that's optimal.

1387
01:12:03,050 --> 01:12:04,800
You can show you
need n squared edges

1388
01:12:04,800 --> 01:12:09,390
to construct certain
polynomials in the plane.

1389
01:12:09,390 --> 01:12:13,560
In three dimensions you're
going to need more, turns out.

1390
01:12:13,560 --> 01:12:15,540
Basically, because the
constructions we used

1391
01:12:15,540 --> 01:12:18,420
want to work in a plane, and
if you live in d dimensions,

1392
01:12:18,420 --> 01:12:20,040
you have to add lots
of edges to force

1393
01:12:20,040 --> 01:12:23,370
it to live in two dimensions.

1394
01:12:23,370 --> 01:12:28,010
So you can do the computation,
add the numbers, and so on.

1395
01:12:28,010 --> 01:12:30,398
What other good things?

1396
01:12:30,398 --> 01:12:33,110
All right, for the
mathematicians,

1397
01:12:33,110 --> 01:12:39,450
you can make any compact
semi algebraic set.

1398
01:12:44,020 --> 01:12:46,900
It's basically a
fancy way to say

1399
01:12:46,900 --> 01:12:48,435
you can add polynomial
constraints,

1400
01:12:48,435 --> 01:12:51,610
but you can also take
unions of solution sets.

1401
01:12:51,610 --> 01:12:53,530
So I could take the
union of five curves

1402
01:12:53,530 --> 01:12:57,525
that I like, or I can take the
intersection of five curves

1403
01:12:57,525 --> 01:12:59,030
that I like.

1404
01:12:59,030 --> 01:13:00,990
Bunch of things
you can do, that's

1405
01:13:00,990 --> 01:13:03,010
the meaning of semi
algebraic set, polynomials

1406
01:13:03,010 --> 01:13:04,135
and union and intersection.

1407
01:13:08,090 --> 01:13:12,800
These are all proved in paper
with Tim Avit and Reid Barton

1408
01:13:12,800 --> 01:13:15,230
from this class six years ago.

1409
01:13:15,230 --> 01:13:19,710
Another fun fact
from that paper is

1410
01:13:19,710 --> 01:13:23,440
that it's coNP-hard
to test rigidity.

1411
01:13:26,690 --> 01:13:30,930
Rigidity is the
topic of next class.

1412
01:13:30,930 --> 01:13:33,460
You're given a linkage,
does it move it all?

1413
01:13:33,460 --> 01:13:34,950
And you can use
these constructions

1414
01:13:34,950 --> 01:13:38,160
to mean, to show that something
will move at all if and only

1415
01:13:38,160 --> 01:13:40,440
if all these polynomials
have a solution.

1416
01:13:40,440 --> 01:13:43,520
And that turns out
to be really hard.

1417
01:13:43,520 --> 01:13:46,110
And it's something
called coNP-hard,

1418
01:13:46,110 --> 01:13:48,110
which is almost the
same as NP-hard.

1419
01:13:48,110 --> 01:13:51,170
it's just for rigidity,
it's really easy

1420
01:13:51,170 --> 01:13:53,530
to prove that something is
not rigid, you just move it.

1421
01:13:53,530 --> 01:13:56,020
It's a little harder to show
that it can't move in any way.

1422
01:13:56,020 --> 01:13:59,060
So that's for those who know
coNP, it's coNP, not NP.

1423
01:14:01,820 --> 01:14:02,570
Cool.

1424
01:14:02,570 --> 01:14:05,550
Those are various
generalizations.

1425
01:14:05,550 --> 01:14:09,290
Big open problem
here is what if you

1426
01:14:09,290 --> 01:14:11,639
forbid the edges from crossing.

1427
01:14:11,639 --> 01:14:13,180
This actually came
up in lecture one.

1428
01:14:13,180 --> 01:14:13,950
Someone asked it.

1429
01:14:13,950 --> 01:14:19,186
It was first posed that I know
from Don Shimamoto in 2004,

1430
01:14:19,186 --> 01:14:20,980
I was giving a talk.

1431
01:14:20,980 --> 01:14:22,880
And all these
constructions, especially

1432
01:14:22,880 --> 01:14:25,110
the contra-parallelogram,
need to have

1433
01:14:25,110 --> 01:14:27,247
edges that cross each other.

1434
01:14:27,247 --> 01:14:28,830
Next class will also
be about the case

1435
01:14:28,830 --> 01:14:30,210
where edges cross
each other but soon

1436
01:14:30,210 --> 01:14:31,506
we are going to
enter the case where

1437
01:14:31,506 --> 01:14:32,797
edges are not allowed to cross.

1438
01:14:32,797 --> 01:14:34,650
Can you do any of
this stuff, construct

1439
01:14:34,650 --> 01:14:36,500
any interesting polynomials?

1440
01:14:36,500 --> 01:14:38,820
Even constructing a straight
line, I think is open.

1441
01:14:38,820 --> 01:14:41,750
The peaucellier linkage
definitely doesn't work.

1442
01:14:41,750 --> 01:14:43,250
Maybe there are
some other linkages,

1443
01:14:43,250 --> 01:14:44,570
we'd have to check them.

1444
01:14:44,570 --> 01:14:47,190
But definitely Kempe's
universality result

1445
01:14:47,190 --> 01:14:51,040
is open for non-crossing edges.

1446
01:14:51,040 --> 01:14:52,814
I have some fun
projects like it'd

1447
01:14:52,814 --> 01:14:54,230
be fun to implement
this algorithm

1448
01:14:54,230 --> 01:14:56,277
and see this thing run.

1449
01:14:56,277 --> 01:14:58,360
You could build a sculpture
out of Kempe linkages.

1450
01:14:58,360 --> 01:15:00,170
They're a little
bit tedious I would

1451
01:15:00,170 --> 01:15:02,810
say to make interesting curves,
but there's some cool ones.

1452
01:15:02,810 --> 01:15:07,310
Like in the notes I have a
link to the letter C made out

1453
01:15:07,310 --> 01:15:08,300
of a linkage.

1454
01:15:08,300 --> 01:15:10,860
You could do a whole
alphabet, be kind of fun.

1455
01:15:10,860 --> 01:15:12,810
Have these things cranking
around and spelling

1456
01:15:12,810 --> 01:15:18,060
all sorts of, actually spelling
names would be the idea.

1457
01:15:18,060 --> 01:15:21,760
Another fun application that I
thought I would just throw in.

1458
01:15:21,760 --> 01:15:24,740
If you are, back to origami.

1459
01:15:24,740 --> 01:15:27,660
Origami can in some
sense simulate linkages.

1460
01:15:27,660 --> 01:15:29,431
And the motivation for
this is you've, you

1461
01:15:29,431 --> 01:15:31,930
know you've designed, you just
design all these awesome tree

1462
01:15:31,930 --> 01:15:35,500
maker diagrams for
your problem set.

1463
01:15:35,500 --> 01:15:37,660
And you realize, oh
man, there's points

1464
01:15:37,660 --> 01:15:39,730
all over the place on
this piece of paper.

1465
01:15:39,730 --> 01:15:41,711
How the heck am I
going to find, if I

1466
01:15:41,711 --> 01:15:43,460
had to make origami
diagrams, first I say,

1467
01:15:43,460 --> 01:15:45,800
oh you do this step, then
this step, then this step.

1468
01:15:45,800 --> 01:15:50,490
How am I going to tell someone
oh, just fold to root 3 over

1469
01:15:50,490 --> 01:15:54,780
7 comma root 5
plus root 7 over 4,

1470
01:15:54,780 --> 01:15:56,510
Which you could figure
out in some sense

1471
01:15:56,510 --> 01:15:59,420
from those geometries what
all those coordinates are.

1472
01:15:59,420 --> 01:16:03,210
How do you tell
someone to fold there?

1473
01:16:03,210 --> 01:16:06,540
Well, you could
use a polynomial.

1474
01:16:06,540 --> 01:16:10,350
Could say oh, you've
got to construct root 5.

1475
01:16:10,350 --> 01:16:11,540
Or what's a tough one?

1476
01:16:14,620 --> 01:16:20,440
Let's say you want to compute
the fifth root of 103.

1477
01:16:20,440 --> 01:16:21,420
Is that prime?

1478
01:16:21,420 --> 01:16:23,250
I think so.

1479
01:16:23,250 --> 01:16:25,660
So some nasty thing like this.

1480
01:16:25,660 --> 01:16:28,510
You say oh well, I
would really like

1481
01:16:28,510 --> 01:16:31,890
x to the fifth to equal 103.

1482
01:16:31,890 --> 01:16:34,330
If I could solve
this polynomial,

1483
01:16:34,330 --> 01:16:36,509
then I could
construct this number.

1484
01:16:36,509 --> 01:16:37,800
How do I solve that polynomial?

1485
01:16:37,800 --> 01:16:40,860
Well I'm all about solving
polynomials using linkages.

1486
01:16:40,860 --> 01:16:43,580
I could build a crazy
linkage device out

1487
01:16:43,580 --> 01:16:45,050
of my piece of paper.

1488
01:16:45,050 --> 01:16:47,030
Don't try this at home, please.

1489
01:16:47,030 --> 01:16:51,120
But in theory, you could come up
with these n really long bars,

1490
01:16:51,120 --> 01:16:54,260
fold them around, and
think of them as linkages.

1491
01:16:54,260 --> 01:16:58,740
And you tell people, oh well you
just need to align these dots,

1492
01:16:58,740 --> 01:17:02,990
you may make little pinches on
at the ends of these things,

1493
01:17:02,990 --> 01:17:05,330
those simulate the
edges of your linkage.

1494
01:17:05,330 --> 01:17:08,350
And then you just tell people
to move these creases around

1495
01:17:08,350 --> 01:17:11,640
until this point
lies on some line.

1496
01:17:11,640 --> 01:17:14,250
And then you've solved
your polynomial.

1497
01:17:14,250 --> 01:17:16,140
And this is a natural
generalization

1498
01:17:16,140 --> 01:17:18,790
in some sense to something
called origami axioms, which

1499
01:17:18,790 --> 01:17:20,770
we will probably talk
about at some point.

1500
01:17:20,770 --> 01:17:22,850
Usually you think about
doing one fold at a time.

1501
01:17:22,850 --> 01:17:24,985
You want to make one fold
so that this point maps

1502
01:17:24,985 --> 01:17:27,110
onto this line, and this
point maps onto this line.

1503
01:17:27,110 --> 01:17:29,110
That's the typical
origami axiom.

1504
01:17:29,110 --> 01:17:29,850
It's pretty good.

1505
01:17:29,850 --> 01:17:31,980
It can trisect angles
and it can do fun things.

1506
01:17:31,980 --> 01:17:34,692
But it can really only
solve degree four equations.

1507
01:17:34,692 --> 01:17:36,400
You want to solve a
degree five equation,

1508
01:17:36,400 --> 01:17:38,550
you need some higher
level techniques.

1509
01:17:38,550 --> 01:17:40,840
And one way to do that is
with Kempe's universality.

1510
01:17:40,840 --> 01:17:44,910
It's pretty impractical,
but at least in theory you

1511
01:17:44,910 --> 01:17:48,850
can do anything, you can
locate any point you want,

1512
01:17:48,850 --> 01:17:50,820
any algebraic point,
which are the things you'd

1513
01:17:50,820 --> 01:17:57,192
want from tree maker, using
this technique, little crazy.

1514
01:17:57,192 --> 01:17:58,650
There are other
ways to do it, too.

1515
01:17:58,650 --> 01:18:01,685
But this one way to do it,
as mentioned in the textbook.

1516
01:18:01,685 --> 01:18:02,185
Questions.

1517
01:18:05,760 --> 01:18:06,642
All right.

1518
01:18:06,642 --> 01:18:07,850
That's how to sign your name.

1519
01:18:07,850 --> 01:18:11,670
Next time we'll talk about
rigidity and other fun linkage

1520
01:18:11,670 --> 01:18:13,220
problems.