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PROFESSOR: All
right before we get

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started, I just
wanted to say goodbye

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to Eric Joisel, who
died on Sunday, sadly.

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He's a master origami creator,
and I encourage you all

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to come see "Between
the Folds" on Tuesday,

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showing Tuesday
evening, because there's

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a great interview with him, and
it really captures his spirit.

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And sadly, I never
met him in person.

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The last I corresponded with
him was to get permission

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to show these photos in class.

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So that sucks.

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Today is about rigidity again.

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And in particular, something
called infinitesimal rigidity.

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So last class was mostly
about generic rigidity.

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Today we're going to
talk about a variation,

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although related to
generic rigidity.

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And this is going to
be a very useful tool.

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It's a lot easier to work
with algorithmically.

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It still captures
essentially generic rigidity,

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but in a different
way-- in a linear way.

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And it's going to
be super useful.

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So the idea is we think about
rigidity-- does a linkage

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move or not-- to
the first order.

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First order means
first derivative.

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OK, so I'm going to do some
analysis stuff-- calculus,

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which you should all know,
because is the General

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Institute Requirement.

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Don't worry, you don't
need to know much.

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So in particular
rigidity, is really

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about the lack of motion.

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So let's define the idea
of infinitesimal motion,

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or first-order motion.

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This is, again, suppose
we start somewhere.

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And somewhere would be
a linkage configuration.

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There's a linkage.

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There's a graph,
with edge lengths

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plus configuration maps
every vertex to a point.

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I'm going to assume we're
working in the plane for now.

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But really this works in--
I'll say d dimensions.

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So this is what
I want to define.

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I'm going to say that
it is a valid-- this

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is sort of an informal notion--
a valid first derivative

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of a motion.

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We've already defined motions.

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I want to take such a
motion, essentially,

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take its derivative
with respect to time,

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And evaluate that
derivative at time 0.

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This is a fancy way of saying
the following picture-- I'd

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like a motion, which is some
path in the configuration

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space-- remember,
configuration space

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is some hugely
dimensional object,

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d n dimensions-- and
a point in that space

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corresponds to a configuration.

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We have a motion is
some path in that space.

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And motions are
hard to think about.

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They can be really complicated.

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So I just want to say, well,
how do you get started moving?

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What direction do you
go in in a linear sense,

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to the first order?

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What's the derivative?

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What's the tangent
right at the beginning?

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And then this
notion will, stated

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in terms of motions--
you can think

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about this in the
absence of a motion.

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Say, well, yeah, I don't really
know whether this curve exists,

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but let's just see, does this
initial starting direction

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exist?

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And that's actually a little
weaker than the whole motion

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existing.

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If the motion exists, surely
you can take this derivative

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and evaluate it.

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It's not totally
obvious, but it's true.

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You need some smoothness.

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But if there's a way
to get started moving,

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that doesn't actually mean
you could actually move.

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Why don't I give a more formal
definition, and a more useful

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one-- but that's the intuition.

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So another way to think
of an infinitesimal motion

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is, instead of saying how each
vertex moves in d dimensions,

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you just give a velocity
vector for each vertex.

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For every vertex v, we're going
to get some velocity vector

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d of v-- that's direction
or derivative of v.

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And with motions,
we were supposed

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to preserve edge lengths.

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So here we'd like to
preserve edge lengths,

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but that's actually
kind of hard to do.

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If you take an edge--
so I have two vertices,

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and I assign some velocity
vectors, something like this--

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as soon as you move a little
bit in that direction,

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the edge length won't
be quite the same.

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But it'll be the same
to the first order,

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so this is going to
be a little subtle.

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And in the book, we
take the derivatives.

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We work out all the details.

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But I'll just tell you
what the condition is,

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because ultimately all we
need is this condition.

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I'm just defining an
infinitesimal motion

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to be a velocity vector
for every vertex,

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such that this property holds.

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This is a dot product
between two vectors.

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If you have two vectors, a and
b, you take their dot product,

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this is just
something like this.

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You take the x-coordinates.

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You multiply them together.

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You take the y-coordinates.

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You multiply them
together, and so on.

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Add them up.

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And we're going to need
some linear algebra.

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That's one useful
linearity thing.

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So these are where
the configuration

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places vertex v and vertex w.

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So this should be true for
each edge v w on the graph.

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This is a vector C
of v minus C of w.

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Let me give some labels.

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Let's say this is
w, and this is v.

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Let's say these are the points
that the configuration places

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w and v at.

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Then C of v minus C
of w is this vector.

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It's a vector.

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It doesn't really
have any space.

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It's just been
translated around.

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It's a vector that
would point from w

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to v in the configuration.

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That's the left part.

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The right part has no
real intuitive meaning,

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but it involves this thing--
some velocity vector for w--

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and the velocity
vector for v. So this

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is some relation on this
vector and these two vectors.

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It's actually a very
intuitive relation.

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Let me tell you what it means.

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Basically, we want
to understand how

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this length changes
to the first order.

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So this guy's moving to first
order in this direction.

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In fact, it's moving
on some curve,

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and the initial tangent
is that direction.

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And to understand how that
motion changes this edge

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length, turns out
the right thing

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to do is to project this
vector onto this one.

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Makes sense.

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So as this guy moves
in that direction,

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This is the first-order change.

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This distance here is
the first-order change

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of this length as caused
by this vertex moving.

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OK, that projected length
is actually very simple.

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This is again, some
linear algebra stuff

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interpreted geometrically.

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This is a dot
product in d of w--

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that vector-- dot product
with C of v minus C of w.

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It turns out dot product
corresponds to projection.

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So I'm taking this vector.

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I'm projecting onto this one.

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That's how you'd write
it algebraically.

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And same deal over here.

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If I project this
vector onto this one,

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that's going to be the
first-order change to this edge

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length caused by this vertex
moving in that direction.

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So this is d of v dot
C of v, minus C of w.

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So if I want this
edge length not

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to change at all
to the first order,

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then this projected length
should equal this projected

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length.

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And that's why I
drew it this way.

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So it turns out this is a valid
motion, because that projected

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length equals that one.

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And in general, we want this
thing to equal that thing.

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And if you rearrange
terms, that's

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the same thing as this property.

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I just take this one minus this
one, and set it equal to zero,

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and you get that.

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OK, so this is some intuition.

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Maybe it makes sense, maybe not.

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In the end, this is
all I care about.

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And it's useful to
realize that it's

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a first derivative of a motion,
because what that implies

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is that if there's
a motion-- a real,

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honest to goodness
motion-- then there's

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an infinitesimal motion.

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I'm going to abbreviate
that "inf motion."

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Infinitesimal is pretty long.

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Infinitesimal means
smaller-- very tiny-- smaller

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than anything, I think
is the literal meaning.

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So this is useful to us,
because the contrapositive

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says if a linkage is
infinitesimally rigid-- which

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just means if there is no
infinitesimal motion-- then

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it's rigid.

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So these are the
same statements.

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This you prove just
by taking a motion

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and taking its derivative.

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This is identical to that.

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So this is useful
because if we can ever

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find something and show that
it's infinitesimally rigid,

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then we've determined
that it's rigid.

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Question?

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STUDENT: [INAUDIBLE]
dot product [INAUDIBLE]

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at a left right turn is
0, or that the terms are

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at 90 degrees to each other?

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PROFESSOR: Yeah, this
dot product basically

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means these two vectors are
at 90 degrees to each other.

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That's another
geometric meaning.

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I don't have a
geometric picture.

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Maybe there is one,
but I have trouble

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thinking about d
of v minus d of w.

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That's not an intuitive thing
to me, but it's something.

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And we're saying that
that vector should

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be perpendicular to
C of v minus C of w.

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However you want
to think about, I

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like to think about it this
way, but they're equivalent.

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All right, so how do we
tell whether something's

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infinitesimally rigid?

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Because somehow
that's useful to us.

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00:12:02,290 --> 00:12:04,990
Well, there's a
saying that everything

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is linear to the first
order-- sort of a tautology.

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And all you need to observe
is that this constraint

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is linear in what we don't know.

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So I want to know is there
some set of velocity vectors

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d of v, and d of w,
for all the vertices,

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so that these constraints hold.

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But I already know C of
v, and I know C of w.

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00:12:26,700 --> 00:12:30,050
This is just where are
these vertices currently.

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And I want to know can
I get started moving?

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So these are my
known quantities,

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and these are my
unknown quantities.

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This is just some linear thing.

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And the dot product-- I mean,
this is all just constant.

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I know what they are.

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So this is a linear equation.

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The fancy way to write this
is as a matrix equation,

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if you want.

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So this is the idea
of a rigidity matrix.

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So we have a whole bunch
of these constraints,

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one for every edge.

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And whenever you have a
bunch of linear constraints,

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you can write them as
R-- some big matrix--

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00:13:07,860 --> 00:13:17,530
times the velocity, or
call them d's equals 0.

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00:13:20,880 --> 00:13:22,530
So I can absorb
this into a matrix.

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If you're used to that, great.

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00:13:24,540 --> 00:13:28,100
Otherwise, this is
what it looks like.

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00:13:28,100 --> 00:13:32,260
My matrix R is going
to have a row that

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has a whole bunch of zeroes.

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And then at some point, it
has basically various versions

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of this vector.

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00:13:41,910 --> 00:13:47,030
But because this
is one equation,

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but in two dimensions, there's
two parts to all these vectors.

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00:14:09,690 --> 00:14:13,475
Two dimensions that'll
look something like this--

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00:14:13,475 --> 00:14:14,600
why did I change the order?

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00:14:19,079 --> 00:14:19,870
I think it's right.

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OK, this is not very amazing.

248
00:14:38,580 --> 00:14:42,630
It's just there are going
to be four terms in two

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00:14:42,630 --> 00:14:45,380
dimensions in this
equation, because we've

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00:14:45,380 --> 00:14:50,006
got d of v times this vector.

251
00:14:50,006 --> 00:14:51,380
So there's a
x-coordinate vector,

252
00:14:51,380 --> 00:14:53,421
and the y-coordinate
vector, and in d dimensions,

253
00:14:53,421 --> 00:14:54,510
there will be d of them.

254
00:14:54,510 --> 00:14:57,020
Here in two dimensions, I've got
the x-coordinate of the vector,

255
00:14:57,020 --> 00:14:58,478
and the y-coordinate
of the vector.

256
00:14:58,478 --> 00:15:05,020
So this will end up getting
multiplied on the right hand

257
00:15:05,020 --> 00:15:11,059
side by d of v.
So the two v rows,

258
00:15:11,059 --> 00:15:13,350
and this part will end up
getting multiplied by d of w.

259
00:15:16,962 --> 00:15:18,420
yes, so these guys
switched around.

260
00:15:18,420 --> 00:15:22,060
It's now w minus v in order
to implement that minus sign.

261
00:15:24,960 --> 00:15:28,360
So whatever-- the point
is you get a matrix.

262
00:15:28,360 --> 00:15:32,100
And what we care about, what
we're trying to solve for,

263
00:15:32,100 --> 00:15:32,980
is d.

264
00:15:32,980 --> 00:15:35,150
We know this entire
matrix, because we know c.

265
00:15:35,150 --> 00:15:36,920
We know the configuration.

266
00:15:36,920 --> 00:15:39,910
We just want to know is there
some set of velocity vectors

267
00:15:39,910 --> 00:15:43,720
such that the rigidity matrix
times this is equal to zero?

268
00:15:43,720 --> 00:15:46,180
This is called the set
of d's i for which this

269
00:15:46,180 --> 00:15:51,280
is possible is called
the-- anyone know?

270
00:15:51,280 --> 00:15:53,560
Good old linear
algebra-- it's called

271
00:15:53,560 --> 00:15:57,740
the null space of
this rigidity--

272
00:15:57,740 --> 00:16:02,505
null referring to
that 0, or the kernel.

273
00:16:06,020 --> 00:16:14,840
So the set of all
infinitesimal motions

274
00:16:14,840 --> 00:16:20,199
is the null space of
that matrix, also known

275
00:16:20,199 --> 00:16:21,365
as the kernel of the matrix.

276
00:16:24,310 --> 00:16:25,680
And so what?

277
00:16:25,680 --> 00:16:29,590
Well this lets us use a bunch
of theorems in linear algebra.

278
00:16:29,590 --> 00:16:37,422
So one fun fact is
that the dimension

279
00:16:37,422 --> 00:16:42,444
of this space-- the space of
all infinitesimal motions--

280
00:16:42,444 --> 00:16:43,860
seems like a useful
thing to know.

281
00:16:43,860 --> 00:16:45,700
We want to know how
many-- this is basically

282
00:16:45,700 --> 00:16:48,860
the number of degrees of freedom
in infinitesimal land, which

283
00:16:48,860 --> 00:16:51,850
is not quite reality,
but it's close.

284
00:16:51,850 --> 00:16:54,305
This is called the
nullity of the matrix.

285
00:16:57,140 --> 00:17:00,060
That's just the
definition, but there's

286
00:17:00,060 --> 00:17:04,480
a fun theorem called
the rank nullity

287
00:17:04,480 --> 00:17:07,819
theorem from linear algebra.

288
00:17:18,750 --> 00:17:21,460
Which says if you
have a matrix, you

289
00:17:21,460 --> 00:17:24,740
take its rank-- which is
another quantity-- you take

290
00:17:24,740 --> 00:17:27,890
its nullity, and you
add them together,

291
00:17:27,890 --> 00:17:32,050
it will be the number of
columns in your matrix.

292
00:17:32,050 --> 00:17:41,600
Now in our case, the number
of columns is d times n.

293
00:17:41,600 --> 00:17:45,760
We have n vertices down
here, and each of them

294
00:17:45,760 --> 00:17:48,140
has d coordinates.

295
00:17:48,140 --> 00:17:49,770
So for us, this is d times n.

296
00:17:53,780 --> 00:17:56,990
So this is like a
conserved quantity.

297
00:17:56,990 --> 00:17:58,910
What we care about
is the nullity.

298
00:17:58,910 --> 00:18:04,320
This is the space of motions--
infinitesimal motions.

299
00:18:04,320 --> 00:18:07,690
And so consequently, we
care about this thing

300
00:18:07,690 --> 00:18:08,460
called the rank.

301
00:18:08,460 --> 00:18:10,680
And if you've ever seen
linear algebra, especially

302
00:18:10,680 --> 00:18:15,210
linear algebra algorithms,
the bread and butter business

303
00:18:15,210 --> 00:18:17,000
is computing the
rank of a matrix.

304
00:18:17,000 --> 00:18:19,405
This can be done
in polynomial time.

305
00:18:19,405 --> 00:18:20,280
So this is good news.

306
00:18:20,280 --> 00:18:22,450
And intuitively what
it corresponds to

307
00:18:22,450 --> 00:18:24,370
is the number of useful edges.

308
00:18:29,410 --> 00:18:32,180
And this thing is the number
of degrees of freedom.

309
00:18:39,760 --> 00:18:43,284
So it turns out these will
always be equal to d times n

310
00:18:43,284 --> 00:18:43,825
in summation.

311
00:18:46,650 --> 00:18:56,010
And what we care about,
infinitesimal rigidity,

312
00:18:56,010 --> 00:18:59,780
means that-- or let's say
it's infinitesimally rigid if

313
00:18:59,780 --> 00:19:07,150
and only if the
nullity of the matrix

314
00:19:07,150 --> 00:19:12,250
is the space of rigid motions,
which I have to look up.

315
00:19:18,070 --> 00:19:22,410
In general, in d dimensions,
it's this-- d plus 1 choose 2.

316
00:19:22,410 --> 00:19:24,870
In two dimensions, there's
three rigid motions.

317
00:19:24,870 --> 00:19:27,050
You've got the two
translations and the rotation.

318
00:19:27,050 --> 00:19:28,670
Those we're not getting rid of.

319
00:19:28,670 --> 00:19:31,240
They still exist in
infinitesimal land.

320
00:19:31,240 --> 00:19:33,950
So in two dimensions,
ideally your nullity

321
00:19:33,950 --> 00:19:38,660
is-- if you want rigidity,
your nullity should be 3.

322
00:19:38,660 --> 00:19:45,140
And what this is telling us
is then the rank of the matrix

323
00:19:45,140 --> 00:19:47,670
should be d n minus that.

324
00:19:51,450 --> 00:19:53,820
In two dimensions,
that is 2n minus 3.

325
00:19:53,820 --> 00:19:55,570
So this is telling us
something we already

326
00:19:55,570 --> 00:19:59,060
knew, but essentially for
free, which is kind of fun,

327
00:19:59,060 --> 00:20:00,560
if you know all
this linear algebra.

328
00:20:00,560 --> 00:20:03,260
Which admittedly, even if you've
learned linear algebra, that

329
00:20:03,260 --> 00:20:05,920
may not be the first thing you
care about in that context,

330
00:20:05,920 --> 00:20:10,180
but for rigidity, this
is a really useful way

331
00:20:10,180 --> 00:20:11,730
of thinking about things.

332
00:20:11,730 --> 00:20:13,954
You've got all these
columns-- d times n of them.

333
00:20:13,954 --> 00:20:16,370
You want to obliterate all the
degrees of freedom you can,

334
00:20:16,370 --> 00:20:19,650
and leave just three of
them in two dimensions--

335
00:20:19,650 --> 00:20:22,140
d times d plus 1 over
2 in d dimensions,

336
00:20:22,140 --> 00:20:24,320
and you do that by
adding useful edges.

337
00:20:24,320 --> 00:20:26,600
And useful edges here means
that the rank is high,

338
00:20:26,600 --> 00:20:29,940
which essentially means none
of the rows in this matrix

339
00:20:29,940 --> 00:20:32,030
cancel out with each other.

340
00:20:32,030 --> 00:20:34,240
So it's another way of
thinking about essentially

341
00:20:34,240 --> 00:20:35,530
everything we've seen.

342
00:20:35,530 --> 00:20:37,363
And if you know linear
algebra, it's useful.

343
00:20:37,363 --> 00:20:40,120
Otherwise, I'll tell
you how we can use it.

344
00:20:45,840 --> 00:20:50,390
OK, one mathematical
thing we can do

345
00:20:50,390 --> 00:20:52,980
is give another definition
of a generic point set.

346
00:20:56,120 --> 00:20:58,300
Last class we talked
about one definition

347
00:20:58,300 --> 00:21:00,400
of generic, which
was that you forbid

348
00:21:00,400 --> 00:21:07,630
all non-trivial polynomial or
algebraic rational equations

349
00:21:07,630 --> 00:21:12,180
on your points, which is very
messy and infinite and hard

350
00:21:12,180 --> 00:21:13,230
to think about.

351
00:21:13,230 --> 00:21:15,500
This definition will
be easy to think about

352
00:21:15,500 --> 00:21:17,520
if you're used to
linear algebra.

353
00:21:17,520 --> 00:21:20,080
So for some, it's
an improvement.

354
00:21:20,080 --> 00:21:23,950
For others, it might be
the reverse-- whatever.

355
00:21:32,260 --> 00:21:34,700
But at least this definition
is finite, which I like.

356
00:22:27,025 --> 00:22:29,275
OK, for this definition we
need the notion of a minor.

357
00:22:32,280 --> 00:22:34,130
So we have some matrix.

358
00:22:34,130 --> 00:22:37,650
Actually, our matrices
are pretty rectangular.

359
00:22:37,650 --> 00:22:41,870
And I just choose, let's say
k columns from the matrix--

360
00:22:41,870 --> 00:22:47,990
let's say 3, and
I choose k rows--

361
00:22:47,990 --> 00:22:49,960
the same number of
rows and columns.

362
00:22:49,960 --> 00:22:53,490
I look at those elements
in the intersection.

363
00:22:53,490 --> 00:22:54,605
That's a square matrix.

364
00:22:57,140 --> 00:23:01,150
That's a minor-- that
3 by 3 square submatrix

365
00:23:01,150 --> 00:23:02,620
is called a minor.

366
00:23:02,620 --> 00:23:04,274
So there's a whole
bunch of minors,

367
00:23:04,274 --> 00:23:05,440
but it's only finitely many.

368
00:23:05,440 --> 00:23:07,140
It's exponential.

369
00:23:07,140 --> 00:23:09,920
It's not easy to check.

370
00:23:09,920 --> 00:23:13,280
You take all the minors
of your rigidity matrix.

371
00:23:13,280 --> 00:23:15,030
And here I'm imagining
the complete graph,

372
00:23:15,030 --> 00:23:16,350
so there's tons of edges.

373
00:23:16,350 --> 00:23:19,390
There's n choose two edges.

374
00:23:19,390 --> 00:23:20,900
So this'll be
independent linkage.

375
00:23:20,900 --> 00:23:22,566
You could do it
specific to the linkage,

376
00:23:22,566 --> 00:23:25,300
but it's better to do it
for the complete graph.

377
00:23:25,300 --> 00:23:27,915
Then we take the
determinant of that matrix,

378
00:23:27,915 --> 00:23:28,790
which I won't define.

379
00:23:28,790 --> 00:23:31,800
It's a number associated
with that matrix.

380
00:23:31,800 --> 00:23:33,960
And if it's non-zero--
now remember,

381
00:23:33,960 --> 00:23:36,300
rigidity matrix is
defined as a function

382
00:23:36,300 --> 00:23:39,570
of C. It depends where
you put the vertices.

383
00:23:39,570 --> 00:23:43,070
So what I'm saying is if it's
non-zero for some choice of C,

384
00:23:43,070 --> 00:23:46,360
then it should be non-zero
for this choice of C.

385
00:23:46,360 --> 00:23:48,350
And if you find
such a configuration

386
00:23:48,350 --> 00:23:51,440
C-- or I guess we call
this a realization C--

387
00:23:51,440 --> 00:23:53,470
then that C is generic.

388
00:23:53,470 --> 00:23:55,370
So if I gave you a
specific configuration,

389
00:23:55,370 --> 00:23:57,535
you could, in principle,
in finite time-- it

390
00:23:57,535 --> 00:24:00,270
won't be efficient-- you could
check all of these things.

391
00:24:00,270 --> 00:24:02,940
And say, oh, I compute
out, I work out

392
00:24:02,940 --> 00:24:06,980
what the determinant of that
submatrix is, that minor.

393
00:24:06,980 --> 00:24:08,470
It's some algebraic function.

394
00:24:08,470 --> 00:24:10,060
If it doesn't
cancel out entirely,

395
00:24:10,060 --> 00:24:13,190
if it's not zero trivially,
it's not zero always,

396
00:24:13,190 --> 00:24:16,160
then I just evaluate it
at this configuration.

397
00:24:16,160 --> 00:24:18,720
And if it's non-zero
there, good.

398
00:24:18,720 --> 00:24:21,230
That's generic.

399
00:24:21,230 --> 00:24:25,920
So it's not useful
in a computable way,

400
00:24:25,920 --> 00:24:27,171
but it's a definition.

401
00:24:27,171 --> 00:24:28,170
It's a little bit nicer.

402
00:24:32,970 --> 00:24:34,600
And now there's a
bunch of fun facts

403
00:24:34,600 --> 00:24:44,740
about infinitesimal
motions and rigidity,

404
00:24:44,740 --> 00:24:46,570
and the relation to
the generic case.

405
00:24:54,610 --> 00:24:59,649
So there as we have
before, this definition

406
00:24:59,649 --> 00:25:01,440
is not quite the same,
but it's effectively

407
00:25:01,440 --> 00:25:03,000
the same as the last one.

408
00:25:03,000 --> 00:25:06,955
Almost every
configuration is generic.

409
00:25:10,160 --> 00:25:15,200
So maybe I should say
realization-- whatever.

410
00:25:15,200 --> 00:25:21,460
I take some point set--
I take some vertex set.

411
00:25:21,460 --> 00:25:24,310
I assign random
points to each vertex.

412
00:25:24,310 --> 00:25:26,422
Then with probability
1, it will be generic

413
00:25:26,422 --> 00:25:27,630
according to this definition.

414
00:25:27,630 --> 00:25:30,280
That's really obvious if
you know linear algebra.

415
00:25:30,280 --> 00:25:32,860
That's a good thing.

416
00:25:32,860 --> 00:25:35,880
But here's what's
really cool-- if we

417
00:25:35,880 --> 00:25:38,610
take any of these
generic configurations--

418
00:25:38,610 --> 00:25:48,010
so just fix one--
rigidity is the same thing

419
00:25:48,010 --> 00:25:54,050
as infinitesimal rigidity,
is the same thing

420
00:25:54,050 --> 00:25:56,130
as generic rigidity
from last class.

421
00:26:00,620 --> 00:26:04,320
Wow, this is cool,
because if I care

422
00:26:04,320 --> 00:26:07,210
about any one of
these three things,

423
00:26:07,210 --> 00:26:10,270
I just have to pick my
favorite generic point, which

424
00:26:10,270 --> 00:26:11,120
I can do randomly.

425
00:26:11,120 --> 00:26:13,530
I can just take a
random point, and if I

426
00:26:13,530 --> 00:26:15,440
can evaluate any one
of these three things,

427
00:26:15,440 --> 00:26:17,940
I learn about all the others.

428
00:26:17,940 --> 00:26:22,950
So if you want to tell whether
a 3D graph is generically rigid,

429
00:26:22,950 --> 00:26:26,560
you take a random
realization in 3D.

430
00:26:26,560 --> 00:26:28,920
You compute the rank of
this matrix generalized

431
00:26:28,920 --> 00:26:30,880
to the 3D case.

432
00:26:30,880 --> 00:26:35,350
You see is it 3 n minus
6, and that will tell you

433
00:26:35,350 --> 00:26:40,390
whether that generic realization
is infinitesimally rigid.

434
00:26:40,390 --> 00:26:45,100
And if you chose it right, which
you will do at probability 1,

435
00:26:45,100 --> 00:26:47,850
then that will tell you whether
the graph is generically rigid.

436
00:26:47,850 --> 00:26:51,020
So when I said last class
generic rigidity in 3D is hard,

437
00:26:51,020 --> 00:26:53,890
it's hard in the sense we don't
have a deterministic algorithm,

438
00:26:53,890 --> 00:26:55,840
but randomized-- super easy.

439
00:26:55,840 --> 00:26:57,735
Choose a random
configuration, check

440
00:26:57,735 --> 00:26:59,360
the rank of the
rigidity matrix-- done.

441
00:27:01,870 --> 00:27:03,580
So that's kind of cool.

442
00:27:03,580 --> 00:27:06,310
And that's why this
infinitesimal rigidity stuff

443
00:27:06,310 --> 00:27:10,320
is so important, because
it's so easy algorithmically.

444
00:27:10,320 --> 00:27:12,980
Things become a lot easier.

445
00:27:12,980 --> 00:27:16,600
Let me draw you one
example just to show

446
00:27:16,600 --> 00:27:18,390
what happens at
non-generic points.

447
00:27:18,390 --> 00:27:20,410
I've actually drawn
this example before.

448
00:27:24,120 --> 00:27:25,370
So here's a non-generic point.

449
00:27:25,370 --> 00:27:27,164
These three points
are collinear.

450
00:27:27,164 --> 00:27:28,830
And you can work out
there is some minor

451
00:27:28,830 --> 00:27:31,479
that is 0, but shouldn't be.

452
00:27:31,479 --> 00:27:34,020
And this thing is rigid, because
we have rigid triangle here.

453
00:27:34,020 --> 00:27:36,130
We have a rigid triangle here.

454
00:27:36,130 --> 00:27:38,730
So these two points are
pinned relative to each other.

455
00:27:38,730 --> 00:27:41,330
And then this thing is
taut, so it can't move.

456
00:27:41,330 --> 00:27:44,200
But to the first
order, it can move.

457
00:27:44,200 --> 00:27:48,330
And it's allowed to move
straight up or down--

458
00:27:48,330 --> 00:27:51,049
anything perpendicular
to this segment.

459
00:27:51,049 --> 00:27:51,840
That's always true.

460
00:27:51,840 --> 00:27:56,570
If you take a segment, and
you move one vertex but not

461
00:27:56,570 --> 00:27:58,330
the other perpendicular
to the segment,

462
00:27:58,330 --> 00:28:01,141
that's a valid
infinitesimal motion.

463
00:28:01,141 --> 00:28:01,640
Why?

464
00:28:01,640 --> 00:28:05,180
Because if I project this
onto the segment, I get 0.

465
00:28:05,180 --> 00:28:08,170
So this length is not
changing to the first order.

466
00:28:08,170 --> 00:28:09,710
And this is really necessary.

467
00:28:09,710 --> 00:28:12,292
To the first order, you can't
really distinguish this guy

468
00:28:12,292 --> 00:28:14,000
actually moving up,
and therefore getting

469
00:28:14,000 --> 00:28:21,020
slightly longer versus this guy
moving along a circle centered

470
00:28:21,020 --> 00:28:23,080
at that point, which
is a valid motion.

471
00:28:23,080 --> 00:28:24,900
To the first order,
they look the same,

472
00:28:24,900 --> 00:28:28,940
because you're going straight
up or down in first derivative.

473
00:28:28,940 --> 00:28:31,780
OK, here, to the
first order, you

474
00:28:31,780 --> 00:28:33,489
can't realize the
fact that if you're

475
00:28:33,489 --> 00:28:35,030
going to go on this
circle and you're

476
00:28:35,030 --> 00:28:37,880
going to go on this circle,
you get a contradiction.

477
00:28:37,880 --> 00:28:39,420
But it's pretty accurate.

478
00:28:39,420 --> 00:28:42,240
And if you perturb this
example, infinitesimal rigidity

479
00:28:42,240 --> 00:28:43,700
will give you the right answer.

480
00:28:43,700 --> 00:28:46,250
This is an example where
infinitesimal rigidity is

481
00:28:46,250 --> 00:28:52,379
a little bit weaker than--
no, infinitesimal flexibility

482
00:28:52,379 --> 00:28:54,795
having infinitesimal motion
is a little weaker than having

483
00:28:54,795 --> 00:28:59,171
a motion, but the reverse holds.

484
00:28:59,171 --> 00:29:01,670
If there's a motion, there's
always an infinitesimal motion.

485
00:29:01,670 --> 00:29:04,300
If you're infinitesimally
rigid, you're always rigid.

486
00:29:04,300 --> 00:29:08,170
That example is rigid, but
not infinitesimally rigid,

487
00:29:08,170 --> 00:29:11,440
so it's sort of strict.

488
00:29:11,440 --> 00:29:13,754
OK so far?

489
00:29:13,754 --> 00:29:15,670
That's the more technical
part of the lecture.

490
00:29:18,200 --> 00:29:19,090
Question?

491
00:29:19,090 --> 00:29:21,970
STUDENT: Will [INAUDIBLE]
be square matricies?

492
00:29:21,970 --> 00:29:24,890
PROFESSOR: The rigidity matrices
are almost never square.

493
00:29:24,890 --> 00:29:26,334
Good question.

494
00:29:26,334 --> 00:29:27,750
The minors are
going to be square,

495
00:29:27,750 --> 00:29:29,380
because we told them to be.

496
00:29:29,380 --> 00:29:33,700
But I think typically,
in the ideal setting,

497
00:29:33,700 --> 00:29:35,500
we're going to
have d n columns--

498
00:29:35,500 --> 00:29:39,990
so let's say 2 times n
columns in two dimensions.

499
00:29:39,990 --> 00:29:44,060
And we're supposed to
have 2n minus 3 rows.

500
00:29:44,060 --> 00:29:46,120
So it's close, but
in general, they

501
00:29:46,120 --> 00:29:49,070
should differ by
about this much.

502
00:29:49,070 --> 00:29:51,970
So they're almost
square, actually.

503
00:29:51,970 --> 00:29:53,750
Glad you asked.

504
00:29:53,750 --> 00:29:55,125
STUDENT: You can
add [INAUDIBLE].

505
00:29:57,599 --> 00:29:59,390
PROFESSOR: You could
always add extra edges

506
00:29:59,390 --> 00:30:00,456
and make it square, yeah.

507
00:30:00,456 --> 00:30:02,080
STUDENT: [INAUDIBLE].

508
00:30:02,080 --> 00:30:04,190
PROFESSOR: Well, no.

509
00:30:04,190 --> 00:30:06,630
So that's another good question.

510
00:30:06,630 --> 00:30:09,360
What happens if I add
more edges than this many?

511
00:30:09,360 --> 00:30:11,470
The rank just won't go up.

512
00:30:11,470 --> 00:30:14,970
Any edges you add will
add redundant constraints,

513
00:30:14,970 --> 00:30:18,542
and so the rank can never
go above this value.

514
00:30:18,542 --> 00:30:20,500
You need at least this
many edges to get there,

515
00:30:20,500 --> 00:30:22,650
and we know we can get
there with that many edges,

516
00:30:22,650 --> 00:30:26,190
but only in the
minimal rigid case.

517
00:30:26,190 --> 00:30:27,350
You could add more edges.

518
00:30:27,350 --> 00:30:30,410
You could make it a
super-tall matrix if you want,

519
00:30:30,410 --> 00:30:34,110
but you're not going
to increase the rank.

520
00:30:34,110 --> 00:30:39,010
All right, let's use this
for something more fun.

521
00:30:39,010 --> 00:30:46,250
So maybe go over here.

522
00:30:52,569 --> 00:30:54,318
I don't need the
rigidity matrix any more.

523
00:31:12,900 --> 00:31:19,500
Tensegrity-- so this
is in some sense

524
00:31:19,500 --> 00:31:24,620
an extension of rigidity
theory to a whole other theory

525
00:31:24,620 --> 00:31:25,870
called tensegrity theory.

526
00:31:28,390 --> 00:31:32,210
The word tensegrity comes from
tensional integrity, which

527
00:31:32,210 --> 00:31:35,226
is a very Buckminster Fuller
term, and he invented it.

528
00:31:35,226 --> 00:31:36,850
I don't think he
invented tensegrities,

529
00:31:36,850 --> 00:31:40,330
but he invented the word.

530
00:31:40,330 --> 00:31:45,540
And they are things
like this-- here's

531
00:31:45,540 --> 00:31:49,760
a tensegrity built and
given to me by Bob Connolly.

532
00:31:49,760 --> 00:31:52,570
And it has well,
I guess, there are

533
00:31:52,570 --> 00:31:55,550
three kinds of edges physically.

534
00:31:55,550 --> 00:31:57,380
There are the springs.

535
00:31:57,380 --> 00:32:03,650
There are these struts, and
there are these wire cables.

536
00:32:03,650 --> 00:32:06,070
And its rigid.

537
00:32:06,070 --> 00:32:08,070
It's a little flexible,
just because the springs

538
00:32:08,070 --> 00:32:12,090
aren't perfectly stiff,
like all springs.

539
00:32:12,090 --> 00:32:13,770
But there are three
kinds of edges here,

540
00:32:13,770 --> 00:32:15,853
and we're going to model
the three kinds of edges.

541
00:32:15,853 --> 00:32:17,640
The edges that
we've modeled so far

542
00:32:17,640 --> 00:32:20,350
are most closely related
to these springs,

543
00:32:20,350 --> 00:32:22,820
so that's not quite the
right physical analogy.

544
00:32:22,820 --> 00:32:26,480
What I would call the
edges and linkages is bars.

545
00:32:26,480 --> 00:32:30,020
Bars are, like,
solid attachments,

546
00:32:30,020 --> 00:32:32,070
like steel rods
that are permanently

547
00:32:32,070 --> 00:32:34,180
attached to the two endpoints.

548
00:32:34,180 --> 00:32:37,140
The lengths of these
edges should never change.

549
00:32:37,140 --> 00:32:40,060
That's been the world
we live in, at least

550
00:32:40,060 --> 00:32:41,630
to the first order, whatever.

551
00:32:41,630 --> 00:32:44,100
Edge lengths of bars
should never change.

552
00:32:44,100 --> 00:32:45,470
Now we have these cables.

553
00:32:45,470 --> 00:32:48,710
Now cables really
can get shorter.

554
00:32:48,710 --> 00:32:51,190
This cable is perfectly
happy to get shorter.

555
00:32:51,190 --> 00:32:54,195
I can pull those two
vertices together.

556
00:32:54,195 --> 00:32:55,820
But what it prevents
is getting longer.

557
00:32:55,820 --> 00:32:57,930
If I pull-- you
can't tell, but I'm

558
00:32:57,930 --> 00:33:00,740
pulling really hard on
these vertices apart.

559
00:33:00,740 --> 00:33:03,860
Nothing happens, because this
cable can't get any longer.

560
00:33:03,860 --> 00:33:06,850
So cables prevent
extension, but they do not

561
00:33:06,850 --> 00:33:08,310
prevent compression.

562
00:33:08,310 --> 00:33:10,942
Struts, on the other hand--
they prevent compression.

563
00:33:10,942 --> 00:33:13,400
If I push on these two vertices,
they don't get any closer.

564
00:33:13,400 --> 00:33:15,900
If I pull, it comes right out.

565
00:33:15,900 --> 00:33:19,910
And you have to realize
that these are eye hooks,

566
00:33:19,910 --> 00:33:22,010
and so really, those
could come apart.

567
00:33:22,010 --> 00:33:25,160
All these struts are
preventing is compression.

568
00:33:25,160 --> 00:33:27,010
They don't prevent expansion.

569
00:33:27,010 --> 00:33:30,150
So that's the thing
we want to model.

570
00:33:32,920 --> 00:33:34,800
This is a tensegrity.

571
00:33:34,800 --> 00:33:39,580
It's a generalization
of a linkage

572
00:33:39,580 --> 00:33:42,545
and where we allow
three kinds of edges.

573
00:33:46,950 --> 00:33:54,540
It can be a bar-- and
this is the old case,

574
00:33:54,540 --> 00:33:56,005
which is that the
length is fixed.

575
00:34:00,150 --> 00:34:08,318
We can have cables, which
is the length can decrease,

576
00:34:08,318 --> 00:34:09,234
but it can't increase.

577
00:34:14,889 --> 00:34:18,809
And there are struts,
which are the reverse--

578
00:34:18,809 --> 00:34:22,224
you can increase
but not decrease.

579
00:34:28,750 --> 00:34:30,250
Seems like a useful
generalization,

580
00:34:30,250 --> 00:34:32,389
especially because
people build things

581
00:34:32,389 --> 00:34:34,780
they actually use
struts and cables.

582
00:34:34,780 --> 00:34:37,710
Like you look at most bridges--
tons of cables around.

583
00:34:37,710 --> 00:34:41,989
I think I have an example here.

584
00:34:41,989 --> 00:34:44,570
There's this guy,
Ken Snelson, who

585
00:34:44,570 --> 00:34:47,940
makes a whole bunch of
cool tensegrity sculptures.

586
00:34:47,940 --> 00:34:50,800
This is a particularly
tall one in Washington, DC

587
00:34:50,800 --> 00:34:53,620
that I saw a couple years ago.

588
00:34:53,620 --> 00:34:55,815
That's to give you
an idea of height.

589
00:34:55,815 --> 00:34:57,190
And there's this
really cool view

590
00:34:57,190 --> 00:34:59,230
if you stand-- here's a person.

591
00:34:59,230 --> 00:35:02,100
You stand in the
center, and you look up.

592
00:35:02,100 --> 00:35:06,250
It has this really
cool, sixfold rotational

593
00:35:06,250 --> 00:35:08,700
reflectional symmetry.

594
00:35:08,700 --> 00:35:10,130
I could stare at this for hours.

595
00:35:10,130 --> 00:35:12,940
I think that's just
the [? zoom ?].

596
00:35:12,940 --> 00:35:16,250
So we want to understand
those structures.

597
00:35:16,250 --> 00:35:17,970
This one's also
rigid, hopefully.

598
00:35:17,970 --> 00:35:22,320
It's been standing for 40 years.

599
00:35:22,320 --> 00:35:25,500
So what can we define in
this generalized setting

600
00:35:25,500 --> 00:35:27,060
of tensegrity?

601
00:35:27,060 --> 00:35:28,210
We want motions.

602
00:35:28,210 --> 00:35:29,700
We want rigidity.

603
00:35:29,700 --> 00:35:31,790
We want generic rigidity.

604
00:35:31,790 --> 00:35:33,260
We want infinitesimal rigidity.

605
00:35:33,260 --> 00:35:34,910
All the good things
we've defined,

606
00:35:34,910 --> 00:35:36,870
hopefully, carry over to this.

607
00:35:36,870 --> 00:35:38,790
Well, all but one of
them carry over to this.

608
00:35:48,930 --> 00:35:51,470
OK, we can define a
configuration space.

609
00:35:51,470 --> 00:35:54,820
It's going to be a little
smaller, or a little different

610
00:35:54,820 --> 00:35:57,855
from our old
configuration space.

611
00:35:57,855 --> 00:36:00,180
It's a space of all possible
configurations where

612
00:36:00,180 --> 00:36:02,810
these three things
hold-- give you

613
00:36:02,810 --> 00:36:04,630
desired lengths for
each of these things.

614
00:36:04,630 --> 00:36:06,460
As long as these
constraints hold,

615
00:36:06,460 --> 00:36:08,460
that the lengths stay the
same here, get smaller

616
00:36:08,460 --> 00:36:11,542
here or the same, or stay
the same or get larger here--

617
00:36:11,542 --> 00:36:12,750
that's a valid configuration.

618
00:36:12,750 --> 00:36:14,710
So it's actually a slightly
larger configuration space

619
00:36:14,710 --> 00:36:15,474
than what we had.

620
00:36:15,474 --> 00:36:16,890
A path through
that space is going

621
00:36:16,890 --> 00:36:19,300
to be a motion,
just like before.

622
00:36:19,300 --> 00:36:23,170
Configuration is just
a bunch of points

623
00:36:23,170 --> 00:36:27,050
where this holds--
blah, blah, blah.

624
00:36:27,050 --> 00:36:31,275
The problem is, there's no
notion of generic rigidity.

625
00:36:39,560 --> 00:36:53,100
For example, this is a
generically flexible linkage.

626
00:36:53,100 --> 00:36:54,798
We all know it.

627
00:36:59,190 --> 00:37:02,125
Now I take my yellow
chalk and add struts.

628
00:37:05,400 --> 00:37:08,880
Maybe I'll put some arrows on
it to make it more obvious.

629
00:37:08,880 --> 00:37:14,675
These are things they can
expand, but cannot contract.

630
00:37:17,280 --> 00:37:21,770
Then this configuration
is rigid or flexible?

631
00:37:24,770 --> 00:37:28,727
Flexible, yeah-- this is a
one degree of freedom linkage.

632
00:37:28,727 --> 00:37:30,810
The white part is a one
degree of freedom linkage,

633
00:37:30,810 --> 00:37:34,100
and if you move this thing
in a particular direction,

634
00:37:34,100 --> 00:37:38,710
indeed, this increases
and that increases.

635
00:37:38,710 --> 00:37:39,770
Check it.

636
00:37:39,770 --> 00:37:42,000
This one is rigid.

637
00:37:42,000 --> 00:37:44,360
The only way to move
the square-- or in fact,

638
00:37:44,360 --> 00:37:47,150
any generic convex
configuration--

639
00:37:47,150 --> 00:37:50,444
is one of these pairs
has to get shorter.

640
00:37:50,444 --> 00:37:52,360
The other one will get
longer, but one of them

641
00:37:52,360 --> 00:37:54,850
will get shorter.

642
00:37:54,850 --> 00:38:00,000
So this is the same graph,
and the same labeling

643
00:38:00,000 --> 00:38:02,376
of which edges are bars
and which are struts,

644
00:38:02,376 --> 00:38:04,000
but in one configuration
it's flexible.

645
00:38:04,000 --> 00:38:04,875
The other it's rigid.

646
00:38:04,875 --> 00:38:06,550
So we lose generic rigidity.

647
00:38:06,550 --> 00:38:09,120
But infinitesimal
rigidity still works,

648
00:38:09,120 --> 00:38:11,840
and generic configurations
are still meaningful.

649
00:38:11,840 --> 00:38:14,210
It's just that they're
not all the same.

650
00:38:14,210 --> 00:38:16,110
There's not, in this
case, for four vertices,

651
00:38:16,110 --> 00:38:19,000
there's two kinds of
generic configurations.

652
00:38:19,000 --> 00:38:21,250
So it makes life
a little harder.

653
00:38:21,250 --> 00:38:26,050
But let's say this
part will work well.

654
00:38:26,050 --> 00:38:31,661
So why don't I be
clever, erase this part.

655
00:38:38,880 --> 00:38:41,415
So this is the idea of
an infinitesimal motion.

656
00:38:45,914 --> 00:38:46,580
of a tensegrity.

657
00:38:54,640 --> 00:38:57,380
Well, I want to take
the same dot product,

658
00:38:57,380 --> 00:38:59,860
and I claim this is really
measuring-- it's a number,

659
00:38:59,860 --> 00:39:03,964
and it's measuring the sign's
change in length of that edge.

660
00:39:03,964 --> 00:39:11,290
So if C of v minus
C of w dot product

661
00:39:11,290 --> 00:39:17,980
with d of v minus d of w is
greater than or equal to 0,

662
00:39:17,980 --> 00:39:20,990
that's the property
I want for struts.

663
00:39:20,990 --> 00:39:22,740
That says it gets
longer or stays

664
00:39:22,740 --> 00:39:24,987
the same to the first order.

665
00:39:24,987 --> 00:39:27,320
Now of course, in reality it
might get slightly shorter,

666
00:39:27,320 --> 00:39:29,970
but not in the first derivative.

667
00:39:29,970 --> 00:39:36,250
And it should be less than
or equal to 0 for cables,

668
00:39:36,250 --> 00:39:39,760
and should be equal
to 0 for bars.

669
00:39:43,380 --> 00:39:45,450
So that's the new notion
of infinitesimal motion.

670
00:39:45,450 --> 00:39:50,920
Now, this is no longer a linear
system of linear equations.

671
00:39:50,920 --> 00:39:55,600
I can no longer write this as a
matrix times a vector equals 0.

672
00:39:55,600 --> 00:39:57,420
But this is this
more general thing

673
00:39:57,420 --> 00:40:03,480
called a linear program,
where I have linear equations.

674
00:40:03,480 --> 00:40:06,020
I also have linear inequalities.

675
00:40:06,020 --> 00:40:09,420
So in fact, I can write it
as some other matrix R prime.

676
00:40:09,420 --> 00:40:12,000
Actually, it's the same matrix.

677
00:40:12,000 --> 00:40:19,220
R prime times d is greater
than or equal to 0.

678
00:40:19,220 --> 00:40:21,670
This is the general form
of a linear program.

679
00:40:21,670 --> 00:40:24,780
And you can write an equality
constraint in this world

680
00:40:24,780 --> 00:40:26,799
just by taking it
and its negation,

681
00:40:26,799 --> 00:40:28,590
setting both of them
greater or equal to 0.

682
00:40:28,590 --> 00:40:31,960
That forces them to
be equal to zero.

683
00:40:31,960 --> 00:40:34,110
All you need to know
is it there are fast

684
00:40:34,110 --> 00:40:36,190
algorithms to solve this, also.

685
00:40:36,190 --> 00:40:39,300
So if I give you a tensegrity
in polynomial time,

686
00:40:39,300 --> 00:40:47,360
you can tell whether there is a
valid infinitesimal motion that

687
00:40:47,360 --> 00:40:50,070
does all things you want on
the struts, cables, and bars.

688
00:40:55,030 --> 00:40:57,410
Cool, questions?

689
00:40:57,410 --> 00:40:59,410
STUDENT: [INAUDIBLE]

690
00:40:59,410 --> 00:41:01,500
PROFESSOR: This R--
this is some matrix,

691
00:41:01,500 --> 00:41:04,665
and this is your
vector of all the d's.

692
00:41:04,665 --> 00:41:05,290
STUDENT: Right.

693
00:41:05,290 --> 00:41:09,780
And saying that it's greater
than or equal to 0 [INAUDIBLE].

694
00:41:09,780 --> 00:41:11,850
PROFESSOR: Oh, that means
element lines, sorry.

695
00:41:11,850 --> 00:41:13,225
You take a matrix
times a vector.

696
00:41:13,225 --> 00:41:14,290
You get a big vector.

697
00:41:14,290 --> 00:41:15,450
This is a vector of zeroes.

698
00:41:15,450 --> 00:41:16,300
I should have said that.

699
00:41:16,300 --> 00:41:18,300
And it's just saying every
element in the vector

700
00:41:18,300 --> 00:41:21,480
is greater than
or equal to zero.

701
00:41:21,480 --> 00:41:25,840
So it's just a fancy way of
rewriting this stuff again.

702
00:41:25,840 --> 00:41:28,390
But this is what we
call linear programs,

703
00:41:28,390 --> 00:41:29,914
and that's what's solvable.

704
00:41:29,914 --> 00:41:32,640
STUDENT: And that's
just for struts, then?

705
00:41:32,640 --> 00:41:34,440
PROFESSOR: Well, for
cables you could just

706
00:41:34,440 --> 00:41:35,964
negate the left-hand side.

707
00:41:35,964 --> 00:41:37,880
And then that's greater
than or equal to zero.

708
00:41:37,880 --> 00:41:40,600
For bars you have to include
the same left-hand side

709
00:41:40,600 --> 00:41:42,290
and its negation.

710
00:41:42,290 --> 00:41:44,650
Yeah, I'm skipping some steps.

711
00:41:44,650 --> 00:41:46,355
That's why I had
to write R prime--

712
00:41:46,355 --> 00:41:47,605
it changes a little bit.

713
00:41:51,910 --> 00:41:54,670
How many people know
about linear programming?

714
00:41:54,670 --> 00:41:58,640
OK, about 2/3, so for
you guys, you also

715
00:41:58,640 --> 00:42:01,020
know that there's this thing
called linear programming

716
00:42:01,020 --> 00:42:03,490
duality.

717
00:42:03,490 --> 00:42:05,740
If you have a linear program,
the first thing everyone

718
00:42:05,740 --> 00:42:08,260
should do when they have a
linear program is take the dual

719
00:42:08,260 --> 00:42:09,330
and see what you get.

720
00:42:09,330 --> 00:42:11,250
I'm just going to
tell you what you get.

721
00:42:15,020 --> 00:42:17,650
But that's the next
trick I want to pull.

722
00:42:21,440 --> 00:42:26,870
So dual notion-- essentially
what prevents you

723
00:42:26,870 --> 00:42:30,580
from solving that system--
in other words, what

724
00:42:30,580 --> 00:42:32,980
makes something
rigid-- is something

725
00:42:32,980 --> 00:42:34,530
called equilibrium stress.

726
00:42:40,410 --> 00:42:44,440
So stress, for me,
is just a real number

727
00:42:44,440 --> 00:42:45,864
assigned to every edge.

728
00:42:45,864 --> 00:42:47,280
And here I'm going
to use the word

729
00:42:47,280 --> 00:42:49,460
edge to mean bar,
strut, or cable.

730
00:42:52,550 --> 00:43:02,930
So a real number s
of e for each edge e,

731
00:43:02,930 --> 00:43:06,060
and have to have
three properties.

732
00:43:11,020 --> 00:43:21,610
If I have a cable, then that
number should be non-negative.

733
00:43:21,610 --> 00:43:24,960
If I have a strut, it
should be non-positive.

734
00:43:29,870 --> 00:43:32,620
If I have a bar, I don't care--
could be positive, or negative,

735
00:43:32,620 --> 00:43:34,510
or zero.

736
00:43:34,510 --> 00:43:37,310
And then there's the
equilibrium condition,

737
00:43:37,310 --> 00:43:44,320
which is that if I take the
sum over all vertices w, where

738
00:43:44,320 --> 00:43:55,030
v w is an edge,
the weighted vector

739
00:43:55,030 --> 00:43:57,595
C of v minus C of
w-- so I forgot

740
00:43:57,595 --> 00:43:58,970
to mention we're
assuming we have

741
00:43:58,970 --> 00:44:03,600
some configuration
of our tensegrity.

742
00:44:03,600 --> 00:44:08,190
This is the zero
vector for all vertices

743
00:44:08,190 --> 00:44:11,940
v. Let me draw a picture.

744
00:44:21,100 --> 00:44:22,930
I'll draw a real
example, how's that?

745
00:44:31,530 --> 00:44:36,340
Here's a rigid linkage.

746
00:44:36,340 --> 00:44:39,610
I'm not going to say yet which
edges are struts and cables.

747
00:44:39,610 --> 00:44:42,310
In some sense, I
can do that later.

748
00:44:42,310 --> 00:44:47,420
And I'm going to put numbers
1 along all the outside edges

749
00:44:47,420 --> 00:44:53,430
and the numbers 3 along all
the inside edges-- sorry,

750
00:44:53,430 --> 00:44:59,720
negative 1 and positive 3.

751
00:44:59,720 --> 00:45:03,240
And then if you work
out what this is saying,

752
00:45:03,240 --> 00:45:05,690
I'm taking the
vectors-- so I'm looking

753
00:45:05,690 --> 00:45:09,120
at each vertex like this
one, and saying OK, well, I

754
00:45:09,120 --> 00:45:14,270
take all of the
incoming, whatever.

755
00:45:14,270 --> 00:45:19,450
Let's think of all the edges
incident to that vertex.

756
00:45:19,450 --> 00:45:20,945
I weight them by this number.

757
00:45:20,945 --> 00:45:23,164
In this case, they're
all multiplied by 3.

758
00:45:23,164 --> 00:45:24,080
So I take this vector.

759
00:45:24,080 --> 00:45:25,050
I multiply it by 3.

760
00:45:25,050 --> 00:45:26,430
It gets three times as long.

761
00:45:26,430 --> 00:45:29,040
Take this vector, multiply by
3, this vector multiplied by 3.

762
00:45:29,040 --> 00:45:29,920
I add them up.

763
00:45:29,920 --> 00:45:31,344
I should get the zero vector.

764
00:45:31,344 --> 00:45:33,260
And if this is really
an equilateral triangle,

765
00:45:33,260 --> 00:45:36,150
I will get the zero
vector, so that's good.

766
00:45:36,150 --> 00:45:39,540
And I'm going to
draw arrows here

767
00:45:39,540 --> 00:45:43,330
to signify what this
means intuitively

768
00:45:43,330 --> 00:45:47,480
is that this edge is pushing
with a really big arrow.

769
00:45:47,480 --> 00:45:48,950
This is an arrowhead of size 3.

770
00:45:51,900 --> 00:45:54,270
Try to draw it to scale.

771
00:45:54,270 --> 00:45:55,840
Then, of course,
this vertex will not

772
00:45:55,840 --> 00:45:58,470
move subject to those forces.

773
00:45:58,470 --> 00:46:00,610
These forces balance out.

774
00:46:00,610 --> 00:46:02,630
On the other hand,
this guy, I have

775
00:46:02,630 --> 00:46:04,760
these edges of
weight negative 1.

776
00:46:04,760 --> 00:46:07,550
That means it's going
to pull, and it's

777
00:46:07,550 --> 00:46:10,280
going to pull with the
arrowhead of size 1.

778
00:46:22,770 --> 00:46:27,860
OK, and I claim also that
these vertices do not

779
00:46:27,860 --> 00:46:30,390
move subject to
those three forces.

780
00:46:30,390 --> 00:46:31,750
How do you check that?

781
00:46:31,750 --> 00:46:33,330
The proper way to
check it is to draw

782
00:46:33,330 --> 00:46:34,590
what's called a force polygon.

783
00:46:34,590 --> 00:46:37,470
Those who have done structural
engineering know this.

784
00:46:37,470 --> 00:46:41,860
So I take, let's see, there's
this vector, which is length

785
00:46:41,860 --> 00:46:45,630
3 times that, so it would
be something like that long.

786
00:46:45,630 --> 00:46:50,960
And let's say I'm pushing in
that direction, so I go there.

787
00:46:50,960 --> 00:46:55,290
Then I follow this thing
with the weight of 1,

788
00:46:55,290 --> 00:46:56,910
but in sort of the
opposite direction,

789
00:46:56,910 --> 00:46:59,720
so it will be like this.

790
00:46:59,720 --> 00:47:02,010
This is my attempt at
copying that edge length

791
00:47:02,010 --> 00:47:03,520
and getting what I want.

792
00:47:03,520 --> 00:47:05,690
And then the other
force is this one.

793
00:47:05,690 --> 00:47:08,780
It doesn't actually matter
which order I do these in.

794
00:47:08,780 --> 00:47:10,930
And if I end up back
where I started,

795
00:47:10,930 --> 00:47:15,041
that means the total
force is 0 as a vector,

796
00:47:15,041 --> 00:47:16,415
and so this vertex
is stationary.

797
00:47:19,110 --> 00:47:22,270
That's the force polygon.

798
00:47:22,270 --> 00:47:23,400
So you can just check this.

799
00:47:23,400 --> 00:47:26,589
It's a geometric
condition, if you will,

800
00:47:26,589 --> 00:47:27,880
that each vertex is stationary.

801
00:47:27,880 --> 00:47:30,550
That is the notion of
being in equilibrium.

802
00:47:30,550 --> 00:47:33,360
And this is just the
algebra to write that down.

803
00:47:33,360 --> 00:47:36,640
Now what the heck does this
thing mean intuitively?

804
00:47:36,640 --> 00:47:41,870
If your tensegrity
is going to be rigid,

805
00:47:41,870 --> 00:47:48,100
somehow the edges
can prevent a motion.

806
00:47:48,100 --> 00:47:50,135
And intuitively,
what that means--

807
00:47:50,135 --> 00:47:56,320
I'm going to say this right--
so this is like an edge pushing

808
00:47:56,320 --> 00:48:02,620
really hard away from
itself, against the vertices.

809
00:48:02,620 --> 00:48:04,490
That's like a strut.

810
00:48:04,490 --> 00:48:06,680
A strut prevents compression.

811
00:48:06,680 --> 00:48:09,740
So that the force that it
can actuate effectively

812
00:48:09,740 --> 00:48:12,240
is an outward force.

813
00:48:12,240 --> 00:48:15,070
Whereas cables
prevent extension,

814
00:48:15,070 --> 00:48:17,530
so they can sort of
pull back, say, no,

815
00:48:17,530 --> 00:48:19,070
you can't get any longer.

816
00:48:19,070 --> 00:48:21,280
I'm going to pull towards me.

817
00:48:21,280 --> 00:48:22,820
So this is like a cable.

818
00:48:22,820 --> 00:48:24,290
This is like a strut.

819
00:48:24,290 --> 00:48:25,680
Now bars can do both.

820
00:48:25,680 --> 00:48:28,170
That's why there's no
constraint on bars.

821
00:48:28,170 --> 00:48:30,635
But assuming I've got
this right-- and I didn't.

822
00:48:30,635 --> 00:48:32,050
I got it backwards.

823
00:48:34,560 --> 00:48:41,660
Minus 3 plus 1-- it's
symmetric so it doesn't matter

824
00:48:41,660 --> 00:48:47,440
in some sense, but the intuition
matches when cables can pull,

825
00:48:47,440 --> 00:48:55,350
and struts can push,
and bars can do both,

826
00:48:55,350 --> 00:48:58,340
because they prevent
motion in both directions.

827
00:48:58,340 --> 00:49:00,679
All right, now, it would be
nice to say you're rigid if

828
00:49:00,679 --> 00:49:02,470
and only if you have
an equilibrium stress.

829
00:49:02,470 --> 00:49:03,732
But that's not true.

830
00:49:03,732 --> 00:49:04,940
Let me tell you what is true.

831
00:49:21,040 --> 00:49:21,920
So this is duality.

832
00:49:24,870 --> 00:49:32,850
This goes back to Roth and
Whiteley, 1981-- good year.

833
00:49:32,850 --> 00:49:42,770
So let's see,
here's what's true--

834
00:49:42,770 --> 00:49:46,270
if I look at all the
equilibrium stresses,

835
00:49:46,270 --> 00:49:54,810
and I find some stress
that is non-zero

836
00:49:54,810 --> 00:50:00,980
on a particular structure
or cable-- I should mention

837
00:50:00,980 --> 00:50:02,720
there's always a
trivial stress, which

838
00:50:02,720 --> 00:50:05,520
is you set all of
these S's to zero.

839
00:50:05,520 --> 00:50:07,410
Then everything will hold.

840
00:50:07,410 --> 00:50:10,787
So what we're interested
in are nontrivial stresses,

841
00:50:10,787 --> 00:50:13,120
where you're nonzero somewhere,
and the interesting case

842
00:50:13,120 --> 00:50:18,750
is when you're nonzero on some
strut or cable, let's say.

843
00:50:18,750 --> 00:50:21,690
So if there is such
an equilibrium stress,

844
00:50:21,690 --> 00:50:24,960
then it's not quite the
case that you're rigid,

845
00:50:24,960 --> 00:50:26,445
but you're rigid on that edge.

846
00:50:29,410 --> 00:50:39,460
Every infinitesimal motion holds
the length of that edge fixed.

847
00:50:43,160 --> 00:50:44,920
So this is a strut or a cable.

848
00:50:44,920 --> 00:50:46,980
It could get longer or
it could get smaller,

849
00:50:46,980 --> 00:50:48,396
depending which case we're in.

850
00:50:48,396 --> 00:50:50,770
But if there's a stress that's
nonzero on that edge, then

851
00:50:50,770 --> 00:50:52,480
in fact, it can't
change length at all.

852
00:50:52,480 --> 00:50:55,860
It effectively acts like a
bar in this configuration.

853
00:50:55,860 --> 00:50:57,110
That's what this theorem says.

854
00:50:57,110 --> 00:50:58,194
And these are equivalent.

855
00:50:58,194 --> 00:50:59,610
If you want to
know can I possibly

856
00:50:59,610 --> 00:51:01,290
make the strut any longer?

857
00:51:01,290 --> 00:51:03,224
You just see can I find
an equilibrium stress

858
00:51:03,224 --> 00:51:04,515
for that thing that's non-zero?

859
00:51:07,070 --> 00:51:11,420
Another thing you can say--
almost equivalent to this

860
00:51:11,420 --> 00:51:15,730
or a consequence of that--
if I have some tensegrity,

861
00:51:15,730 --> 00:51:21,920
and I want to know is it
infinitesimally rigid,

862
00:51:21,920 --> 00:51:26,450
that is the case if and
only if two conditions hold.

863
00:51:26,450 --> 00:51:35,400
One is that every strut
or cable is nonzero

864
00:51:35,400 --> 00:51:48,400
in some stress-- in some
equilibrium stress--

865
00:51:48,400 --> 00:52:03,740
and I'll say the corresponding
linkage is rigid.

866
00:52:07,430 --> 00:52:10,320
So all this is saying is if
you can find a stress that

867
00:52:10,320 --> 00:52:13,434
is nonzero on every
particular strut or cable,

868
00:52:13,434 --> 00:52:15,350
then all of those things
are effectively bars.

869
00:52:15,350 --> 00:52:18,380
So you can convert your
tensegrity into a linkage

870
00:52:18,380 --> 00:52:19,957
where all the edges are bars.

871
00:52:19,957 --> 00:52:21,540
And then it's just
a matter of testing

872
00:52:21,540 --> 00:52:24,640
rigidity or infinitesimal
rigidity of that linkage, which

873
00:52:24,640 --> 00:52:27,710
is something we
already know how to do.

874
00:52:27,710 --> 00:52:31,620
So in some sense, we already
know how to do all of this,

875
00:52:31,620 --> 00:52:33,250
because this is just
a linear program.

876
00:52:33,250 --> 00:52:34,010
We can solve it.

877
00:52:34,010 --> 00:52:36,352
But this is a more intuitive
way of proving this.

878
00:52:36,352 --> 00:52:38,310
If you want to prove to
someone that tensegrity

879
00:52:38,310 --> 00:52:41,340
is infinitesimally rigid, you
give them a bunch of stresses,

880
00:52:41,340 --> 00:52:43,890
one for each strut
or cable, or maybe

881
00:52:43,890 --> 00:52:46,092
you just need one stress
to hit all of them.

882
00:52:46,092 --> 00:52:48,300
And then you show that the
linkage is infinitesimally

883
00:52:48,300 --> 00:52:49,630
rigid, which you
could do by computing

884
00:52:49,630 --> 00:52:50,963
the rank of the rigidity matrix.

885
00:52:55,130 --> 00:52:59,850
Strut can have zero
stress-- the weight is zero.

886
00:53:02,800 --> 00:53:06,430
OK, let's keep going.

887
00:53:06,430 --> 00:53:09,600
Lots of a cool things to
talk about, maybe over here.

888
00:53:17,380 --> 00:53:23,230
I want to do a little
diversion for fun,

889
00:53:23,230 --> 00:53:34,752
something called
spider webs, like this.

890
00:53:34,752 --> 00:53:36,085
You might have seen them before.

891
00:53:36,085 --> 00:53:40,895
A spiderweb is something
where every edge is basically

892
00:53:40,895 --> 00:53:42,130
a cable.

893
00:53:42,130 --> 00:53:45,250
You can see under
wind and tension,

894
00:53:45,250 --> 00:53:48,080
they get a little bit shorter
than they're supposed to be.

895
00:53:48,080 --> 00:53:49,455
But they can't
really get longer.

896
00:53:49,455 --> 00:53:53,140
They are pretty strong
at preventing expansion.

897
00:53:53,140 --> 00:53:54,960
So a spiderweb is
something that can

898
00:53:54,960 --> 00:53:57,810
be stressed with all
positive weights.

899
00:53:57,810 --> 00:54:00,720
And the reason it holds
in tension-- positive,

900
00:54:00,720 --> 00:54:04,470
yeah, cables-- all of the edges
can have a positive stress,

901
00:54:04,470 --> 00:54:07,120
and it will be in equilibrium.

902
00:54:07,120 --> 00:54:09,920
And in spiderwebs, in
fact, this is the same--

903
00:54:09,920 --> 00:54:13,079
these two-- rigidity
and positive stresses

904
00:54:13,079 --> 00:54:13,870
are the same thing.

905
00:54:20,300 --> 00:54:26,130
This is sort of a special
case, and Connolly

906
00:54:26,130 --> 00:54:30,630
studied these in 1982.

907
00:54:30,630 --> 00:54:37,420
There's some special
points in a spiderweb.

908
00:54:37,420 --> 00:54:38,976
Here they're kind
of off the page.

909
00:54:38,976 --> 00:54:41,350
There's a point down here
where it attaches to something,

910
00:54:41,350 --> 00:54:42,690
and up there, and up there.

911
00:54:42,690 --> 00:54:44,440
Those guys do not
satisfy equilibrium,

912
00:54:44,440 --> 00:54:48,820
or you can say those edges
have some negative stresses.

913
00:54:48,820 --> 00:54:51,310
Or another way to
think of it is there's

914
00:54:51,310 --> 00:54:53,942
a point off to infinity,
which is how this is drawn,

915
00:54:53,942 --> 00:54:55,400
and you can have
a positive stress.

916
00:54:55,400 --> 00:54:58,070
And at infinity, you don't
have to satisfy equilibrium.

917
00:54:58,070 --> 00:55:00,410
So there's a little something
special at the boundary,

918
00:55:00,410 --> 00:55:02,870
but everywhere else-- interior
and all these vertices

919
00:55:02,870 --> 00:55:06,820
in here-- you have equilibrium,
and all the stresses

920
00:55:06,820 --> 00:55:08,780
are positive.

921
00:55:08,780 --> 00:55:11,290
And it turns out in
this case, you're

922
00:55:11,290 --> 00:55:14,110
automatically
infinitesimally rigid.

923
00:55:14,110 --> 00:55:16,784
That tensegrity is
infinitesimally rigid.

924
00:55:16,784 --> 00:55:18,700
Normally, we need to
also check the underlying

925
00:55:18,700 --> 00:55:22,830
linkage is rigid, but for
spiderwebs, you don't have to.

926
00:55:22,830 --> 00:55:24,800
Why do I care about spiderwebs?

927
00:55:24,800 --> 00:55:30,490
Because they give us something
called origami tesselations.

928
00:55:30,490 --> 00:55:32,470
This is a brand-new
result from this summer

929
00:55:32,470 --> 00:55:35,730
at [? OSMI ?] by Robert
Lang and Alex Bateman.

930
00:55:35,730 --> 00:55:39,260
Here's a little example of an
origami tessellation, this fold

931
00:55:39,260 --> 00:55:41,990
by Jenny [? Ramsier ?],
just the one

932
00:55:41,990 --> 00:55:43,460
I had hanging
around in my office.

933
00:55:43,460 --> 00:55:45,090
It looks a little
cooler that way.

934
00:55:45,090 --> 00:55:48,060
Let me hold it up to the light.

935
00:55:48,060 --> 00:55:49,400
I can see it great.

936
00:55:49,400 --> 00:55:50,620
You don't see the light.

937
00:55:50,620 --> 00:55:51,790
It looks pretty cool.

938
00:55:51,790 --> 00:55:55,577
It's folded from
one square, and it's

939
00:55:55,577 --> 00:55:57,410
folded in some sense
with a regular pattern.

940
00:55:57,410 --> 00:56:01,750
You can see regular
hexagons, hexagonal pattern.

941
00:56:01,750 --> 00:56:04,375
And these guys have been around
for a long time, probably

942
00:56:04,375 --> 00:56:07,340
at least the '60s,
maybe earlier--

943
00:56:07,340 --> 00:56:11,080
a lot of Japanese artists,
lot of American artists.

944
00:56:11,080 --> 00:56:12,580
And there's been a
lot of excitement

945
00:56:12,580 --> 00:56:15,720
about origami tesselations
since the mid '90s.

946
00:56:15,720 --> 00:56:18,910
I think Chris Palmer revitalized
them, and then tons of people

947
00:56:18,910 --> 00:56:20,030
are looking at them.

948
00:56:20,030 --> 00:56:23,220
And there's this cool
algorithm for building them.

949
00:56:30,400 --> 00:56:33,500
So you take some tessellation
like these squares,

950
00:56:33,500 --> 00:56:37,460
and triangles, and hexagons--
some collection of polygons.

951
00:56:37,460 --> 00:56:40,220
You shrink all of
them a little bit,

952
00:56:40,220 --> 00:56:42,150
but all by the same amount.

953
00:56:42,150 --> 00:56:46,580
And then you twist them a
little bit, all the same amount.

954
00:56:46,580 --> 00:56:48,540
So here they are twisted.

955
00:56:48,540 --> 00:56:51,150
And then you just
connect the dots.

956
00:56:51,150 --> 00:56:53,472
So these two edges used to
be joined to each other.

957
00:56:53,472 --> 00:56:55,930
So I'm going to connect these
guys, and connect these guys.

958
00:56:55,930 --> 00:56:57,190
That's the green edges.

959
00:56:57,190 --> 00:56:58,481
And that's your crease pattern.

960
00:56:58,481 --> 00:57:02,330
You erase the red stuff, and
you get your crease pattern,

961
00:57:02,330 --> 00:57:06,460
and you fold it, and you get
this cool tessellation-- magic.

962
00:57:06,460 --> 00:57:09,630
Now the question is,
when does this work?

963
00:57:09,630 --> 00:57:12,120
This algorithm has been around
forever, as far as I know.

964
00:57:12,120 --> 00:57:16,420
I don't know exactly how old
it is, or who invented it,

965
00:57:16,420 --> 00:57:18,510
but it's pretty natural.

966
00:57:18,510 --> 00:57:20,520
It's how most
tesselations are folded.

967
00:57:20,520 --> 00:57:22,760
But when does it work?

968
00:57:22,760 --> 00:57:28,030
And there is a brand-new
result from the summer

969
00:57:28,030 --> 00:57:32,200
that it works for some
choice of those angles.

970
00:57:32,200 --> 00:57:34,230
So I had some flexibility--
how much do I shrink

971
00:57:34,230 --> 00:57:35,880
and how much do I turn?

972
00:57:35,880 --> 00:57:38,210
But this will work for
some choice of turn

973
00:57:38,210 --> 00:57:42,030
and shrink amount, if
and only if your graph

974
00:57:42,030 --> 00:57:48,960
or your initial
tessellation is a spiderweb.

975
00:57:48,960 --> 00:57:52,350
Pretty cool-- spiderwebs
have this neat property

976
00:57:52,350 --> 00:57:56,190
that you can do the shrink and
twist tessellation algorithm,

977
00:57:56,190 --> 00:57:57,120
and it works.

978
00:57:57,120 --> 00:58:01,800
And here is an example of
that-- a spiderweb, so to speak,

979
00:58:01,800 --> 00:58:03,710
but drawn visually.

980
00:58:03,710 --> 00:58:06,330
And this is actually a proof
that-- these are basically

981
00:58:06,330 --> 00:58:08,180
all the force polygons
showing that you

982
00:58:08,180 --> 00:58:11,300
can get a positive
equilibrium stress.

983
00:58:11,300 --> 00:58:14,430
And you turn it into a
crease pattern, fold it flat,

984
00:58:14,430 --> 00:58:15,274
and it works.

985
00:58:15,274 --> 00:58:16,690
This is, of course,
in simulation,

986
00:58:16,690 --> 00:58:21,510
but you can really fold
these things, and they work.

987
00:58:21,510 --> 00:58:24,400
So that's kind of fun--
a little bit origami out

988
00:58:24,400 --> 00:58:27,635
of linkages and tensegrities.

989
00:58:27,635 --> 00:58:28,135
Questions?

990
00:58:31,250 --> 00:58:33,870
All right, then we move on.

991
00:58:33,870 --> 00:58:37,010
I want to tell you one more
cool thing about stresses

992
00:58:37,010 --> 00:58:52,427
and tensegrities-- yet another
characterization of stresses,

993
00:58:52,427 --> 00:58:53,635
which is polyhedral liftings.

994
00:59:07,450 --> 00:59:09,990
A polyhedral lifting
is going to work

995
00:59:09,990 --> 00:59:13,530
if I have a non-crossing
configuration.

996
00:59:21,620 --> 00:59:27,450
So I don't want
to have this-- two

997
00:59:27,450 --> 00:59:31,010
edges that cross each other.

998
00:59:31,010 --> 00:59:33,182
I only want to have
pictures like this,

999
00:59:33,182 --> 00:59:34,640
where edges don't
cross each other.

1000
00:59:34,640 --> 00:59:37,850
They only share endpoints.

1001
00:59:37,850 --> 00:59:40,550
Then I can define what's
called a polyhedral lifting.

1002
00:59:40,550 --> 00:59:44,970
So this is a
planograph, if you will.

1003
00:59:44,970 --> 00:59:47,400
Polyhedral lifting
is going to be

1004
00:59:47,400 --> 00:59:56,270
an assignment of
z-coordinates to the vertices.

1005
00:59:56,270 --> 01:00:01,330
So it's going to be
a z-coordinate z of v

1006
01:00:01,330 --> 01:00:14,370
for each vertex v such that
each face remains planar.

1007
01:00:23,660 --> 01:00:26,250
Because I forbid
crossings, I end up

1008
01:00:26,250 --> 01:00:31,825
decomposing space-- the
plane here-- this only works

1009
01:00:31,825 --> 01:00:32,575
in two dimensions.

1010
01:00:35,311 --> 01:00:37,060
This is one thing that
does not generalize

1011
01:00:37,060 --> 01:00:39,140
to arbitrary dimension.

1012
01:00:39,140 --> 01:00:41,910
So I get a bunch
of triangles here.

1013
01:00:41,910 --> 01:00:42,890
And so I want to lift.

1014
01:00:42,890 --> 01:00:44,380
I want to assign
some z-coordinate

1015
01:00:44,380 --> 01:00:47,660
to each of these vertices, such
that each of these triangles

1016
01:00:47,660 --> 01:00:48,350
remains planar.

1017
01:00:48,350 --> 01:00:49,766
Well, actually,
that always holds.

1018
01:00:49,766 --> 01:00:51,049
Triangles are always planar.

1019
01:00:51,049 --> 01:00:53,590
But there's another face, which
is the outside face out here.

1020
01:00:53,590 --> 01:00:55,320
That's a quadrilateral.

1021
01:00:55,320 --> 01:00:56,950
That one also has
to remain planar.

1022
01:00:56,950 --> 01:01:03,090
So these four vertices have
to be lifted onto some plane.

1023
01:01:03,090 --> 01:01:04,485
This guy can be
lifted however it

1024
01:01:04,485 --> 01:01:05,860
wants in this
particular picture.

1025
01:01:05,860 --> 01:01:08,290
In general, you're going to
have a lot of non-triangles.

1026
01:01:08,290 --> 01:01:11,612
Those should all remain planar.

1027
01:01:11,612 --> 01:01:12,320
That's a lifting.

1028
01:01:15,260 --> 01:01:18,290
One thing you can do--
there's some freedom here.

1029
01:01:18,290 --> 01:01:20,029
If you have any lifting
or, for example,

1030
01:01:20,029 --> 01:01:21,320
you could not lift them at all.

1031
01:01:21,320 --> 01:01:23,430
Set all the
z-coordinates to zero.

1032
01:01:23,430 --> 01:01:24,610
That's fine.

1033
01:01:24,610 --> 01:01:26,490
You could also just
lift everything

1034
01:01:26,490 --> 01:01:29,260
onto some other
plane, and generally

1035
01:01:29,260 --> 01:01:31,085
have a rigid motion of freedom.

1036
01:01:33,652 --> 01:01:35,110
You can't translate
around, but you

1037
01:01:35,110 --> 01:01:38,350
have one two-degree
two-rotational freedoms,

1038
01:01:38,350 --> 01:01:40,840
I think.

1039
01:01:40,840 --> 01:01:43,560
And so we would just
like to get rid of those.

1040
01:01:43,560 --> 01:02:03,780
So say remove rigid motions
by forcing the outside face

1041
01:02:03,780 --> 01:02:10,070
to lie in z equals 0.

1042
01:02:10,070 --> 01:02:13,730
So let's say these
guys all stay at 0.

1043
01:02:13,730 --> 01:02:15,450
So in this particular
example, this guy

1044
01:02:15,450 --> 01:02:19,910
can go to any value, positive
or negative, but that's it.

1045
01:02:19,910 --> 01:02:24,470
Now, this is a lot like
stresses-- in fact,

1046
01:02:24,470 --> 01:02:28,100
it's identical to stresses.

1047
01:02:28,100 --> 01:02:29,570
It's probably not
obvious, but just

1048
01:02:29,570 --> 01:02:31,653
like stresses where you
could set everything to 0,

1049
01:02:31,653 --> 01:02:33,520
here you can also
set everything to 0.

1050
01:02:36,720 --> 01:02:39,000
It's not a big connection,
but it turns out

1051
01:02:39,000 --> 01:02:42,134
they really are
the same thing for

1052
01:02:42,134 --> 01:02:43,300
non-crossing configurations.

1053
01:02:52,560 --> 01:02:55,865
This is called
Maxwell-Cremona Theorem.

1054
01:02:59,730 --> 01:03:01,550
This is probably one
of the oldest theorems

1055
01:03:01,550 --> 01:03:06,670
we will cover here--
I guess anywhere.

1056
01:03:06,670 --> 01:03:09,470
Well, it's not that old-- 1864.

1057
01:03:09,470 --> 01:03:11,000
That might be the oldest.

1058
01:03:11,000 --> 01:03:19,380
Maxwell proved this,
or claimed it in 1864.

1059
01:03:19,380 --> 01:03:27,420
Theorem is there's a
one-to-one correspondence

1060
01:03:27,420 --> 01:03:38,120
between the equilibrium stresses
and polyhedral liftings.

1061
01:03:46,990 --> 01:03:56,530
And fun fact-- negative
stresses correspond to valleys.

1062
01:04:00,300 --> 01:04:05,750
So valley is
something like this.

1063
01:04:05,750 --> 01:04:08,582
You know what valleys are, but
it's maybe a little bit subtle.

1064
01:04:08,582 --> 01:04:10,040
We're not folding
all the way here,

1065
01:04:10,040 --> 01:04:12,340
so this would also be
considered a valley,

1066
01:04:12,340 --> 01:04:17,170
because it has sort of
negative dihedral triangle.

1067
01:04:17,170 --> 01:04:18,849
It's really hard to see.

1068
01:04:18,849 --> 01:04:20,390
What I meant was
something like this.

1069
01:04:30,030 --> 01:04:33,120
That's a valley edge here.

1070
01:04:33,120 --> 01:04:38,170
This is another valley
edge here-- whatever.

1071
01:04:38,170 --> 01:04:41,570
A positive stress
corresponds to a mountain,

1072
01:04:41,570 --> 01:04:45,530
which is the same
picture upside down.

1073
01:04:45,530 --> 01:04:52,540
A zero stress corresponds
to a flat angle--

1074
01:04:52,540 --> 01:04:55,310
doesn't have to be horizontal,
but it has to be flat.

1075
01:04:55,310 --> 01:04:57,990
So it could be like this.

1076
01:05:02,760 --> 01:05:06,250
I'm going to need this in a
moment proving the next thing.

1077
01:05:09,730 --> 01:05:12,240
I'm not going to
prove this theorem,

1078
01:05:12,240 --> 01:05:14,390
but it would be fun to
actually see it in action.

1079
01:05:14,390 --> 01:05:18,820
I've never seen it implemented,
so another fun project.

1080
01:05:18,820 --> 01:05:21,214
I give you tensegrity.

1081
01:05:21,214 --> 01:05:22,880
You compute an
equilibrium stress, which

1082
01:05:22,880 --> 01:05:28,570
is easy to do by linear
programming, and draw-- you

1083
01:05:28,570 --> 01:05:31,859
can directly construct
from that the 3D lifting,

1084
01:05:31,859 --> 01:05:33,025
which would be kind of cool.

1085
01:05:36,510 --> 01:05:38,135
I'm going to use
basically everything I

1086
01:05:38,135 --> 01:05:41,452
proved today to
prove a big theorem--

1087
01:05:41,452 --> 01:05:42,660
the Carpenter's Rule Theorem.

1088
01:05:42,660 --> 01:05:44,584
It's why I'm wearing
this t-shirt that

1089
01:05:44,584 --> 01:05:46,750
has nothing to do with the
Carpenter's Rule Theorem.

1090
01:05:49,280 --> 01:05:52,390
And it's about
non-crossing linkages.

1091
01:05:52,390 --> 01:05:53,640
So this is the beginning.

1092
01:05:53,640 --> 01:05:56,794
So far we've been
allowing crossings,

1093
01:05:56,794 --> 01:05:58,210
except for this
very last theorem.

1094
01:06:00,870 --> 01:06:03,240
Now we begin this section
where I forbid crossings

1095
01:06:03,240 --> 01:06:09,090
between the edges, just
like what that picture was.

1096
01:06:09,090 --> 01:06:14,370
Prevent this, and this is a
constraint on the configuration

1097
01:06:14,370 --> 01:06:15,129
space now.

1098
01:06:15,129 --> 01:06:16,920
I'm going to look at
all the configurations

1099
01:06:16,920 --> 01:06:18,450
where no two edges cross.

1100
01:06:18,450 --> 01:06:21,470
This is a smaller version of
our old configuration space.

1101
01:06:21,470 --> 01:06:22,970
I'm not going to
define it formally.

1102
01:06:22,970 --> 01:06:24,370
It's a bit messy.

1103
01:06:24,370 --> 01:06:26,610
It's not too nasty.

1104
01:06:26,610 --> 01:06:28,970
It's still defined by
polynomial inequalities.

1105
01:06:28,970 --> 01:06:29,800
Question?

1106
01:06:29,800 --> 01:06:31,800
STUDENT: What if you have
one of those diagonals

1107
01:06:31,800 --> 01:06:38,080
and switch it to be in relation
to each other [INAUDIBLE]?

1108
01:06:38,080 --> 01:06:39,972
PROFESSOR: You want
this picture and--

1109
01:06:39,972 --> 01:06:41,580
STUDENT: Either one.

1110
01:06:41,580 --> 01:06:43,940
PROFESSOR: And you
want to add this?

1111
01:06:43,940 --> 01:06:49,000
Or-- that's crossing--
not allowed.

1112
01:06:49,000 --> 01:06:50,900
Whatever you're
saying is not allowed.

1113
01:06:50,900 --> 01:06:52,220
That's the easy answer.

1114
01:06:52,220 --> 01:06:55,974
I don't quite know what
it is, but what we allow--

1115
01:06:55,974 --> 01:06:57,390
and in this case
I'm just thinking

1116
01:06:57,390 --> 01:06:59,430
about linkages for the moment.

1117
01:06:59,430 --> 01:07:03,420
It's just you can hinge
at common vertices.

1118
01:07:03,420 --> 01:07:06,930
Two edges either
share a vertex or not.

1119
01:07:06,930 --> 01:07:09,670
And that's it.

1120
01:07:09,670 --> 01:07:12,070
But they're not allowed
to touch anywhere else.

1121
01:07:15,220 --> 01:07:18,070
And now what we care
about is something

1122
01:07:18,070 --> 01:07:19,135
called a locked linkage.

1123
01:07:24,700 --> 01:07:30,970
And a linkage is locked
if its configuration

1124
01:07:30,970 --> 01:07:36,165
space is disconnected.

1125
01:07:40,720 --> 01:07:44,650
So the configuration space
would look something like this.

1126
01:07:44,650 --> 01:07:46,410
Maybe it has three components.

1127
01:07:46,410 --> 01:07:48,000
If I have some
configuration here,

1128
01:07:48,000 --> 01:07:50,400
I can move to any other
configuration here,

1129
01:07:50,400 --> 01:07:52,272
and same over here,
but there's no way

1130
01:07:52,272 --> 01:07:54,480
to get from this configuration
to that configuration.

1131
01:07:54,480 --> 01:07:57,360
They're disconnected
from each other.

1132
01:07:57,360 --> 01:07:58,930
So this would be
kind of sad news

1133
01:07:58,930 --> 01:07:59,960
if you're looking for motions.

1134
01:07:59,960 --> 01:08:02,210
It means you cannot get from
everywhere to everywhere.

1135
01:08:02,210 --> 01:08:04,820
There are some locked
configurations that cannot get

1136
01:08:04,820 --> 01:08:07,020
back to start.

1137
01:08:07,020 --> 01:08:09,400
However you define start
and locked doesn't matter,

1138
01:08:09,400 --> 01:08:12,062
but the linkage is locked when
there's two things that cannot

1139
01:08:12,062 --> 01:08:12,770
reach each other.

1140
01:08:16,020 --> 01:08:31,850
And the Carpenter's Rule Theorem
is about one case of that.

1141
01:08:38,880 --> 01:08:41,520
So this was essentially
my Ph.D. thesis,

1142
01:08:41,520 --> 01:08:46,649
so paper in 2000 by Bob Connolly
who made that tensegrity,

1143
01:08:46,649 --> 01:08:48,569
[? Gunta ?] [? Rota ?] and me.

1144
01:08:48,569 --> 01:08:57,270
And it says that if
we have a linkage,

1145
01:08:57,270 --> 01:09:00,290
and say we take a configuration
of that linkage, and let's

1146
01:09:00,290 --> 01:09:07,210
say it's of maximum degree
2, so every vertex has

1147
01:09:07,210 --> 01:09:10,920
at most two edges
coming out of it.

1148
01:09:10,920 --> 01:09:25,100
Then there's a motion of that
linkage that straightens out--

1149
01:09:25,100 --> 01:09:28,910
so if you have maximum
degree 2, you have paths,

1150
01:09:28,910 --> 01:09:30,884
and you might have some cycles.

1151
01:09:30,884 --> 01:09:32,050
You might have many of them.

1152
01:09:39,560 --> 01:09:41,430
And it's going to
straighten all the paths

1153
01:09:41,430 --> 01:09:42,749
and convexify all the cycles.

1154
01:09:55,160 --> 01:09:58,270
And it's also expansive.

1155
01:10:02,420 --> 01:10:04,440
But before I get to
that, this is essentially

1156
01:10:04,440 --> 01:10:06,260
showing that these
configuration's bases

1157
01:10:06,260 --> 01:10:07,119
are connected.

1158
01:10:07,119 --> 01:10:09,160
It doesn't quite imply it
the way I've worded it.

1159
01:10:09,160 --> 01:10:13,659
But for example, for
paths, I take some path.

1160
01:10:13,659 --> 01:10:15,700
I want to know, can I get
from this configuration

1161
01:10:15,700 --> 01:10:16,866
to some other configuration?

1162
01:10:19,320 --> 01:10:20,890
Well, I could take
this one, find

1163
01:10:20,890 --> 01:10:22,886
a motion that
straightens the chain.

1164
01:10:22,886 --> 01:10:24,386
I could take this
one, find a motion

1165
01:10:24,386 --> 01:10:26,110
that straightens the chain.

1166
01:10:26,110 --> 01:10:29,180
So here I've drawn it straight.

1167
01:10:29,180 --> 01:10:30,000
I take this motion.

1168
01:10:30,000 --> 01:10:30,840
I apply it forwards.

1169
01:10:30,840 --> 01:10:31,631
I take this motion.

1170
01:10:31,631 --> 01:10:34,430
I apply it backwards, and
I get from here to here.

1171
01:10:34,430 --> 01:10:37,300
So if I can straighten
all the paths,

1172
01:10:37,300 --> 01:10:42,330
I can definitely get from
anywhere to anywhere.

1173
01:10:42,330 --> 01:10:44,800
For convex cycles, it's
a little less obvious,

1174
01:10:44,800 --> 01:10:46,260
but it's also true.

1175
01:10:46,260 --> 01:10:48,990
There's one catch which
I didn't say here.

1176
01:10:48,990 --> 01:10:50,845
I need to add outermost.

1177
01:10:54,630 --> 01:10:57,750
When you have nesting like
this, you're in trouble.

1178
01:10:57,750 --> 01:11:00,050
This guy is not going
to get straightened out.

1179
01:11:00,050 --> 01:11:01,010
It could be super long.

1180
01:11:01,010 --> 01:11:03,020
It may not have room
to straighten out

1181
01:11:03,020 --> 01:11:05,839
inside that convex chamber.

1182
01:11:05,839 --> 01:11:07,630
So these guys will come
along for the ride,

1183
01:11:07,630 --> 01:11:09,338
but they won't actually
get straightened.

1184
01:11:09,338 --> 01:11:14,700
The outermost guys will get
straightened and convexified.

1185
01:11:14,700 --> 01:11:16,446
The other fun
property-- and this

1186
01:11:16,446 --> 01:11:18,070
is how we actually
proved the theorem--

1187
01:11:18,070 --> 01:11:19,445
so this was open
for a long time.

1188
01:11:19,445 --> 01:11:21,030
People thought this was true.

1189
01:11:21,030 --> 01:11:28,690
And the key to proving it is
by making the theorem stronger,

1190
01:11:28,690 --> 01:11:39,410
which is that all pairwise
distances between vertices

1191
01:11:39,410 --> 01:11:44,010
increase or stay the same.

1192
01:11:47,004 --> 01:11:49,420
Some of them have to stay the
same, because they're edges.

1193
01:11:49,420 --> 01:11:50,560
They are bars in there.

1194
01:11:50,560 --> 01:11:52,890
They can't change in length.

1195
01:11:52,890 --> 01:11:56,000
But everything
else will increase.

1196
01:11:56,000 --> 01:11:59,190
Now there's a couple
reasons why this is useful.

1197
01:11:59,190 --> 01:12:04,880
One is that it says you don't
have to worry about crossings.

1198
01:12:04,880 --> 01:12:08,440
If I have some edges that
initially don't cross,

1199
01:12:08,440 --> 01:12:11,790
and I'm worried
about them crossing,

1200
01:12:11,790 --> 01:12:13,490
that can't happen
if you're expansive.

1201
01:12:13,490 --> 01:12:16,800
If I say, well, this
distance has to increase,

1202
01:12:16,800 --> 01:12:18,374
and this distance
has to increase,

1203
01:12:18,374 --> 01:12:19,790
there's no way
that this vertex is

1204
01:12:19,790 --> 01:12:22,170
going to cross over that edge.

1205
01:12:22,170 --> 01:12:24,330
One of them would
have to get shorter.

1206
01:12:24,330 --> 01:12:25,970
So this essentially
says, everything's

1207
01:12:25,970 --> 01:12:26,940
flying away from each other.

1208
01:12:26,940 --> 01:12:28,440
So if initially
you're non-crossing,

1209
01:12:28,440 --> 01:12:30,400
you will be
non-crossing forever.

1210
01:12:30,400 --> 01:12:32,400
That really makes life
easier, because crossings

1211
01:12:32,400 --> 01:12:34,040
are hard to think about.

1212
01:12:34,040 --> 01:12:35,880
This lets you not
worry about crossings.

1213
01:12:35,880 --> 01:12:37,530
You just have to
worry about distances.

1214
01:12:37,530 --> 01:12:39,280
How do I worry about
distances increasing?

1215
01:12:49,420 --> 01:12:55,820
Tensegrities--
struts-- struts will

1216
01:12:55,820 --> 01:12:58,210
force things to be expansive.

1217
01:12:58,210 --> 01:13:03,270
If I want to model this
picture, all I need to do

1218
01:13:03,270 --> 01:13:06,310
is add a whole of struts.

1219
01:13:06,310 --> 01:13:07,650
Let's go over here.

1220
01:13:09,624 --> 01:13:11,540
So we're going to prove
this theorem in, like,

1221
01:13:11,540 --> 01:13:13,550
five minutes-- easy.

1222
01:13:13,550 --> 01:13:18,300
Once you build all this
technology, which was existing,

1223
01:13:18,300 --> 01:13:19,930
you can prove this
theorem, once you

1224
01:13:19,930 --> 01:13:21,200
realize this is the
theorem you want

1225
01:13:21,200 --> 01:13:22,449
to prove-- this expansiveness.

1226
01:13:27,291 --> 01:13:29,290
So what we're going to
do is build a tensegrity.

1227
01:13:35,580 --> 01:13:43,586
And it's going to have a bar
for every edge of the given

1228
01:13:43,586 --> 01:13:44,085
linkage.

1229
01:13:47,910 --> 01:13:50,530
So that's just the same
thing as we had given.

1230
01:13:53,750 --> 01:13:57,375
It's also going to have a
strut between every other pair

1231
01:13:57,375 --> 01:14:01,415
of vertices-- heck,
every pair of vertices.

1232
01:14:01,415 --> 01:14:03,940
It's kind of redundant
for the bars,

1233
01:14:03,940 --> 01:14:07,770
but every pair of vertices.

1234
01:14:07,770 --> 01:14:11,770
That says that all pairwise
distances must increase or stay

1235
01:14:11,770 --> 01:14:14,150
the same.

1236
01:14:14,150 --> 01:14:15,170
So this is a tensegrity.

1237
01:14:15,170 --> 01:14:18,220
What I want to prove is that
this tensegrity is flexible.

1238
01:14:18,220 --> 01:14:20,794
If I can prove that
it's always flexible--

1239
01:14:20,794 --> 01:14:23,085
well, until the end, which
is when everybody's straight

1240
01:14:23,085 --> 01:14:25,043
or convex-- all the
outermost guys are straight

1241
01:14:25,043 --> 01:14:29,490
or convex-- then I'll
have proved this theorem.

1242
01:14:29,490 --> 01:14:31,980
So while this theorem is
about actual connectivity

1243
01:14:31,980 --> 01:14:34,870
of configuration spaces,
it says paths and cycles

1244
01:14:34,870 --> 01:14:36,370
have connected
configuration spaces.

1245
01:14:36,370 --> 01:14:37,790
You get from everywhere
to everywhere.

1246
01:14:37,790 --> 01:14:39,456
It even gives you an
algorithm for doing

1247
01:14:39,456 --> 01:14:41,580
that, which we'll
talk about next class.

1248
01:14:41,580 --> 01:14:45,920
We use just, is this thing
rigid or flexible at all?

1249
01:14:45,920 --> 01:14:49,460
And that's the power of all this
rigidity stuff, just telling

1250
01:14:49,460 --> 01:14:51,790
whether something can
move a little bit,

1251
01:14:51,790 --> 01:14:54,490
and then just doing that
over and over again,

1252
01:14:54,490 --> 01:14:56,630
you will be able to find
that you can actually

1253
01:14:56,630 --> 01:15:00,444
go all the way, and get
to straight or convex.

1254
01:15:00,444 --> 01:15:02,110
But how do we actually
prove this thing?

1255
01:15:06,040 --> 01:15:15,190
So the point is, claim this
tensegrity is infinitesimally

1256
01:15:15,190 --> 01:15:20,800
flexible, meaning it is
not infinitesimally rigid.

1257
01:15:23,860 --> 01:15:26,450
And that will imply
that there's at least

1258
01:15:26,450 --> 01:15:28,200
an infinitesimal
expansive motion.

1259
01:15:28,200 --> 01:15:31,926
And then you have to use some
fancy tricks-- not fancy,

1260
01:15:31,926 --> 01:15:33,550
some tricks I don't
want to talk about.

1261
01:15:33,550 --> 01:15:36,340
You basically integrate
that vector field,

1262
01:15:36,340 --> 01:15:40,000
and that will give you an
actual expansive motion.

1263
01:15:40,000 --> 01:15:41,340
Let's not worry about that.

1264
01:15:41,340 --> 01:15:44,260
The interesting part is show
that at least infinitesimally,

1265
01:15:44,260 --> 01:15:48,330
this tensegrity
moves a little bit.

1266
01:15:48,330 --> 01:15:50,640
So what do know?

1267
01:15:50,640 --> 01:15:55,809
Duality-- if I want to
show not that it's rigid--

1268
01:15:55,809 --> 01:15:57,225
I want to show
that it's flexible.

1269
01:16:00,190 --> 01:16:05,780
But it's rigid if and only if
this is true and this is true.

1270
01:16:05,780 --> 01:16:07,590
This will actually
always be true--

1271
01:16:07,590 --> 01:16:09,450
the corresponding
linkage is rigid--

1272
01:16:09,450 --> 01:16:12,080
because I put in every
edge is in there.

1273
01:16:12,080 --> 01:16:15,260
If I turn them all into bars,
this thing's not moving at all.

1274
01:16:15,260 --> 01:16:18,170
So fact, this second condition
doesn't really matter.

1275
01:16:18,170 --> 01:16:20,860
What matters is the stresses.

1276
01:16:20,860 --> 01:16:24,236
If I want to show that
it's flexible, not rigid.

1277
01:16:24,236 --> 01:16:25,985
I want to show these
stresses don't exist.

1278
01:16:25,985 --> 01:16:27,830
Now stresses always exist.

1279
01:16:27,830 --> 01:16:29,380
I could set all the
stresses to be 0.

1280
01:16:29,380 --> 01:16:32,310
But I claim actually
pretty much all the

1281
01:16:32,310 --> 01:16:35,470
stresses have to be 0, at least
on the cables and the struts,

1282
01:16:35,470 --> 01:16:37,760
which is what I care about here.

1283
01:16:37,760 --> 01:16:40,390
They have to be 0.

1284
01:16:40,390 --> 01:16:48,040
So this will be implied
by every equilibrium

1285
01:16:48,040 --> 01:16:59,200
stress is 0 on all
struts and cables.

1286
01:17:02,000 --> 01:17:04,290
So you can never get
this thing to hold

1287
01:17:04,290 --> 01:17:06,650
for any of the struts
and the cables.

1288
01:17:06,650 --> 01:17:09,570
And therefore, in some sense,
none of them are fixed,

1289
01:17:09,570 --> 01:17:12,730
and therefore, you're
actually flexible.

1290
01:17:12,730 --> 01:17:16,850
That's implied by this
duality statement.

1291
01:17:16,850 --> 01:17:20,050
OK, stresses are a little
hard to think about.

1292
01:17:20,050 --> 01:17:24,400
I prefer to think about
polyhedral liftings, let's say.

1293
01:17:24,400 --> 01:17:27,630
OK, the key here is to plug
together all the right results.

1294
01:17:31,050 --> 01:17:38,090
This will be true if and only
if every polyhedron lifting

1295
01:17:38,090 --> 01:17:38,600
is flat.

1296
01:17:46,180 --> 01:17:48,070
So I guess in our
case, it will be 0,

1297
01:17:48,070 --> 01:17:51,340
because the outside face sets
all the z-coordinates to 0,

1298
01:17:51,340 --> 01:17:53,910
so everything else will
have to be in that plane.

1299
01:17:53,910 --> 01:17:58,662
So all the
z-coordinates will be 0.

1300
01:17:58,662 --> 01:18:00,120
So these are
equivalent statements.

1301
01:18:00,120 --> 01:18:03,170
All I need to prove is that
every polyhedral lifting

1302
01:18:03,170 --> 01:18:03,685
is flat.

1303
01:18:05,941 --> 01:18:08,190
So we're going to prove this
theorem by contradiction.

1304
01:18:20,670 --> 01:18:24,570
Suppose there was a
non-flat polyhedron lifting.

1305
01:18:24,570 --> 01:18:25,570
What could it look like?

1306
01:18:31,390 --> 01:18:33,120
Did I just erase what I wanted?

1307
01:18:33,120 --> 01:18:34,960
Probably.

1308
01:18:34,960 --> 01:18:40,372
Greater than or equal to
0 for cables, less than

1309
01:18:40,372 --> 01:18:41,714
or equal to 0 for struts.

1310
01:18:41,714 --> 01:18:42,880
That's the one I care about.

1311
01:18:42,880 --> 01:18:43,838
Who cares about cables?

1312
01:18:48,020 --> 01:18:50,580
All right, so this
is the stress.

1313
01:18:50,580 --> 01:18:53,170
And equilibrium stress
has to be less than

1314
01:18:53,170 --> 01:18:56,130
or equal to 0 for every strut.

1315
01:18:56,130 --> 01:18:58,750
Less than or equal
to 0 corresponds

1316
01:18:58,750 --> 01:19:02,670
to a valley edge or flat edge.

1317
01:19:02,670 --> 01:19:06,340
Now in this linkage, almost
all the edges are struts.

1318
01:19:06,340 --> 01:19:08,240
There's a few that
are bars, this,

1319
01:19:08,240 --> 01:19:10,190
like, little path is a bar.

1320
01:19:10,190 --> 01:19:11,690
Those could be
valleys or mountains.

1321
01:19:11,690 --> 01:19:12,980
We don't know.

1322
01:19:12,980 --> 01:19:16,070
But everything else has
to be a valley or flat.

1323
01:19:16,070 --> 01:19:17,130
For flat, we're good.

1324
01:19:17,130 --> 01:19:17,640
We're happy.

1325
01:19:17,640 --> 01:19:19,530
We want all the
edges to be flat.

1326
01:19:19,530 --> 01:19:23,700
So how could it be that almost
every edge in this picture

1327
01:19:23,700 --> 01:19:26,470
is a valley?

1328
01:19:26,470 --> 01:19:29,540
That seems a little tricky.

1329
01:19:29,540 --> 01:19:32,770
And we're going to
show it can't happen.

1330
01:19:32,770 --> 01:19:35,640
If everything's a
valley, something's

1331
01:19:35,640 --> 01:19:38,020
going to go wrong
at the top picture,

1332
01:19:38,020 --> 01:19:41,360
because you've got to have
mountains, like, turn around.

1333
01:19:41,360 --> 01:19:43,660
So let's look at the top.

1334
01:19:43,660 --> 01:19:45,370
Look at the maximum
z-coordinate.

1335
01:19:48,997 --> 01:19:51,330
Now this could actually look
like many different things,

1336
01:19:51,330 --> 01:19:54,690
but I'm going to consider one
case-- the easy case-- where

1337
01:19:54,690 --> 01:19:56,940
you have, in the
maximum z-coordinate,

1338
01:19:56,940 --> 01:19:58,640
you have one point.

1339
01:19:58,640 --> 01:20:02,250
And then around that,
everything descends.

1340
01:20:02,250 --> 01:20:04,610
OK, these are all the edges
incident to that point.

1341
01:20:04,610 --> 01:20:13,580
You form some kind of--
OK, so this is maximum z.

1342
01:20:13,580 --> 01:20:14,520
This is another plane.

1343
01:20:14,520 --> 01:20:19,730
I'm going to slice that, which
is maximum z minus epsilon--

1344
01:20:19,730 --> 01:20:22,940
slightly below the
maximum z-coordinate.

1345
01:20:22,940 --> 01:20:27,550
And in a sort of typical case,
we're going to get a polygon.

1346
01:20:27,550 --> 01:20:30,730
Now note that this polygon
has this nice property,

1347
01:20:30,730 --> 01:20:35,050
that wherever it has a convex
angle, you get a mountain here.

1348
01:20:35,050 --> 01:20:37,770
Wherever you have a reflex
angle, like this one,

1349
01:20:37,770 --> 01:20:39,360
you get a valley edge here.

1350
01:20:39,360 --> 01:20:42,920
You can see that
3-dimensional picture.

1351
01:20:42,920 --> 01:20:47,260
Every polygon has at least
three convex vertices.

1352
01:20:47,260 --> 01:20:49,710
That means there are at least
three mountains incident

1353
01:20:49,710 --> 01:20:53,650
to this maximum
z-coordinate But mountains

1354
01:20:53,650 --> 01:20:55,960
have to correspond
to positive stresses.

1355
01:20:55,960 --> 01:20:59,450
Positive stresses have
to correspond to bars,

1356
01:20:59,450 --> 01:21:03,280
because struts can't
carry positive stresses.

1357
01:21:03,280 --> 01:21:06,410
So that means you have
at least three bars

1358
01:21:06,410 --> 01:21:08,000
incident to that edge.

1359
01:21:08,000 --> 01:21:13,360
But I assumed I was maximum
degree 2-- contradiction.

1360
01:21:13,360 --> 01:21:15,429
Isn't that cool?

1361
01:21:15,429 --> 01:21:17,095
It's very easy when
you set it up right.

1362
01:21:20,820 --> 01:21:24,750
This works almost always, but
there is another situation

1363
01:21:24,750 --> 01:21:27,270
which can happen, which
is, like, you have--

1364
01:21:27,270 --> 01:21:29,560
in the maximum
z-coordinate you might

1365
01:21:29,560 --> 01:21:32,400
have a couple of
bars at the top.

1366
01:21:32,400 --> 01:21:33,210
That's a worry.

1367
01:21:33,210 --> 01:21:35,402
This is all in maximum z.

1368
01:21:35,402 --> 01:21:37,360
And so when you slice
below, you don't actually

1369
01:21:37,360 --> 01:21:38,790
get a whole polygon.

1370
01:21:38,790 --> 01:21:44,530
You might actually only
get something around here.

1371
01:21:44,530 --> 01:21:46,200
But again, you
can't get from here

1372
01:21:46,200 --> 01:21:49,770
to around there
without some mountain.

1373
01:21:49,770 --> 01:21:52,010
It's hard to even imagine,
because it can't happen.

1374
01:21:52,010 --> 01:21:53,968
But this is a mountain,
and this is a mountain,

1375
01:21:53,968 --> 01:21:57,310
and you want to somehow slope
a surface below this plane,

1376
01:21:57,310 --> 01:22:02,230
and get around to over here with
only valleys-- ain't happening.

1377
01:22:02,230 --> 01:22:05,020
You've got to have
another mountain in here.

1378
01:22:05,020 --> 01:22:09,270
And so that's the
general picture.

1379
01:22:09,270 --> 01:22:12,050
And the only thing
that can happen

1380
01:22:12,050 --> 01:22:15,750
is you actually prove that all
of this stuff has to be flat.

1381
01:22:15,750 --> 01:22:17,250
So that's the one
way it can happen.

1382
01:22:17,250 --> 01:22:20,980
You can't have strict valleys,
but you could have them all 0.

1383
01:22:20,980 --> 01:22:25,900
So in fact, the one case
where you can have stress

1384
01:22:25,900 --> 01:22:29,600
is when you have all of
this outside stuff flat.

1385
01:22:29,600 --> 01:22:31,640
And then inside you don't know.

1386
01:22:31,640 --> 01:22:34,280
I can prove, using
this generalized lemma,

1387
01:22:34,280 --> 01:22:36,231
that all of these
regions are locally flat.

1388
01:22:36,231 --> 01:22:38,480
And therefore, this whole
outside face has to be flat.

1389
01:22:38,480 --> 01:22:39,563
Therefore, I'm particular.

1390
01:22:39,563 --> 01:22:43,219
I can't have a reflex vertex,
because then this side

1391
01:22:43,219 --> 01:22:44,510
would have to all be flattened.

1392
01:22:44,510 --> 01:22:46,160
So then the whole
thing is flattened.

1393
01:22:46,160 --> 01:22:48,020
The only way the whole
thing can't be flat

1394
01:22:48,020 --> 01:22:52,570
is if your polygon is already
convex, and then we're done.

1395
01:22:52,570 --> 01:22:55,320
Also straight this can happen.

1396
01:22:55,320 --> 01:22:57,730
I think with straight,
you don't get any stress.

1397
01:22:57,730 --> 01:22:59,517
Here you can actually
have a lifting.

1398
01:22:59,517 --> 01:23:01,600
Actually, all these guys
would be below the plane,

1399
01:23:01,600 --> 01:23:03,200
because everything's a valley.

1400
01:23:03,200 --> 01:23:08,260
This would be like
an actual valley,

1401
01:23:08,260 --> 01:23:10,644
where you have a
village or whatever.

1402
01:23:10,644 --> 01:23:13,060
That can happen, but only when
the boundary of your valley

1403
01:23:13,060 --> 01:23:13,720
is convex.

1404
01:23:16,330 --> 01:23:18,180
And so that proves the theorem.

1405
01:23:18,180 --> 01:23:20,260
And that is the
Carpenter's Rule Theorem.

1406
01:23:20,260 --> 01:23:22,676
Next time, we're going to talk
about the other situations.

1407
01:23:22,676 --> 01:23:27,020
This was two dimensions,
and maximum degree 2.

1408
01:23:27,020 --> 01:23:28,939
We can think about what
happens with degree 3.

1409
01:23:28,939 --> 01:23:30,230
Then you can get locked things.

1410
01:23:30,230 --> 01:23:31,340
What happens in
three dimensions?

1411
01:23:31,340 --> 01:23:32,210
Then you can get locked things.

1412
01:23:32,210 --> 01:23:33,920
What happens in four dimensions?

1413
01:23:33,920 --> 01:23:36,290
Then you can't get locked
things, and other fun things

1414
01:23:36,290 --> 01:23:36,790
like that.

1415
01:23:39,130 --> 01:23:43,380
Oh, and why is it called the
Carpenter's Rule Theorem?

1416
01:23:43,380 --> 01:23:46,490
Because this is a
carpenter's rule.

1417
01:23:46,490 --> 01:23:49,260
And in a carpenter's rule,
actually, all the edge lengths

1418
01:23:49,260 --> 01:23:50,024
are the same.

1419
01:23:50,024 --> 01:23:51,440
But as far as we
know that doesn't

1420
01:23:51,440 --> 01:23:53,550
make this theorem
any easier to prove.

1421
01:23:53,550 --> 01:23:55,950
And also in the carpenter's
rule, you can have crossings,

1422
01:23:55,950 --> 01:23:59,910
but other than that, it's
just like a carpenter's rule.

1423
01:23:59,910 --> 01:24:01,990
So there you go.

1424
01:24:01,990 --> 01:24:03,620
So if you've never
played with one,

1425
01:24:03,620 --> 01:24:08,077
this is what people used
before flexible measuring tape.

1426
01:24:08,077 --> 01:24:09,910
They still use them a
lot in Europe, but not

1427
01:24:09,910 --> 01:24:11,440
so much in the US,
but you can still

1428
01:24:11,440 --> 01:24:13,850
buy them at the hardware store.

1429
01:24:13,850 --> 01:24:15,540
That's it.