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PROFESSOR: Alright, lecture
12 is about tensegrities,

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like this one which you saw.

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And when they're rigid--
infinitesimal rigidity,

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and carpenter's rule theorem,
all in one quick lecture.

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So, just a couple questions
about tensegrities.

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Well first is about
infinitesimal rigidity

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in general.

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This is sort of extra, bonus.

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I talked about one reason why
this dot product condition

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is the right thing, which
is based on projection.

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But there are other
ways to think about it.

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So let's say we
have an edge, vw--

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actually I think I'll
draw it w on the left.

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So here's vw.

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We would think of
this point is being

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c of w, that's
your configuration.

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This point is c of e.

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Then we were talking about
when, if you have a velocity

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vector, d of w, and
a velocity vector--

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Say something like
this, d of v, when

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it preserves the length of
this bar to the first order.

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And the condition we had was c
of v minus c of w dot product

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with d of v minus
d of w equals 0.

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So the question--
so the way we said

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in class this works is if
you look at the projected

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length of this vector
onto the segment,

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that's essentially
how much shorter,

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in this case it gets
shorter, in this case

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this is how much
longer the segment

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gets to the first order.

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And that corresponds to d of v
dot product with this vector,

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because this vector,
c of v minus c of w

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is this direction here.

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c of v minus c of w.

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So if you take the dot
product with this vector,

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you get that projected length.

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You want those two dot
products to be equal.

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And so you want the
difference to be 0.

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That's one way to
think about it.

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But, in fact, this quantity
also has an intuitive notion,

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if you've done some
basic mechanics.

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This is the relative motion
of v with respect to w.

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So it's like-- well
imagine w is not moving.

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To do that you just subtract
d of w from all motions.

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Then w will remain stationary
to the first order,

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so all first order motions.

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So then what's the motion of v?

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Well, it's going to be
d of v minus d of w,

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because everything get's
subtracted by d of w.

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So that it is-- minus d of
w corresponds to drawing

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that vector in the
other direction,

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and then adding that
to d of w corresponds

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to this kind of picture.

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This is minus d of w.

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And so this vector is
d of v minus d of w.

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That's the sum of
those two vectors.

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And the claim is what we
want is for these two vectors

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to be perpendicular
to each other.

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The edge and the motion.

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And this makes a lot of sense,
because we're imagining w

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as not moving, so
it's just v moving.

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And we want the motion of v
to be perpendicular locally,

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to the segment, because that's
going to preserve the length.

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So that's actually, I think,
an even more intuitive way

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to see it.

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And since this video,
this lecture video,

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has been online for
two years, every year

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so I get an email saying
hey, you asked what is d of w

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minus d of w mean?

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And, yes, I didn't
realize it at the time,

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but it's just relative motion.

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And so, yeah.

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It's like if you're
moving along this circle,

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centered at w, that's where you
want to be, either straight up

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or straight down.

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Clear?

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Some bonus intuition.

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It's a little harder to
see the other conditions

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from this perspective, at
least I find it harder to see.

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But I guess I shouldn't
claim that, or else I'm

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going to get more
emails in the future.

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Let's see, if you
want it to be--

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what does the dot product being
greater or equal to 0 mean?

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It probably means
that this angle

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is obtuse, which means that
the lengths getting longer.

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And less than or
equal to 0 means

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that the angle is
getting non-obtuse.

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And if the dot product rule
of the cosine of the angle,

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you should be able
to figure that out.

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But it's a little bit
less memorized in my head.

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Whereas dot product
equals 0, everyone

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remembers that's
being perpendicular.

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So this was struts and cables.

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Next question was, a
couple questions here.

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One is how can you say the
tensegrity you showed is rigid?

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Because you can flex it.

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This doesn't look very
rigid, this guy here.

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It looks quite flexible.

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And then there was another
question, by someone else,

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saying why did you use
springs to represent bars?

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And these are sort
of the same question.

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This is flexible because
the springs are flexible.

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The idea with the
springs, I mean

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I'm just guessing why this
model is made this way,

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springs are nice because they do
have a natural resting length.

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So in that sense, they
want to be a given length.

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And if you pull on them they try
to restore the original length.

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Also if you push on
them, you can do it,

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but they try to restore
to there resting length.

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And right now these guys are
all on their resting length.

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And what's cool
about this is you

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could kind of feel
the resistance.

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I mean this is what happens in
any material, you pull on it,

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it can maybe pull a
little bit, but it

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gives a lot of force going
back to its original state.

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These-- I don't know,
these cables, probably you

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could stretch that material
and very tiny amount.

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I could feel the force,
it's a little less visible.

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Here you can actually
see the force

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and, because it's
so much smaller,

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it's easier to feel
what's going on in here.

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I think that's the intuition.

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You could, of course, construct
these with steel bars.

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It's just then it
wouldn't move, it

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would be a little less exciting.

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So it's up to you, of course.

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But this is just one way to
build models of tensegrities.

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But it's also why
it's so flexible,

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because these bars
are not super strong.

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But what's kind of fun is you
feel the equilibrium stress

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here, that things go back to
where they were originally.

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They kind of hold
in position there,

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because all the stresses
here balance out.

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And one more kind of question is
about-- sculpture-- tensegrity

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sculptures are cool so I
thought I'd show you a few more

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examples.

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The sort of-- the master
here is Kenneth Snelson.

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I think he possibly
invented tensegrities.

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And I showed one
example in lecture,

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but these are many more.

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He's been doing
it since the '60s.

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Some of them are very big, I
have some measurements here.

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This guy is 85 feet long and
rests on these three posts.

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And it's a little hard
to see the cables here,

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but it's all struts and cables.

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So using these bars
to be-- actually

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they might be bars
in this picture.

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I think they might be rigid
even when they're struts.

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And some more.

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This is one of the
longest, at 72 feet.

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And this is the
tallest, at 90 feet.

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Pretty impressive.

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This is a taller version of the
one that I showed last time,

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or in lecture.

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So I'm not sure how he designs
them, he's an artist by trade.

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Whether he uses computational
tools, or gadgets,

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or I'm not sure, it would
be interesting to talk

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to him, actually.

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But if you want to see more
examples of this sculpture,

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this is just a piece of
it, go to his website.

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On the design side, our
good friend, Tomohio Tachi,

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who do the origami
Origamizer Bunny,

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has also done a
tensegrity bunny.

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And he didn't design this by
hand, as you might imagine.

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He designed it with a new piece
of software that hasn't yet

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been released, called
Freeform Tensegrity.

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You start with a
polyhedron, and there's

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a few different initial
constructions that kind of

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set that thing up as
a kind of tensegrity.

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And then it solves
to make it balance,

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to have an equilibrium stress.

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And on the right is the stress.

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These are what a
structural engineer

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would call force polygons.

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It's essentially the-- it's
like if you take this graph

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and you rotate-- this is all
in 3D, by the way, so a little

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tricky.

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That's rotating.

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In this case he's
pulling on things,

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trying to force vertices
to come together.

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And the whole thing is updating.

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And as long as this number
down here is close to 0,

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here it's 10 to
the minus 20, it's

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a pretty good
approximation of 0,

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then this thing
is in equilibrium.

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It's again, solving
all those constraints

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like we have before.

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So force polygons.

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You take every edge, you
rotate it 90 degrees,

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and then sort of con-- imagine
constructing a little polygon

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around each face.

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So you scale the edge
by the stress in there,

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and those polygons should close
up to be 0, if, at the vertex,

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you satisfy equilibrium.

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And you can draw
all those polygons.

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Those polygons can
be joined together

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to make one kind of graph.

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And if that graph
closes up, then you

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have an equilibrium stress.

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So it's kind of a
neat way to visualize

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that the thing is rigid.

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Or rather that it would hold all
of these edges at fixed length,

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and you have to verify that
the bar structure is rigid,

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but this is
presumably constructed

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to have that property.

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So hopefully this software
will be released at some point.

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It definitely looks
like a cool way

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to play around with
tensegrities and design things.

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Question?

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AUDIENCE: Is there
any way to see

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what happens to the equilibrium
stresses [INAUDIBLE]?

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Uh.

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I see, yeah.

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That's a good question.

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So can you see what happens if
I-- like when that one, when

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I tweak it and then wobbles.

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Yeah.

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It would be nice to see that
in simulation, definitely.

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This software is based on
free form origami designer,

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and that has the
two modes, right?

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One where you can change
the crease pattern,

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and one where it tries
to stay, keep it fixed.

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So I'm not sure whether the
software has the same two

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modes, but it certainly could,
that's definitely computable.

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You could basically ignore
the concerns for awhile,

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pull on something, then let go,
let it restore the constraints

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without it changing
the tensegrity,

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like it's doing here.

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And you should see it wobble.

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In fact, in that
case you'd probably

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want to use a less
good numerical method.

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Because you don't want it
to stabilize as quickly,

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you'd like to see it
jiggle for a while.

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So it should be doable.

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Could be an interesting
project to extend the software,

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if it hasn't been
implemented yet.

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I could ask him.

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I asked him last night, he say
that the plan is to release it

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at some point, but not quite
ready for prime time yet.

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Other questions about this?

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So I think this would
be fun to play with.

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If you want to build
some tensegrities

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there's a couple descriptions
of how to do this out

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of various, easy to find,
household objects like straws,

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or with these wove rubber
bands, things like that.

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So you can check
out-- George Hart

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has two construction
web pages about them.

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They're linked from this slide,
if you go to the lecture.

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That was all I
had for questions.

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Any more questions
about this lecture?

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00:12:26,830 --> 00:12:28,580
All perfectly clear?

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00:12:28,580 --> 00:12:31,182
We'll be seeing a lot more
stuff about locked linkages,

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00:12:31,182 --> 00:12:32,640
the carpenter's
rule part, which is

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at the end of class in the
next two-three lectures.

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00:12:35,920 --> 00:12:41,520
AUDIENCE: Can you explain
how infinitesimal motions--

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00:12:41,520 --> 00:12:46,090
Well, in general,
a linear program

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is something like--
you have a matrix

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and you multiply it by a vector.

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00:12:50,803 --> 00:12:53,386
And then you have, let's say,
in our case we have greater than

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or equal to 0.

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00:12:54,880 --> 00:12:59,547
So when I say 0 I mean 0 0 0.

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00:13:03,620 --> 00:13:05,570
The dual of a linear
program is what

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00:13:05,570 --> 00:13:07,760
you get by transposing
the matrix, basically.

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00:13:07,760 --> 00:13:14,190
So you get-- you
rotate it 90 degrees,

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or actually flip around that
diagonal, so all the columns

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become rows, rows
become columns.

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00:13:20,620 --> 00:13:23,200
So now you've got
some other thing here,

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which is going to be this big.

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00:13:27,115 --> 00:13:30,630
It's probably going to be less
than or equal to something

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00:13:30,630 --> 00:13:31,350
that big.

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00:13:33,921 --> 00:13:37,650
So this is a
transpose, this is y,

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and this is usually
called c, this is,

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in general called little b.

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00:13:43,240 --> 00:13:46,330
And they are relations
between these two things,

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00:13:46,330 --> 00:13:48,870
is the short version.

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00:13:48,870 --> 00:13:51,704
But you can kind of
see-- so in principle,

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00:13:51,704 --> 00:13:53,120
if we take-- this
is, in our case,

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this is basically
the rigidity matrix.

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00:13:55,550 --> 00:13:58,360
This is rigidity matrix
prime, in the lecture.

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00:13:58,360 --> 00:14:00,720
Because if we want to write
greater than or equal to 0,

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00:14:00,720 --> 00:14:02,030
struts are just fine.

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00:14:02,030 --> 00:14:05,030
Cables we negate
everything in the row.

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00:14:05,030 --> 00:14:08,480
Bars, where we have equality, we
need both the original version

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00:14:08,480 --> 00:14:10,995
and the negated
version of that row.

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00:14:10,995 --> 00:14:12,870
But if we just imagine
struts for the moment,

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because I mean subsets
everything-- well,

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can be simulated by struts
and negative struts, struts

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and cables.

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00:14:19,640 --> 00:14:21,450
Then we had the
number of rows here,

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this was essentially the
number of edges in our linkage.

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00:14:26,910 --> 00:14:30,111
And the number of columns
here was d times n,

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00:14:30,111 --> 00:14:31,860
this was the number
of degrees of freedom.

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00:14:31,860 --> 00:14:34,770
These are the coordinates
of all of our things.

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00:14:34,770 --> 00:14:40,335
And then x here was actually
our velocity vectors, and so on.

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00:14:40,335 --> 00:14:42,210
Well, actually, I should
probably write that.

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00:14:42,210 --> 00:14:45,090
So this, what we're calling
velocity vector's, d.

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00:14:45,090 --> 00:14:47,537
This is our-- It's
like conflict.

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00:14:47,537 --> 00:14:48,370
This is different d.

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00:14:48,370 --> 00:14:52,479
These are the d vectors,
the derivatives.

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00:14:52,479 --> 00:14:53,770
So, what would these things be?

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00:14:53,770 --> 00:14:58,935
Well the columns here
are going to be edges.

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00:15:02,190 --> 00:15:05,460
And the rows are going
to be coordinates.

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00:15:10,540 --> 00:15:14,020
And remember equilibrium
stress looks like this.

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00:15:14,020 --> 00:15:18,060
Basically for every
vertex we have

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00:15:18,060 --> 00:15:21,650
sum over all other
vertices, let's say

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00:15:21,650 --> 00:15:34,610
v w is an edge of, what
is it, stress of vw times

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00:15:34,610 --> 00:15:41,020
c of-- I forget whether c of
v minus c of w or the reverse,

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00:15:41,020 --> 00:15:41,860
equals 0.

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00:15:44,800 --> 00:15:49,510
So this should look good.

310
00:15:49,510 --> 00:15:52,810
The number of these
constraints is-- well here

311
00:15:52,810 --> 00:15:54,590
it says it's the
number of vertices.

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00:15:54,590 --> 00:15:58,440
But in fact, when you say equals
zero, this is a vector sum,

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00:15:58,440 --> 00:15:59,790
so this has d coordinates.

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00:15:59,790 --> 00:16:02,490
When you say that equals
0, that's d constraints.

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00:16:02,490 --> 00:16:05,530
So they're d times
n constraints.

316
00:16:05,530 --> 00:16:08,820
And how many things are
involved in the constraints?

317
00:16:08,820 --> 00:16:13,400
Well essentially the edges,
every edge has a term in here.

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00:16:13,400 --> 00:16:16,954
So that's kind of
why that looks right.

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00:16:16,954 --> 00:16:18,370
You have to go
through the algebra

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00:16:18,370 --> 00:16:21,180
and exactly what's written here,
to see that when you transpose

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00:16:21,180 --> 00:16:23,940
it, you do exactly
get this constraint.

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00:16:23,940 --> 00:16:27,420
And there's the issue
of equals 0, versus

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00:16:27,420 --> 00:16:28,990
greater than or equal to 0.

324
00:16:28,990 --> 00:16:31,380
So that's a little
bit more subtle.

325
00:16:35,200 --> 00:16:36,570
But that's the short version.

326
00:16:36,570 --> 00:16:40,400
And you also have to check that
the-- when your strut or cable,

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00:16:40,400 --> 00:16:43,450
you get just a sign
constraint on this thing.

328
00:16:43,450 --> 00:16:44,990
That's maybe a
little less obvious.

329
00:16:44,990 --> 00:16:47,560
But at least a high
level, this looks right.

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00:16:47,560 --> 00:16:50,420
And the relations
about the primal

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00:16:50,420 --> 00:16:52,760
and the dual linear
program about--

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00:16:52,760 --> 00:16:55,140
So for example,
this linear program,

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00:16:55,140 --> 00:16:57,290
which is characterizing
all infinitesimal motions,

334
00:16:57,290 --> 00:17:01,150
if you find an infinitesimal
motion, twice that vector

335
00:17:01,150 --> 00:17:03,520
or the set of vectors is
also an infinitesimal motion.

336
00:17:03,520 --> 00:17:05,990
So the solution space
to this linear program

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00:17:05,990 --> 00:17:10,050
is what's called a
convex polyhedral cone.

338
00:17:10,050 --> 00:17:14,140
Meaning-- I mean this is sort
of the beginning of a cone,

339
00:17:14,140 --> 00:17:15,390
but it goes off to infinity.

340
00:17:15,390 --> 00:17:17,240
You could take any
motion you find

341
00:17:17,240 --> 00:17:18,520
and scale it off to infinity.

342
00:17:18,520 --> 00:17:21,099
You always include
the origin 0, 0, 0.

343
00:17:21,099 --> 00:17:23,430
Because you can
always do no motion.

344
00:17:23,430 --> 00:17:25,450
But any motion can be scaled up.

345
00:17:25,450 --> 00:17:29,200
So if this thing has a solution
at all, other than the 0, 0, 0,

346
00:17:29,200 --> 00:17:34,370
motion, it's an unbounded lp,
meaning you go off to infinity.

347
00:17:34,370 --> 00:17:37,010
And when the primal
lp is unbounded

348
00:17:37,010 --> 00:17:42,210
you know things about
the dual, which I forget.

349
00:17:42,210 --> 00:17:45,670
But you cannot have one being
unbound and the other being

350
00:17:45,670 --> 00:17:46,440
something else.

351
00:17:46,440 --> 00:17:49,700
And so that guarantees that
there's an equilibrium stress.

352
00:17:49,700 --> 00:17:52,900
That's roughly how it works.

353
00:17:52,900 --> 00:17:54,620
Are there questions?