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PROFESSOR: All right, welcome.

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We have some fun origami
topics today, mostly

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related to a topic
called pleat folding.

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General idea of pleat
folding is to fold

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alternating mountains and
valleys back and forth.

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And you can do a lot of cool
things with pleat folding,

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and that's what I want
to talk about today.

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In particular, this
model is usually

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called the hyperbolic
paraboloid.

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It's pretty simple.

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You start with a square paper.

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You fold this crease
pattern, which

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is concentric squares
alternating mountain

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and valley, and you
fold the diagonals,

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alternating mountain and valley.

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You crush it all
together, you'll

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get this sort of x-shape.

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And you sort of let
go, give it a twist,

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and it pops into this
saddle form, which

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looks a lot like a
hyperbolic paraboloid, which

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is one of the standard
saddle surfaces.

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It's approximating a hyperbolic
paraboloid, we think,

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or we thought, and that's
why it's called that.

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So you should all
try this at home.

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It's a great model.

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It's very cool.

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It's an example of
something I like

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to call self-folding origami
because the paper basically

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folds itself.

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You put in a very simple crease
pattern, concentric squares.

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What could be simpler than that?

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And yet you get this really
cool 3D form automatically.

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Physics, in some sense, is
finding this form for us.

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You can take this
pleating idea and apply it

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to many different
crease patterns.

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Another fun one if you have
a compass lying around,

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to score circles is you just
crease concentric circles,

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alternating mountain and valley.

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In this case, you
also need to cut out

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a circular hole in the center.

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Then you get a
different saddle form,

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which we don't know
exactly what it is.

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Let's see, I have some various
fun things to talk about here.

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So what's going on with paper--
if I take a piece of paper,

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paper is an elastic material.

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It remembers that
it was made flat.

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It likes to be flat.

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If I curl a paper a
little bit, for example,

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it flattens back out.

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That's elastic memory.

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If however, I crease
a piece of paper,

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then that's called a
plastic deformation

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beyond the yield
point of the material.

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This works in sheet metal
also, lots of materials.

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I've effectively changed
the memory of the paper.

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It now wants to stay bent.

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If I try to unfold it, it
goes back to some angle.

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Depending on how
hard I crease it,

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it will go to a sharper angle.

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So that's basically
what's going on.

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In these surfaces, where
we crease the paper,

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it wants to stay bent.

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Where we don't crease the
paper, it wants to stay flat.

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It can't do all of those
things, but physics

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finds an equilibrium
among all those forces.

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So we simulated
that some years ago.

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This is with architecture
undergrad Jenna Fizel

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and Professor John
Ochsendorf, architecture.

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So on the right of
each of these examples,

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we have photographs of real
models, pleated foldings,

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a hyperbolic paraboloid,
squares, hexagon,

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octagon, just not so pretty.

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But we re-created that physical
model with a virtual model

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just using some
spring approximation

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to the forces I described.

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And here's an approximation
to a circle, where we just

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took a really big regular n-gon.

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This is real circles
and the simulation.

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So this confirms that what
I said is actually true.

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Those are the only
forces you need to model,

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and you get an
approximation of what

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really happens in real life.

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So instead of folding
real paper in this way,

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you could simulate it.

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Cool.

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Here is that simulator in
action with pleated hexagons,

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just to show you
what it looks like.

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So the creases are trying
to become more bent,

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at this point have
reached equilibrium,

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and you get your 3D form.

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Once you have that
3D form, so you

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have a virtual model of a
physical piece of paper,

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the natural thing to do
is build a physical model

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of the virtual model of the
physical piece of paper.

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And you've seen this
on the mailing list.

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This is what we're
calling origami skeletons.

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Here the vertices, which
you can see bigger here,

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are 3D printed plastic spheres,
and there's aluminum rods.

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The spheres have holes
at just the right angles

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to make exactly the 3D form
that was made by that simulator.

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So this is something you
could really only build

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if you had a virtual
model of that thing.

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So this is a fun example
of taking mathematics

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and our sort of the
computational understanding

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of how paper folds
and turning it

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into sculpture that requires
that mathematical basis.

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On the topic of sculpture,
at the very beginning--

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so this is early on in my PhD,
with my dad and my PhD advisor

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Anna Lubiw, same authors as
like the folding problem,

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one of our early explorations on
the mathematical sculpture side

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was to take
hyperbolic paraboloids

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and take a lot of them
and join them together

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to make polyhedral surfaces.

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These are what we
call hyperhedra

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because hyperbolic paraboloids
are also called hypars,

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I think, originally by
the architect community.

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So these are
algorithmically-generated

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sculpture.

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The input to the algorithm is
a polyhedron, like the cube,

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and the output is a way
to join hypars together

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to represent that cube.

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And the algorithm
is very simple.

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For each face of the
polyhedron-- so here it

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has four sides, you take
four hyperbolic paraboloids

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and join them together in a
cycle just sharing one edge.

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So you do that for each of
the six faces, so in all

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you have 24 hyperbolic
paraboloids.

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Then to join two
faces together, you

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join them along a pair
of edges like this.

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And when you do that,
you get an enclosure.

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It's a closed solid.

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And you can do that
for every-- here

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we're doing it for all
the Platonic solids.

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You can do it with any
polyhedron in theory.

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So it's like an infinite
family of sculptures here,

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which is kind of fun.

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Here is a simulation of
that cube just for fun.

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That works, too.

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And back to the curve creases.

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These are some examples I
showed way back in lecture 1.

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These are sculptures in the
permanent collection at MOMA.

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And the idea here
is instead of taking

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just one concentric circle
which goes around 360 degrees,

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we take a circular ramp
which goes around twice,

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and then join the ends.

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So we have 720
degrees of material

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there, which you might think
of as negative curvature.

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But it's a little
weird because there's

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the hole cut out in the center.

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So it's a little hard
to measure curvature,

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but you can actually define it.

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And these are three
different foldings

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with the same idea, slightly
different parameters

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in how big those ramps are, and
different numbers of creases.

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And you get very different
equilibrium forms.

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Again, this all sort
of self-folding paper

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wants to live in these
three configurations.

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Here's some that you
may not have seen.

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They're on the web.

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But these are taking
regular 360-degree circles

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but taking two or
three or four of them

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and joining them at a
couple of key points,

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and the rest is self-folding.

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So the big-- I mean, the sort of
powerful scientific engineering

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idea here is that you
could deploy complicated

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3D structures just
by manufacturing

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very simple flat
structures, maybe

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joining a few points together,
and then just say go.

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If you can manufacture
these things

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so that every crease
locally wants to bend,

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then you'll get these
3D forms automatically.

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This is especially powerful
at the nano scale, where

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you can't have your fingers
moving around and pressing

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things.

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But you can probably
use materials

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that, say, when heated
or you add some chemical,

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cause everything to fold.

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It's very easy for us from
all the chip fabrication

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we do to manufacture
flat crease patterns,

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and that will let us manufacture
3D things at the nano scale.

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You could also imagine it a
much larger scale like a space

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station-- like a space
station that looks like that.

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That would be pretty
cool-- and something

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where you don't want
to have to physically

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fold everything by hand.

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But if it could be
done automatically,

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life would be good.

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Now, we don't have
any algorithms

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for the reverse
engineering problem--

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If I give you a 3D
curved surface like this,

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find the crease pattern and
the joins that make it happen.

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But that's the goal.

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And towards that goal,
we make sculpture

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to explore the space
of what you can make.

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All right.

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I think I have one more
sculptural example here,

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which is combining these curve
creases with glassblowing.

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And to make glassblowing
more tactical,

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Marty here, our cameraman,
is blowing glass blindfold.

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Don't try this at home.

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So paper folding is a
very tactile experience.

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To make glassblowing more about
touching the material, which

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you're not usually supposed
to do because it's over 1,000

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degrees Fahrenheit-- blindfolds.

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AUDIENCE: 1,000
degrees Centigrade.

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PROFESSOR: 1,000
degrees centigrade.

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Wow.

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The temperature of
an erupting volcano.

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[MUSIC PLAYING]

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This is in the glassblowing
studio in 4.003 near here.

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He's made a whole
bunch of those.

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And here's what it looks like to
fold from scratch, so to speak,

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a concentric circle model.

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Of course, it's
accelerated in movie time.

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It only takes 10 seconds.

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In reality, it takes 10 minutes
after pre-scoring, maybe more.

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And now we get some
ship-in-the-bottle action.

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And this is a lot of fun
for us because not only

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do you have the
self-folding constraint,

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but you have this
enclosure constraint.

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These forms would
not look as exciting

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if they could
sprawl out, and they

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didn't have any constraints
to live inside these bubbles.

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So you get yet
another collection

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of forms from that glass.

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All right.

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I think that's the end of
our little sculpture tour.

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We'll come back to sculpture
at the end of this lecture.

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But I want to talk more
about-- oh, question.

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AUDIENCE: What was the
idea behind the blindfold?

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PROFESSOR: What was the
idea behind the blindfold?

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So paper folding is all
about touching material.

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Glassblowing is
usually very visual.

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It's about you look
at the material,

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and you don't usually touch it.

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So we wanted to unify the two
in order to put them together.

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Plus it was just a crazy idea.

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I think what happened
is I was in my office.

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My dad calls me up.

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He's blowing glass.

252
00:12:03,411 --> 00:12:04,890
He's like, I got
this great idea.

253
00:12:04,890 --> 00:12:06,260
Come over with a camera.

254
00:12:06,260 --> 00:12:09,260
So in that video, I
was the cameraman.

255
00:12:09,260 --> 00:12:11,904
And I was like,
all right, you want

256
00:12:11,904 --> 00:12:13,195
to burn yourself blowing glass.

257
00:12:13,195 --> 00:12:14,528
He didn't actually burn himself.

258
00:12:14,528 --> 00:12:15,480
Usually, he does.

259
00:12:15,480 --> 00:12:16,938
But blindfold, he
was more careful.

260
00:12:16,938 --> 00:12:20,305

261
00:12:20,305 --> 00:12:22,760
Yeah, blindfold glassblowing,
it's pretty crazy.

262
00:12:22,760 --> 00:12:24,960
That was his first
try ever blindfold

263
00:12:24,960 --> 00:12:29,300
glassblowing in that video.

264
00:12:29,300 --> 00:12:29,800
All right.

265
00:12:29,800 --> 00:12:31,800
So this hyperbolic
paraboloid, it's

266
00:12:31,800 --> 00:12:32,982
been around for a long time.

267
00:12:32,982 --> 00:12:34,190
I didn't mention the history.

268
00:12:34,190 --> 00:12:36,850
It goes back to the
Bauhaus in the late '20s.

269
00:12:36,850 --> 00:12:40,110
Albers, who I'm sure
many of you know of,

270
00:12:40,110 --> 00:12:43,780
taught a class about design.

271
00:12:43,780 --> 00:12:46,630
And he liked using
paper as a material that

272
00:12:46,630 --> 00:12:49,955
would force you to focus--
not worry about material

273
00:12:49,955 --> 00:12:53,500
in the sort of architectural
scale and what would stand up--

274
00:12:53,500 --> 00:12:55,260
and just think about design.

275
00:12:55,260 --> 00:13:00,000
And paper folding was really
tactile and good for that.

276
00:13:00,000 --> 00:13:01,880
And I'm not sure exactly
whether it was him

277
00:13:01,880 --> 00:13:05,930
or a student of his, it's
somebody in that period, 1927,

278
00:13:05,930 --> 00:13:09,810
'28, came up with this
model and the circular one.

279
00:13:09,810 --> 00:13:13,005
And then it's been taught
many times since then.

280
00:13:13,005 --> 00:13:19,310
Then it really hit it big in the
origami community in the 1980s

281
00:13:19,310 --> 00:13:23,760
by [INAUDIBLE], and since then
everybody's been folding it.

282
00:13:23,760 --> 00:13:24,330
Question?

283
00:13:24,330 --> 00:13:28,654
AUDIENCE: Is it not rigid,
but paper [INAUDIBLE]?

284
00:13:28,654 --> 00:13:29,820
PROFESSOR: Ah, you made one.

285
00:13:29,820 --> 00:13:31,260
Can I show it?

286
00:13:31,260 --> 00:13:34,570

287
00:13:34,570 --> 00:13:35,480
Thanks, [INAUDIBLE].

288
00:13:35,480 --> 00:13:40,850
Simple little example, but
here is concentric squares,

289
00:13:40,850 --> 00:13:43,490
and you crease it down.

290
00:13:43,490 --> 00:13:47,760
Normally you get this
kind of X. You let go,

291
00:13:47,760 --> 00:13:49,360
and it pops into
a little saddle.

292
00:13:49,360 --> 00:13:52,300
So this is a sort of
low-resolution one.

293
00:13:52,300 --> 00:13:54,510
If you spend twice
as much time, you

294
00:13:54,510 --> 00:13:57,190
can double the
resolution and so on.

295
00:13:57,190 --> 00:13:58,860
Your question, is it rigid?

296
00:13:58,860 --> 00:13:59,730
Probably not rigid.

297
00:13:59,730 --> 00:14:02,310
There's like this
degree of freedom.

298
00:14:02,310 --> 00:14:04,331
AUDIENCE: Rigid in
the sense of that

299
00:14:04,331 --> 00:14:06,894
between creases [INAUDIBLE].

300
00:14:06,894 --> 00:14:07,560
PROFESSOR: Yeah.

301
00:14:07,560 --> 00:14:10,830
What's happening in
between the creases?

302
00:14:10,830 --> 00:14:11,580
AUDIENCE: Exactly.

303
00:14:11,580 --> 00:14:13,913
PROFESSOR: That is exactly
the topic of today's lecture.

304
00:14:13,913 --> 00:14:16,840
In fact, we have a paper called
"How Does Paper Fold Between

305
00:14:16,840 --> 00:14:18,790
Creases" to address
exactly this question.

306
00:14:18,790 --> 00:14:19,682
I'm going to hang
onto it, if you

307
00:14:19,682 --> 00:14:21,910
don't mind because it'll
be useful to point at.

308
00:14:21,910 --> 00:14:24,780

309
00:14:24,780 --> 00:14:28,110
In fact, it is
completely impossible

310
00:14:28,110 --> 00:14:31,412
to fold that crease
pattern into this shape.

311
00:14:31,412 --> 00:14:34,540
And so your idea that there's
something weird going on

312
00:14:34,540 --> 00:14:36,980
is an idea-- we've
known for a long time

313
00:14:36,980 --> 00:14:40,030
that these faces
could not stay planar.

314
00:14:40,030 --> 00:14:44,170
I mean, at least visually it's
hard to see unless you stare

315
00:14:44,170 --> 00:14:45,680
at it with all the right angles.

316
00:14:45,680 --> 00:14:48,670
But the faces look
like they twist.

317
00:14:48,670 --> 00:14:49,390
Now, that's OK.

318
00:14:49,390 --> 00:14:53,770
Paper can do all sorts of
curving stuff without creasing.

319
00:14:53,770 --> 00:14:58,540
I mean, this is sort of like
a twist, no creases involved.

320
00:14:58,540 --> 00:15:02,520
So we thought this was
possible, but in fact, it's not.

321
00:15:02,520 --> 00:15:04,780
So this is the big surprise.

322
00:15:04,780 --> 00:15:08,460
It's something we
discovered just last year,

323
00:15:08,460 --> 00:15:10,620
and I'm going to prove
that to you today.

324
00:15:10,620 --> 00:15:13,860
So the theorem is, if I have
this concentric square crease

325
00:15:13,860 --> 00:15:16,370
pattern, even ignoring
the mountains and valleys,

326
00:15:16,370 --> 00:15:19,370
it is impossible to
fold that crease pattern

327
00:15:19,370 --> 00:15:21,430
into anything that is not flat.

328
00:15:21,430 --> 00:15:23,180
So you can, of course,
not fold it at all,

329
00:15:23,180 --> 00:15:24,680
then none of the
creases get folded.

330
00:15:24,680 --> 00:15:27,890
You can collapse it
all the way down.

331
00:15:27,890 --> 00:15:29,610
That probably
isn't even allowed.

332
00:15:29,610 --> 00:15:31,030
But never mind.

333
00:15:31,030 --> 00:15:33,490
But what we really
want is a 3D form

334
00:15:33,490 --> 00:15:35,840
where every crease is
bent by a non-zero angle

335
00:15:35,840 --> 00:15:38,500
and also not by 180 degrees.

336
00:15:38,500 --> 00:15:40,810
So we'll call that a
proper folding, something

337
00:15:40,810 --> 00:15:44,854
where every crease is
strictly between 0 and 180,

338
00:15:44,854 --> 00:15:46,520
which is what we want
in these 3D forms.

339
00:15:46,520 --> 00:15:48,620
And the theorem is,
that is impossible

340
00:15:48,620 --> 00:15:51,560
for the hyperbolic paraboloid.

341
00:15:51,560 --> 00:15:52,422
There is none.

342
00:15:52,422 --> 00:15:57,250

343
00:15:57,250 --> 00:16:00,060
So that's weird because
we fold them all the time.

344
00:16:00,060 --> 00:16:02,425
We've been folding
them for like 11 years,

345
00:16:02,425 --> 00:16:07,210
and other people have been
folding them for 80 years.

346
00:16:07,210 --> 00:16:10,940
What's happening with
the real piece of paper?

347
00:16:10,940 --> 00:16:13,310
Well, one possible answer
is that there's more creases

348
00:16:13,310 --> 00:16:15,330
that you don't
see in this model.

349
00:16:15,330 --> 00:16:17,410
And if you add
more creases, it is

350
00:16:17,410 --> 00:16:19,590
possible to fold
something that looks

351
00:16:19,590 --> 00:16:22,090
like a hyperbolic paraboloid.

352
00:16:22,090 --> 00:16:24,960
So here we have the regular
crease pattern in black,

353
00:16:24,960 --> 00:16:27,500
and then I've added
some purple diagonals.

354
00:16:27,500 --> 00:16:29,770
Wherever we had a
black trapezoid,

355
00:16:29,770 --> 00:16:32,910
I've added one purple
diagonal to triangulate

356
00:16:32,910 --> 00:16:34,000
the crease pattern.

357
00:16:34,000 --> 00:16:36,200
And here I've chosen what
seems to be an especially

358
00:16:36,200 --> 00:16:38,700
good triangulation
where I zigzag

359
00:16:38,700 --> 00:16:44,240
back and forth
within one quarter.

360
00:16:44,240 --> 00:16:49,600
And then also from around a
ring, I zigzag back and forth.

361
00:16:49,600 --> 00:16:52,370
So that triangulation
folds into this.

362
00:16:52,370 --> 00:16:54,030
That's another theorem.

363
00:16:54,030 --> 00:16:57,350
This is construction on a
computer, obviously, where

364
00:16:57,350 --> 00:17:03,130
we have 16 rings, and
the central crease here

365
00:17:03,130 --> 00:17:06,980
is folded by an
angle of 30 degrees.

366
00:17:06,980 --> 00:17:08,839
That's the theta.

367
00:17:08,839 --> 00:17:12,880
Notice I also had to
remove part of a diagonal.

368
00:17:12,880 --> 00:17:15,619
That's also necessary.

369
00:17:15,619 --> 00:17:17,170
In the center, you
can't crease both

370
00:17:17,170 --> 00:17:22,490
of those, the folds in an
X, by a non-zero amount.

371
00:17:22,490 --> 00:17:24,490
Those are all things we're
going to prove today.

372
00:17:24,490 --> 00:17:26,871
But before we prove
the negative things,

373
00:17:26,871 --> 00:17:28,329
I want to prove
the positive thing,

374
00:17:28,329 --> 00:17:30,920
that this thing can
actually be built.

375
00:17:30,920 --> 00:17:33,320
That may seem obvious
because here it is.

376
00:17:33,320 --> 00:17:33,940
I built one.

377
00:17:33,940 --> 00:17:37,280
But because there's all these
arguments of, well, here it

378
00:17:37,280 --> 00:17:41,010
is, I built one, we've got to
be especially careful about,

379
00:17:41,010 --> 00:17:43,790
does this thing
really truly exist?

380
00:17:43,790 --> 00:17:46,220
So let me tell you
how it's constructed

381
00:17:46,220 --> 00:17:49,624
and how we can actually prove
not only can we build it

382
00:17:49,624 --> 00:17:52,040
approximately on a computer,
but we can prove there really

383
00:17:52,040 --> 00:17:54,655
is one there even though
we don't have it exactly.

384
00:17:54,655 --> 00:17:57,300

385
00:17:57,300 --> 00:18:01,920
So it's actually pretty
cool and pretty easy.

386
00:18:01,920 --> 00:18:08,355
So this is the triangulated
hyperbolic paraboloid.

387
00:18:08,355 --> 00:18:13,280

388
00:18:13,280 --> 00:18:16,600
So the idea is to work
from the inside out.

389
00:18:16,600 --> 00:18:19,140
We're going to start
with this central square

390
00:18:19,140 --> 00:18:25,810
here-- and let me draw
it over here-- fold it

391
00:18:25,810 --> 00:18:28,390
by some angle theta.

392
00:18:28,390 --> 00:18:31,072
That, I think, we
all know how to do.

393
00:18:31,072 --> 00:18:35,200
It's some rotation matrix.

394
00:18:35,200 --> 00:18:36,840
And then I want to
work my way out.

395
00:18:36,840 --> 00:18:39,410
So at this point,
I know the location

396
00:18:39,410 --> 00:18:40,840
of these four vertices.

397
00:18:40,840 --> 00:18:45,290
In general, I know some
square and everything inside.

398
00:18:45,290 --> 00:18:52,510
I want to figure out
everything in the next square.

399
00:18:52,510 --> 00:18:58,690
And the creases are going to
look like this, from my zigzag

400
00:18:58,690 --> 00:18:59,650
pattern.

401
00:18:59,650 --> 00:19:01,935
So there's stuff in
here, which is known.

402
00:19:01,935 --> 00:19:04,550

403
00:19:04,550 --> 00:19:07,665
So I already have figured out
the location of these vertices.

404
00:19:07,665 --> 00:19:09,900
And I want to know,
how do I figure out

405
00:19:09,900 --> 00:19:11,330
these vertices on the outside?

406
00:19:11,330 --> 00:19:14,260
If I can do that, I just repeat,
and I get a bigger and bigger

407
00:19:14,260 --> 00:19:17,300
hyperbolic paraboloid.

408
00:19:17,300 --> 00:19:21,840
So what I do is pretty easy.

409
00:19:21,840 --> 00:19:24,660
I'm going to look at this
vertex first and say,

410
00:19:24,660 --> 00:19:27,720
well, I have these
three points, and I

411
00:19:27,720 --> 00:19:31,860
have known distances
between those three points.

412
00:19:31,860 --> 00:19:34,640
I'm going to assume here,
and we'll see why later,

413
00:19:34,640 --> 00:19:37,250
that each of these
creases remains

414
00:19:37,250 --> 00:19:38,990
a straight line in
three dimensions.

415
00:19:38,990 --> 00:19:43,700
It's not obvious, because
creases might bend around.

416
00:19:43,700 --> 00:19:45,840
But let's assume
that we were going

417
00:19:45,840 --> 00:19:46,966
to fold it in a simple way.

418
00:19:46,966 --> 00:19:48,881
These triangles are going
to remain triangles.

419
00:19:48,881 --> 00:19:50,370
These edges will
remain straight.

420
00:19:50,370 --> 00:19:51,840
And they have to
remain the same length

421
00:19:51,840 --> 00:19:53,006
because we're paper folding.

422
00:19:53,006 --> 00:19:54,480
You can't stretch.

423
00:19:54,480 --> 00:19:56,800
And so I know these
three points in 3D,

424
00:19:56,800 --> 00:19:58,630
and I know these
three distances, just

425
00:19:58,630 --> 00:20:03,120
measuring them on the crease
pattern, flat crease pattern.

426
00:20:03,120 --> 00:20:07,750
So this point is on the
intersection of three spheres,

427
00:20:07,750 --> 00:20:10,250
and the centers of the
spheres are not collinear.

428
00:20:10,250 --> 00:20:13,420
This is a trick we used last
class but in two dimensions.

429
00:20:13,420 --> 00:20:16,590
So I have three
non-collinear points.

430
00:20:16,590 --> 00:20:19,450
I have three spheres--
little hard to draw--

431
00:20:19,450 --> 00:20:20,570
centered at them.

432
00:20:20,570 --> 00:20:22,210
The intersection of
two of the spheres

433
00:20:22,210 --> 00:20:26,990
is going to be a circle,
and the intersection

434
00:20:26,990 --> 00:20:33,040
with the third sphere is going
to be actually two points.

435
00:20:33,040 --> 00:20:36,950
So the intersection of
three spheres is two points.

436
00:20:36,950 --> 00:20:38,330
So a little bit of ambiguity.

437
00:20:38,330 --> 00:20:40,440
But it turns out
one of these points

438
00:20:40,440 --> 00:20:43,020
will set the mountain-valley
assignment correctly,

439
00:20:43,020 --> 00:20:44,770
and the other one will
set it incorrectly.

440
00:20:44,770 --> 00:20:47,080
Like if I want this to be
a mountain, one of them

441
00:20:47,080 --> 00:20:49,260
will be inside and one
of them will be outside.

442
00:20:49,260 --> 00:20:50,760
One of them will
make this mountain,

443
00:20:50,760 --> 00:20:51,820
one will make it valley.

444
00:20:51,820 --> 00:20:53,930
So it turns out if
you actually do this,

445
00:20:53,930 --> 00:20:56,620
it's uniquely determined
at every step of the way.

446
00:20:56,620 --> 00:20:58,620
Because you know what
mountain-valley assignment

447
00:20:58,620 --> 00:21:02,280
you're aiming for, you uniquely
figure out what this point is.

448
00:21:02,280 --> 00:21:04,010
By the same reasoning,
you can figure out

449
00:21:04,010 --> 00:21:08,020
what this point is, intersect
these three spheres.

450
00:21:08,020 --> 00:21:09,710
And then once I know
those two points,

451
00:21:09,710 --> 00:21:11,376
I can figure out this
point because it's

452
00:21:11,376 --> 00:21:14,130
the intersection of
these three spheres,

453
00:21:14,130 --> 00:21:16,070
with now these points are known.

454
00:21:16,070 --> 00:21:20,365
And then I can figure out this
point in the symmetric way.

455
00:21:20,365 --> 00:21:21,250
OK?

456
00:21:21,250 --> 00:21:22,250
That's how you do it.

457
00:21:22,250 --> 00:21:22,990
We repeat.

458
00:21:22,990 --> 00:21:25,890
And that is exactly how
this model is built.

459
00:21:25,890 --> 00:21:29,096

460
00:21:29,096 --> 00:21:33,620
Now, this is still not a
proof that it really folds.

461
00:21:33,620 --> 00:21:35,052
It's a construction method.

462
00:21:35,052 --> 00:21:36,135
I'd say it's an algorithm.

463
00:21:36,135 --> 00:21:38,720

464
00:21:38,720 --> 00:21:41,160
If I start out with
the inner thing here

465
00:21:41,160 --> 00:21:43,640
and I give you some angle,
I can approximate where

466
00:21:43,640 --> 00:21:45,030
these vertices are.

467
00:21:45,030 --> 00:21:47,350
And then I can keep going
and, at each step of the way,

468
00:21:47,350 --> 00:21:49,469
approximate where
the vertices are.

469
00:21:49,469 --> 00:21:52,010
The worry is the approximation
gets worse and worse because I

470
00:21:52,010 --> 00:21:53,980
have this propagating
error effect.

471
00:21:53,980 --> 00:21:56,020
The larger n is,
the number of rings

472
00:21:56,020 --> 00:22:01,830
in my hyperbolic paraboloid,
the lower the accuracy will be.

473
00:22:01,830 --> 00:22:03,490
But I get to
choose, at each step

474
00:22:03,490 --> 00:22:05,140
of the way, how
much precision I use

475
00:22:05,140 --> 00:22:06,790
to compute all of these numbers.

476
00:22:06,790 --> 00:22:10,150
Let me tell you how this
computation actually works.

477
00:22:10,150 --> 00:22:14,210
I did this in Mathematica, or
we did this in Mathematica.

478
00:22:14,210 --> 00:22:15,620
Here's Mathematica.

479
00:22:15,620 --> 00:22:17,636
And I asked
Mathematica, well, here

480
00:22:17,636 --> 00:22:20,790
is the equations for the
intersection of three spheres.

481
00:22:20,790 --> 00:22:24,160
I say, well, squared Euclidean
distance between this point

482
00:22:24,160 --> 00:22:26,724
and some unknown thing x, y,
z is this distance squared.

483
00:22:26,724 --> 00:22:28,890
And here's the second one,
and here's the third one.

484
00:22:28,890 --> 00:22:31,180
It's kind of tiny.

485
00:22:31,180 --> 00:22:32,340
And then here's the answer.

486
00:22:32,340 --> 00:22:36,615

487
00:22:36,615 --> 00:22:38,650
It's a little messy.

488
00:22:38,650 --> 00:22:41,780
In fact, you can ask how
many terms are in this thing.

489
00:22:41,780 --> 00:22:46,750
It's 444,000 terms
in the solution.

490
00:22:46,750 --> 00:22:50,025
Don't try this at
home, I guess, by hand.

491
00:22:50,025 --> 00:22:51,900
Intersection of three
spheres is a bit messy.

492
00:22:51,900 --> 00:22:54,880
There's probably a cleaner way
than what Mathematica does,

493
00:22:54,880 --> 00:22:56,600
but this is an easy
way to get it in.

494
00:22:56,600 --> 00:22:59,150

495
00:22:59,150 --> 00:23:02,640
Now, in theory-- so if you
look at all of this stuff,

496
00:23:02,640 --> 00:23:04,210
all that's happening
is you have all

497
00:23:04,210 --> 00:23:08,920
the various inputs, x1,
y1, x2, z2, and so on.

498
00:23:08,920 --> 00:23:13,152
You have some numbers
like 4 and minus and 2.

499
00:23:13,152 --> 00:23:14,860
All you're doing is
taking these numbers,

500
00:23:14,860 --> 00:23:18,640
adding them, multiplying them,
dividing them, squaring them,

501
00:23:18,640 --> 00:23:20,210
subtracting them.

502
00:23:20,210 --> 00:23:22,290
And at some point,
taking square roots.

503
00:23:22,290 --> 00:23:24,110
I don't know if
we'll ever see that.

504
00:23:24,110 --> 00:23:26,735
I mean, if we search through,
there are some square roots,

505
00:23:26,735 --> 00:23:28,735
but they're going to be
in very specific places.

506
00:23:28,735 --> 00:23:31,782

507
00:23:31,782 --> 00:23:35,140
Yeah, so I probably
won't find one instantly.

508
00:23:35,140 --> 00:23:37,990
Oh, there's one, square root.

509
00:23:37,990 --> 00:23:39,927
So all I'm claiming
at this point

510
00:23:39,927 --> 00:23:41,760
is you can compute the
intersection of three

511
00:23:41,760 --> 00:23:44,570
spheres just using basic
arithmetic and square roots.

512
00:23:44,570 --> 00:23:46,745
This is called a
radical expression,

513
00:23:46,745 --> 00:23:50,670
not because it's so
amazing and controversial.

514
00:23:50,670 --> 00:23:54,790
But it's radical because that
symbol is called rad, I guess,

515
00:23:54,790 --> 00:23:56,960
and radical refers
to square roots.

516
00:23:56,960 --> 00:24:00,160
I guess they were
crazy at the time.

517
00:24:00,160 --> 00:24:01,440
All right.

518
00:24:01,440 --> 00:24:04,160
Actually, I want to
stay with Mathematica.

519
00:24:04,160 --> 00:24:06,770
So that's the idea.

520
00:24:06,770 --> 00:24:10,040
Still not a proof, just
a construction method.

521
00:24:10,040 --> 00:24:12,640
But here's a trick,
which you can

522
00:24:12,640 --> 00:24:15,580
use to turn a construction
into an actual proof

523
00:24:15,580 --> 00:24:17,460
that this thing exists.

524
00:24:17,460 --> 00:24:22,240
It's called interval arithmetic,
little computer science lesson.

525
00:24:22,240 --> 00:24:26,410

526
00:24:26,410 --> 00:24:29,640
How many people have heard
of interval arithmetic?

527
00:24:29,640 --> 00:24:31,720
No one-- one person, all right.

528
00:24:31,720 --> 00:24:34,200
Yeah, even computer scientists
probably don't necessarily

529
00:24:34,200 --> 00:24:37,200
know this unless they've
seen some numerical analysis.

530
00:24:37,200 --> 00:24:40,590
So it's an idea that
instead of computing

531
00:24:40,590 --> 00:24:42,950
an approximate location
for this point,

532
00:24:42,950 --> 00:24:45,740
I want to get not only an
approximate location, but also

533
00:24:45,740 --> 00:24:49,270
an error bound on how
much I don't know it.

534
00:24:49,270 --> 00:24:52,240
So there are three
coordinates to every point.

535
00:24:52,240 --> 00:24:55,240
For each coordinate, every
number I want to represent I'm

536
00:24:55,240 --> 00:24:57,770
going to represent
as an interval

537
00:24:57,770 --> 00:25:01,196
from some lower bound
to some upper bound.

538
00:25:01,196 --> 00:25:03,070
So say, well, I don't
know what the value is,

539
00:25:03,070 --> 00:25:08,710
but I know that it is somewhere
in between these two numbers.

540
00:25:08,710 --> 00:25:11,800
So represent every
number like this.

541
00:25:11,800 --> 00:25:14,831

542
00:25:14,831 --> 00:25:16,580
And then you just need
to define, how do I

543
00:25:16,580 --> 00:25:18,871
add numbers, subtract them,
multiply them, divide them,

544
00:25:18,871 --> 00:25:20,460
and take square roots?

545
00:25:20,460 --> 00:25:23,910
And the answer is carefully.

546
00:25:23,910 --> 00:25:25,980
I'll show you one example.

547
00:25:25,980 --> 00:25:30,150
If I have two numbers,
L1, U1, and L2, U2,

548
00:25:30,150 --> 00:25:33,450
and I want to add
them together, then I

549
00:25:33,450 --> 00:25:35,200
believe it's pretty simple.

550
00:25:35,200 --> 00:25:40,060
It's just L1 plus
L2, and U1 plus U2.

551
00:25:40,060 --> 00:25:42,440
This is in perfect
mathematical world

552
00:25:42,440 --> 00:25:43,820
where there's no round off.

553
00:25:43,820 --> 00:25:45,540
In reality, when
you add two numbers,

554
00:25:45,540 --> 00:25:47,067
you lose a bit of precision.

555
00:25:47,067 --> 00:25:49,150
And you make sure that
when you add these numbers,

556
00:25:49,150 --> 00:25:51,820
you always round down, and
when you add these numbers

557
00:25:51,820 --> 00:25:53,600
you always round up.

558
00:25:53,600 --> 00:25:57,680
So that way, these intervals,
the accumulation error

559
00:25:57,680 --> 00:26:00,750
is realized by these
intervals getting wider.

560
00:26:00,750 --> 00:26:02,280
You can start with
them super tiny,

561
00:26:02,280 --> 00:26:03,780
maybe, in fact, 0
length because you

562
00:26:03,780 --> 00:26:07,430
know exactly where these points
are, so they're 0 intervals.

563
00:26:07,430 --> 00:26:10,150
But then, because of the
error in every operation

564
00:26:10,150 --> 00:26:14,040
you do on the computer,
the intervals will widen.

565
00:26:14,040 --> 00:26:17,040
But as long as you
do this computation

566
00:26:17,040 --> 00:26:21,820
with enough bits of precision,
they'll widen slowly.

567
00:26:21,820 --> 00:26:24,390
If you do it slowly
enough-- if you do it

568
00:26:24,390 --> 00:26:26,870
with enough precision, and
the errors accumulate slowly

569
00:26:26,870 --> 00:26:31,070
enough, you can build your
n-ring hyperbolic paraboloid

570
00:26:31,070 --> 00:26:34,660
without any error, or
without too much error

571
00:26:34,660 --> 00:26:37,030
to make things go wrong.

572
00:26:37,030 --> 00:26:41,840
What could go wrong is that
these three spheres might not

573
00:26:41,840 --> 00:26:42,960
intersect.

574
00:26:42,960 --> 00:26:44,260
That's really the worry here.

575
00:26:44,260 --> 00:26:45,320
There are two things
that could go wrong.

576
00:26:45,320 --> 00:26:46,986
One is that the spheres
don't intersect.

577
00:26:46,986 --> 00:26:49,590
The other is that the
surface intersects itself.

578
00:26:49,590 --> 00:26:51,480
Both would be bad.

579
00:26:51,480 --> 00:26:54,270
How do you tell when the
spheres don't intersect?

580
00:26:54,270 --> 00:26:58,704
Well, we have this formula,
this ginormous formula.

581
00:26:58,704 --> 00:27:00,120
The only thing
that could go wrong

582
00:27:00,120 --> 00:27:03,056
is that you take a square
root of a negative number.

583
00:27:03,056 --> 00:27:04,930
Now, hopefully that
never happens in reality.

584
00:27:04,930 --> 00:27:06,890
But what could
happen for us is we

585
00:27:06,890 --> 00:27:10,110
have such a poor
approximation of our numbers.

586
00:27:10,110 --> 00:27:13,200
And we try to take the
square root of some interval,

587
00:27:13,200 --> 00:27:16,440
and L is less than 0.

588
00:27:16,440 --> 00:27:19,310
Now, probably the actual
number is more than 0,

589
00:27:19,310 --> 00:27:21,000
but we can't tell,
and we just know

590
00:27:21,000 --> 00:27:22,708
the number is somewhere
in this interval.

591
00:27:22,708 --> 00:27:25,060
If the lower bound
is negative, then we

592
00:27:25,060 --> 00:27:27,260
can't take the square root.

593
00:27:27,260 --> 00:27:30,750
We don't know what the
right answer would be.

594
00:27:30,750 --> 00:27:33,940
So Mathematica conveniently
can do all of this for you.

595
00:27:33,940 --> 00:27:36,360
If you just plug in
intervals instead of numbers,

596
00:27:36,360 --> 00:27:38,610
it will do interval
arithmetic correctly,

597
00:27:38,610 --> 00:27:40,850
and you can tell it
what precision to do.

598
00:27:40,850 --> 00:27:44,370
And that is how we
found this example.

599
00:27:44,370 --> 00:27:51,840
In fact, I will show
you the notebook,

600
00:27:51,840 --> 00:27:54,500
and there's all this
computation and stuff.

601
00:27:54,500 --> 00:27:56,770
This is all, I believe,
it's in our paper,

602
00:27:56,770 --> 00:27:58,006
so if you want the code.

603
00:27:58,006 --> 00:27:59,380
So we compute the
crease patterns

604
00:27:59,380 --> 00:28:01,450
so I can measure
all the distances.

605
00:28:01,450 --> 00:28:03,580
And there's some collision
detection and stuff

606
00:28:03,580 --> 00:28:05,270
to make sure
everything's working.

607
00:28:05,270 --> 00:28:06,680
And it computes some rings.

608
00:28:06,680 --> 00:28:09,920
And then, actually, the one I
want to show you is this one.

609
00:28:09,920 --> 00:28:11,420
This is model we've
been looking at,

610
00:28:11,420 --> 00:28:13,216
and here it is in
three dimensions.

611
00:28:13,216 --> 00:28:15,860

612
00:28:15,860 --> 00:28:19,690
The way it's being rendered is
a bit odd, but you get the idea.

613
00:28:19,690 --> 00:28:21,900
That is the hyperbolic
paraboloid we construct.

614
00:28:21,900 --> 00:28:24,120
Now, in reality,
each of these points

615
00:28:24,120 --> 00:28:25,500
is actually a little interval.

616
00:28:25,500 --> 00:28:26,958
It's hard to draw
that because it's

617
00:28:26,958 --> 00:28:29,780
smaller than the
resolution of the screen.

618
00:28:29,780 --> 00:28:34,230
I think I compute
up here the-- this

619
00:28:34,230 --> 00:28:36,670
is like how much we've
messed up the edge lengths.

620
00:28:36,670 --> 00:28:38,170
It's like all of
these 9's, and then

621
00:28:38,170 --> 00:28:40,692
this little area at the end.

622
00:28:40,692 --> 00:28:42,850
OK.

623
00:28:42,850 --> 00:28:44,210
Cool.

624
00:28:44,210 --> 00:28:46,970
So the point is, you just do
this with enough precision.

625
00:28:46,970 --> 00:28:48,550
As long as you don't
end up computing

626
00:28:48,550 --> 00:28:50,610
any negative square
roots, you get a surface

627
00:28:50,610 --> 00:28:52,880
where every point is
actually a little box.

628
00:28:52,880 --> 00:28:55,470
You don't know where the
point is exactly in that box.

629
00:28:55,470 --> 00:28:57,366
Then you check for
collision, just knowing

630
00:28:57,366 --> 00:28:59,240
that the points are
somewhere in those boxes.

631
00:28:59,240 --> 00:29:01,460
You try to intersect
two triangles.

632
00:29:01,460 --> 00:29:04,300
As long as they intersect away
from the boxes, you're OK.

633
00:29:04,300 --> 00:29:07,350
Then you know there's
no actual intersection.

634
00:29:07,350 --> 00:29:10,540
And we have done that
for this triangulation up

635
00:29:10,540 --> 00:29:15,540
to-- get it right--
up to n equals

636
00:29:15,540 --> 00:29:20,400
100, 100 rings, and where
the theta angle here

637
00:29:20,400 --> 00:29:27,970
is any even number between 2
and 178 degrees, so 2 degrees,

638
00:29:27,970 --> 00:29:31,380
4 degrees, up to 178 degrees.

639
00:29:31,380 --> 00:29:33,982
We're not interested in 180
because that would be flat.

640
00:29:33,982 --> 00:29:34,940
That's not interesting.

641
00:29:34,940 --> 00:29:36,464
We're not interested in 0.

642
00:29:36,464 --> 00:29:38,130
Every even number of
degrees in between.

643
00:29:38,130 --> 00:29:39,240
Why even?

644
00:29:39,240 --> 00:29:40,840
Just because of
the representation.

645
00:29:40,840 --> 00:29:42,090
I'm sure it works for the odd.

646
00:29:42,090 --> 00:29:43,530
In fact, I'm pretty
sure it should

647
00:29:43,530 --> 00:29:46,790
work for any theta and any n.

648
00:29:46,790 --> 00:29:49,060
But this technique will
only let us to prove it

649
00:29:49,060 --> 00:29:51,501
for specific theta
and specific n.

650
00:29:51,501 --> 00:29:53,000
Because we're just
using a computer,

651
00:29:53,000 --> 00:29:54,730
it's only going to
check one example.

652
00:29:54,730 --> 00:29:58,030
We don't have a nice way
to do them all at once.

653
00:29:58,030 --> 00:30:00,590
In case you're
interested-- oh, here I've

654
00:30:00,590 --> 00:30:06,184
built it out of sheet
aluminum, I think?

655
00:30:06,184 --> 00:30:07,100
AUDIENCE: Galvanized--

656
00:30:07,100 --> 00:30:08,480
PROFESSOR: Galvanized steel?

657
00:30:08,480 --> 00:30:09,380
All right.

658
00:30:09,380 --> 00:30:11,780
Water jet cut along the creases.

659
00:30:11,780 --> 00:30:14,180
Now, steel's a little tricky
to not add extra pieces

660
00:30:14,180 --> 00:30:16,300
by accident, so
there's a few defects.

661
00:30:16,300 --> 00:30:21,160
But this is also some kind of
verification that it works.

662
00:30:21,160 --> 00:30:25,750
In the computer here, the
number of digits of precision.

663
00:30:25,750 --> 00:30:27,670
So for whatever
reason, Mathematica

664
00:30:27,670 --> 00:30:29,450
speaks base 10
instead of base 2.

665
00:30:29,450 --> 00:30:34,000
Which is weird to me, but maybe
intuitive to everyone else,

666
00:30:34,000 --> 00:30:35,370
non-computer scientist, I guess.

667
00:30:35,370 --> 00:30:38,090
So there's normal sort
of floating points

668
00:30:38,090 --> 00:30:40,420
like 16 digits of
precision or so.

669
00:30:40,420 --> 00:30:43,380
And there you can build this
thing up to n equals 3, 3

670
00:30:43,380 --> 00:30:44,680
rings.

671
00:30:44,680 --> 00:30:47,780
But you go up to about
1,000 digits of precision,

672
00:30:47,780 --> 00:30:49,520
and then we can get beyond 100.

673
00:30:49,520 --> 00:30:52,610
It depends, though, on what
the angle is that you fold.

674
00:30:52,610 --> 00:30:56,160
So for angles that are
very big, close to 90,

675
00:30:56,160 --> 00:30:59,220
we had to go up to 2,000
digits of precision.

676
00:30:59,220 --> 00:31:01,150
And I don't know how
far n can go here.

677
00:31:01,150 --> 00:31:04,760
I think maybe a couple 100,
but I'm not sure exactly.

678
00:31:04,760 --> 00:31:07,340
The larger n is, of course,
the more accumulation you have,

679
00:31:07,340 --> 00:31:08,960
and so you have to
do every operation

680
00:31:08,960 --> 00:31:11,410
with more digits of precision.

681
00:31:11,410 --> 00:31:15,030
But the conjecture would be,
any specific data and n, there's

682
00:31:15,030 --> 00:31:17,590
some precision for which
this approach will work,

683
00:31:17,590 --> 00:31:21,790
but we don't actually
know how to prove that.

684
00:31:21,790 --> 00:31:24,151
All right.

685
00:31:24,151 --> 00:31:27,380
So that's a triangulated hypar.

686
00:31:27,380 --> 00:31:29,880
Here's another
triangulation, which we also

687
00:31:29,880 --> 00:31:34,130
studied in this paper, where
instead of zigzagging up

688
00:31:34,130 --> 00:31:36,130
one quarter of the
square, we just

689
00:31:36,130 --> 00:31:38,765
do all the diagonals
in the same direction.

690
00:31:38,765 --> 00:31:44,520
We still zigzag around a
ring but not between rings.

691
00:31:44,520 --> 00:31:47,486
And this can also fold,
as the other model

692
00:31:47,486 --> 00:31:48,860
that was in the
Mathematica file.

693
00:31:48,860 --> 00:31:50,090
I won't bother going there.

694
00:31:50,090 --> 00:31:54,710
This is for a small fold angle
of just 8 degrees in here,

695
00:31:54,710 --> 00:31:55,480
n equals 16.

696
00:31:55,480 --> 00:31:57,480
And there's a reason I
did it for a small angle,

697
00:31:57,480 --> 00:32:00,170
because it doesn't
work for large angles.

698
00:32:00,170 --> 00:32:03,040
So that triangulation we
started with was a good one.

699
00:32:03,040 --> 00:32:06,010
It's actually not the
first one we tried,

700
00:32:06,010 --> 00:32:09,170
but it is sort of natural.

701
00:32:09,170 --> 00:32:11,210
If you fold by a
very small amount,

702
00:32:11,210 --> 00:32:13,090
you can get all the
way up to 133 rings,

703
00:32:13,090 --> 00:32:14,560
but then it fails after that.

704
00:32:14,560 --> 00:32:18,910
The spheres just don't
intersect anymore.

705
00:32:18,910 --> 00:32:21,810
Depending on how much you fold,
like if I fold by 22 degrees,

706
00:32:21,810 --> 00:32:23,590
I can get up to 13 rings.

707
00:32:23,590 --> 00:32:25,510
But here I wanted
to get to 16 rings

708
00:32:25,510 --> 00:32:29,010
so I only went up to, what was
it, 8 degrees or something.

709
00:32:29,010 --> 00:32:32,360
I could have gotten a
little beyond but not a lot.

710
00:32:32,360 --> 00:32:34,840
And up at 178, you can
still get three rings,

711
00:32:34,840 --> 00:32:36,350
but it doesn't fold after that.

712
00:32:36,350 --> 00:32:39,450

713
00:32:39,450 --> 00:32:43,920
So triangulation matters,
which triangulation you do.

714
00:32:43,920 --> 00:32:47,280
Going back to the
one that works well,

715
00:32:47,280 --> 00:32:48,910
natural question
is, is it actually

716
00:32:48,910 --> 00:32:51,040
a hyperbolic paraboloid?

717
00:32:51,040 --> 00:32:53,540
It's called a hyperbolic
paraboloid because--

718
00:32:53,540 --> 00:32:56,850
or there is a surface called
a hyperbolic paraboloid, where

719
00:32:56,850 --> 00:33:00,280
if you take a cross-section
this way, you get a parabola.

720
00:33:00,280 --> 00:33:01,980
That's the paraboloid part.

721
00:33:01,980 --> 00:33:04,817
And if you take a cross-section
this way, you get a hyperbola.

722
00:33:04,817 --> 00:33:06,400
That's a little
harder to draw, but it

723
00:33:06,400 --> 00:33:08,570
goes around here
and around here.

724
00:33:08,570 --> 00:33:13,330
Hyperbolas have two
connecting components.

725
00:33:13,330 --> 00:33:16,565
So let's look at the
parabolic part here.

726
00:33:16,565 --> 00:33:18,690
We're supposed to be
approximating a parabola if we

727
00:33:18,690 --> 00:33:22,280
look at all the points on
the top here and also maybe

728
00:33:22,280 --> 00:33:24,010
on the bottom.

729
00:33:24,010 --> 00:33:25,290
So that's what I've drawn.

730
00:33:25,290 --> 00:33:27,420
The green lines, which
are very light so a little

731
00:33:27,420 --> 00:33:29,800
tricky to see, they
zigzag back and forth,

732
00:33:29,800 --> 00:33:32,520
and that is what we compute.

733
00:33:32,520 --> 00:33:35,970
Again, it's approximate, but
it's so close to accurate

734
00:33:35,970 --> 00:33:39,520
that I just draw them
as single points.

735
00:33:39,520 --> 00:33:42,150
The error is much smaller than
the thickness of these lines.

736
00:33:42,150 --> 00:33:44,550
And then the blue
and purple lines

737
00:33:44,550 --> 00:33:48,520
here are fits of
the best parabola

738
00:33:48,520 --> 00:33:49,880
that matches these points.

739
00:33:49,880 --> 00:33:52,319
Actually, it's not
even the best parabola.

740
00:33:52,319 --> 00:33:53,110
It's kind of funny.

741
00:33:53,110 --> 00:33:55,930
I just take the last three
points-- three points

742
00:33:55,930 --> 00:33:57,210
determine a parabola.

743
00:33:57,210 --> 00:33:59,410
I take the parabola that
fits through those three

744
00:33:59,410 --> 00:34:00,700
points and bam!

745
00:34:00,700 --> 00:34:02,360
It is almost a perfect fit.

746
00:34:02,360 --> 00:34:04,300
These are the error
charts, I guess.

747
00:34:04,300 --> 00:34:07,450
This is relative errors,
probably the most informative.

748
00:34:07,450 --> 00:34:14,000
Or actually, it's the ratio--
it's not really an error--

749
00:34:14,000 --> 00:34:15,620
the fit value over
the actual value.

750
00:34:15,620 --> 00:34:18,080
So when it's 1, that
means it's perfect.

751
00:34:18,080 --> 00:34:22,520
And yeah, this is at the end
of the chain where I fit it,

752
00:34:22,520 --> 00:34:24,710
so of course it's going
to be perfect out here.

753
00:34:24,710 --> 00:34:28,580
At the center, at
the very beginning--

754
00:34:28,580 --> 00:34:32,630
so I'm only looking at one
quarter of this thing-- yeah,

755
00:34:32,630 --> 00:34:35,210
the error is a little bit.

756
00:34:35,210 --> 00:34:39,989
The ratio is not 1,
but it's 0.9997 or so,

757
00:34:39,989 --> 00:34:42,719
maybe a little bit less, so
that I had to write it down.

758
00:34:42,719 --> 00:34:49,260
It's like 0.003% error,
if I got it right,

759
00:34:49,260 --> 00:34:52,726
maybe just 0.03% error.

760
00:34:52,726 --> 00:34:54,060
Fix the notes.

761
00:34:54,060 --> 00:34:55,510
Something like that.

762
00:34:55,510 --> 00:34:57,260
It's very small is the point.

763
00:34:57,260 --> 00:35:00,092
But it is non-zero.

764
00:35:00,092 --> 00:35:01,550
I mean, initially
we thought, well,

765
00:35:01,550 --> 00:35:04,080
maybe if you increase
the resolution-- you

766
00:35:04,080 --> 00:35:06,080
make a finer and finer
hyperbolic paraboloid--

767
00:35:06,080 --> 00:35:10,870
it will more closely approximate
a hyperbolic paraboloid.

768
00:35:10,870 --> 00:35:13,800
That does not seem to
be true because, really,

769
00:35:13,800 --> 00:35:16,530
making it finer is really
just like making it bigger.

770
00:35:16,530 --> 00:35:18,826
You never really change
the center behavior.

771
00:35:18,826 --> 00:35:19,700
It's always the same.

772
00:35:19,700 --> 00:35:21,060
And this construction proves it.

773
00:35:21,060 --> 00:35:24,160
We build the center, and we
can go as far out as we want.

774
00:35:24,160 --> 00:35:25,939
It's just changing the
scale of the thing.

775
00:35:25,939 --> 00:35:27,730
But the center will
always remain the same,

776
00:35:27,730 --> 00:35:29,750
and it will always
remain off the parabola,

777
00:35:29,750 --> 00:35:31,250
but super, super close.

778
00:35:31,250 --> 00:35:34,960
Really, it's all right to call
this a hyperbolic paraboloid,

779
00:35:34,960 --> 00:35:39,520
but you should
triangulate, especially

780
00:35:39,520 --> 00:35:43,460
if you're making something
out of more rigid material.

781
00:35:43,460 --> 00:35:44,510
Cool.

782
00:35:44,510 --> 00:35:45,630
Let's go on.

783
00:35:45,630 --> 00:35:49,335

784
00:35:49,335 --> 00:35:50,251
AUDIENCE: [INAUDIBLE].

785
00:35:50,251 --> 00:35:51,451
PROFESSOR: Yeah, question.

786
00:35:51,451 --> 00:35:53,375
AUDIENCE: Do the creases
have to be uniform?

787
00:35:53,375 --> 00:35:57,230
I mean, could you
tighten up inside?

788
00:35:57,230 --> 00:35:59,410
PROFESSOR: The creases do
not have to be uniform.

789
00:35:59,410 --> 00:36:01,400
Most of the hyperbolic
paraboloids we've made,

790
00:36:01,400 --> 00:36:03,520
we do evenly space
all the squares,

791
00:36:03,520 --> 00:36:06,320
but they don't have to be.

792
00:36:06,320 --> 00:36:09,490
I say that in that we've
made them out of paper,

793
00:36:09,490 --> 00:36:11,160
and they fold to
something, and you

794
00:36:11,160 --> 00:36:13,190
get other kinds of surfaces.

795
00:36:13,190 --> 00:36:16,550
Probably not going to be a
hyperbolic paraboloid anymore.

796
00:36:16,550 --> 00:36:18,490
I have not done it
with Mathematica

797
00:36:18,490 --> 00:36:22,155
and checked that it really is
possible, but it should be.

798
00:36:22,155 --> 00:36:24,280
That would be a fun thing
to explore at some point.

799
00:36:24,280 --> 00:36:27,000

800
00:36:27,000 --> 00:36:28,960
Other questions?

801
00:36:28,960 --> 00:36:29,460
All right.

802
00:36:29,460 --> 00:36:32,090
So this is the end
of the positive news

803
00:36:32,090 --> 00:36:33,360
for hyperbolic paraboloids.

804
00:36:33,360 --> 00:36:35,720
Now we're going to go
to the negative stuff,

805
00:36:35,720 --> 00:36:39,300
showing that it is
impossible to fold

806
00:36:39,300 --> 00:36:41,590
this with this crease pattern.

807
00:36:41,590 --> 00:36:45,540
Hyperbolic paraboloids don't
exist without triangulation.

808
00:36:45,540 --> 00:36:51,320
For this we need a
little bit of math tools

809
00:36:51,320 --> 00:37:05,020
in this paper, "How Paper
Folds Between Creases."

810
00:37:05,020 --> 00:37:08,120
So in the study
that we just did,

811
00:37:08,120 --> 00:37:11,000
I assumed that all of the
creases stayed straight.

812
00:37:11,000 --> 00:37:15,430
And therefore that all
of the faces-- therefore?

813
00:37:15,430 --> 00:37:18,470
Yeah, I guess, because
triangles are rigid.

814
00:37:18,470 --> 00:37:20,905
If I forced the edges of the
triangle to stay straight,

815
00:37:20,905 --> 00:37:22,280
then the interior
of the triangle

816
00:37:22,280 --> 00:37:24,990
must stay flat just
to preserve distances.

817
00:37:24,990 --> 00:37:29,560
So I assumed that every face of
the piece of paper stayed flat.

818
00:37:29,560 --> 00:37:30,980
Why did I assume that?

819
00:37:30,980 --> 00:37:33,110
Because paper, real
paper, does not

820
00:37:33,110 --> 00:37:37,946
have to stay flat in
between the creases.

821
00:37:37,946 --> 00:37:40,510
This is real stuff here.

822
00:37:40,510 --> 00:37:45,730
But paper is constrained on how
it can curve without creases,

823
00:37:45,730 --> 00:37:49,780
and that is the purpose of this
little mathematical endeavour.

824
00:37:49,780 --> 00:37:54,130

825
00:37:54,130 --> 00:37:56,790
There's some technical stuff,
and I want to get to the meat

826
00:37:56,790 --> 00:37:58,500
as quickly as possible.

827
00:37:58,500 --> 00:38:01,950
But I'll just mention
some assumptions.

828
00:38:01,950 --> 00:38:04,230
We assume that the
thing that we fold,

829
00:38:04,230 --> 00:38:07,140
the folding of our piece
of paper, is piecewise C2.

830
00:38:07,140 --> 00:38:09,750
C2 means you can
take two derivatives,

831
00:38:09,750 --> 00:38:13,290
and it's still good and
continuous and sort of smooth

832
00:38:13,290 --> 00:38:14,790
up to the second level.

833
00:38:14,790 --> 00:38:17,580
Piecewise means, of
course, we have creases.

834
00:38:17,580 --> 00:38:20,250
Those are not C2, not even C1.

835
00:38:20,250 --> 00:38:22,820
So we have some crease
pattern on our piece of paper.

836
00:38:22,820 --> 00:38:24,120
It's whatever.

837
00:38:24,120 --> 00:38:28,020
And what we mean is that inside
one of these regions, it's C2.

838
00:38:28,020 --> 00:38:29,860
On the creases, it's nothing.

839
00:38:29,860 --> 00:38:30,450
It's C0.

840
00:38:30,450 --> 00:38:31,460
It's continuous.

841
00:38:31,460 --> 00:38:32,790
We don't rip the paper.

842
00:38:32,790 --> 00:38:35,330
But we don't necessarily
have derivatives everywhere.

843
00:38:35,330 --> 00:38:36,840
So that's the assumption.

844
00:38:36,840 --> 00:38:39,170
Now, one annoying
thing here is you

845
00:38:39,170 --> 00:38:44,460
can have something called
a semi-crease, something

846
00:38:44,460 --> 00:38:49,770
we call a semi-crease,
which is C1 but not C2.

847
00:38:49,770 --> 00:38:50,970
Yeah.

848
00:38:50,970 --> 00:38:53,730
So you have to
divide into pieces,

849
00:38:53,730 --> 00:38:55,856
but it's not
technically a crease.

850
00:38:55,856 --> 00:38:58,230
And this is some of the worry
of what might be happening.

851
00:38:58,230 --> 00:38:59,970
Maybe you don't
need creases here,

852
00:38:59,970 --> 00:39:01,220
you only need semi-creases.

853
00:39:01,220 --> 00:39:03,595
Maybe you only need to violate
the second derivative, not

854
00:39:03,595 --> 00:39:04,890
the first.

855
00:39:04,890 --> 00:39:06,940
Creases should be violation
of first derivative.

856
00:39:06,940 --> 00:39:08,510
Those are sharp things.

857
00:39:08,510 --> 00:39:12,690
Semi-creases are just
kind of little sharp.

858
00:39:12,690 --> 00:39:16,310
It's like-- what's a good
example of a semi-sharp thing?

859
00:39:16,310 --> 00:39:18,730
I guess if I take a
parabola, and then

860
00:39:18,730 --> 00:39:23,194
I take a more shallow
parabola, then at this point

861
00:39:23,194 --> 00:39:25,610
there's no second derivative
because the first derivatives

862
00:39:25,610 --> 00:39:26,110
don't meet.

863
00:39:26,110 --> 00:39:27,920
This one has a first
derivative of this.

864
00:39:27,920 --> 00:39:30,000
This one has a first
derivative of that.

865
00:39:30,000 --> 00:39:32,910
They're not the same.

866
00:39:32,910 --> 00:39:35,270
Is that true?

867
00:39:35,270 --> 00:39:36,670
Parabolas maybe not.

868
00:39:36,670 --> 00:39:37,780
But you get the idea.

869
00:39:37,780 --> 00:39:40,500
This is a discontinuity
in the second derivative,

870
00:39:40,500 --> 00:39:42,010
not the first.

871
00:39:42,010 --> 00:39:42,510
All right.

872
00:39:42,510 --> 00:39:44,300
I'm going to basically
ignore semi-creases here,

873
00:39:44,300 --> 00:39:46,120
though, because they're
just kind of a technicality.

874
00:39:46,120 --> 00:39:47,286
They don't end up mattering.

875
00:39:47,286 --> 00:39:50,170

876
00:39:50,170 --> 00:39:53,050
So we have piecewise C2.

877
00:39:53,050 --> 00:39:56,820
And we assume that our
surface is intrinsically flat,

878
00:39:56,820 --> 00:40:02,166
meaning it came from a
piece of paper-- trinsic--

879
00:40:02,166 --> 00:40:07,830
oh, boy, intrinsically flat.

880
00:40:07,830 --> 00:40:11,690
Now this is something we
know as curvature zero.

881
00:40:11,690 --> 00:40:15,956

882
00:40:15,956 --> 00:40:18,330
And I want to tell you a little
bit more about curvature.

883
00:40:18,330 --> 00:40:20,650
We have defined curvature
in a few different-- well,

884
00:40:20,650 --> 00:40:22,680
I guess in one
particular way, which

885
00:40:22,680 --> 00:40:27,030
is you add up the material,
and you take 360 minus that.

886
00:40:27,030 --> 00:40:28,790
So curvature zero
means you have 360

887
00:40:28,790 --> 00:40:30,765
of material, which is good news.

888
00:40:30,765 --> 00:40:34,067

889
00:40:34,067 --> 00:40:35,650
But there are other
ways to define it,

890
00:40:35,650 --> 00:40:37,830
in particular, the
way Gauss defined it.

891
00:40:37,830 --> 00:40:40,365
This is usually called
Gaussian curvature

892
00:40:40,365 --> 00:40:41,880
because he invented it.

893
00:40:41,880 --> 00:40:46,270
Gauss defined curvature
of a 3D surface.

894
00:40:46,270 --> 00:40:49,930
You have some weird
curved 3D thing,

895
00:40:49,930 --> 00:40:53,720
like this shape,
the orange part.

896
00:40:53,720 --> 00:40:56,580
And there's actually a lot of
natural notions of curvature

897
00:40:56,580 --> 00:40:57,160
here.

898
00:40:57,160 --> 00:40:59,219
Gaussian curvature
is just one of them.

899
00:40:59,219 --> 00:41:01,010
So let me tell you a
little bit about that.

900
00:41:01,010 --> 00:41:02,670
We're looking at
this point, and I

901
00:41:02,670 --> 00:41:05,880
want to compute the
curvature of this point.

902
00:41:05,880 --> 00:41:09,000
Well, if I look
in any direction,

903
00:41:09,000 --> 00:41:11,290
like say this red
direction-- so it's

904
00:41:11,290 --> 00:41:15,660
like I slice this world with
a plane, this blue plane,

905
00:41:15,660 --> 00:41:17,730
I get a nice
one-dimensional curve.

906
00:41:17,730 --> 00:41:23,250
Then the curvature is how bent
that curve is at that point.

907
00:41:23,250 --> 00:41:24,820
That's like a
directional curvature.

908
00:41:24,820 --> 00:41:28,880
In this direction, how bent
is-- what does how bent mean?

909
00:41:28,880 --> 00:41:31,530
It just means you try to
nestle a circle in there,

910
00:41:31,530 --> 00:41:34,290
in that plane, and you
take 1 over the radius.

911
00:41:34,290 --> 00:41:35,281
That's a curvature.

912
00:41:35,281 --> 00:41:35,780
OK?

913
00:41:35,780 --> 00:41:38,350
So if it's flat, the radius
could be infinite, and so

914
00:41:38,350 --> 00:41:39,680
curvature zero.

915
00:41:39,680 --> 00:41:42,630
If it's very sharp, then
the radius is very small.

916
00:41:42,630 --> 00:41:45,760
And so 1 over the radius is
very large, big curvature.

917
00:41:45,760 --> 00:41:48,350
So it's some thing here.

918
00:41:48,350 --> 00:41:49,880
Let's say it's a
positive number.

919
00:41:49,880 --> 00:41:53,260
And then if I take, for example,
this other blue plane, so

920
00:41:53,260 --> 00:41:55,490
this cross-section,
in that direction

921
00:41:55,490 --> 00:41:57,250
the curvature is
bent the other way.

922
00:41:57,250 --> 00:41:59,555
So you say, well, there
the directional curvature

923
00:41:59,555 --> 00:42:01,195
is negative.

924
00:42:01,195 --> 00:42:02,320
It depends which way is up.

925
00:42:02,320 --> 00:42:04,720
One of them is positive,
one of them is negative.

926
00:42:04,720 --> 00:42:08,420
Somewhere in between
it's going to be 0.

927
00:42:08,420 --> 00:42:11,250
But you have all these different
directional curvatures.

928
00:42:11,250 --> 00:42:14,500
The Gaussian curvature,
which is the one

929
00:42:14,500 --> 00:42:17,930
that we sort of know and love
and have used a lot, mostly

930
00:42:17,930 --> 00:42:24,390
for polyhedra, is the
min directional curvature

931
00:42:24,390 --> 00:42:28,330
times the max
directional curvature.

932
00:42:28,330 --> 00:42:30,110
Now, I really mean min and max.

933
00:42:30,110 --> 00:42:32,250
So the smallest one, in
this case, is negative.

934
00:42:32,250 --> 00:42:34,600
The largest one is positive.

935
00:42:34,600 --> 00:42:37,490
You take the product, and
that's the Gaussian curvature.

936
00:42:37,490 --> 00:42:39,564
It's a weird thing.

937
00:42:39,564 --> 00:42:40,980
But in particular,
because there's

938
00:42:40,980 --> 00:42:42,605
a negative one and
a positive one here,

939
00:42:42,605 --> 00:42:45,200
that means the
product is negative,

940
00:42:45,200 --> 00:42:47,269
and that's because
this is a saddle.

941
00:42:47,269 --> 00:42:49,060
Still the case, negative
Gaussian curvature

942
00:42:49,060 --> 00:42:49,920
means saddle.

943
00:42:49,920 --> 00:42:51,590
Positive means a convex cone.

944
00:42:51,590 --> 00:42:54,880
0 means intrinsically flat.

945
00:42:54,880 --> 00:42:57,355
So this is still
the thing we know,

946
00:42:57,355 --> 00:42:59,230
but this is a weird way
of thinking about it.

947
00:42:59,230 --> 00:43:00,790
This is how Gauss defined it.

948
00:43:00,790 --> 00:43:03,240
And he proved that even
if you fold the surface,

949
00:43:03,240 --> 00:43:05,160
the Gaussian curvature
never changes.

950
00:43:05,160 --> 00:43:10,560
That is called the
theorema egregium.

951
00:43:10,560 --> 00:43:12,540
Cool name.

952
00:43:12,540 --> 00:43:15,200
So Gaussian curvature
doesn't change under folding.

953
00:43:15,200 --> 00:43:18,000
We start with something
that's flat, zero curvature.

954
00:43:18,000 --> 00:43:20,776
Because if you
take a flat plane,

955
00:43:20,776 --> 00:43:22,900
you take any directional
curvature, everything's 0.

956
00:43:22,900 --> 00:43:24,560
So it's the product
of min and mix.

957
00:43:24,560 --> 00:43:25,840
They're both 0.

958
00:43:25,840 --> 00:43:29,970
So we start with something zero
curvature, it will remain so.

959
00:43:29,970 --> 00:43:31,940
Now, if I have a
product of two things,

960
00:43:31,940 --> 00:43:34,260
and I know this is
equal to 0, that

961
00:43:34,260 --> 00:43:36,955
means one of the two
things is 0, maybe both.

962
00:43:36,955 --> 00:43:40,610
If it's still a plane,
both of them will be 0.

963
00:43:40,610 --> 00:43:42,200
But one of them
still has to be 0.

964
00:43:42,200 --> 00:43:45,490
What this means is, basically,
locally at any point,

965
00:43:45,490 --> 00:43:49,980
we have a cylinder, some
kind of generalized cylinder.

966
00:43:49,980 --> 00:43:53,189
But it really only
curves in one direction

967
00:43:53,189 --> 00:43:55,730
because there's some direction
where it doesn't curve at all.

968
00:43:55,730 --> 00:43:58,049
Everything's straight.

969
00:43:58,049 --> 00:43:59,590
In all the other
directions, yeah, it

970
00:43:59,590 --> 00:44:00,990
curves different amounts.

971
00:44:00,990 --> 00:44:04,040
The orthogonal direction will
be where it curves the most.

972
00:44:04,040 --> 00:44:09,340
That will be the max, I
guess, and the min will be 0.

973
00:44:09,340 --> 00:44:12,490
So that is what a
folded sheet looks like.

974
00:44:12,490 --> 00:44:14,230
And it's maybe not
so obvious, but when

975
00:44:14,230 --> 00:44:16,230
I did all this
contorting and what not,

976
00:44:16,230 --> 00:44:21,115
really I was only bending in
one dimension, like a cylinder.

977
00:44:21,115 --> 00:44:22,740
That's not exactly
a cylinder because I

978
00:44:22,740 --> 00:44:27,880
can change the radius of the
cylinder all over the place,

979
00:44:27,880 --> 00:44:30,740
but I'm really only
bending in one direction.

980
00:44:30,740 --> 00:44:34,550
I was reading on Wikipedia,
this explains how we eat pizza.

981
00:44:34,550 --> 00:44:36,840
Because you take
a piece of pizza,

982
00:44:36,840 --> 00:44:38,710
which is basically
like a piece of paper,

983
00:44:38,710 --> 00:44:42,030
and if you bend it a little
bit, like you push in the center

984
00:44:42,030 --> 00:44:44,382
and push up on the
sides, you give it

985
00:44:44,382 --> 00:44:45,715
some curvature in one direction.

986
00:44:45,715 --> 00:44:47,923
And therefore it has to
remain straight in the other,

987
00:44:47,923 --> 00:44:53,219
so it kind of supports
the piece of pizza.

988
00:44:53,219 --> 00:44:55,010
Never thought of it
that way, but there you

989
00:44:55,010 --> 00:44:57,920
go, practical applications
for this stuff.

990
00:44:57,920 --> 00:45:00,070
All right.

991
00:45:00,070 --> 00:45:03,280
These things we
call planar points.

992
00:45:03,280 --> 00:45:07,440
If I have a point p and it's
locally flat like a plane,

993
00:45:07,440 --> 00:45:08,840
call it planar.

994
00:45:08,840 --> 00:45:12,285
These things, for some crazy
reason, we call parabolic.

995
00:45:12,285 --> 00:45:16,190

996
00:45:16,190 --> 00:45:18,100
This is to be consistent
with other notation.

997
00:45:18,100 --> 00:45:20,990
I know it looks-- cylindrical
would be another fine term.

998
00:45:20,990 --> 00:45:23,010
But there's a third
kind called a elliptic,

999
00:45:23,010 --> 00:45:24,460
which would be like this stuff.

1000
00:45:24,460 --> 00:45:25,390
That doesn't happen.

1001
00:45:25,390 --> 00:45:27,320
So we just worry about
locally cylinder, which

1002
00:45:27,320 --> 00:45:29,820
we're going to call parabolic
because it's also like locally

1003
00:45:29,820 --> 00:45:35,190
a parabola, doesn't
matter, and locally planar.

1004
00:45:35,190 --> 00:45:37,670
Those are the two kinds
of points we can have.

1005
00:45:37,670 --> 00:45:44,020
Now, I'm going to give you some
accelerated facts we're just

1006
00:45:44,020 --> 00:45:49,230
going to take as given
from differential geometry.

1007
00:45:49,230 --> 00:45:51,150
Well, really, we had
to take some facts that

1008
00:45:51,150 --> 00:45:53,430
were about differential
geometry and sort of port

1009
00:45:53,430 --> 00:45:55,600
them to our context.

1010
00:45:55,600 --> 00:45:57,590
They didn't give us
exactly what we wanted.

1011
00:45:57,590 --> 00:46:00,510
We had to generalize
them a little.

1012
00:46:00,510 --> 00:46:03,200
Differential geometry, if you
were at Sunday's lectures,

1013
00:46:03,200 --> 00:46:08,090
that was one of the main
tools being used there.

1014
00:46:08,090 --> 00:46:12,150
Differential geometry is about
smooth things, like C4 usually.

1015
00:46:12,150 --> 00:46:13,960
Now, sometimes about C2 things.

1016
00:46:13,960 --> 00:46:17,270
It is almost never about
piecewise C2 things.

1017
00:46:17,270 --> 00:46:19,334
So you got to worry
about the pieces.

1018
00:46:19,334 --> 00:46:20,750
Now, most the time,
we're thinking

1019
00:46:20,750 --> 00:46:24,300
about one little region
here, and that's nice in C2.

1020
00:46:24,300 --> 00:46:25,480
And you can check.

1021
00:46:25,480 --> 00:46:28,940
Most of the differential
geometry still applies there.

1022
00:46:28,940 --> 00:46:29,986
Blah, blah, blah.

1023
00:46:29,986 --> 00:46:31,110
Let me tell you some facts.

1024
00:46:31,110 --> 00:46:33,720

1025
00:46:33,720 --> 00:46:39,170
If I take a smooth point, so
that means not on a crease

1026
00:46:39,170 --> 00:46:47,110
and not a semi-crease,
then it lies on something

1027
00:46:47,110 --> 00:46:52,790
called a rule segment,
which is a line segment,

1028
00:46:52,790 --> 00:46:56,180
also called a rule line.

1029
00:46:56,180 --> 00:46:57,740
And the endpoints
of that segment,

1030
00:46:57,740 --> 00:47:07,650
here's the interesting part,
are on creases or the boundary.

1031
00:47:07,650 --> 00:47:14,630

1032
00:47:14,630 --> 00:47:17,430
So we already know
from this picture

1033
00:47:17,430 --> 00:47:20,470
that any point
lies on a segment,

1034
00:47:20,470 --> 00:47:23,440
like an actual 3D
line segment, because

1035
00:47:23,440 --> 00:47:25,470
of this parabolic nature.

1036
00:47:25,470 --> 00:47:28,240
What's interesting is that
segment, it can't stop.

1037
00:47:28,240 --> 00:47:32,010
Just keeps on going,
like the Energizer bunny,

1038
00:47:32,010 --> 00:47:33,370
until it hits a crease.

1039
00:47:33,370 --> 00:47:34,980
At that point,
things aren't smooth.

1040
00:47:34,980 --> 00:47:36,340
We don't know what happens.

1041
00:47:36,340 --> 00:47:38,640
But really, these
creases go straight.

1042
00:47:38,640 --> 00:47:40,820
Those are rule lines.

1043
00:47:40,820 --> 00:47:43,770
And here, actually, we have
a choice, many different rule

1044
00:47:43,770 --> 00:47:44,470
lines.

1045
00:47:44,470 --> 00:47:48,660
But in the parabolic case,
that rule line is unique.

1046
00:47:48,660 --> 00:47:54,735
It's going to be unique
for parabolic points.

1047
00:47:54,735 --> 00:48:00,640

1048
00:48:00,640 --> 00:48:08,980
And so we get what's called a
ruled surface, which is just

1049
00:48:08,980 --> 00:48:14,900
the union of a whole bunch
of line segments, rule lines,

1050
00:48:14,900 --> 00:48:15,980
around any point.

1051
00:48:15,980 --> 00:48:23,750

1052
00:48:23,750 --> 00:48:30,590
Maybe any smooth
point, just to be safe.

1053
00:48:30,590 --> 00:48:32,545
So you may have heard
of ruled surfaces.

1054
00:48:32,545 --> 00:48:33,420
They're quite common.

1055
00:48:33,420 --> 00:48:36,460
They're fun because you can
build them out of strings.

1056
00:48:36,460 --> 00:48:39,450
So each string, if you hold
taut, it's a line segment.

1057
00:48:39,450 --> 00:48:41,950
Take a whole bunch of them
and just imagine the envelope

1058
00:48:41,950 --> 00:48:42,449
there.

1059
00:48:42,449 --> 00:48:44,491
That is a 3D surface, and
that's a ruled surface.

1060
00:48:44,491 --> 00:48:46,240
Now this one does not
have zero curvature.

1061
00:48:46,240 --> 00:48:47,850
It has negative
curvature everywhere.

1062
00:48:47,850 --> 00:48:49,020
So that can't happen.

1063
00:48:49,020 --> 00:48:50,860
This actually can happen.

1064
00:48:50,860 --> 00:48:54,240
This is like you take
a helix, and so there's

1065
00:48:54,240 --> 00:48:57,550
a blue curve in
3D, a space curve.

1066
00:48:57,550 --> 00:49:01,619
And then you imagine taking
the tangent at every point.

1067
00:49:01,619 --> 00:49:04,160
So just like if you just went
straight at every point instead

1068
00:49:04,160 --> 00:49:07,050
of turning, then you
get a bunch of lines.

1069
00:49:07,050 --> 00:49:08,430
Those are rule lines.

1070
00:49:08,430 --> 00:49:10,950
And you get this cool surface.

1071
00:49:10,950 --> 00:49:13,420
Now, that is a valid
folding of a piece of paper.

1072
00:49:13,420 --> 00:49:15,380
Doesn't look like a
cylinder, does it?

1073
00:49:15,380 --> 00:49:18,150
But locally, each of these
lines looks like a cylinder.

1074
00:49:18,150 --> 00:49:20,175
It's just the radius
of the cylinder

1075
00:49:20,175 --> 00:49:24,188
is changing all over the place.

1076
00:49:24,188 --> 00:49:28,070
But that's a valid folding
of a piece of paper, I think.

1077
00:49:28,070 --> 00:49:31,366
Has zero curvature
everywhere, if I did it right.

1078
00:49:31,366 --> 00:49:36,630
Or I didn't do it, but if
I imagine it correctly.

1079
00:49:36,630 --> 00:49:37,130
All right.

1080
00:49:37,130 --> 00:49:38,810
So it's a ruled surface, great.

1081
00:49:38,810 --> 00:49:42,680
I mean, I can really create
my whole surface locally

1082
00:49:42,680 --> 00:49:46,229
around a point by a whole
bunch of rule segments.

1083
00:49:46,229 --> 00:49:48,520
They're not all going to be
parallel or anything, which

1084
00:49:48,520 --> 00:49:50,410
is what we imagined
from the cylinder.

1085
00:49:50,410 --> 00:49:51,940
They can turn around.

1086
00:49:51,940 --> 00:49:54,790
But these segments do keep going
until they hit another crease.

1087
00:49:54,790 --> 00:49:56,480
Maybe the blue line's a crease.

1088
00:49:56,480 --> 00:49:59,760
And the boundary here could
be the boundary of the paper,

1089
00:49:59,760 --> 00:50:01,255
could be a crease boundary.

1090
00:50:01,255 --> 00:50:03,080
But that's what they look like.

1091
00:50:03,080 --> 00:50:09,110
Now, it's also what we
call torsal, torsal ruled.

1092
00:50:09,110 --> 00:50:13,190
Here we get to somewhat
more obscure terminology

1093
00:50:13,190 --> 00:50:14,570
but some useful things.

1094
00:50:14,570 --> 00:50:15,495
It's quite restricted.

1095
00:50:15,495 --> 00:50:18,970

1096
00:50:18,970 --> 00:50:27,961
I want to have a common
tangent plane throughout a rule

1097
00:50:27,961 --> 00:50:28,460
segment.

1098
00:50:28,460 --> 00:50:33,030

1099
00:50:33,030 --> 00:50:36,620
So if I look at any one
of these rule segments,

1100
00:50:36,620 --> 00:50:38,500
there's one plane,
which is going

1101
00:50:38,500 --> 00:50:42,540
to be like this, that is tangent
to the surface at every point

1102
00:50:42,540 --> 00:50:44,629
along that segment.

1103
00:50:44,629 --> 00:50:46,670
So in general, you might
imagine that the tangent

1104
00:50:46,670 --> 00:50:49,370
plane turns as we go along.

1105
00:50:49,370 --> 00:50:52,090
But these segments are kind of--
they're locally cylindrical.

1106
00:50:52,090 --> 00:50:55,030
So you can really make a
tangent plane all the way along.

1107
00:50:55,030 --> 00:50:59,640
Whereas here, that's
probably not true.

1108
00:50:59,640 --> 00:51:02,219
Imagine the tangent plane
starts-- can you see my hand?

1109
00:51:02,219 --> 00:51:03,760
It starts to bend
like this, and it's

1110
00:51:03,760 --> 00:51:05,880
going to bend around like that.

1111
00:51:05,880 --> 00:51:08,252
It twists along a single line.

1112
00:51:08,252 --> 00:51:09,960
In our case where we
have zero curvature,

1113
00:51:09,960 --> 00:51:12,020
it's actually going
to be torsal ruled,

1114
00:51:12,020 --> 00:51:14,625
meaning the tangent plane
just goes straight along each

1115
00:51:14,625 --> 00:51:15,250
of these lines.

1116
00:51:15,250 --> 00:51:18,120
We're going to need this,
that's why I mention it.

1117
00:51:18,120 --> 00:51:21,930
Another fun fact
along the same spirit

1118
00:51:21,930 --> 00:51:28,130
is that the points
along a rule line

1119
00:51:28,130 --> 00:51:30,410
are all the same in
terms of whether they

1120
00:51:30,410 --> 00:51:33,270
are planar or parabolic.

1121
00:51:33,270 --> 00:51:36,750
So I'll call them uniformly
planar slash parabolic.

1122
00:51:36,750 --> 00:51:40,190

1123
00:51:40,190 --> 00:51:42,510
Remember, planar just
means it lies in a plane.

1124
00:51:42,510 --> 00:51:44,280
Parabolic is the other case.

1125
00:51:44,280 --> 00:51:46,356
So here everything's parabolic.

1126
00:51:46,356 --> 00:51:48,355
But you can't like suddenly
switch in the middle

1127
00:51:48,355 --> 00:51:49,420
and become flat.

1128
00:51:49,420 --> 00:51:53,234
That's not allowed,
not possible.

1129
00:51:53,234 --> 00:51:54,400
So those are some fun facts.

1130
00:51:54,400 --> 00:51:55,650
We're not going to prove them.

1131
00:51:55,650 --> 00:51:58,607
If you want to see the proofs,
they're a bit technical.

1132
00:51:58,607 --> 00:52:01,190
You have to read our paper, and
then the differential geometry

1133
00:52:01,190 --> 00:52:02,500
books we cite.

1134
00:52:02,500 --> 00:52:06,482
But a bunch of these things, you
can understand the proofs just

1135
00:52:06,482 --> 00:52:08,440
assuming a little bit of
differential geometry,

1136
00:52:08,440 --> 00:52:09,270
and it's not too hard.

1137
00:52:09,270 --> 00:52:11,394
It's in our paper, but it's
a little bit technical.

1138
00:52:11,394 --> 00:52:15,180
So I want to look at the
things that are more related

1139
00:52:15,180 --> 00:52:18,730
to paper folding,
how paper folds.

1140
00:52:18,730 --> 00:52:20,480
This is obviously
related, but it's

1141
00:52:20,480 --> 00:52:24,630
like the foundation on which
we build what I care about.

1142
00:52:24,630 --> 00:52:27,210
What I care about
most-- all right,

1143
00:52:27,210 --> 00:52:30,580
let me tell you a fun fact,
what we're going to prove.

1144
00:52:30,580 --> 00:52:35,020
We want this nice proper folding
of the hyperbolic paraboloid,

1145
00:52:35,020 --> 00:52:38,140
meaning every crease is bent
by a non-zero angle and not 180

1146
00:52:38,140 --> 00:52:39,990
degrees.

1147
00:52:39,990 --> 00:52:42,710
In that situation,
first claim is

1148
00:52:42,710 --> 00:52:45,420
every crease remains straight.

1149
00:52:45,420 --> 00:52:50,590
Second claim is every
face remains rigid,

1150
00:52:50,590 --> 00:52:55,260
can't bend anywhere--
every interior face.

1151
00:52:55,260 --> 00:52:58,220
These guys on the boundary,
they can do crazy things.

1152
00:52:58,220 --> 00:52:59,930
And you see that
in the model, where

1153
00:52:59,930 --> 00:53:02,460
the outside gets
kind of all wiggly.

1154
00:53:02,460 --> 00:53:03,867
That's allowed.

1155
00:53:03,867 --> 00:53:05,450
The outside faces,
the ones that share

1156
00:53:05,450 --> 00:53:08,650
the boundary with the paper,
can wiggle, can curve.

1157
00:53:08,650 --> 00:53:11,130
But everything else
has to remain flat

1158
00:53:11,130 --> 00:53:13,520
if there are no
creases in there.

1159
00:53:13,520 --> 00:53:16,260
And that's not at all obvious.

1160
00:53:16,260 --> 00:53:19,425
So we're going to prove it.

1161
00:53:19,425 --> 00:53:21,389
And I need some water.

1162
00:53:21,389 --> 00:53:28,737

1163
00:53:28,737 --> 00:53:29,236
All right.

1164
00:53:29,236 --> 00:53:32,460

1165
00:53:32,460 --> 00:53:39,640
First claim, I'll call it
polygonal implies flat.

1166
00:53:39,640 --> 00:53:43,710
So what I mean is, suppose
I have some region of paper.

1167
00:53:43,710 --> 00:53:45,660
It's in the middle somewhere.

1168
00:53:45,660 --> 00:53:47,680
Let's say it's smooth.

1169
00:53:47,680 --> 00:53:51,817
And I look at the
boundary of the region.

1170
00:53:51,817 --> 00:53:53,275
Doesn't actually
have to be smooth.

1171
00:53:53,275 --> 00:53:54,540
It could have semi-creases.

1172
00:53:54,540 --> 00:53:56,790
But it has no creases inside.

1173
00:53:56,790 --> 00:54:01,390
Suppose the boundary
in 3D is a polygon,

1174
00:54:01,390 --> 00:54:04,964
so it's piecewise straight.

1175
00:54:04,964 --> 00:54:06,880
I don't know how that
corresponds to anything,

1176
00:54:06,880 --> 00:54:09,120
but say the boundary
is straight.

1177
00:54:09,120 --> 00:54:12,990
Then the inside must be planar.

1178
00:54:12,990 --> 00:54:17,760
So if I have a
polygonal boundary,

1179
00:54:17,760 --> 00:54:22,060
then I have a flat
planar inside.

1180
00:54:22,060 --> 00:54:25,830

1181
00:54:25,830 --> 00:54:27,210
This is assuming no creases.

1182
00:54:27,210 --> 00:54:31,371

1183
00:54:31,371 --> 00:54:31,870
OK?

1184
00:54:31,870 --> 00:54:36,100
So what that means is that
every point inside this region

1185
00:54:36,100 --> 00:54:38,280
must be planar, not parabolic.

1186
00:54:38,280 --> 00:54:40,030
That's what we want to prove.

1187
00:54:40,030 --> 00:54:42,110
So let's do a proof
by contradiction,

1188
00:54:42,110 --> 00:54:44,290
and suppose that we
have a parabolic point.

1189
00:54:44,290 --> 00:54:46,910

1190
00:54:46,910 --> 00:54:52,710
So we have some point,
and locally it is curved,

1191
00:54:52,710 --> 00:54:54,610
and we know there's a
rule line through it.

1192
00:54:54,610 --> 00:54:57,640
I know by smoothness,
by continuity,

1193
00:54:57,640 --> 00:55:05,830
that if this guy is parabolic,
it's bent, then, in fact,

1194
00:55:05,830 --> 00:55:08,650
all the points nearby
should also be bent,

1195
00:55:08,650 --> 00:55:12,942
because you can't go instantly
from bent to straight.

1196
00:55:12,942 --> 00:55:14,150
You've got to do that slowly.

1197
00:55:14,150 --> 00:55:15,566
So there's some
little region-- we

1198
00:55:15,566 --> 00:55:20,190
call this a neighborhood--
around the point that

1199
00:55:20,190 --> 00:55:20,965
is all parabolic.

1200
00:55:20,965 --> 00:55:23,752

1201
00:55:23,752 --> 00:55:25,460
So I'll call this
parabolic neighborhood.

1202
00:55:25,460 --> 00:55:28,661

1203
00:55:28,661 --> 00:55:29,160
All right.

1204
00:55:29,160 --> 00:55:32,270
All of those points
have unique rule lines.

1205
00:55:32,270 --> 00:55:34,100
That's what we've been saying.

1206
00:55:34,100 --> 00:55:36,430
So I take this
little neighborhood,

1207
00:55:36,430 --> 00:55:38,660
and each one of them
defines some rule line.

1208
00:55:38,660 --> 00:55:41,390

1209
00:55:41,390 --> 00:55:43,110
Those rule lines
go all the way out

1210
00:55:43,110 --> 00:55:47,860
to the boundary
something like this.

1211
00:55:47,860 --> 00:55:50,550
Now, this boundary
is the boundary.

1212
00:55:50,550 --> 00:55:53,820
That means it's polygonal,
can't be curved like this.

1213
00:55:53,820 --> 00:55:56,820
In fact, it looks
something like this.

1214
00:55:56,820 --> 00:55:58,130
Maybe this is straight.

1215
00:55:58,130 --> 00:55:59,470
Maybe this has two segments.

1216
00:55:59,470 --> 00:56:01,050
We don't know how many segments.

1217
00:56:01,050 --> 00:56:03,680
I just want to look at
one of these segments here

1218
00:56:03,680 --> 00:56:05,650
and all the rule lines
coming out of it.

1219
00:56:05,650 --> 00:56:08,692
So what I have is I have
the boundary of the paper.

1220
00:56:08,692 --> 00:56:11,330
That's a poor imitation
of a straight line.

1221
00:56:11,330 --> 00:56:13,030
Straight line.

1222
00:56:13,030 --> 00:56:14,720
I have some rule
lines coming out.

1223
00:56:14,720 --> 00:56:17,412
Now, I don't really know
what they look like.

1224
00:56:17,412 --> 00:56:19,120
I want to understand
what they look like.

1225
00:56:19,120 --> 00:56:20,480
This is in 3D.

1226
00:56:20,480 --> 00:56:21,872
Imagine.

1227
00:56:21,872 --> 00:56:24,900
OK, this is straight.

1228
00:56:24,900 --> 00:56:28,940
First thing I want to look at
is the normal to the surface.

1229
00:56:28,940 --> 00:56:33,690
So normal is like
perpendicular to the surface.

1230
00:56:33,690 --> 00:56:36,940
It's easy to define normals in
the interior of the surface,

1231
00:56:36,940 --> 00:56:40,180
but I can actually extend
that out to the boundary.

1232
00:56:40,180 --> 00:56:41,970
So I want to look
at a normal here.

1233
00:56:41,970 --> 00:56:44,720

1234
00:56:44,720 --> 00:56:49,040
Maybe let me put it here,
something like that.

1235
00:56:49,040 --> 00:56:51,270
If I took normals really
close to the boundary,

1236
00:56:51,270 --> 00:56:53,150
I just took the limit
out to the boundary,

1237
00:56:53,150 --> 00:56:54,316
I'll get some normal vector.

1238
00:56:54,316 --> 00:56:56,265
And by smoothness, that exists.

1239
00:56:56,265 --> 00:57:00,060
Just wave my hands
there, but that's true.

1240
00:57:00,060 --> 00:57:01,810
OK, what's true
about this normal?

1241
00:57:01,810 --> 00:57:04,550
Well, it's perpendicular
to the surface.

1242
00:57:04,550 --> 00:57:06,310
Now locally, the
surface here, it's

1243
00:57:06,310 --> 00:57:10,920
defined by the plane of this
line segment and this line

1244
00:57:10,920 --> 00:57:12,110
segment.

1245
00:57:12,110 --> 00:57:15,130
So in fact, it's
perpendicular to the boundary,

1246
00:57:15,130 --> 00:57:20,720
and it's perpendicular
here to the rule line.

1247
00:57:20,720 --> 00:57:22,470
Buy that?

1248
00:57:22,470 --> 00:57:25,565
If I take some other normal,
like this one-- sorry,

1249
00:57:25,565 --> 00:57:29,060
like that, it's also
perpendicular to this,

1250
00:57:29,060 --> 00:57:32,060
but it's perpendicular to
some other rule line now.

1251
00:57:32,060 --> 00:57:35,550
So these guys are
perpendicular to a common line,

1252
00:57:35,550 --> 00:57:39,680
but they may not be
the same direction.

1253
00:57:39,680 --> 00:57:43,910
What we do know, though,
is that it's torsal.

1254
00:57:43,910 --> 00:57:46,230
There is a single
tangent plane that's

1255
00:57:46,230 --> 00:57:48,710
perpendicular to this
entire rule line, which

1256
00:57:48,710 --> 00:57:50,530
means the normal is
the thing perpendicular

1257
00:57:50,530 --> 00:57:51,700
to that tangent plane.

1258
00:57:51,700 --> 00:57:53,840
So in fact, all of the
normals along this line

1259
00:57:53,840 --> 00:57:56,900
are identical, which
is kind of neat.

1260
00:57:56,900 --> 00:57:59,880
These guys are all parallel.

1261
00:57:59,880 --> 00:58:00,905
Same thing here.

1262
00:58:00,905 --> 00:58:01,655
I won't draw that.

1263
00:58:01,655 --> 00:58:03,900
It's going to get too messy.

1264
00:58:03,900 --> 00:58:05,141
So what?

1265
00:58:05,141 --> 00:58:05,640
All right.

1266
00:58:05,640 --> 00:58:09,630
Well, here's a crazy
thing to imagine.

1267
00:58:09,630 --> 00:58:12,310
I will look at the
derivative of the normal.

1268
00:58:12,310 --> 00:58:14,451
So I have this point p.

1269
00:58:14,451 --> 00:58:16,950
Imagine you have a point p, and
it's moving along this curve

1270
00:58:16,950 --> 00:58:18,210
continuously.

1271
00:58:18,210 --> 00:58:19,060
Sorry-- not curve.

1272
00:58:19,060 --> 00:58:20,350
This is a straight line.

1273
00:58:20,350 --> 00:58:24,950
It's moving along this edge
of the boundary just once.

1274
00:58:24,950 --> 00:58:26,810
It goes like this.

1275
00:58:26,810 --> 00:58:30,600
And I'm going to define n of
p is the normal at that point.

1276
00:58:30,600 --> 00:58:34,275
So it starts at something,
and then maybe it's changing.

1277
00:58:34,275 --> 00:58:36,275
I get to this normal,
then I get to this normal,

1278
00:58:36,275 --> 00:58:38,820
and then who knows
what it looks like.

1279
00:58:38,820 --> 00:58:41,347
I want to understand,
can it change at all?

1280
00:58:41,347 --> 00:58:42,930
So to understand its
change, I'm going

1281
00:58:42,930 --> 00:58:45,890
to take the derivative,
n prime of p.

1282
00:58:45,890 --> 00:58:47,870
So this is as p
moves along here,

1283
00:58:47,870 --> 00:58:50,390
how does the normal change?

1284
00:58:50,390 --> 00:58:53,500
I claim, in fact, it
can't change at all.

1285
00:58:53,500 --> 00:58:54,780
Why?

1286
00:58:54,780 --> 00:58:59,710
Well, the derivative of
the normal, first of all,

1287
00:58:59,710 --> 00:59:03,670
must be perpendicular to-- I
haven't given anything a name

1288
00:59:03,670 --> 00:59:05,030
here.

1289
00:59:05,030 --> 00:59:08,120
This thing is the boundary edge.

1290
00:59:08,120 --> 00:59:14,350

1291
00:59:14,350 --> 00:59:15,840
So here's my boundary edge.

1292
00:59:15,840 --> 00:59:18,700
Every single normal here--
I didn't draw the greatest

1293
00:59:18,700 --> 00:59:21,480
picture-- of those have to
be perpendicular to this one

1294
00:59:21,480 --> 00:59:22,110
common edge.

1295
00:59:22,110 --> 00:59:24,660
Here's the fact where we use
that this is a straight line.

1296
00:59:24,660 --> 00:59:27,770
If it curves, then this
is not a consistent thing.

1297
00:59:27,770 --> 00:59:31,590
But because it's straight for
a while, all of these guys

1298
00:59:31,590 --> 00:59:33,870
are perpendicular
to the same thing.

1299
00:59:33,870 --> 00:59:36,530
All of the ends are
perpendicular to the boundary

1300
00:59:36,530 --> 00:59:37,069
edge.

1301
00:59:37,069 --> 00:59:38,860
This is just little
perpendicular notation.

1302
00:59:38,860 --> 00:59:43,230

1303
00:59:43,230 --> 00:59:45,610
And if all of the n's are
perpendicular to the boundary

1304
00:59:45,610 --> 00:59:47,989
edge, the change in n must
also be perpendicular.

1305
00:59:47,989 --> 00:59:49,780
Otherwise, it would
change in such in a way

1306
00:59:49,780 --> 00:59:51,100
it's longer perpendicular.

1307
00:59:51,100 --> 00:59:51,600
OK?

1308
00:59:51,600 --> 00:59:53,675
So that's just
sort of intuitive.

1309
00:59:53,675 --> 00:59:56,990

1310
00:59:56,990 --> 00:59:58,800
Great.

1311
00:59:58,800 --> 00:59:59,510
What else?

1312
00:59:59,510 --> 01:00:01,790
I need it to be perpendicular
to something else.

1313
01:00:01,790 --> 01:00:06,470

1314
01:00:06,470 --> 01:00:09,730
I claim that n
prime of p is also

1315
01:00:09,730 --> 01:00:13,950
perpendicular to the rule line.

1316
01:00:13,950 --> 01:00:14,780
Ah, yes.

1317
01:00:14,780 --> 01:00:21,089
Because this normal-- all of
these points along a rule line

1318
01:00:21,089 --> 01:00:22,380
have the same normal direction.

1319
01:00:22,380 --> 01:00:24,830
These guys were all parallel.

1320
01:00:24,830 --> 01:00:28,835
And so if I look
at the change in n,

1321
01:00:28,835 --> 01:00:32,350
in order to make all
these guys be the same,

1322
01:00:32,350 --> 01:00:39,180
I also cannot change n in such a
way that it has a non-trivial--

1323
01:00:39,180 --> 01:00:40,942
I'm not going to
say this too well.

1324
01:00:40,942 --> 01:00:44,620
Do you believe it, more or less?

1325
01:00:44,620 --> 01:00:45,800
Let me try to say it once.

1326
01:00:45,800 --> 01:00:49,384

1327
01:00:49,384 --> 01:00:51,050
I'm looking at the
change in the normal.

1328
01:00:51,050 --> 01:00:53,060
I don't want the normal
to change in such a way

1329
01:00:53,060 --> 01:00:55,850
that it will change
along this axis.

1330
01:00:55,850 --> 01:00:58,390
So for that to be true,
the change in the normal

1331
01:00:58,390 --> 01:01:00,155
must be perpendicular
to this direction.

1332
01:01:00,155 --> 01:01:03,840

1333
01:01:03,840 --> 01:01:06,700
This is kind of weird
because the normal is also

1334
01:01:06,700 --> 01:01:08,270
perpendicular to this direction.

1335
01:01:08,270 --> 01:01:10,320
I mean, the normal
satisfies the same things.

1336
01:01:10,320 --> 01:01:11,986
It's perpendicular
to the boundary edge.

1337
01:01:11,986 --> 01:01:14,190
It's also perpendicular
to the rule line.

1338
01:01:14,190 --> 01:01:20,632
That means that n, the
normal, and its derivative,

1339
01:01:20,632 --> 01:01:21,840
they have the same direction.

1340
01:01:21,840 --> 01:01:26,910

1341
01:01:26,910 --> 01:01:28,810
That's weird.

1342
01:01:28,810 --> 01:01:34,460
In fact, it means that
the derivative must be 0.

1343
01:01:34,460 --> 01:01:36,170
Because the normal is
always unit length.

1344
01:01:36,170 --> 01:01:38,670
So this would be saying that
the normal is getting bigger

1345
01:01:38,670 --> 01:01:40,810
or shorter, but it
can't do either.

1346
01:01:40,810 --> 01:01:43,051
So in fact, normal's
not changing at all.

1347
01:01:43,051 --> 01:01:44,800
So in fact, all of
these guys are parallel

1348
01:01:44,800 --> 01:01:46,940
along the whole segment.

1349
01:01:46,940 --> 01:01:50,490
That means this
whole region is flat,

1350
01:01:50,490 --> 01:01:53,020
and that's a contradiction.

1351
01:01:53,020 --> 01:01:53,520
All right?

1352
01:01:53,520 --> 01:01:57,200
So I waved my hands a little
bit, especially on this step,

1353
01:01:57,200 --> 01:01:58,620
but believe me.

1354
01:01:58,620 --> 01:02:01,650

1355
01:02:01,650 --> 01:02:02,470
All right.

1356
01:02:02,470 --> 01:02:05,360
That's one fun fact,
but we want more.

1357
01:02:05,360 --> 01:02:08,080

1358
01:02:08,080 --> 01:02:10,580
Because how do we know that the
boundary would be polygonal?

1359
01:02:10,580 --> 01:02:13,210

1360
01:02:13,210 --> 01:02:19,476
For that we need that straight
creases remain straight.

1361
01:02:19,476 --> 01:02:22,075
And something's not true
when you have curve creases.

1362
01:02:22,075 --> 01:02:25,407

1363
01:02:25,407 --> 01:02:27,240
Curve creases obviously
don't remain curved.

1364
01:02:27,240 --> 01:02:29,031
But curve creases don't
even remain planar.

1365
01:02:29,031 --> 01:02:31,680
I mean, they can
do crazy things.

1366
01:02:31,680 --> 01:02:38,850
But straight creases
stay straight.

1367
01:02:38,850 --> 01:02:45,980

1368
01:02:45,980 --> 01:02:48,350
So what I mean is, I have
my flat piece of paper.

1369
01:02:48,350 --> 01:02:51,005
I take a straight line
in the piece of paper,

1370
01:02:51,005 --> 01:02:52,970
and I say that's
going to be a crease.

1371
01:02:52,970 --> 01:02:54,975
It has no first derivative
along the crease.

1372
01:02:54,975 --> 01:02:58,460

1373
01:02:58,460 --> 01:03:02,440
Then when I fold that,
I get a straight line

1374
01:03:02,440 --> 01:03:07,460
and some crazy stuff
on the other side,

1375
01:03:07,460 --> 01:03:08,890
but that crease
remains straight.

1376
01:03:08,890 --> 01:03:10,005
So I was in the
middle of the paper,

1377
01:03:10,005 --> 01:03:11,420
so it actually looks like this.

1378
01:03:11,420 --> 01:03:16,240

1379
01:03:16,240 --> 01:03:18,860
Remains a straight segment,
obviously of the same length.

1380
01:03:18,860 --> 01:03:24,762
But the point is it can't
curl, and it can't even kink.

1381
01:03:24,762 --> 01:03:28,680
Now, just any good theorem
needs a counter-example.

1382
01:03:28,680 --> 01:03:34,030
So if we take a piece of paper
and we make a straight crease,

1383
01:03:34,030 --> 01:03:37,810
look, I can curl that crease.

1384
01:03:37,810 --> 01:03:40,230
So this theorem is
not true unless I

1385
01:03:40,230 --> 01:03:43,300
say that the crease is proper.

1386
01:03:43,300 --> 01:03:47,050
So here I had to make the crease
all the way to 180 degrees,

1387
01:03:47,050 --> 01:03:48,050
then I can curl it.

1388
01:03:48,050 --> 01:03:51,350
Also, if I don't fold it
at all, then I can curl it.

1389
01:03:51,350 --> 01:03:54,700
But if it's folded something
in the middle, I cannot curl.

1390
01:03:54,700 --> 01:03:57,170
Paper will not be happy with me.

1391
01:03:57,170 --> 01:03:59,760
I need extra creases
in order to curl.

1392
01:03:59,760 --> 01:04:01,325
Well, yeah, it's
just not possible.

1393
01:04:01,325 --> 01:04:04,410

1394
01:04:04,410 --> 01:04:17,090
So proper crease, which
means the fold angle

1395
01:04:17,090 --> 01:04:25,540
is not equal to 0 or
plus or minus 180.

1396
01:04:25,540 --> 01:04:26,720
So that's the theorem.

1397
01:04:26,720 --> 01:04:27,480
Let's prove it.

1398
01:04:27,480 --> 01:04:41,049

1399
01:04:41,049 --> 01:04:42,090
Maybe go a little faster.

1400
01:04:42,090 --> 01:04:47,050

1401
01:04:47,050 --> 01:04:48,794
All right.

1402
01:04:48,794 --> 01:04:50,710
So we have our crease,
and in three dimensions

1403
01:04:50,710 --> 01:04:52,030
it might look curved.

1404
01:04:52,030 --> 01:04:55,390
We take some point.

1405
01:04:55,390 --> 01:04:57,380
We know that the paper is flat.

1406
01:04:57,380 --> 01:04:59,510
So if you look at the left
side and the right side

1407
01:04:59,510 --> 01:05:05,520
of the piece of paper,
locally, at least, it's flat.

1408
01:05:05,520 --> 01:05:07,900
And I want to think
of there being

1409
01:05:07,900 --> 01:05:10,470
some tangent plane
on the right side

1410
01:05:10,470 --> 01:05:13,370
and some tangent plane
on the left side.

1411
01:05:13,370 --> 01:05:14,940
Wow, that's not a good picture.

1412
01:05:14,940 --> 01:05:17,360
Let me draw the planes first.

1413
01:05:17,360 --> 01:05:21,230
So here's two planes, and
then here's my point p.

1414
01:05:21,230 --> 01:05:24,790
And locally the surface actually
lies in these two planes

1415
01:05:24,790 --> 01:05:26,730
and kinks here.

1416
01:05:26,730 --> 01:05:28,490
The surface kinks.

1417
01:05:28,490 --> 01:05:30,300
We want the crease
to be straight here,

1418
01:05:30,300 --> 01:05:35,530
but maybe the crease is
bent something like that.

1419
01:05:35,530 --> 01:05:36,510
All right.

1420
01:05:36,510 --> 01:05:39,520
Here's what we do.

1421
01:05:39,520 --> 01:05:42,180
Again, I want to think
of-- in this case,

1422
01:05:42,180 --> 01:05:44,410
I'm going to look at
tangents, not normals.

1423
01:05:44,410 --> 01:05:46,910
So I'm going to say, well,
every point on the curve

1424
01:05:46,910 --> 01:05:48,370
here has some tangent.

1425
01:05:48,370 --> 01:05:51,290
In fact, the tangent
at this point

1426
01:05:51,290 --> 01:05:52,837
must lie right along
the intersection

1427
01:05:52,837 --> 01:05:53,670
of these two planes.

1428
01:05:53,670 --> 01:05:55,860
I'm going to give
these planes names.

1429
01:05:55,860 --> 01:05:56,600
This is p.

1430
01:05:56,600 --> 01:05:58,640
This is going to be
tangent plane Tp,

1431
01:05:58,640 --> 01:06:00,600
and this is going to be
other tangent plane Sp,

1432
01:06:00,600 --> 01:06:03,170
so on the left and
the right side.

1433
01:06:03,170 --> 01:06:07,846
So the normal at p-- sorry, not
the normal, the tangent at p--

1434
01:06:07,846 --> 01:06:11,512
did I give it a name-- p prime.

1435
01:06:11,512 --> 01:06:12,970
That's what you'd
normally call it.

1436
01:06:12,970 --> 01:06:14,950
That's the derivative.

1437
01:06:14,950 --> 01:06:20,780
Tangent of p lies
along the intersection

1438
01:06:20,780 --> 01:06:22,680
of those two planes.

1439
01:06:22,680 --> 01:06:24,731
We don't really need
that, but it's true.

1440
01:06:24,731 --> 01:06:25,730
Give you some intuition.

1441
01:06:25,730 --> 01:06:27,120
Now, what I'm
really going to use

1442
01:06:27,120 --> 01:06:30,170
is the second derivative,
little more extreme here.

1443
01:06:30,170 --> 01:06:33,775
Second derivative is
curvature on the curve.

1444
01:06:33,775 --> 01:06:36,670

1445
01:06:36,670 --> 01:06:44,180
Now, we know that in
the unfolded surface,

1446
01:06:44,180 --> 01:06:46,820
there's exactly 180
degrees of material here.

1447
01:06:46,820 --> 01:06:48,640
Here's some point p.

1448
01:06:48,640 --> 01:06:50,410
We know there's 180
degrees of material.

1449
01:06:50,410 --> 01:06:56,220
I mean, locally, everything's
flat in terms of the surface.

1450
01:06:56,220 --> 01:07:00,530
So here's where I'm going to
wave my hands a little bit.

1451
01:07:00,530 --> 01:07:03,820
I claim that the
curvature vector

1452
01:07:03,820 --> 01:07:11,200
must be perpendicular to
Tp and perpendicular to Sp.

1453
01:07:11,200 --> 01:07:15,810
Because if I take this
curve, and I project it

1454
01:07:15,810 --> 01:07:19,770
into one of these
planes, say Tp,

1455
01:07:19,770 --> 01:07:26,060
it should be straight
at p in projection.

1456
01:07:26,060 --> 01:07:29,511
It also must be straight
when I project into Sp.

1457
01:07:29,511 --> 01:07:31,260
And that actually means
that it's straight

1458
01:07:31,260 --> 01:07:32,944
in three dimensions also.

1459
01:07:32,944 --> 01:07:34,360
So the straightness
and projection

1460
01:07:34,360 --> 01:07:37,166
corresponds to straightness
when flattened basically.

1461
01:07:37,166 --> 01:07:39,930
I'm waving my
hands a little bit.

1462
01:07:39,930 --> 01:07:43,870
But if I need the
curvature vector

1463
01:07:43,870 --> 01:07:45,480
to be perpendicular
to this plane--

1464
01:07:45,480 --> 01:07:48,220
so it's something like this.

1465
01:07:48,220 --> 01:07:50,190
It's coming straight
out of this plane.

1466
01:07:50,190 --> 01:07:53,170
But that uniquely
determines its direction.

1467
01:07:53,170 --> 01:07:55,500
It also needs to be
perpendicular to Sp.

1468
01:07:55,500 --> 01:07:58,090
These two vectors can't
be the same thing,

1469
01:07:58,090 --> 01:08:03,890
and yet they have to be, unless
these planes are the same.

1470
01:08:03,890 --> 01:08:06,700
So it could be there's a zero
fold angle, then it's fine.

1471
01:08:06,700 --> 01:08:08,460
Or it could be it's
180-degree fold angle,

1472
01:08:08,460 --> 01:08:11,442
then again the
planes are the same.

1473
01:08:11,442 --> 01:08:14,180
Can go in one
direction or the other.

1474
01:08:14,180 --> 01:08:20,040
But if I have a proper crease,
then there's no such vector.

1475
01:08:20,040 --> 01:08:21,188
And so we're done.

1476
01:08:21,188 --> 01:08:22,979
That proves straight
creases stay straight.

1477
01:08:22,979 --> 01:08:28,790

1478
01:08:28,790 --> 01:08:29,290
Cool.

1479
01:08:29,290 --> 01:08:32,430

1480
01:08:32,430 --> 01:08:33,820
Any questions about that?

1481
01:08:33,820 --> 01:08:41,600

1482
01:08:41,600 --> 01:08:43,670
So I'm trying to hit a
middle ground of not too

1483
01:08:43,670 --> 01:08:48,850
technical but also
not too shallow.

1484
01:08:48,850 --> 01:08:53,020
And so you're left in the middle
state of sort of make sense.

1485
01:08:53,020 --> 01:08:56,166
But I did wave my hands,
and it's not super rigorous.

1486
01:08:56,166 --> 01:08:58,040
Because to be rigorous
it would take forever,

1487
01:08:58,040 --> 01:09:00,330
and I need to teach you
differential geometry and so

1488
01:09:00,330 --> 01:09:01,520
on.

1489
01:09:01,520 --> 01:09:04,029
But now that we are
armed with these tools--

1490
01:09:04,029 --> 01:09:07,670
straight creases stay straight
and any polygonal boundary

1491
01:09:07,670 --> 01:09:11,120
stays flat, planar--
that tells us

1492
01:09:11,120 --> 01:09:13,370
that if we have
some crease pattern,

1493
01:09:13,370 --> 01:09:20,819
any crease pattern made
out of straight edges,

1494
01:09:20,819 --> 01:09:24,010
so no curve creases allowed,
if I make straight creases,

1495
01:09:24,010 --> 01:09:29,050
every single interior
face must stay rigid

1496
01:09:29,050 --> 01:09:32,510
in any proper folding.

1497
01:09:32,510 --> 01:09:33,844
That's pretty cool.

1498
01:09:33,844 --> 01:09:36,260
Because we've talked about
rigid origami and said not much

1499
01:09:36,260 --> 01:09:37,468
is known about rigid origami.

1500
01:09:37,468 --> 01:09:38,547
It's tough.

1501
01:09:38,547 --> 01:09:40,880
But in fact, you really need
to understand rigid origami

1502
01:09:40,880 --> 01:09:43,700
if you want to fold something
like a hyperbolic paraboloid.

1503
01:09:43,700 --> 01:09:47,500
Now, when you do flat
folding stuff-- well,

1504
01:09:47,500 --> 01:09:51,960
I guess those are
kind of rigid also.

1505
01:09:51,960 --> 01:09:53,600
But any kind of
3-dimensional folding,

1506
01:09:53,600 --> 01:09:56,860
where you fold every crease
somewhat but not all the way,

1507
01:09:56,860 --> 01:09:59,942
all the faces need to be
rigid in the final form.

1508
01:09:59,942 --> 01:10:01,900
Now, you could get there
by all sorts of means.

1509
01:10:01,900 --> 01:10:03,620
But we want the
hyperbolic paraboloid

1510
01:10:03,620 --> 01:10:07,000
to exist with creases
only along these lines.

1511
01:10:07,000 --> 01:10:10,790
And now we know each of these
trapezoids must remain planar.

1512
01:10:10,790 --> 01:10:14,140
Also, each of these
triangles must remain planar.

1513
01:10:14,140 --> 01:10:17,560
Now, now we get to
the contradictions.

1514
01:10:17,560 --> 01:10:19,430
If you have this piece
of paper and those

1515
01:10:19,430 --> 01:10:22,560
are your only creases,
you can fold along

1516
01:10:22,560 --> 01:10:25,930
one of the creases or
the other, but not both.

1517
01:10:25,930 --> 01:10:28,600
Unless you fold one crease
all the way to 180-- here,

1518
01:10:28,600 --> 01:10:30,180
we can do it.

1519
01:10:30,180 --> 01:10:35,940
I have one diagonal, and
I have another diagonal.

1520
01:10:35,940 --> 01:10:37,720
So everything's
meeting at 180 degrees.

1521
01:10:37,720 --> 01:10:39,790
I can fold one and
then the other.

1522
01:10:39,790 --> 01:10:42,180
I can fold one, or I
can fold the other.

1523
01:10:42,180 --> 01:10:44,932
But I cannot fold a
little bit of one,

1524
01:10:44,932 --> 01:10:46,390
and then try to
fold the other too.

1525
01:10:46,390 --> 01:10:48,710
It's not possible.

1526
01:10:48,710 --> 01:10:52,180
Therefore, this crease
pattern doesn't fold at all.

1527
01:10:52,180 --> 01:10:52,680
OK?

1528
01:10:52,680 --> 01:10:54,280
Kind of trivial.

1529
01:10:54,280 --> 01:10:55,980
The center is messed up.

1530
01:10:55,980 --> 01:10:57,930
But keep in mind,
again, these faces

1531
01:10:57,930 --> 01:11:00,620
we don't know anything about
because this edge could

1532
01:11:00,620 --> 01:11:02,290
do crazy things.

1533
01:11:02,290 --> 01:11:04,250
But the interior faces
of the crease pattern

1534
01:11:04,250 --> 01:11:07,240
must all stay flat
and stay exactly

1535
01:11:07,240 --> 01:11:08,730
as they were in the original.

1536
01:11:08,730 --> 01:11:11,440
So it's really like we
have rigid panels here

1537
01:11:11,440 --> 01:11:12,590
and hinges between them.

1538
01:11:12,590 --> 01:11:15,520
The outside, though, we
have no idea what happens.

1539
01:11:15,520 --> 01:11:16,020
OK.

1540
01:11:16,020 --> 01:11:17,000
So the inside sucks.

1541
01:11:17,000 --> 01:11:20,010
So I think, at this point,
we get to this picture.

1542
01:11:20,010 --> 01:11:21,670
Here's the regular
crease pattern.

1543
01:11:21,670 --> 01:11:24,220
So let's just make life
easier and say, well, we'll

1544
01:11:24,220 --> 01:11:26,150
cut out a hole in the center.

1545
01:11:26,150 --> 01:11:28,980
That surely-- that
must be better.

1546
01:11:28,980 --> 01:11:30,180
But it's no better.

1547
01:11:30,180 --> 01:11:33,890
That thing still
can't fold because you

1548
01:11:33,890 --> 01:11:37,830
take any of the rings
that goes around,

1549
01:11:37,830 --> 01:11:42,406
like a square annulus--
here's a ring.

1550
01:11:42,406 --> 01:11:44,960
The creases in that ring
are just those four.

1551
01:11:44,960 --> 01:11:47,520
So I'm just looking at
these four trapezoids.

1552
01:11:47,520 --> 01:11:50,230
Again, each one
must stay planar.

1553
01:11:50,230 --> 01:11:52,220
And these are the
only hinges I have.

1554
01:11:52,220 --> 01:11:55,217
It's actually really
just like this.

1555
01:11:55,217 --> 01:11:57,300
I've cut a hole in the
center, but it won't really

1556
01:11:57,300 --> 01:12:01,940
matter because the extension
of these lines meet at a point.

1557
01:12:01,940 --> 01:12:04,870
So effectively, you are
folding this whole triangle.

1558
01:12:04,870 --> 01:12:06,892
You can fold just the trapezoid.

1559
01:12:06,892 --> 01:12:08,350
But however you
fold the trapezoid,

1560
01:12:08,350 --> 01:12:10,350
you could just extend it
and make the triangle,

1561
01:12:10,350 --> 01:12:12,016
and these triangles
will remain meeting.

1562
01:12:12,016 --> 01:12:15,680
So it's actually the same
thing as the center diagram.

1563
01:12:15,680 --> 01:12:18,150
Because this doesn't fold,
this also won't fold.

1564
01:12:18,150 --> 01:12:20,140
It doesn't matter how
many layers you cut out.

1565
01:12:20,140 --> 01:12:21,890
As long as there's one
interior layer that

1566
01:12:21,890 --> 01:12:26,530
doesn't touch the boundary,
it's going to be screwed.

1567
01:12:26,530 --> 01:12:28,440
OK.

1568
01:12:28,440 --> 01:12:31,390
That is why
hyperbolic paraboloid

1569
01:12:31,390 --> 01:12:33,910
doesn't fold without diagonals.

1570
01:12:33,910 --> 01:12:37,110
Once you know the creases are
straight, it's kind of obvious.

1571
01:12:37,110 --> 01:12:41,160
But let me give you a
more general picture.

1572
01:12:41,160 --> 01:12:43,770
This makes things a little
more interesting and tells you,

1573
01:12:43,770 --> 01:12:45,270
really, all those
straight crease

1574
01:12:45,270 --> 01:12:48,180
pleatings that I've shown you,
like hexagons and octagons,

1575
01:12:48,180 --> 01:12:52,560
none of them are possible,
which was a surprise.

1576
01:12:52,560 --> 01:12:54,790
So here I'm taking concentric
shapes in such a way

1577
01:12:54,790 --> 01:12:56,950
that the diagonals
meet at a point.

1578
01:12:56,950 --> 01:12:58,080
I still need that fact.

1579
01:12:58,080 --> 01:13:01,581
If they don't, in fact,
it's possible to fold.

1580
01:13:01,581 --> 01:13:03,350
And we have a little
example of that.

1581
01:13:03,350 --> 01:13:10,910

1582
01:13:10,910 --> 01:13:13,515
OK, now I actually
need two interior rings

1583
01:13:13,515 --> 01:13:16,110
to get a contradiction.

1584
01:13:16,110 --> 01:13:23,900
So I take some ring like
this, and I take another ring,

1585
01:13:23,900 --> 01:13:26,254
and I suppose none of
them touch the boundary.

1586
01:13:26,254 --> 01:13:27,920
As long as I know
there's at least four,

1587
01:13:27,920 --> 01:13:30,325
there's going to be two
consecutive rings like this.

1588
01:13:30,325 --> 01:13:39,480
And the hinges are the diagonals
and the inner ring boundaries.

1589
01:13:39,480 --> 01:13:40,360
OK.

1590
01:13:40,360 --> 01:13:48,460
If I just look locally at
this, I claim it must be

1591
01:13:48,460 --> 01:13:52,470
mountain-mountain-mountain-valley
 or the reverse.

1592
01:13:52,470 --> 01:13:54,810
This is something we would
normally prove by saying,

1593
01:13:54,810 --> 01:13:57,177
OK, this is locally smallest,
therefore these guys

1594
01:13:57,177 --> 01:13:57,760
are different.

1595
01:13:57,760 --> 01:13:59,176
And this is also
locally smallest,

1596
01:13:59,176 --> 01:14:01,264
therefore these guys are
different, and so on.

1597
01:14:01,264 --> 01:14:02,930
Except that only works
for flat folding.

1598
01:14:02,930 --> 01:14:04,780
Here I need to prove
it for 3D folding.

1599
01:14:04,780 --> 01:14:07,440
I'm not going to bother,
but you can do it.

1600
01:14:07,440 --> 01:14:11,390
So in fact, let's say this is
 mountain-mountain-mountain-valley,

1601
01:14:11,390 --> 01:14:13,040
then this must be
mountain, this must,

1602
01:14:13,040 --> 01:14:15,510
all the way around
must be mountain.

1603
01:14:15,510 --> 01:14:16,010
OK?

1604
01:14:16,010 --> 01:14:17,350
That's not such a big deal.

1605
01:14:17,350 --> 01:14:21,090
This must all be valley,
and this must all

1606
01:14:21,090 --> 01:14:24,590
be mountain, or
the entire reverse.

1607
01:14:24,590 --> 01:14:27,665
But the point is,
all of these guys

1608
01:14:27,665 --> 01:14:32,190
out here must be mountains.

1609
01:14:32,190 --> 01:14:34,530
So really if I extend
all of these guys

1610
01:14:34,530 --> 01:14:37,120
and they meet at a single
point, if I was lucky,

1611
01:14:37,120 --> 01:14:40,740
that's what I'm supposed to
assume, really what I'm doing

1612
01:14:40,740 --> 01:14:44,890
is folding a whole
bunch of triangles

1613
01:14:44,890 --> 01:14:47,810
where all of the
creases are mountains.

1614
01:14:47,810 --> 01:14:49,400
That's not possible.

1615
01:14:49,400 --> 01:14:52,564
So if you didn't
believe this thing,

1616
01:14:52,564 --> 01:14:53,980
it follows from
the same argument.

1617
01:14:53,980 --> 01:14:56,535
They can't all be mountains,
and therefore some crease

1618
01:14:56,535 --> 01:14:58,530
is not getting folded.

1619
01:14:58,530 --> 01:15:01,350
So it's the same
argument again, you just

1620
01:15:01,350 --> 01:15:03,055
have to be a little
more precise.

1621
01:15:03,055 --> 01:15:05,680

1622
01:15:05,680 --> 01:15:09,870
And that is not folding
hyperbolic paraboloids

1623
01:15:09,870 --> 01:15:11,120
and things like that.

1624
01:15:11,120 --> 01:15:13,390
But if you had
diagonals, it all works.

1625
01:15:13,390 --> 01:15:16,240
Now, in the last few minutes, I
want to show you some more fun

1626
01:15:16,240 --> 01:15:20,000
things, in particular, pillows.

1627
01:15:20,000 --> 01:15:22,610
Everybody likes pillows.

1628
01:15:22,610 --> 01:15:23,600
But how are they made?

1629
01:15:23,600 --> 01:15:26,840
Well, it's like two
squares of material joined

1630
01:15:26,840 --> 01:15:29,875
along the edges, and then
you stuff stuff inside.

1631
01:15:29,875 --> 01:15:32,000
That kind of weird, because
doubly-covered square I

1632
01:15:32,000 --> 01:15:35,230
think of as flat, and yet
you can put material inside.

1633
01:15:35,230 --> 01:15:39,540
What is the maximum volume
you can stuff into a pillow?

1634
01:15:39,540 --> 01:15:41,090
This is called the
tea bag problem

1635
01:15:41,090 --> 01:15:44,520
because it also works
for square tea bags.

1636
01:15:44,520 --> 01:15:45,320
Open.

1637
01:15:45,320 --> 01:15:45,960
We don't know.

1638
01:15:45,960 --> 01:15:49,890
Lots of practical
explorations, but open.

1639
01:15:49,890 --> 01:15:51,700
Another version of
this is Mylar balloons.

1640
01:15:51,700 --> 01:15:54,740
Mylar balloons can't
stretch, more or less.

1641
01:15:54,740 --> 01:15:57,980
And here is two
circles glued together.

1642
01:15:57,980 --> 01:16:00,310
Again flat, but you
can pump air into them.

1643
01:16:00,310 --> 01:16:03,180
You get this weird behavior,
these ripples on the outside.

1644
01:16:03,180 --> 01:16:04,880
That's real stuff.

1645
01:16:04,880 --> 01:16:07,320
But we don't know what the
maximum volume shape is,

1646
01:16:07,320 --> 01:16:11,970
but it should look something
like what we see in reality.

1647
01:16:11,970 --> 01:16:13,920
Let me tell you
some fun theorems.

1648
01:16:13,920 --> 01:16:20,470
If you take any convex
polyhedron, like say a cube,

1649
01:16:20,470 --> 01:16:23,610
you can inflate it and
increase its volume.

1650
01:16:23,610 --> 01:16:26,260
Also, I guess, works for
a doubly-covered square.

1651
01:16:26,260 --> 01:16:27,930
This is the tea
bag open problem,

1652
01:16:27,930 --> 01:16:31,260
where these edges are joined.

1653
01:16:31,260 --> 01:16:32,510
How do you do it?

1654
01:16:32,510 --> 01:16:34,540
I'm going to quote
from the paper--

1655
01:16:34,540 --> 01:16:37,157
"by simultaneously
delivering karate chops

1656
01:16:37,157 --> 01:16:38,490
to the edges of the polyhedron."

1657
01:16:38,490 --> 01:16:39,960
That's in the abstract.

1658
01:16:39,960 --> 01:16:45,380
So you take an edge like
this-- here's an edge

1659
01:16:45,380 --> 01:16:48,846
and the two incident
faces-- and I go like this.

1660
01:16:48,846 --> 01:16:51,070
Wow, that was exciting.

1661
01:16:51,070 --> 01:16:57,250
So what happens when you go like
this-- the whole board shakes,

1662
01:16:57,250 --> 01:16:59,304
I'll probably break
it eventually--

1663
01:16:59,304 --> 01:17:00,470
you get something like that.

1664
01:17:00,470 --> 01:17:03,040
So there's a valley
here, and then

1665
01:17:03,040 --> 01:17:05,970
mountains there to
replace this mountain.

1666
01:17:05,970 --> 01:17:09,640
And you can prove, if you do
that with suitable parameters

1667
01:17:09,640 --> 01:17:11,525
here, at every edge
simultaneously you

1668
01:17:11,525 --> 01:17:14,740
increase the volume,
which is pretty neat.

1669
01:17:14,740 --> 01:17:17,100
Now, how far can you go?

1670
01:17:17,100 --> 01:17:19,230
What happens when you keep
increasing the volume?

1671
01:17:19,230 --> 01:17:23,475
Is the limit polyhedral, or
is it smooth with ripples?

1672
01:17:23,475 --> 01:17:25,100
Conjecture is it's
smooth with ripples.

1673
01:17:25,100 --> 01:17:26,891
In fact, you can prove
it's not polyhedral.

1674
01:17:26,891 --> 01:17:28,810
Because not only for
convex polyhedral,

1675
01:17:28,810 --> 01:17:33,800
you take any polyhedron, it
is possible to inflate it

1676
01:17:33,800 --> 01:17:35,150
by at least some volume.

1677
01:17:35,150 --> 01:17:37,860
So the limit has to be curvy.

1678
01:17:37,860 --> 01:17:40,714
It's not going to be polyhedral.

1679
01:17:40,714 --> 01:17:42,130
Now, exactly what
it looks like we

1680
01:17:42,130 --> 01:17:46,280
don't know, but lots of
experiments to that effect.

1681
01:17:46,280 --> 01:17:48,950

1682
01:17:48,950 --> 01:17:52,690
A big open question in this
field is, does this exist?

1683
01:17:52,690 --> 01:17:53,440
We think so.

1684
01:17:53,440 --> 01:17:55,250
It seems curve creases--
none of this stuff

1685
01:17:55,250 --> 01:17:56,780
works for curve creases.

1686
01:17:56,780 --> 01:17:58,952
Curve creases seem
a lot more powerful.

1687
01:17:58,952 --> 01:18:00,660
We haven't been able
to prove this exists

1688
01:18:00,660 --> 01:18:02,000
because it seems very flexible.

1689
01:18:02,000 --> 01:18:03,500
There's a lot of
degrees of freedom.

1690
01:18:03,500 --> 01:18:05,740
It's hard to figure out
where the rule lines are.

1691
01:18:05,740 --> 01:18:09,210
But we think so.

1692
01:18:09,210 --> 01:18:10,440
So what about curved creases?

1693
01:18:10,440 --> 01:18:11,940
I haven't talked
about curve creases

1694
01:18:11,940 --> 01:18:14,210
really at all
until this lecture.

1695
01:18:14,210 --> 01:18:21,090
I would say the most seminal
work in curve creases

1696
01:18:21,090 --> 01:18:23,160
was done by David Huffman.

1697
01:18:23,160 --> 01:18:24,810
Huffman is super famous.

1698
01:18:24,810 --> 01:18:30,370
He was a grad student
and a professor

1699
01:18:30,370 --> 01:18:32,112
here at MIT for many years.

1700
01:18:32,112 --> 01:18:33,820
In particular, when
he was a grad student

1701
01:18:33,820 --> 01:18:35,319
he came up with
these Huffman codes,

1702
01:18:35,319 --> 01:18:38,160
which are used in
every MP3 player.

1703
01:18:38,160 --> 01:18:40,460
Every device you use that
uses any kind of compression

1704
01:18:40,460 --> 01:18:41,665
has Huffman codes in it.

1705
01:18:41,665 --> 01:18:43,420
Super cool from the '60s.

1706
01:18:43,420 --> 01:18:46,670
But he also did a lot
of curve crease origami,

1707
01:18:46,670 --> 01:18:48,310
curve crease folding.

1708
01:18:48,310 --> 01:18:51,980
And over the last
couple of years,

1709
01:18:51,980 --> 01:18:56,690
we've been working with
his family, so his wife

1710
01:18:56,690 --> 01:19:00,160
and two daughters,
Linda and Elise.

1711
01:19:00,160 --> 01:19:02,788
And this is me,
that's Marty, and this

1712
01:19:02,788 --> 01:19:04,162
is Duks Koschitz,
who many of you

1713
01:19:04,162 --> 01:19:06,650
know is a PhD student
in architecture.

1714
01:19:06,650 --> 01:19:09,660
And we were visiting there
back in May in particular.

1715
01:19:09,660 --> 01:19:12,600
And what we're doing is
taking his work, which

1716
01:19:12,600 --> 01:19:17,810
is all in their houses and
almost no one has seen,

1717
01:19:17,810 --> 01:19:19,412
and figuring out
how he made them.

1718
01:19:19,412 --> 01:19:20,870
What is the underlying
mathematics?

1719
01:19:20,870 --> 01:19:23,720
What are the crease patterns
that make it possible?

1720
01:19:23,720 --> 01:19:26,590
And then recreating them to
check that we did it right.

1721
01:19:26,590 --> 01:19:30,380
So what I have here are just a
few examples of our recreations

1722
01:19:30,380 --> 01:19:30,880
so far.

1723
01:19:30,880 --> 01:19:32,421
So these are not
the original models,

1724
01:19:32,421 --> 01:19:34,310
but they look just like
the original models,

1725
01:19:34,310 --> 01:19:37,339
made out of the same material
and more or less the same way.

1726
01:19:37,339 --> 01:19:39,380
Although we do it a little
more high-tech than he

1727
01:19:39,380 --> 01:19:45,760
did because we want to draw
perfect computational diagrams

1728
01:19:45,760 --> 01:19:47,762
in CAD, and then
reproduce them exactly

1729
01:19:47,762 --> 01:19:49,470
on the paper with no
reproduction errors.

1730
01:19:49,470 --> 01:19:56,650
So we use fancy
robotically-controlled devices

1731
01:19:56,650 --> 01:19:58,050
to do that.

1732
01:19:58,050 --> 01:20:00,000
So there's some
fun curve creases.

1733
01:20:00,000 --> 01:20:02,660
You get some really
nice shadow patterns.

1734
01:20:02,660 --> 01:20:07,130
All of those are circular
arcs, all the creases.

1735
01:20:07,130 --> 01:20:08,270
Here are some more.

1736
01:20:08,270 --> 01:20:12,140
These are actually parabolic
arcs and some straight creases

1737
01:20:12,140 --> 01:20:13,095
in between.

1738
01:20:13,095 --> 01:20:14,940
Get some cool 3D relief effect.

1739
01:20:14,940 --> 01:20:16,690
These are like
tessellations like Tom Hull

1740
01:20:16,690 --> 01:20:18,398
was talking about but
with curve creases.

1741
01:20:18,398 --> 01:20:23,260

1742
01:20:23,260 --> 01:20:24,390
This is pretty awesome.

1743
01:20:24,390 --> 01:20:27,860
Here the creases are quite
complicated to figure out.

1744
01:20:27,860 --> 01:20:31,130
But what's happening is
you're taking a cone like this

1745
01:20:31,130 --> 01:20:33,930
and pleating it back and forth
but with different angles,

1746
01:20:33,930 --> 01:20:36,600
and so the whole thing twists.

1747
01:20:36,600 --> 01:20:38,870
We have a physical one
of these in our offices.

1748
01:20:38,870 --> 01:20:41,580
But this is the 3D model
of what's going on.

1749
01:20:41,580 --> 01:20:44,210

1750
01:20:44,210 --> 01:20:47,050
Here's some particularly
awesome crease patterns.

1751
01:20:47,050 --> 01:20:50,050
David Huffman made a
whole variety of these.

1752
01:20:50,050 --> 01:20:51,950
These are just a
couple of examples.

1753
01:20:51,950 --> 01:20:53,485
Got some crazy circular arcs.

1754
01:20:53,485 --> 01:20:55,530
You've got some-- I
guess these are also

1755
01:20:55,530 --> 01:20:58,020
circular arcs and some
straight segments.

1756
01:20:58,020 --> 01:21:01,490
And then you wrap this
around to make a cylinder,

1757
01:21:01,490 --> 01:21:04,800
and you get this.

1758
01:21:04,800 --> 01:21:07,660
That one-- there's a bunch of
photos of the original David

1759
01:21:07,660 --> 01:21:11,750
Huffman model on the web
if you Google around.

1760
01:21:11,750 --> 01:21:16,590
Not looking exactly like that,
but there's a whole variety.

1761
01:21:16,590 --> 01:21:18,314
Here's another one,
a bit of a maybe

1762
01:21:18,314 --> 01:21:19,480
surprise what it looks like.

1763
01:21:19,480 --> 01:21:22,620
Most of the center just gets
eaten away from these creases.

1764
01:21:22,620 --> 01:21:25,020
Here the creases are
elliptical, and you

1765
01:21:25,020 --> 01:21:27,280
get something like that.

1766
01:21:27,280 --> 01:21:30,261
So there's some hidden
structure beneath.

1767
01:21:30,261 --> 01:21:32,100
I think just one more example.

1768
01:21:32,100 --> 01:21:34,640
This is probably the
most coolest puzzle.

1769
01:21:34,640 --> 01:21:38,720
You say, well, you float along
these parabolas, just one

1770
01:21:38,720 --> 01:21:40,930
valley and one mountain.

1771
01:21:40,930 --> 01:21:45,140
But you fold them a
lot, you get that.

1772
01:21:45,140 --> 01:21:47,670
And Huffman composed
these to make

1773
01:21:47,670 --> 01:21:50,600
various kinds of tubes
and really cool things.

1774
01:21:50,600 --> 01:21:51,826
So this is work in progress.

1775
01:21:51,826 --> 01:21:53,200
Just wanted to
give you a sample.

1776
01:21:53,200 --> 01:21:55,710
There's a paper about this
which shows more examples.

1777
01:21:55,710 --> 01:21:58,850
But our goal is to recreate
all or most of them

1778
01:21:58,850 --> 01:22:03,140
and document how he
did it and figure out

1779
01:22:03,140 --> 01:22:04,790
how he designed these models.

1780
01:22:04,790 --> 01:22:06,789
He had mathematical
tools to do this,

1781
01:22:06,789 --> 01:22:08,580
and we're still figuring
out what they are.

1782
01:22:08,580 --> 01:22:13,500
The hope is it will lead to
more great mathematics and art

1783
01:22:13,500 --> 01:22:15,280
about curve creases.

1784
01:22:15,280 --> 01:22:16,490
And that is it.

1785
01:22:16,490 --> 01:22:18,270
This is my last lecture.

1786
01:22:18,270 --> 01:22:21,980
Next class we have-- or
next three classes we

1787
01:22:21,980 --> 01:22:24,001
have a whole bunch of
student presentations.

1788
01:22:24,001 --> 01:22:24,500
Please come.

1789
01:22:24,500 --> 01:22:26,320
There will be lots of
awesome things there.

1790
01:22:26,320 --> 01:22:27,740
Send me your slides
ahead of time,

1791
01:22:27,740 --> 01:22:31,820
and you'll have 10 minutes
to do it, to give a talk.

1792
01:22:31,820 --> 01:22:35,980
And then the last lecture,
Wednesday two weeks from now,

1793
01:22:35,980 --> 01:22:37,720
is Tomohiro Tachi.

1794
01:22:37,720 --> 01:22:38,920
And thanks very much.

1795
01:22:38,920 --> 01:22:40,830
It's been a lot of
fun doing this class

1796
01:22:40,830 --> 01:22:42,121
and having you all as students.

1797
01:22:42,121 --> 01:22:44,920
[APPLAUSE]

1798
01:22:44,920 --> 01:22:58,961