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PROFESSOR: All right.

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Let's get started.

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So we have a fun lecture today
about efficient origami design.

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Last Monday, we did
inefficient origami design,

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but it was universal.

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We could fold anything.

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And let's see,
Thursday, we talked

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about some basic
foldability, crease patterns,

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what make the valid.

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That'll ground us
a little bit today

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when designing some
crease patterns.

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Although, we're going to
stay fairly high level today

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because there are two big
methods I want to talk about.

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One is tree method
which is hit it pretty

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big in the impractical
origami design.

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Lot of modern complex
origami designers

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use it either in their head
or occasionally on a computer.

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I demoed it quickly last time.

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So we are going to see some
level of detail how that works.

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And then, I want to
talk about Orgamizer

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which is one of the latest
techniques for designing

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crazy, arbitrary,
three-dimensional shapes that

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seems to be pretty efficient.

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Although we don't have a formal
sense in which it is efficient,

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it has some nice properties.

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And it's pretty cool,
and I can also demo it.

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It's also freely downloadable
software for Windows.

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Good.

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So just to get you
motivated a little bit,

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I brought a bunch of examples.

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I'll show you more later.

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But this is the
sort of thing you

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can do with the tree method.

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It's not going to be the tree
method as I presented here.

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This is a variation
on it called box

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pleating which
you can read about

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in Origami Design Secrets.

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And I don't think Jason
will talk about that either.

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But it's a variation on
what we'll be talking about.

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It lets you do crazy things like
these two praying mantises, one

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eating the other.

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This is a design by Robert Lang.

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Fairly new.

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I don't have a year
here, but I think

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it's last year or something.

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And that's the sort
of thing you can

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do getting all the limbs, all
the right proportions, even

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multiple characters
by representing

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your model as a stick figure.

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And that's what
the tree method is

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all about and doing
that efficiently.

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So this is a statement
last time of the theorem.

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There's some catches to this.

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It's an algorithm.

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Find a folding of
the smallest square

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possible into and origami
base with the desired

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tree as a shadow
or as a projection.

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So you remember,
this kind of picture.

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You want to make a lizard.

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You specify the lengths
of each of these limbs

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and how they're connected
together into a tree.

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And then, you want to build an
origami model on top of that,

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so to speak.

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So that it looks
something like this.

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And you want to find a square
the folds into such a shape.

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This projection is
exactly that tree.

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Now, say it's an algorithm, and
it finds the smallest square.

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But to do that, essentially
requires exponential time.

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We'll prove in the next class
that this problem, in general,

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is NP-complete.

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So it's really hard.

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But there is an
exponential time algorithm,

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and I didn't say efficient here.

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It's efficient in terms
of design, quality,

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or in terms of algorithm.

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But you have to
pick one of the two.

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So in TreeMaker the
program, there's

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an efficient
algorithm, which finds

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a reasonably good-sized square.

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But it's not guaranteed
to be optimal.

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It's just a local optimum.

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In principle, you could
spend exponential time here.

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So slow algorithm and
get the smallest square.

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So it depends.

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The other catch is this folding.

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We're still working on proving
that this does not actually

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self-intersect in
the folded state.

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I checked the dates.

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We've been working on
that for six years.

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But it's closing in.

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Maybe next year we'll have
a draft of this proof.

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It's quite-- it's
many, many pages.

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Good.

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So those are the catches.

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Now, let me tell you
about this term uniaxial.

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Essentially, it just
means tree shapes.

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But I'd like to be a little
bit more formal about that.

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And last time, I showed you
the standard origami bases.

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All of these are
uniaxial, I think,

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except the pinwheel
which we folded.

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So the pinwheel-- so let
me tell you intuitively

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what uniaxial means.

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It means you can take
all these flaps of paper

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and lie them, place
them along a line.

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And the hinges
between those flaps

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are all perpendicular
to that line.

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So this is the axis.

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Whereas something like
this, essentially there

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are four axes.

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The flaps are here,
or two axes I guess.

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But definitely not one.

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So these cannot be
lined up along a line,

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even if you've flapped
them around some other way.

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That's intuitive definition.

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Multiaxial is not a
formally defined thing.

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But uniaxial we can
formally define.

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And it will capture things
like this water bomb base,

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all the other bases there,
as well as bases like this.

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And it's defined
by Robert Lang, I

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think probably around '94
was the first publication.

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And it's just a
bunch of conditions.

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And a bunch of them are just
technical to make things work

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out mathematically.

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First thing I'd like to say
is that the entire base-- base

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just means origami
for our purposes.

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It's sort of
practical distinction

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not a mathematical one.

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Is that everything
lies above the floor.

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So the floor is equal
to zero, and we'll just

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say everything's above that.

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And the action is going
to be in the floor.

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That's where I've
drawn it that way.

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Here, there's a floor.

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And the tree is going
to lie on the floor,

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and everything
else is above that.

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Second property.

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Sort of a shadow property.

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If I look at where
the base meets

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the floor is equals to
zero, that's the same thing

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as if I look at the
shadow onto the floor.

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This is essentially saying
that this base does not

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have any overhang.

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So if it had, for
example, some feature

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like this that hung over
its shadow-- was more--

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went out here.

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The shadow goes out here,
but the base does not.

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That's not allowed.

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So I want everything--
actually want

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things to get smaller
as you go up in z.

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This is a stronger
statement of property two.

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And then, I want to
define this notion flaps.

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And the basic idea
is that you have

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faces of the crease pattern.

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so the faces are
just the regions

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we get out of the creases, all
these triangles for example.

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I can divide them,
partition them

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into groups which I call flaps.

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So for example, these two
guys over here form one flap.

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They fold together.

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They're going to be
manipulated together.

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And so in this case,
I'll get four flaps.

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Anything I want to say here?

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Yeah.

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Each flap is going to
project to a line segment.

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It's going to be one of
the edges of the tree.

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So then, there's the
notion of a hinge crease.

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And these are just creases
shared by two flaps.

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So they're the creases that
separate one flap from another.

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These will always require
that they projects to a point.

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So this is equivalent to saying
the hinge crease is vertical.

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It's perpendicular to the floor.

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I'm always projecting
straight down onto the floor

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orthographically, just
setting z to zero.

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And so that's saying
these are the hinges.

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They should be vertical.

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So projection is a point.

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And then from those
two properties,

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I can define a graph
which I want to be a tree.

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So each flap I want to
make an edge of my graph.

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And that edge is going
to be the line segment

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that the flap projects,
each flat projects to.

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And I'm going to connect
those edges together

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at vertices when the flaps
share the hinge crease.

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All right.

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That's a graph which
you can define.

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And that graph is a tree.

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That's the constraint.

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And I think I have even more.

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I've got one more property.

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I think I actually
want projects here.

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Let's try that.

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All right.

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This is a bunch of formalism to
state what's pretty intuitive.

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I want all the flaps of
paper to be vertical,

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so they project
to a line segment.

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When I look from the-- when
I look at the projection,

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I can define a
graph where there's

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an edge for each flap,
where it's projecting.

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And I join those edges together.

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Here, I'm joinging
four them at a vertex.

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Because if you unfold it,
they all share hinge creases.

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Hinge creases in this case
are the perpendicular.

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These four guys.

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So because-- it's
hard to manipulate.

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I've got a flap over here.

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A flap over here.

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They share a hinge, so I connect
them together in the graph.

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It's just a formal way to
make the graph correct.

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It may seem tedious,
but this definition

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sidesteps some issues which
would occur if you defined it

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in the more obvious way which
is just take the projection,

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call it a tree.

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But I don't want
to get into why you

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need to do it this way exactly.

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Maybe, we'll see
it at some point.

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Essentially, some flaps can
be hidden inside others,

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so you need this definition
for it to really work.

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And then, there's
this extra constraint

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which is that my base
is pointy at the leaves.

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Leaves are the
vertices of the tree

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to have only one incident edge.

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And so I want there
to be only one

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point that lives at the leaf.

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Obviously, elsewhere
in the tree,

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there's a whole bunch of points,
a whole vertical segment,

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that all projects to that point.

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Here, I just want one.

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That's important
because I want to think

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about where the leaves are.

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And the whole idea
in the tree method

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is to think about how to
place the leaves on your piece

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of paper so that
this folding exists.

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So that's what
we're going to do.

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The tree method is kind of
surprising in its simplicity.

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There's a bunch of
details to make it work.

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But the idea is
actually very simple.

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Let's suppose you want
ability uniaxial base.

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I'll tell you something
that must be satisfied

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by your uniaxial base,
a necessary condition.

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Assuming you're starting from
a convex piece of paper, which

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is the case we
usually care about.

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Actually, we're starting
from a square, a rectangle,

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or something convex.

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Here's what has to be true.

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I didn't give a
name, but this graph

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that's supposed
to be a tree, I'm

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going to call the shadow
tree for obvious reasons.

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00:16:05,850 --> 00:16:09,200
And now, I want to take two
points in the shadow tree,

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measure their distance
in a tree sense.

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So I have some tree like this.

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I have two points like, say,
this point and that point.

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The distance between
them is the distance as

253
00:16:22,910 --> 00:16:26,850
measured if you had to walk
in the tree, how far is it

254
00:16:26,850 --> 00:16:28,490
to go from here to here.

255
00:16:28,490 --> 00:16:30,100
And because our tree
is a metric tree,

256
00:16:30,100 --> 00:16:31,849
because we specified
all the edge lengths,

257
00:16:31,849 --> 00:16:34,850
we can just add up those
lengths, measure them.

258
00:16:34,850 --> 00:16:38,160
And that's the distance
between two points in the tree.

259
00:16:38,160 --> 00:16:42,330
That must be less than
or equal to the distance

260
00:16:42,330 --> 00:16:46,200
between those two points
on the piece of paper.

261
00:17:06,960 --> 00:17:08,730
What does that mean?

262
00:17:08,730 --> 00:17:12,160
So on piece of paper
that's convex--

263
00:17:12,160 --> 00:17:17,089
so it might not be a square, but
square's easier picture draw.

264
00:17:17,089 --> 00:17:20,060
The distance between
them is that.

265
00:17:20,060 --> 00:17:22,260
Pretty simple.

266
00:17:22,260 --> 00:17:24,599
So what does this mean?

267
00:17:24,599 --> 00:17:25,750
I'm taking this square.

268
00:17:25,750 --> 00:17:28,119
Somehow, I'm folding
it into a base

269
00:17:28,119 --> 00:17:30,650
whose projection is the tree.

270
00:17:30,650 --> 00:17:36,630
So I look at these
two points, p and q,

271
00:17:36,630 --> 00:17:38,940
I fold them somewhere
in the 3D picture

272
00:17:38,940 --> 00:17:40,980
which is not drawn up here.

273
00:17:40,980 --> 00:17:46,070
Those points-- so maybe there's
a p up here and a q up here.

274
00:17:46,070 --> 00:17:49,380
I project those points
down onto the floor which

275
00:17:49,380 --> 00:17:53,000
is going to fall on the
tree by this definition.

276
00:17:53,000 --> 00:17:56,390
Call that, let's say, p prime
for the projected version of p,

277
00:17:56,390 --> 00:17:58,160
q prime.

278
00:17:58,160 --> 00:18:00,360
I measure the distance here.

279
00:18:00,360 --> 00:18:04,186
That has to be-- the
distance between p prime

280
00:18:04,186 --> 00:18:05,560
and q prime in
the tree should be

281
00:18:05,560 --> 00:18:07,950
less than or equal to the
distance between p and q

282
00:18:07,950 --> 00:18:11,110
in the piece of paper, for
every pair points p and q.

283
00:18:11,110 --> 00:18:13,870
That's the condition.

284
00:18:13,870 --> 00:18:19,650
It's almost trivial
to show because when

285
00:18:19,650 --> 00:18:23,490
I take this segment of paper,
I fold the piece of paper.

286
00:18:23,490 --> 00:18:26,250
But in particular, I
fold p and q somehow.

287
00:18:26,250 --> 00:18:29,280
I can't get p and q farther
away from each other

288
00:18:29,280 --> 00:18:32,650
because folding only
makes things closer.

289
00:18:32,650 --> 00:18:35,530
There, I'm assuming that the
piece of paper is convex.

290
00:18:35,530 --> 00:18:37,704
There's no way to
fold and stretch pq

291
00:18:37,704 --> 00:18:39,120
because that's a
segment of paper.

292
00:18:39,120 --> 00:18:41,670
It can only contract.

293
00:18:41,670 --> 00:18:44,950
I mean, you can fold the
segment something like this.

294
00:18:44,950 --> 00:18:47,170
Then, the distance
between p and q

295
00:18:47,170 --> 00:18:50,890
gets smaller than the
length of this segment.

296
00:18:50,890 --> 00:18:55,460
Because if I took this-- this
line segment of paper that got

297
00:18:55,460 --> 00:18:56,650
folded.

298
00:18:56,650 --> 00:18:58,240
If I project it
onto the line here,

299
00:18:58,240 --> 00:19:01,120
it's only going to get shorter.

300
00:19:01,120 --> 00:19:02,030
So I fold p and q.

301
00:19:02,030 --> 00:19:05,750
They get closer in three-space.

302
00:19:05,750 --> 00:19:08,430
And then, I project
them down to the floor.

303
00:19:08,430 --> 00:19:11,750
They can also only get
closer when I do that.

304
00:19:16,170 --> 00:19:18,476
So that's essentially the proof.

305
00:19:18,476 --> 00:19:20,220
Do I need to spell that out?

306
00:19:20,220 --> 00:19:22,850
So you have the line
segment on the paper.

307
00:19:22,850 --> 00:19:23,410
You fold it.

308
00:19:23,410 --> 00:19:24,420
It gets shorter.

309
00:19:24,420 --> 00:19:25,880
You project it onto the floor.

310
00:19:25,880 --> 00:19:26,730
It also get shorter.

311
00:19:26,730 --> 00:19:28,465
Therefore, whatever
this distance

312
00:19:28,465 --> 00:19:31,320
is on the tree has to be less
than or equal to the distance

313
00:19:31,320 --> 00:19:32,310
you started with.

314
00:19:32,310 --> 00:19:34,400
So this may seem
kind of trivial.

315
00:19:34,400 --> 00:19:40,180
But the surprising thing is it
this is really all you need.

316
00:19:40,180 --> 00:19:43,227
So this is true between any
two points in the shadow tree.

317
00:19:43,227 --> 00:19:45,060
In fact, we're going
to focus on the leaves.

318
00:19:45,060 --> 00:19:47,750
We'll say, all right,
so in particular, I've

319
00:19:47,750 --> 00:19:51,750
got a place this leaf, and
each of these six leaves here,

320
00:19:51,750 --> 00:19:54,430
I have to place them somewhere
on the piece of paper.

321
00:19:54,430 --> 00:19:58,310
I better do it so that that
condition is satisfied.

322
00:19:58,310 --> 00:20:01,150
I have to place these two
leaves and the piece of paper--

323
00:20:01,150 --> 00:20:06,637
let's say this distance is
one, and this distance is one.

324
00:20:06,637 --> 00:20:08,970
These two leaves have to be
placed on the piece of paper

325
00:20:08,970 --> 00:20:12,110
such that their distance
is at least two.

326
00:20:12,110 --> 00:20:14,870
And the distance between these
two guys has to be at least two

327
00:20:14,870 --> 00:20:17,300
and between these two guys
has to be at least two.

328
00:20:17,300 --> 00:20:18,620
And same over here.

329
00:20:18,620 --> 00:20:20,270
Let's say all the
edge lengths are one.

330
00:20:20,270 --> 00:20:24,940
And the distance between,
say, this leaf and this leaf

331
00:20:24,940 --> 00:20:27,840
has to be at least three
because the distance in the tree

332
00:20:27,840 --> 00:20:29,510
is three.

333
00:20:29,510 --> 00:20:32,945
So at the very least, we should
place the points on the paper

334
00:20:32,945 --> 00:20:34,570
so that those conditions
are satisfied,

335
00:20:34,570 --> 00:20:36,767
and it turns out,
that's enough as long

336
00:20:36,767 --> 00:20:39,350
as you find a placement of the
points such as those conditions

337
00:20:39,350 --> 00:20:40,260
are satisfied.

338
00:20:40,260 --> 00:20:43,230
There will be a folding where
those leaves actually come

339
00:20:43,230 --> 00:20:44,710
from those points of paper.

340
00:20:44,710 --> 00:20:46,500
That's the crazy part.

341
00:20:46,500 --> 00:20:50,070
But this idea is actually
kind of obvious in some sense.

342
00:20:50,070 --> 00:20:52,390
I mean, once you know
it, it's really obvious.

343
00:20:52,390 --> 00:20:57,010
But what's surprising is it this
is all you need to worry about.

344
00:20:57,010 --> 00:21:01,490
There's a lot of details that
make that work, but you can.

345
00:21:01,490 --> 00:21:06,940
So let me just
mention one detail

346
00:21:06,940 --> 00:21:09,960
which is the scale factor.

347
00:21:09,960 --> 00:21:12,730
If you fix the size,
the edge lengths

348
00:21:12,730 --> 00:21:14,510
on the tree which
is the usual, which

349
00:21:14,510 --> 00:21:17,430
is one way to think
about it, and you

350
00:21:17,430 --> 00:21:19,750
start with some square--
like if I start with a one

351
00:21:19,750 --> 00:21:22,166
by one square, there's no way
I'm going to fold that tree.

352
00:21:22,166 --> 00:21:24,737
There's just not enough
distance in the square.

353
00:21:24,737 --> 00:21:26,820
So what I'd like to do is
find the smallest square

354
00:21:26,820 --> 00:21:28,970
that can fold into this thing.

355
00:21:28,970 --> 00:21:31,810
Or equivalently find--
you can think of scaling

356
00:21:31,810 --> 00:21:34,110
the piece of paper, or
you can think of scaling

357
00:21:34,110 --> 00:21:37,150
the tree with a
fixed piece of paper.

358
00:21:37,150 --> 00:21:38,850
Doesn't really matter.

359
00:21:38,850 --> 00:21:42,930
In general, you get
this problem which

360
00:21:42,930 --> 00:21:44,259
I'll call scale optimization.

361
00:21:44,259 --> 00:21:45,300
This is the hard problem.

362
00:21:53,970 --> 00:22:19,000
So let's say-- just
defining some variables.

363
00:22:19,000 --> 00:22:22,470
So P i, I'm going to
maybe number the leaves

364
00:22:22,470 --> 00:22:25,150
or label label them
somehow, various letters.

365
00:22:25,150 --> 00:22:27,800
And then, P i is going
to be the point where

366
00:22:27,800 --> 00:22:30,200
that-- of paper
that actually forms

367
00:22:30,200 --> 00:22:32,020
that leaf in the folded state.

368
00:22:32,020 --> 00:22:36,924
That leaf which corresponds
to a single point of paper

369
00:22:36,924 --> 00:22:37,840
projects to that leaf.

370
00:22:40,900 --> 00:22:45,190
And then, my goal is
to maximize some scale

371
00:22:45,190 --> 00:22:46,755
factor which I'll call lambda.

372
00:22:50,990 --> 00:22:56,150
Subject to a bunch of
constraints which are just

373
00:22:56,150 --> 00:23:00,910
those constraints, except
that I add a scale factor.

374
00:23:10,100 --> 00:23:14,705
So for every pair
of leaves, i and j,

375
00:23:14,705 --> 00:23:16,080
I'm going to
measure the distance

376
00:23:16,080 --> 00:23:18,190
between those
leaves in the tree.

377
00:23:18,190 --> 00:23:19,930
This as a tree distance.

378
00:23:19,930 --> 00:23:21,870
Compare that to the
distance and the piece

379
00:23:21,870 --> 00:23:25,450
of paper between those two
points, the Euclidean distance.

380
00:23:25,450 --> 00:23:26,980
And instead of
requiring that this

381
00:23:26,980 --> 00:23:28,820
is greater than or equal to
this, which is the usual one,

382
00:23:28,820 --> 00:23:30,950
I'm going to add in
the scale factor which

383
00:23:30,950 --> 00:23:35,110
you can think of as shrinking
this or expanding that.

384
00:23:35,110 --> 00:23:36,540
It doesn't matter.

385
00:23:36,540 --> 00:23:41,630
But I want to-- because here I'm
sort of shrinking this amount.

386
00:23:41,630 --> 00:23:44,840
I want to maximize
that factor, so I

387
00:23:44,840 --> 00:23:47,360
shrink it the least possible.

388
00:23:47,360 --> 00:23:51,560
You can formulate it this way
or maybe a more intuitive way.

389
00:23:51,560 --> 00:23:54,640
But this is the standard set up.

390
00:23:54,640 --> 00:23:57,500
And this is something-- this is
called a nonlinear optimization

391
00:23:57,500 --> 00:23:58,000
problem.

392
00:23:58,000 --> 00:24:01,060
It's something that lots
of people think about.

393
00:24:01,060 --> 00:24:02,770
There are heuristics
to solve it.

394
00:24:02,770 --> 00:24:04,815
You can solve in an
exponential time.

395
00:24:04,815 --> 00:24:06,440
In general, it's
NP-complete, and we'll

396
00:24:06,440 --> 00:24:10,190
see next class that actually
this problem of origami design

397
00:24:10,190 --> 00:24:11,540
is NP-complete.

398
00:24:11,540 --> 00:24:14,850
So there's not going to be
anything better than heuristics

399
00:24:14,850 --> 00:24:16,175
and and slow algorithms.

400
00:24:18,730 --> 00:24:20,670
So the idea is, you solve that.

401
00:24:20,670 --> 00:24:25,610
Now, you have your leaves on
your piece of paper somewhere.

402
00:24:25,610 --> 00:24:27,090
Now what?

403
00:24:27,090 --> 00:24:31,730
Now, you have to figure
out how everything folds.

404
00:24:31,730 --> 00:24:34,430
That's where we get to
some real combinatorial,

405
00:24:34,430 --> 00:24:37,656
some discrete geometry.

406
00:24:37,656 --> 00:24:38,570
Fun stuff.

407
00:24:44,770 --> 00:24:45,270
Yeah.

408
00:24:45,270 --> 00:24:48,750
I have one extra
motivation here.

409
00:24:48,750 --> 00:24:50,200
Origami design is
fun, but here's

410
00:24:50,200 --> 00:24:51,810
a puzzle you can solve, too.

411
00:24:56,170 --> 00:24:57,550
Which we can already see.

412
00:25:05,020 --> 00:25:07,860
Margulis napkin problem.

413
00:25:07,860 --> 00:25:09,990
Origin of this problem
is not entirely clear,

414
00:25:09,990 --> 00:25:13,690
but I think it came
from Russia originally.

415
00:25:13,690 --> 00:25:18,060
And the problem, the puzzle
is usually stated as follows.

416
00:25:18,060 --> 00:25:21,400
Prove that if you take
a unit square paper--

417
00:25:21,400 --> 00:25:25,400
so it has perimeter
four that you can,

418
00:25:25,400 --> 00:25:29,770
no matter how you fold it, the
perimeter always gets smaller.

419
00:25:29,770 --> 00:25:31,684
Never bigger than four.

420
00:25:31,684 --> 00:25:33,100
We used a very
similar thing here.

421
00:25:33,100 --> 00:25:34,766
We said, if you have
two points, they're

422
00:25:34,766 --> 00:25:36,690
distance can only get smaller.

423
00:25:36,690 --> 00:25:38,230
That's true.

424
00:25:38,230 --> 00:25:41,300
Margulis napkin
puzzle is not true.

425
00:25:41,300 --> 00:25:42,330
That's the difference.

426
00:25:42,330 --> 00:25:44,481
Perimeter is different
from distance.

427
00:25:44,481 --> 00:25:46,230
And in fact, you can
fold a piece of paper

428
00:25:46,230 --> 00:25:49,610
to make the perimeter
arbitrarily large,

429
00:25:49,610 --> 00:25:50,567
which is pretty crazy.

430
00:25:50,567 --> 00:25:52,150
And this is something
that Robert Lang

431
00:25:52,150 --> 00:25:55,720
proved few years ago, using--

432
00:25:55,720 --> 00:25:58,530
It's sort of easy once
you have the fact-- which

433
00:25:58,530 --> 00:26:02,150
I haven't quite written down
here, but I've been saying.

434
00:26:02,150 --> 00:26:05,520
As long as you place your
points subject to this property,

435
00:26:05,520 --> 00:26:08,290
there is a folding that
has that shadow tree.

436
00:26:10,890 --> 00:26:13,620
And so the idea with the
Margulis napkin problem

437
00:26:13,620 --> 00:26:17,080
is let's make a really
spiky tree, a star.

438
00:26:20,810 --> 00:26:23,720
I want to fold the
smallest square possible,

439
00:26:23,720 --> 00:26:25,780
so that projection
is this thing.

440
00:26:25,780 --> 00:26:31,750
Let's say that it has-- I won't
say how many limbs it has.

441
00:26:31,750 --> 00:26:35,190
But the idea is, if you're using
paper efficiently, in fact,

442
00:26:35,190 --> 00:26:37,600
the folding will be very narrow.

443
00:26:37,600 --> 00:26:41,060
It'll be a pretty efficient
use of paper, hopefully.

444
00:26:41,060 --> 00:26:43,640
And so the actual
3D state will just

445
00:26:43,640 --> 00:26:46,960
be a little bit
taller than that tree.

446
00:26:46,960 --> 00:26:48,510
And then, you just wash it.

447
00:26:48,510 --> 00:26:50,330
And the idea is that then
the perimeter is really big.

448
00:26:50,330 --> 00:26:51,788
You've got a-- the
perimeter as you

449
00:26:51,788 --> 00:26:53,370
walk around the
edges of that tree.

450
00:26:53,370 --> 00:26:56,760
So how big a tree can I get?

451
00:26:56,760 --> 00:26:59,330
I'd like to somehow
place these leaves-- now,

452
00:26:59,330 --> 00:27:01,640
what's the constraint
on the leaves?

453
00:27:01,640 --> 00:27:04,480
Let's say all of
these are length one.

454
00:27:04,480 --> 00:27:06,700
Then, this says it
every pair of leaves

455
00:27:06,700 --> 00:27:10,150
must be at least distance
two way from each other.

456
00:27:10,150 --> 00:27:12,900
So I got to place these
dots in the square

457
00:27:12,900 --> 00:27:15,100
so that every pair has
distance at least two.

458
00:27:15,100 --> 00:27:18,350
This is like saying--
here's my square.

459
00:27:18,350 --> 00:27:22,384
--I'd like to place dots
so their distance is

460
00:27:22,384 --> 00:27:23,300
at least distance two.

461
00:27:23,300 --> 00:27:25,150
That's like saying
if, I drew a unit

462
00:27:25,150 --> 00:27:29,910
disk around two points--
I got to remember.

463
00:27:29,910 --> 00:27:32,530
You should always draw the
disk first and then the center.

464
00:27:32,530 --> 00:27:33,900
Much easier.

465
00:27:33,900 --> 00:27:36,460
Those disks should
not be overlapping.

466
00:27:36,460 --> 00:27:41,750
If this is length one,
and this is length one,

467
00:27:41,750 --> 00:27:43,650
the disks will be
overlapping if and only

468
00:27:43,650 --> 00:27:46,140
if this distance is
smaller than two.

469
00:27:46,140 --> 00:27:48,420
I want it always to be
greater than or equal to two.

470
00:27:48,420 --> 00:27:52,677
So I just have to place a whole
bunch of disks in the square

471
00:27:52,677 --> 00:27:54,010
so that they're not overlapping.

472
00:27:54,010 --> 00:27:57,690
So how big a square
do I need to do that?

473
00:27:57,690 --> 00:28:00,850
This is a well-studied problem,
is the disk packing problem.

474
00:28:00,850 --> 00:28:02,670
A lot of results known about it.

475
00:28:02,670 --> 00:28:04,450
It's quite difficult.

476
00:28:04,450 --> 00:28:07,910
But we don't need to be super
smart here to get a good bound.

477
00:28:07,910 --> 00:28:12,830
Let's put a point-- let's
put points along a grid.

478
00:28:12,830 --> 00:28:14,660
I'm going to regret
making such a big grid.

479
00:28:18,960 --> 00:28:21,640
Let's say, an n by n grid.

480
00:28:27,380 --> 00:28:30,545
And I'm going to set
the size of my disks

481
00:28:30,545 --> 00:28:33,305
right so that these
guys just barely touch.

482
00:28:38,290 --> 00:28:40,870
This is actually not a
terribly good packing.

483
00:28:40,870 --> 00:28:43,810
You should do a triangular
grid instead of a square grid.

484
00:28:43,810 --> 00:28:47,610
But it'll be good
enough asymptotically.

485
00:28:47,610 --> 00:28:49,070
You get the idea.

486
00:28:49,070 --> 00:28:52,840
If I set the size of
my paper to be n by n,

487
00:28:52,840 --> 00:28:58,030
I can fit about n squared
unit disks in there.

488
00:28:58,030 --> 00:29:05,340
N by n paper folds something
like n plus 1 squared.

489
00:29:05,340 --> 00:29:10,320
But let's just say,
approximately n squared disks.

490
00:29:10,320 --> 00:29:16,170
So that means I can make a star
with about n squared limbs.

491
00:29:16,170 --> 00:29:17,990
It's insane.

492
00:29:17,990 --> 00:29:19,400
It's like super efficient.

493
00:29:19,400 --> 00:29:21,100
Each of these little
portions of paper

494
00:29:21,100 --> 00:29:22,599
ends up being one
of these segments.

495
00:29:22,599 --> 00:29:24,620
That's the claim is,
you could fold that.

496
00:29:24,620 --> 00:29:27,230
So once you fold this thing,
I have an n by n square.

497
00:29:27,230 --> 00:29:30,100
You started with
perimeter about four n

498
00:29:30,100 --> 00:29:33,200
And now, I have perimeter
about n squared.

499
00:29:33,200 --> 00:29:35,480
That's huge with
respect to four n.

500
00:29:35,480 --> 00:29:42,316
So this is much bigger than four
n, for n sufficiently large.

501
00:29:42,316 --> 00:29:44,759
AUDIENCE: [INAUDIBLE]
about the length flaps.

502
00:29:44,759 --> 00:29:47,300
PROFESSOR: Here, I was assuming
all the flaps are length one.

503
00:29:47,300 --> 00:29:50,600
So the disks are size one,
and so it's an n by n square.

504
00:29:54,540 --> 00:29:55,040
Clear?

505
00:29:55,040 --> 00:29:56,990
So this is more
motivation for why

506
00:29:56,990 --> 00:29:58,210
this theorem is interesting.

507
00:29:58,210 --> 00:30:01,280
It lets you solve
this fun math puzzle

508
00:30:01,280 --> 00:30:06,020
and show not only does a
perimeter not go-- not only

509
00:30:06,020 --> 00:30:08,716
does the perimeter
not only go down,

510
00:30:08,716 --> 00:30:11,229
but it can go arbitrarily high.

511
00:30:11,229 --> 00:30:12,520
It just takes a lot of folding.

512
00:30:16,370 --> 00:30:20,440
So let's say
something about how we

513
00:30:20,440 --> 00:30:22,820
prove that once you have a
valid placement of the points,

514
00:30:22,820 --> 00:30:26,405
you can actually fill in
the creases, find folding.

515
00:30:41,970 --> 00:30:43,145
Let me bring up an example.

516
00:30:46,966 --> 00:30:48,340
So this is actually
the example I

517
00:30:48,340 --> 00:30:51,110
keep using which is, you
want to make a lizard

518
00:30:51,110 --> 00:30:58,090
or some generic four-legged
tail and head kind of creature.

519
00:30:58,090 --> 00:31:00,900
This is the output
from TreeMaker,

520
00:31:00,900 --> 00:31:02,810
complete with crease
pattern and everything.

521
00:31:02,810 --> 00:31:05,080
But here, I've labeled
all the-- or actually

522
00:31:05,080 --> 00:31:07,920
Robert Lang, I think, has
labeled all of-- this a figure

523
00:31:07,920 --> 00:31:08,920
from our book.

524
00:31:08,920 --> 00:31:13,270
--all the vertices of
the tree and the shadow.

525
00:31:13,270 --> 00:31:15,480
And then, we're
labeling where they

526
00:31:15,480 --> 00:31:16,890
come from on the piece of paper.

527
00:31:19,750 --> 00:31:22,730
So in particular, you see
something like a leaf h.

528
00:31:22,730 --> 00:31:26,410
And it comes from this
one point on the paper.

529
00:31:26,410 --> 00:31:29,380
This leaf d comes
from this point,

530
00:31:29,380 --> 00:31:31,030
and g comes from that point.

531
00:31:31,030 --> 00:31:33,710
It's actually kind of
similarly oriented to this guy.

532
00:31:33,710 --> 00:31:36,730
The interior vertices, they
come from several points.

533
00:31:36,730 --> 00:31:38,440
It's a little messy.

534
00:31:38,440 --> 00:31:41,310
But let's-- one of the things
is to try to locate where those

535
00:31:41,310 --> 00:31:43,146
points ought to be.

536
00:31:43,146 --> 00:31:48,810
s So there's this idea
of an active path which

537
00:31:48,810 --> 00:31:55,720
is a path in the tree
between two leaves.

538
00:31:55,720 --> 00:31:57,227
I'll call them
shadow leaves to say

539
00:31:57,227 --> 00:31:58,560
that they're in the shadow tree.

540
00:32:03,650 --> 00:32:12,850
And the length of
that path equals

541
00:32:12,850 --> 00:32:19,370
the distance in the paper.

542
00:32:22,120 --> 00:32:25,020
So in the case of
making a star graph,

543
00:32:25,020 --> 00:32:28,140
this is exactly
when the disks kiss,

544
00:32:28,140 --> 00:32:32,130
when the just touch each
other on the boundary.

545
00:32:32,130 --> 00:32:35,860
So in other words, we
have this inequality,

546
00:32:35,860 --> 00:32:38,110
saying the distance
between in the paper should

547
00:32:38,110 --> 00:32:40,630
be greater than or equal
to distance in the tree.

548
00:32:40,630 --> 00:32:43,090
If that inequality is
actually an equality,

549
00:32:43,090 --> 00:32:46,640
if they're the same thing,
then it's kind of critical.

550
00:32:46,640 --> 00:32:50,730
I can't get those points
any closer in the paper.

551
00:32:50,730 --> 00:32:52,970
Those things I
call active paths.

552
00:32:52,970 --> 00:32:57,440
And that is some of
the lines up here.

553
00:32:57,440 --> 00:33:01,510
I guess the black
dashed line, actually,

554
00:33:01,510 --> 00:33:03,880
in a lot of the dash lines.

555
00:33:03,880 --> 00:33:06,360
All of the dash lines, I think.

556
00:33:06,360 --> 00:33:12,650
So for example, d to h, that's
a distance between two leaves.

557
00:33:12,650 --> 00:33:14,770
And if you measure the
distance here, it's two.

558
00:33:14,770 --> 00:33:17,410
And just imagine,
this example has

559
00:33:17,410 --> 00:33:19,130
been set up so this
is exactly two.

560
00:33:19,130 --> 00:33:19,910
So this is tight.

561
00:33:19,910 --> 00:33:22,950
I can't move h any closer
to d or vice versa.

562
00:33:22,950 --> 00:33:26,210
And also from h to
a, a is actually

563
00:33:26,210 --> 00:33:30,180
in the middle of the paper
and corresponds to that flap.

564
00:33:30,180 --> 00:33:33,540
That's all of those
green, actually

565
00:33:33,540 --> 00:33:37,190
it's just the green lines,
green dashed lines are active.

566
00:33:37,190 --> 00:33:38,960
They're kind of critical.

567
00:33:38,960 --> 00:33:41,480
And what's nice is that
subdivides my piece of paper

568
00:33:41,480 --> 00:33:43,800
into a bunch of smaller shapes.

569
00:33:43,800 --> 00:33:45,567
So I have a little
triangle out here.

570
00:33:45,567 --> 00:33:46,650
That turns out to be junk.

571
00:33:46,650 --> 00:33:48,070
We're not going to need
it because the sort

572
00:33:48,070 --> 00:33:49,260
of outside the diagram.

573
00:33:49,260 --> 00:33:50,690
You could folder underneath.

574
00:33:50,690 --> 00:33:53,420
Get rid of it.

575
00:33:53,420 --> 00:33:56,738
You've got a quadrilateral
here between the green lines.

576
00:33:56,738 --> 00:33:58,404
We've got a triangle
up here, a triangle

577
00:33:58,404 --> 00:34:00,700
at the top, triangle
on the left.

578
00:34:00,700 --> 00:34:04,190
All we need to do is fill
in those little parts.

579
00:34:04,190 --> 00:34:05,190
Fill in that triangle.

580
00:34:05,190 --> 00:34:06,410
Fill in that quadrilateral.

581
00:34:06,410 --> 00:34:08,868
Of course, in general, there
might not be any active paths,

582
00:34:08,868 --> 00:34:10,865
and we haven't simplified
the diagram at all.

583
00:34:10,865 --> 00:34:12,239
But if there are
no active paths,

584
00:34:12,239 --> 00:34:13,988
you're really probably
not very efficient.

585
00:34:13,988 --> 00:34:17,080
That means none of these
constraints are tight.

586
00:34:17,080 --> 00:34:21,960
That means you could increase
the scale factor lambda,

587
00:34:21,960 --> 00:34:23,544
make a better model.

588
00:34:23,544 --> 00:34:25,460
You can increase lambda
at least a little bit.

589
00:34:25,460 --> 00:34:27,909
If all of these are
strictly greater,

590
00:34:27,909 --> 00:34:30,846
you can increase lambda until
one of them becomes equal.

591
00:34:30,846 --> 00:34:32,679
So you should have at
least one active path.

592
00:34:32,679 --> 00:34:34,179
And in fact if
you're efficient, you

593
00:34:34,179 --> 00:34:37,300
should have lots
of active paths.

594
00:34:37,300 --> 00:34:40,620
I don't think I need to
be too formal about that.

595
00:34:46,080 --> 00:34:46,760
But it's true.

596
00:34:49,380 --> 00:34:53,469
And here's one thing you
can show about active paths.

597
00:34:53,469 --> 00:34:55,750
So what would be really
nice, in this example,

598
00:34:55,750 --> 00:34:57,450
I have triangles
and quadrilaterals.

599
00:34:57,450 --> 00:34:59,241
In general, I'm going
to have a whole bunch

600
00:34:59,241 --> 00:35:00,369
of different shapes.

601
00:35:00,369 --> 00:35:01,910
Some of them could
even be non-convex

602
00:35:01,910 --> 00:35:03,512
which would be annoying.

603
00:35:03,512 --> 00:35:05,470
I would really just like
to deal with triangles

604
00:35:05,470 --> 00:35:07,670
because I like
triangles-- geometer.

605
00:35:07,670 --> 00:35:09,087
And triangles are simple.

606
00:35:09,087 --> 00:35:11,170
And it looks like the
crease pattern in a triangle

607
00:35:11,170 --> 00:35:12,500
is pretty simple.

608
00:35:12,500 --> 00:35:14,440
In fact, it's just
angular bisectors

609
00:35:14,440 --> 00:35:19,940
of the triangle plus a few
extra perpendicular folds.

610
00:35:19,940 --> 00:35:22,380
So that would be
kind of nice if I

611
00:35:22,380 --> 00:35:24,220
could get everything triangles.

612
00:35:24,220 --> 00:35:26,640
To do that, I need
lots of active paths.

613
00:35:26,640 --> 00:35:30,360
So how can I guarantee that
there's lots of active paths?

614
00:35:30,360 --> 00:35:33,420
I'm going wave my hands a little
bit about how this is done.

615
00:35:37,580 --> 00:35:43,200
But the idea is to
augment the tree.

616
00:35:43,200 --> 00:35:46,060
So I have some tree that
I actually want to make,

617
00:35:46,060 --> 00:35:47,490
like lizard.

618
00:35:47,490 --> 00:35:49,930
And I'm going to add
some extra stuff.

619
00:35:49,930 --> 00:35:52,655
Like maybe I'll add a branch
here and a branch here

620
00:35:52,655 --> 00:35:53,700
or whatever.

621
00:35:53,700 --> 00:35:54,450
Whatever it takes.

622
00:35:57,130 --> 00:35:59,830
I got to do so carefully.

623
00:35:59,830 --> 00:36:01,160
So let me say what that means.

624
00:36:05,400 --> 00:36:15,360
So I'm going to add extra
leaves to the shadow tree.

625
00:36:19,340 --> 00:36:47,300
My goal is to make the active
paths triangulate the paper

626
00:36:47,300 --> 00:36:49,030
without changing
the scale factor.

627
00:36:57,040 --> 00:36:58,365
So this is kind of a cheat.

628
00:36:58,365 --> 00:37:00,740
And most of the time, you
don't actually need this cheat.

629
00:37:00,740 --> 00:37:06,330
But for proving things, it
makes life a little easier.

630
00:37:06,330 --> 00:37:08,890
So we want to show that
it's enough to place

631
00:37:08,890 --> 00:37:10,800
the vertices subject
to this, the leaves

632
00:37:10,800 --> 00:37:14,080
subject to this constraint.

633
00:37:14,080 --> 00:37:16,090
So ideally, we make our tree.

634
00:37:16,090 --> 00:37:18,790
But if we make an even
more complicated tree,

635
00:37:18,790 --> 00:37:22,551
like with these extra little
limbs, we can get rid of them

636
00:37:22,551 --> 00:37:23,050
at the end.

637
00:37:23,050 --> 00:37:24,930
You just fold them
over and collapse

638
00:37:24,930 --> 00:37:27,770
this flap against
an adjacent flap.

639
00:37:27,770 --> 00:37:31,872
So if we make our life
harder, that's OK, too.

640
00:37:31,872 --> 00:37:33,580
If we could fold a
more complicated tree,

641
00:37:33,580 --> 00:37:35,540
in particular we folded
the tree we wanted.

642
00:37:35,540 --> 00:37:38,850
If we can do that without
changing the scale factor,

643
00:37:38,850 --> 00:37:41,270
then great.

644
00:37:41,270 --> 00:37:43,480
Then, we did what
we wanted to do.

645
00:37:43,480 --> 00:37:47,845
We folded our piece of paper
with the desired scale factor.

646
00:37:47,845 --> 00:37:49,970
In reality, we're actually
going to move the leaves

647
00:37:49,970 --> 00:37:52,709
around a little bit
so that we have to do.

648
00:37:52,709 --> 00:37:54,250
We're going to move
around the leaves

649
00:37:54,250 --> 00:37:56,125
that you already placed
in order to make room

650
00:37:56,125 --> 00:37:57,400
for the new leaves.

651
00:37:57,400 --> 00:37:58,520
But here's the idea.

652
00:37:58,520 --> 00:38:01,600
We have these leaves.

653
00:38:01,600 --> 00:38:04,470
There's some active
paths, these green lines.

654
00:38:04,470 --> 00:38:07,610
And we'd really-- we have this
quadrilateral in the center.

655
00:38:07,610 --> 00:38:09,380
We'd really like
to subdivide it.

656
00:38:09,380 --> 00:38:11,420
Like this black line is
kind of asking for it.

657
00:38:11,420 --> 00:38:13,140
It would be really
nice if we could just

658
00:38:13,140 --> 00:38:16,590
add in an active paths there.

659
00:38:16,590 --> 00:38:17,930
And you can do it.

660
00:38:17,930 --> 00:38:21,170
Let's see if I can identify
what we're talking about here.

661
00:38:21,170 --> 00:38:23,570
So a fun thing
about active paths,

662
00:38:23,570 --> 00:38:27,130
you look at two leaves like
g d here, which corresponds

663
00:38:27,130 --> 00:38:31,140
to this path g d here,
because it's active,

664
00:38:31,140 --> 00:38:35,490
you know this length is exactly
the length traced right here.

665
00:38:35,490 --> 00:38:37,100
So that means, this
segment has to be

666
00:38:37,100 --> 00:38:39,080
folded right along
the tree here.

667
00:38:39,080 --> 00:38:41,340
You know that this
segment is that.

668
00:38:41,340 --> 00:38:44,790
And so in particular, you know
where c is on that segment.

669
00:38:44,790 --> 00:38:47,360
C actually comes from multiple
points in this diagram.

670
00:38:47,360 --> 00:38:52,345
But you know that this point
right here must fold to c.

671
00:38:52,345 --> 00:38:55,260
And you know this point
must fold here and so on.

672
00:38:55,260 --> 00:38:56,520
These guys correspond.

673
00:39:00,500 --> 00:39:01,730
So that's good.

674
00:39:01,730 --> 00:39:04,590
So if I look at
this quadrilateral,

675
00:39:04,590 --> 00:39:13,450
it corresponds so g to c to d
to c to h to c to b to a back

676
00:39:13,450 --> 00:39:16,220
to b back to c.

677
00:39:16,220 --> 00:39:25,300
And so my guess is if you
add a little limb in here--

678
00:39:25,300 --> 00:39:26,670
I think I can draw on this.

679
00:39:26,670 --> 00:39:30,050
That would be nice.

680
00:39:30,050 --> 00:39:35,100
Should really tell you
about-- is this going to work?

681
00:39:35,100 --> 00:39:35,600
Yes.

682
00:39:35,600 --> 00:39:37,190
It's kind of white,
but there we go.

683
00:39:39,940 --> 00:39:44,020
So great.

684
00:39:44,020 --> 00:39:45,380
Draw a fun diagram here.

685
00:39:48,450 --> 00:39:50,760
This is how I make my lecture
notes if you're curious.

686
00:39:50,760 --> 00:39:51,730
This is a tablet PC.

687
00:39:56,100 --> 00:40:02,261
Now, I've got
some-- Tell me if I

688
00:40:02,261 --> 00:40:04,260
make a mistake, those who
know what I'm drawing.

689
00:40:08,088 --> 00:40:10,060
What the hell is this?

690
00:40:10,060 --> 00:40:12,916
I think it goes there.

691
00:40:12,916 --> 00:40:15,320
There.

692
00:40:15,320 --> 00:40:16,150
There.

693
00:40:16,150 --> 00:40:18,880
I'll explain what I'm
drawing once I've drawn it.

694
00:40:18,880 --> 00:40:21,920
It's easier.

695
00:40:21,920 --> 00:40:23,180
Something like that.

696
00:40:23,180 --> 00:40:27,330
This is a bunch of disks
and a bunch of other things,

697
00:40:27,330 --> 00:40:30,490
there's only one
here called rivers.

698
00:40:30,490 --> 00:40:34,260
And this is a geometric way to
think about the constraints.

699
00:40:34,260 --> 00:40:41,180
If you look at this structure--
so I have a disk down

700
00:40:41,180 --> 00:40:42,290
here corresponding to d.

701
00:40:42,290 --> 00:40:44,540
I have a disk corresponding
to h, a disk corresponding

702
00:40:44,540 --> 00:40:48,392
to g, a river corresponding
to the segment b c.

703
00:40:48,392 --> 00:40:49,850
The reason I only
have one river is

704
00:40:49,850 --> 00:40:52,420
there's only one interior
edge in this tree.

705
00:40:52,420 --> 00:40:54,160
Everything else is a leaf edge.

706
00:40:54,160 --> 00:40:56,330
So leaf edges are
going to be disks.

707
00:40:56,330 --> 00:40:58,760
All non leaf edges are
going to be rivers.

708
00:40:58,760 --> 00:41:01,300
And the structure, the way
that those things connect

709
00:41:01,300 --> 00:41:04,560
to each other is the same as
the structure in this tree.

710
00:41:04,560 --> 00:41:07,020
So you've got the
three disks down here,

711
00:41:07,020 --> 00:41:08,930
which corresponds
to these leaf edges.

712
00:41:08,930 --> 00:41:11,830
They all touch a common river
because all of those edges

713
00:41:11,830 --> 00:41:16,096
are incident to that
edge in the center.

714
00:41:16,096 --> 00:41:17,720
And there's three
disks on the top that

715
00:41:17,720 --> 00:41:21,082
correspond to the three
leaf edges up here.

716
00:41:21,082 --> 00:41:22,540
This is really just
the same thing.

717
00:41:22,540 --> 00:41:25,100
It's saying that if
you want to look, say,

718
00:41:25,100 --> 00:41:28,440
at the distance
between h and a here.

719
00:41:28,440 --> 00:41:31,140
The distance between h and
a should be length three.

720
00:41:31,140 --> 00:41:32,940
And those three
lengths are represented

721
00:41:32,940 --> 00:41:35,400
by the size of
this disk, followed

722
00:41:35,400 --> 00:41:37,410
by the width of
this river, followed

723
00:41:37,410 --> 00:41:39,660
by the size of the a disk.

724
00:41:39,660 --> 00:41:42,510
It's say exactly the same
constraints, just represented

725
00:41:42,510 --> 00:41:43,500
geometrically.

726
00:41:43,500 --> 00:41:45,900
Now, if I'm lucky,
these regions actually

727
00:41:45,900 --> 00:41:47,530
kiss, they touch at points.

728
00:41:47,530 --> 00:41:49,550
That's when things are active.

729
00:41:49,550 --> 00:41:53,940
And you could draw straight
across from a to h and never go

730
00:41:53,940 --> 00:41:56,050
in these outside regions.

731
00:41:56,050 --> 00:41:58,640
If you're not lucky,
they won't touch.

732
00:41:58,640 --> 00:42:01,620
If they don't touch,
make them touch.

733
00:42:01,620 --> 00:42:04,210
That's all I want to do.

734
00:42:04,210 --> 00:42:08,240
And so I just want to
blow up these regions,

735
00:42:08,240 --> 00:42:13,280
make them longer, for
example, until things touch.

736
00:42:13,280 --> 00:42:15,580
When they touch enough,
if you do it right,

737
00:42:15,580 --> 00:42:17,670
you can actually get
them to triangulate.

738
00:42:17,670 --> 00:42:20,270
That's my very
hand wavy argument.

739
00:42:20,270 --> 00:42:22,110
It's proved formally
in the book,

740
00:42:22,110 --> 00:42:23,740
and it's a little bit technical.

741
00:42:23,740 --> 00:42:31,137
So I think I will
move on and tell you

742
00:42:31,137 --> 00:42:32,220
what to do with triangles.

743
00:43:29,610 --> 00:43:34,310
So suppose you
have some triangle.

744
00:43:34,310 --> 00:43:36,970
And each of these edges
is an active path.

745
00:43:36,970 --> 00:43:38,160
So there's some leaf here.

746
00:43:41,130 --> 00:43:44,700
We'll call them a, b, and c.

747
00:43:44,700 --> 00:43:49,670
And this segment we know will
map right along the floor

748
00:43:49,670 --> 00:43:56,430
to make up that path, that
active path in the tree.

749
00:43:56,430 --> 00:43:59,720
Like I said, we're going to
follow along angular bisectors.

750
00:44:05,260 --> 00:44:09,560
You may know the angular
bisectors of a triangle

751
00:44:09,560 --> 00:44:12,750
meet at a single point.

752
00:44:12,750 --> 00:44:15,200
And then, we're going to
make some perpendicular

753
00:44:15,200 --> 00:44:27,705
folds like that.

754
00:44:30,990 --> 00:44:33,280
Where the
perpendicular folds go,

755
00:44:33,280 --> 00:44:38,140
well, they go whenever
there's a shadow

756
00:44:38,140 --> 00:44:40,680
vertex along this segment.

757
00:44:40,680 --> 00:44:44,920
Remember this edge,
b c corresponds

758
00:44:44,920 --> 00:44:49,620
to some path between b
and c in the tree which

759
00:44:49,620 --> 00:44:52,134
looks like whatever.

760
00:44:52,134 --> 00:44:53,550
And so for each
of these branching

761
00:44:53,550 --> 00:44:56,684
points that we visit along
that, we can just measure.

762
00:44:56,684 --> 00:44:59,350
As we move along here, we get to
some vertex then another vertex

763
00:44:59,350 --> 00:45:04,630
then another vertex then c,
except I did it backwards.

764
00:45:04,630 --> 00:45:07,079
And so for each of
these guys, I know

765
00:45:07,079 --> 00:45:08,870
that I need to be able
to articulate there.

766
00:45:08,870 --> 00:45:10,510
I need a hinge crease.

767
00:45:10,510 --> 00:45:13,480
And so I just put in a
hinge grace perpendicular

768
00:45:13,480 --> 00:45:15,100
to the floor,
essentially, because we

769
00:45:15,100 --> 00:45:17,370
know this is mapping
to the floor.

770
00:45:17,370 --> 00:45:19,560
And conveniently,
those will all line up.

771
00:45:19,560 --> 00:45:22,730
So if I have some vertex
here-- let's call it d.

772
00:45:22,730 --> 00:45:24,150
--d will be here.

773
00:45:24,150 --> 00:45:25,900
But d will also be here.

774
00:45:25,900 --> 00:45:27,720
Because if I follow
the path from b to a,

775
00:45:27,720 --> 00:45:30,580
a is some other
guy, maybe this one,

776
00:45:30,580 --> 00:45:32,960
I also have to go through d.

777
00:45:32,960 --> 00:45:35,464
And so these things will
conveniently line up perfectly.

778
00:45:35,464 --> 00:45:36,880
I'm not going to
prove that again.

779
00:45:36,880 --> 00:45:39,420
But it's true.

780
00:45:39,420 --> 00:45:42,730
And you just get this really
nice simple to fold thing.

781
00:45:45,421 --> 00:45:46,920
Shoot, I'll fold
one if you haven't.

782
00:45:46,920 --> 00:45:50,545
This is a standard rabbit ear
molecule in making origami.

783
00:45:50,545 --> 00:45:51,670
You have a little triangle.

784
00:45:51,670 --> 00:45:53,080
You want to make it an ear.

785
00:45:53,080 --> 00:45:55,260
You squeeze along the
angular bisectors,

786
00:45:55,260 --> 00:45:58,150
and it makes a cute rabbit ear.

787
00:45:58,150 --> 00:46:00,310
And you can see it also,
the crease pattern,

788
00:46:00,310 --> 00:46:05,980
in here like in this
triangle in the upper right.

789
00:46:05,980 --> 00:46:08,940
You've got the red lines which
are the angular bisectors.

790
00:46:08,940 --> 00:46:11,470
And then, you've got all
those perpendicular folds.

791
00:46:11,470 --> 00:46:14,680
And they go exactly
where those letters go.

792
00:46:14,680 --> 00:46:18,430
And the triangle at
the top is similar.

793
00:46:18,430 --> 00:46:22,280
It's a little different because
the very top edge of the paper

794
00:46:22,280 --> 00:46:23,550
is not actually active.

795
00:46:23,550 --> 00:46:27,030
So there's really a
special case there.

796
00:46:27,030 --> 00:46:28,410
Upper right is also not active.

797
00:46:28,410 --> 00:46:30,460
Oh, that's annoying.

798
00:46:30,460 --> 00:46:33,012
Yeah.

799
00:46:33,012 --> 00:46:34,470
There's a little
bit of extra stuff

800
00:46:34,470 --> 00:46:35,850
that happens at the
boundary of the paper

801
00:46:35,850 --> 00:46:37,450
where you don't
have active paths.

802
00:46:37,450 --> 00:46:39,590
But it's, as you can see
from the crease pattern,

803
00:46:39,590 --> 00:46:41,770
it's basically the same.

804
00:46:41,770 --> 00:46:46,200
In fact, I could
call it the same.

805
00:46:46,200 --> 00:46:48,595
It's a little bit less pretty
because this is not green.

806
00:46:48,595 --> 00:46:50,511
And so you don't actually
know that c is here.

807
00:46:50,511 --> 00:46:52,030
And you don't know
that b is there.

808
00:46:52,030 --> 00:46:54,542
But you know about
all the other edges.

809
00:46:54,542 --> 00:46:56,500
There's just one edge
you might not know about.

810
00:46:56,500 --> 00:46:57,890
And so you can figure
out what the right edge

811
00:46:57,890 --> 00:47:00,700
is based on the other edges
of the triangle, the other two

812
00:47:00,700 --> 00:47:01,200
edges.

813
00:47:04,210 --> 00:47:05,190
That's just a feature.

814
00:47:05,190 --> 00:47:09,000
You can triangulate everything
except the boundary.

815
00:47:09,000 --> 00:47:11,700
You may not be able to get
active paths in this step.

816
00:47:16,070 --> 00:47:18,300
That kind of does
the tree method

817
00:47:18,300 --> 00:47:20,465
in a super abbreviated version.

818
00:47:23,980 --> 00:47:27,930
I showed you a demo last
time, just in case you forgot.

819
00:47:27,930 --> 00:47:30,135
You draw your favorite tree.

820
00:47:30,135 --> 00:47:31,760
See if I can get it
to do the same one.

821
00:47:37,360 --> 00:47:41,645
And you optimize,
generate a crease pattern.

822
00:47:41,645 --> 00:47:43,350
Oh, it's a different one.

823
00:47:43,350 --> 00:47:44,150
Fun.

824
00:47:44,150 --> 00:47:46,930
There it is.

825
00:47:46,930 --> 00:47:49,129
And here, TreeMaker knows
how to draw the disks.

826
00:47:49,129 --> 00:47:51,670
It doesn't currently know how
to draw the rivers because it's

827
00:47:51,670 --> 00:47:55,790
kind of tricky to make a snakey
path in a computer program.

828
00:47:55,790 --> 00:47:59,380
But you see the three disks down
here, the three disks up there,

829
00:47:59,380 --> 00:48:01,290
and you can imagine
the one river

830
00:48:01,290 --> 00:48:07,437
in the middle representing the
central segment of your tree.

831
00:48:07,437 --> 00:48:09,270
And one of the problems
on the problems set,

832
00:48:09,270 --> 00:48:11,490
problem set one is
released, is to just make

833
00:48:11,490 --> 00:48:12,769
something using TreeMaker.

834
00:48:12,769 --> 00:48:14,310
I would encourage
you to start simple

835
00:48:14,310 --> 00:48:15,420
unless you know
what you're doing.

836
00:48:15,420 --> 00:48:16,750
You don't have to
use the program.

837
00:48:16,750 --> 00:48:18,380
You could do it by
hand, placing disks.

838
00:48:18,380 --> 00:48:20,760
That's how most
origamists actually do it.

839
00:48:20,760 --> 00:48:23,850
I'm sure Jason will
do it that way.

840
00:48:23,850 --> 00:48:27,642
You can use the program,
print out a crease pattern,

841
00:48:27,642 --> 00:48:28,600
see what it looks like.

842
00:48:31,760 --> 00:48:32,440
Next thing.

843
00:48:36,320 --> 00:48:38,200
If you want to do
this in reality--

844
00:48:38,200 --> 00:48:40,962
and what TreeMaker is doing
is not this triangulation.

845
00:48:40,962 --> 00:48:42,670
Doing a triangulation
is a bit of a pain,

846
00:48:42,670 --> 00:48:44,474
but you could keep
modifying your tree

847
00:48:44,474 --> 00:48:45,390
until it triangulates.

848
00:48:45,390 --> 00:48:47,860
The alternative is you
just deal with polygons

849
00:48:47,860 --> 00:48:49,654
that are bigger than triangles.

850
00:48:49,654 --> 00:48:51,820
And there's this thing
called the universal molecule

851
00:48:51,820 --> 00:48:52,980
by Robert Lang.

852
00:48:52,980 --> 00:48:55,610
Here it is for a quadrilateral.

853
00:48:55,610 --> 00:49:01,070
And it makes it-- this works
for any convex polygon.

854
00:49:01,070 --> 00:49:03,430
Now sometimes, you're
active paths don't decompose

855
00:49:03,430 --> 00:49:05,360
your shape into convex polygons.

856
00:49:05,360 --> 00:49:06,710
And this still doesn't work.

857
00:49:06,710 --> 00:49:08,370
You still have to
do something here.

858
00:49:08,370 --> 00:49:10,080
You need to add some
extra leaf edges

859
00:49:10,080 --> 00:49:13,790
to the tree to just
fill things up.

860
00:49:13,790 --> 00:49:15,910
But you don't have to stop.

861
00:49:15,910 --> 00:49:18,440
You have to go all the way to
the point of triangulation.

862
00:49:18,440 --> 00:49:21,370
You can stop at the point which
happens most the time when

863
00:49:21,370 --> 00:49:23,150
all of the faces are convex.

864
00:49:23,150 --> 00:49:27,440
And then, it's a slightly more
general picture what happens.

865
00:49:27,440 --> 00:49:29,060
Intuitively, what
you want to do is,

866
00:49:29,060 --> 00:49:33,010
this is the tree you want
to make among those leaves.

867
00:49:33,010 --> 00:49:35,305
All the boundary edges
here are active paths.

868
00:49:35,305 --> 00:49:37,502
You have g d h a.

869
00:49:37,502 --> 00:49:38,960
Those are active
paths, so you know

870
00:49:38,960 --> 00:49:43,679
where all of those branching
points are in the middle.

871
00:49:43,679 --> 00:49:44,720
You'd like to build that.

872
00:49:44,720 --> 00:49:47,460
And so what we're going to
do is build it bottom up

873
00:49:47,460 --> 00:49:52,050
in the literal sense from z
equals zero, increasing z.

874
00:49:52,050 --> 00:49:54,650
And what that corresponds
to in this picture

875
00:49:54,650 --> 00:49:59,580
is shrinking or offsetting
these edges inward.

876
00:49:59,580 --> 00:50:01,790
So you offset these
all by the same amount.

877
00:50:01,790 --> 00:50:04,014
That's like traveling
up over here.

878
00:50:04,014 --> 00:50:05,680
So you see the red
lines here correspond

879
00:50:05,680 --> 00:50:07,890
to the red cross sections.

880
00:50:07,890 --> 00:50:10,010
So I just see what happens
in cross section is

881
00:50:10,010 --> 00:50:11,460
I shrink things in.

882
00:50:11,460 --> 00:50:15,580
And the first thing that happens
at this first critical red

883
00:50:15,580 --> 00:50:20,890
drawing is that the path
from d to a becomes critical,

884
00:50:20,890 --> 00:50:22,860
becomes active.

885
00:50:22,860 --> 00:50:25,364
Before it was
inactive that-- that

886
00:50:25,364 --> 00:50:26,530
was kind of annoying for me.

887
00:50:26,530 --> 00:50:28,680
I wanted it to be
triangulated, but it wasn't.

888
00:50:28,680 --> 00:50:32,070
The distance from a to
d in the piece of paper

889
00:50:32,070 --> 00:50:40,180
was bigger than the distance
between the leaves in the tree.

890
00:50:40,180 --> 00:50:41,264
I wanted them to be equal.

891
00:50:41,264 --> 00:50:43,138
Well, it turns out, if
you shrink this thing,

892
00:50:43,138 --> 00:50:44,650
eventually they
might become equal.

893
00:50:44,650 --> 00:50:45,650
And that's what happens.

894
00:50:45,650 --> 00:50:47,109
And that's what
TreeMaker computes.

895
00:50:47,109 --> 00:50:48,816
And what you should
do if you're building

896
00:50:48,816 --> 00:50:49,800
the universal molecule.

897
00:50:49,800 --> 00:50:53,680
If you discover, oh,
now a d is active,

898
00:50:53,680 --> 00:50:55,730
now, I subdivide
into two triangles.

899
00:50:55,730 --> 00:50:57,960
And then, I do the thing
in the two triangles.

900
00:50:57,960 --> 00:51:00,167
And generally, you start
with some convex polygon.

901
00:51:00,167 --> 00:51:00,750
You shrink it.

902
00:51:00,750 --> 00:51:03,370
At some point, some diagonal
might become active.

903
00:51:03,370 --> 00:51:07,240
You split it into two,
just keep going in the two.

904
00:51:07,240 --> 00:51:10,510
And there's one other
thing which can happen,

905
00:51:10,510 --> 00:51:12,640
which is what's happening
at the end of a triangle.

906
00:51:12,640 --> 00:51:13,585
You shrink.

907
00:51:13,585 --> 00:51:15,710
And then, it could be two
vertices actually collide

908
00:51:15,710 --> 00:51:16,704
with each other.

909
00:51:16,704 --> 00:51:18,620
And then, you just think
of them as one vertex

910
00:51:18,620 --> 00:51:20,360
and keep shrinking.

911
00:51:20,360 --> 00:51:23,250
So that's the general universal
molecule construction.

912
00:51:23,250 --> 00:51:29,380
You see in a sort of-- these are
the cross sections from above.

913
00:51:29,380 --> 00:51:32,290
You see that as you go up,
things are getting smaller.

914
00:51:32,290 --> 00:51:36,870
That is one of the statements of
the uniaxial base as you go up.

915
00:51:36,870 --> 00:51:40,467
Cross sections get tinier.

916
00:51:40,467 --> 00:51:42,050
And that gives you
the crease pattern.

917
00:51:42,050 --> 00:51:44,780
If you follow along
where the vertices go

918
00:51:44,780 --> 00:51:46,290
during this process,
and you draw in

919
00:51:46,290 --> 00:51:48,760
and all the active path that
you create along the way,

920
00:51:48,760 --> 00:51:49,884
that's your crease pattern.

921
00:51:54,490 --> 00:51:56,579
So that's how you do
it more practically is

922
00:51:56,579 --> 00:51:57,870
you use the universal molecule.

923
00:51:57,870 --> 00:51:59,786
But to prove it, you
don't actually need that.

924
00:52:02,560 --> 00:52:03,450
All right.

925
00:52:03,450 --> 00:52:10,725
I have now some more real
examples by Robert Lang

926
00:52:10,725 --> 00:52:13,030
and by Jason Ku.

927
00:52:13,030 --> 00:52:15,850
So here is Roosevelt elk.

928
00:52:15,850 --> 00:52:21,900
And Rob is all about
getting very realistic form.

929
00:52:21,900 --> 00:52:25,070
So all of the branching
measurements and-- I'm

930
00:52:25,070 --> 00:52:26,839
sure if you knew
a lot about elks,

931
00:52:26,839 --> 00:52:28,380
you could recognizes
this a Roosevelt

932
00:52:28,380 --> 00:52:30,380
elk not some other elk.

933
00:52:30,380 --> 00:52:33,506
And you can achieve that
level of detail and realism

934
00:52:33,506 --> 00:52:34,880
using the tree
method because you

935
00:52:34,880 --> 00:52:38,080
can control all of the relative
lengths of those segments

936
00:52:38,080 --> 00:52:39,719
and get perfect
branching structure

937
00:52:39,719 --> 00:52:41,843
and get the right proportions
for the legs and tail

938
00:52:41,843 --> 00:52:43,310
and so on.

939
00:52:43,310 --> 00:52:46,730
And you can see
here, the-- and you

940
00:52:46,730 --> 00:52:50,640
can go to Robert Lang's webpage,
landorigami.com and print this

941
00:52:50,640 --> 00:52:51,190
out.

942
00:52:51,190 --> 00:52:52,790
And try it out if you want.

943
00:52:52,790 --> 00:52:56,110
This will fold not this but
the base for that model.

944
00:52:56,110 --> 00:52:57,570
And you could see the disks.

945
00:52:57,570 --> 00:52:59,740
And you can see some
approximation of the rivers

946
00:52:59,740 --> 00:53:01,220
here.

947
00:53:01,220 --> 00:53:06,220
But they're not quite drawn
in in this particular diagram.

948
00:53:06,220 --> 00:53:07,302
But a lot of detail.

949
00:53:07,302 --> 00:53:09,010
And if you look
carefully, you can really

950
00:53:09,010 --> 00:53:10,360
read off what the tree is here.

951
00:53:10,360 --> 00:53:12,500
You can see how these
things are separated,

952
00:53:12,500 --> 00:53:16,850
and it will correspond to the
branching structure over there.

953
00:53:16,850 --> 00:53:19,390
Here's a more complicated one.

954
00:53:19,390 --> 00:53:23,010
Scorpion varleg
which you can also

955
00:53:23,010 --> 00:53:27,450
fold at lifesize if
you're really crazy.

956
00:53:27,450 --> 00:53:32,420
And you can also see from
these kinds of diagrams

957
00:53:32,420 --> 00:53:36,530
that paper usage is super
efficient in these designs.

958
00:53:36,530 --> 00:53:40,200
And presumably that's
how Robert design them.

959
00:53:40,200 --> 00:53:42,340
The only paper we're
wasting in some sense

960
00:53:42,340 --> 00:53:47,910
is the little regions between
the disks and the rivers

961
00:53:47,910 --> 00:53:49,341
which is quite small.

962
00:53:49,341 --> 00:53:51,465
Most of the papers getting
absorbed into the flaps.

963
00:53:54,570 --> 00:53:57,810
Here's one of the first
models by Jason Ku that I saw,

964
00:53:57,810 --> 00:54:00,380
the Nazgul from
Lord of the Rings.

965
00:54:00,380 --> 00:54:02,890
And pretty complicated.

966
00:54:02,890 --> 00:54:05,760
So here, the bold lines
show you essentially

967
00:54:05,760 --> 00:54:07,900
where the disks and the
rivers are that have been--

968
00:54:07,900 --> 00:54:09,320
AUDIENCE: Those are
actually the hinge creases.

969
00:54:09,320 --> 00:54:10,944
PROFESSOR: Oh, those
are hinge creases.

970
00:54:10,944 --> 00:54:12,200
Yeah.

971
00:54:12,200 --> 00:54:12,820
Good.

972
00:54:12,820 --> 00:54:16,150
And the top is the
actual crease pattern.

973
00:54:16,150 --> 00:54:17,860
And it's pretty awesome.

974
00:54:17,860 --> 00:54:20,400
You've got a horse and rider
out of one square paper.

975
00:54:24,110 --> 00:54:26,260
Here's a shrimp.

976
00:54:26,260 --> 00:54:29,440
He's super complicated
and super realistic.

977
00:54:29,440 --> 00:54:32,010
It looks very shrimpy.

978
00:54:32,010 --> 00:54:34,210
I know some people who
are freaked out by shrimp.

979
00:54:34,210 --> 00:54:37,320
And so this should really
elicit that similar response.

980
00:54:37,320 --> 00:54:42,140
Or other people get really
hungry at this point, I guess.

981
00:54:42,140 --> 00:54:45,180
But you could see the
tree is pretty dense here,

982
00:54:45,180 --> 00:54:49,980
lots of little features
getting that branching right.

983
00:54:49,980 --> 00:54:52,770
And one last example
is this butterfly

984
00:54:52,770 --> 00:54:56,790
which is pretty
awesome in its realism.

985
00:54:56,790 --> 00:55:00,160
And I guess the tree
is a lot simpler here.

986
00:55:00,160 --> 00:55:03,120
But there's a lot of
extra creases here.

987
00:55:03,120 --> 00:55:07,130
You see just for getting
the flaps nice and narrow.

988
00:55:07,130 --> 00:55:11,740
So in general, these
kinds of constructions

989
00:55:11,740 --> 00:55:14,590
will make this guy
rather pointy and tall.

990
00:55:14,590 --> 00:55:16,540
And you can just squash it back.

991
00:55:16,540 --> 00:55:18,430
And it's called a
sync fold and make

992
00:55:18,430 --> 00:55:23,180
it tinier like-- you
have something like this.

993
00:55:23,180 --> 00:55:25,280
The flaps are you
think are too tall.

994
00:55:25,280 --> 00:55:30,300
You just fold here.

995
00:55:30,300 --> 00:55:32,600
Which, if you look at
the crease pattern,

996
00:55:32,600 --> 00:55:35,840
makes just an offset
version of the original.

997
00:55:35,840 --> 00:55:38,050
And hey, now your
flaps are half is tall.

998
00:55:38,050 --> 00:55:41,140
And if you're a
proper origamist,

999
00:55:41,140 --> 00:55:45,878
you-- I shouldn't do this live.

1000
00:55:50,760 --> 00:55:53,400
You change the mountain valley
assignment a little bit,

1001
00:55:53,400 --> 00:55:55,840
and you sync everything
on the inside

1002
00:55:55,840 --> 00:55:57,470
instead of just folding it over.

1003
00:56:02,728 --> 00:56:05,370
It's not going to
look super pretty.

1004
00:56:05,370 --> 00:56:11,680
But same tree structure, just
the flaps are half as tall.

1005
00:56:11,680 --> 00:56:14,740
So that's all this
pleating here.

1006
00:56:14,740 --> 00:56:18,310
And I think that's it
for my little tour.

1007
00:56:18,310 --> 00:56:21,350
And Jason Ku next.

1008
00:56:21,350 --> 00:56:23,220
Next Monday we'll
be talking more

1009
00:56:23,220 --> 00:56:26,290
about the artistic side,
history of origami design,

1010
00:56:26,290 --> 00:56:29,300
and what it takes to
really make something

1011
00:56:29,300 --> 00:56:31,150
real by these approaches.

1012
00:56:31,150 --> 00:56:33,690
That should be lots of fun.

1013
00:56:33,690 --> 00:56:37,380
I want to move on to other kinds
of efficient origami design.

1014
00:56:37,380 --> 00:56:43,510
Less directly applicable
to real origami design

1015
00:56:43,510 --> 00:56:46,400
so to speak, at least currently.

1016
00:56:46,400 --> 00:56:48,370
But mathematically
more powerful.

1017
00:56:48,370 --> 00:56:50,700
Uniaxial bases are
nice, but it's not

1018
00:56:50,700 --> 00:56:52,840
everything you
might want to fold.

1019
00:56:52,840 --> 00:56:56,940
So what if we want
to fold other stuff.

1020
00:57:03,180 --> 00:57:06,290
And to a geometer,
most natural version

1021
00:57:06,290 --> 00:57:08,650
of folding other stuff
or folding anything

1022
00:57:08,650 --> 00:57:09,740
is a polyhedron.

1023
00:57:09,740 --> 00:57:13,300
You have a bunch of
polygons, flat panels in 3D,

1024
00:57:13,300 --> 00:57:15,570
somehow joined together
to make some surface.

1025
00:57:15,570 --> 00:57:17,130
How do I fold that?

1026
00:57:17,130 --> 00:57:20,410
And let's start with a super
simple example which is I

1027
00:57:20,410 --> 00:57:25,600
want to fold a
square into a cube.

1028
00:57:25,600 --> 00:57:30,590
How big a square do I
need to fold a unit cube?

1029
00:57:30,590 --> 00:57:33,850
Or how big cube can I
fold for a unit square?

1030
00:57:33,850 --> 00:57:36,450
Either way.

1031
00:57:36,450 --> 00:57:43,700
And I'm going to make
it a one by one square.

1032
00:57:43,700 --> 00:57:51,060
And I'm going to fold it
into a cube of dimension x.

1033
00:57:51,060 --> 00:57:55,160
And I want to know--
it looks funny.

1034
00:57:55,160 --> 00:57:56,930
It's the quintuple x cubed.

1035
00:57:59,460 --> 00:58:00,586
It's the x-coordinate.

1036
00:58:00,586 --> 00:58:01,460
That's my motivation.

1037
00:58:06,800 --> 00:58:09,660
So we talked-- one
thing we can think about

1038
00:58:09,660 --> 00:58:13,330
is what makes the
corners of the cubes

1039
00:58:13,330 --> 00:58:15,910
and how far away should they be.

1040
00:58:15,910 --> 00:58:18,650
So if I want to fold
this cube, I look at,

1041
00:58:18,650 --> 00:58:22,420
let's say, the opposite
corners of the cube.

1042
00:58:22,420 --> 00:58:24,920
They're pretty far
away on the cube.

1043
00:58:24,920 --> 00:58:26,560
And I know that by
folding I could only

1044
00:58:26,560 --> 00:58:28,400
make distance is smaller.

1045
00:58:28,400 --> 00:58:33,370
So somehow, if I measure the
shortest path on the cube,

1046
00:58:33,370 --> 00:58:36,460
from this point to
this point, it's

1047
00:58:36,460 --> 00:58:40,980
that if you believe-- when
you unfold this thing,

1048
00:58:40,980 --> 00:58:42,720
it should be flat.

1049
00:58:42,720 --> 00:58:45,240
If I unfolds to just
those two squares,

1050
00:58:45,240 --> 00:58:48,686
it's a straight line
between the two.

1051
00:58:48,686 --> 00:58:50,560
And so that goes to the
midpoint of this edge

1052
00:58:50,560 --> 00:58:51,610
and then over there.

1053
00:58:51,610 --> 00:58:53,700
And you measure that length.

1054
00:58:53,700 --> 00:58:54,960
And oh, trigonometry.

1055
00:58:57,510 --> 00:58:59,730
Root five, that's
not what I wanted.

1056
00:59:04,090 --> 00:59:09,460
So we have x here, 2x here.

1057
00:59:09,460 --> 00:59:18,310
So this distance
is-- yeah, I see.

1058
00:59:23,959 --> 00:59:26,125
Why is that different from
what I have written down?

1059
00:59:33,674 --> 00:59:35,590
Because that was not the
diameter of the cube.

1060
00:59:35,590 --> 00:59:37,710
I see.

1061
00:59:37,710 --> 00:59:40,287
AUDIENCE: You want
them equidistant.

1062
00:59:40,287 --> 00:59:40,870
PROFESSOR: No.

1063
00:59:40,870 --> 00:59:44,750
I do want this but,
I think if I go

1064
00:59:44,750 --> 00:59:48,120
from the center of this
square-- this is hard to draw.

1065
00:59:48,120 --> 00:59:54,570
--to the center of the back
square, which is back here,

1066
00:59:54,570 --> 00:59:58,980
that distance is going
to be wrapping around.

1067
00:59:58,980 --> 01:00:02,550
Which is just going
to be like 2x.

1068
01:00:02,550 --> 01:00:04,812
Is that bigger or
smaller than root 5x?

1069
01:00:04,812 --> 01:00:05,770
AUDIENCE: It's smaller.

1070
01:00:05,770 --> 01:00:07,560
PROFESSOR: Smaller.

1071
01:00:07,560 --> 01:00:08,060
Interesting.

1072
01:00:11,676 --> 01:00:16,100
One, two, three, four.

1073
01:00:16,100 --> 01:00:17,317
What did I do wrong?

1074
01:00:21,610 --> 01:00:22,410
Oh, I see.

1075
01:00:22,410 --> 01:00:23,710
OK.

1076
01:00:23,710 --> 01:00:25,600
Here's a fun fact.

1077
01:00:25,600 --> 01:00:29,280
This is actually the
smallest antipodal distance.

1078
01:00:29,280 --> 01:00:30,470
Get this right.

1079
01:00:30,470 --> 01:00:32,530
So if you take some
point on the cube,

1080
01:00:32,530 --> 01:00:34,290
and you look at the
point farthest away

1081
01:00:34,290 --> 01:00:35,873
from it on the other
side of the cube,

1082
01:00:35,873 --> 01:00:38,780
it will always be
at least 2x away.

1083
01:00:38,780 --> 01:00:39,830
So here, it's bigger.

1084
01:00:39,830 --> 01:00:41,450
This is probably the diameter.

1085
01:00:41,450 --> 01:00:44,130
It's bigger than 2x, but it
will always be at least 2x away.

1086
01:00:44,130 --> 01:00:48,380
This is actually the smallest
situation you can get.

1087
01:00:48,380 --> 01:00:50,850
And so I want to think
about the point that

1088
01:00:50,850 --> 01:00:53,260
corresponds to the
center of the square.

1089
01:00:53,260 --> 01:00:53,870
Right?

1090
01:00:53,870 --> 01:00:55,880
Yes.

1091
01:00:55,880 --> 01:00:59,810
Now maybe that maps to
the center like this.

1092
01:00:59,810 --> 01:01:03,390
And the antipodal points is
2x away, or maybe it's bigger.

1093
01:01:03,390 --> 01:01:09,060
But at least I know that this
length is greater than or equal

1094
01:01:09,060 --> 01:01:15,200
to 2x because that's-- the
antipodal point has to be made

1095
01:01:15,200 --> 01:01:16,540
from that.

1096
01:01:16,540 --> 01:01:18,290
I need to think
about all situation

1097
01:01:18,290 --> 01:01:20,873
because I really want to think
about the center of the square.

1098
01:01:20,873 --> 01:01:23,260
Once that is at
least 2x, then I know

1099
01:01:23,260 --> 01:01:29,570
that the side of the square
is at least 2 root 2x.

1100
01:01:29,570 --> 01:01:30,070
Yes.

1101
01:01:35,450 --> 01:01:40,780
And so I know that this is one.

1102
01:01:40,780 --> 01:01:41,880
And you work it out.

1103
01:01:41,880 --> 01:01:43,740
And x is root 2 over 4.

1104
01:01:46,910 --> 01:01:49,370
Or it's at most that.

1105
01:01:49,370 --> 01:01:52,235
And so that gives you some
bound on what it takes.

1106
01:01:52,235 --> 01:01:54,110
So this is actually
really the only technique

1107
01:01:54,110 --> 01:01:57,550
we know to prove lower
bounds on how much-- how big

1108
01:01:57,550 --> 01:01:59,520
a square you need
to make something.

1109
01:01:59,520 --> 01:02:02,540
It's this kind of distance
increasing argument.

1110
01:02:02,540 --> 01:02:05,390
And it turns out you can
actually achieve x equals this.

1111
01:02:05,390 --> 01:02:07,230
So this is what I
call lower bound.

1112
01:02:07,230 --> 01:02:09,440
It says, you can't do
any better than this.

1113
01:02:09,440 --> 01:02:16,310
But there's also a matching
upper bound which achieves this

1114
01:02:16,310 --> 01:02:29,840
and not going to
draw it perfectly.

1115
01:02:40,950 --> 01:02:42,760
So there are the six
sides of the cube.

1116
01:02:42,760 --> 01:02:45,220
You've got one, two,
three, four, five.

1117
01:02:45,220 --> 01:02:47,470
And the sixth one is
split into quarters.

1118
01:02:47,470 --> 01:02:51,390
And you can see, you just
actually fold here, here,

1119
01:02:51,390 --> 01:02:53,880
here, and here to get
rid of that excess.

1120
01:02:53,880 --> 01:02:55,440
And it will come
together as a cube.

1121
01:02:55,440 --> 01:02:57,320
You also fold along
the edges of the cube.

1122
01:02:57,320 --> 01:03:00,180
And it perfectly
achieves this property.

1123
01:03:00,180 --> 01:03:02,870
That from the
center of the paper,

1124
01:03:02,870 --> 01:03:07,400
you have exactly one
this distance 2 root

1125
01:03:07,400 --> 01:03:10,590
2x to the antipodal
point which is

1126
01:03:10,590 --> 01:03:12,630
the center of the opposite face.

1127
01:03:12,630 --> 01:03:13,130
Question?

1128
01:03:13,130 --> 01:03:13,796
AUDIENCE: Sorry.

1129
01:03:13,796 --> 01:03:17,670
Can you explain where the
2x came from [INAUDIBLE]?

1130
01:03:17,670 --> 01:03:20,440
PROFESSOR: I wave my hands.

1131
01:03:20,440 --> 01:03:22,530
So I'm thinking
about an arbitrary

1132
01:03:22,530 --> 01:03:24,410
point on the
surface of the cube.

1133
01:03:24,410 --> 01:03:26,560
Here, it should
be clear it's 2x.

1134
01:03:26,560 --> 01:03:28,350
There's x right here.

1135
01:03:28,350 --> 01:03:30,290
And there's 1/2x here.

1136
01:03:30,290 --> 01:03:32,800
And there's 1/2x on the back.

1137
01:03:32,800 --> 01:03:34,860
And I looked at
another situation

1138
01:03:34,860 --> 01:03:38,335
which is when it was at a
corner that bigger than 2x.

1139
01:03:38,335 --> 01:03:40,210
And I claim if you
interpolate in the middle,

1140
01:03:40,210 --> 01:03:41,668
you'll get something
in the middle,

1141
01:03:41,668 --> 01:03:44,920
in between 2x and root 5x.

1142
01:03:44,920 --> 01:03:46,830
For example, if
take a point here

1143
01:03:46,830 --> 01:03:50,530
that's closer to the
corner, then that

1144
01:03:50,530 --> 01:03:53,300
point-- you should probably
also think about the edge case.

1145
01:03:53,300 --> 01:03:55,780
But you check all of them,
and they're at least 2x.

1146
01:03:55,780 --> 01:03:57,480
That's what I'm claiming.

1147
01:03:57,480 --> 01:03:59,330
So I didn't really
prove that formally.

1148
01:03:59,330 --> 01:04:03,820
But claim is 2x is the smallest
antepodal pair you could get.

1149
01:04:03,820 --> 01:04:05,480
AUDIENCE: What does
antipodal mean?

1150
01:04:05,480 --> 01:04:07,750
PROFESSOR: Antipodal simple
means on the other side.

1151
01:04:07,750 --> 01:04:10,980
The anti pode, the opposite
pole, like from North Pole

1152
01:04:10,980 --> 01:04:11,600
to South Pole.

1153
01:04:11,600 --> 01:04:12,516
AUDIENCE: [INAUDIBLE].

1154
01:04:14,532 --> 01:04:15,990
PROFESSOR: Right
now, we know we're

1155
01:04:15,990 --> 01:04:18,430
taking whatever point is
the center of the square.

1156
01:04:18,430 --> 01:04:20,350
It maps somewhere in the cube.

1157
01:04:20,350 --> 01:04:21,760
I take the antipode from there.

1158
01:04:21,760 --> 01:04:26,540
I know that has to
be at least 2x away.

1159
01:04:26,540 --> 01:04:29,850
And if you look at
the distance map here,

1160
01:04:29,850 --> 01:04:32,480
the farthest away point in
the squared from the center

1161
01:04:32,480 --> 01:04:34,230
is the corner point.

1162
01:04:34,230 --> 01:04:36,760
So I know that that distance
can only get smaller.

1163
01:04:36,760 --> 01:04:38,800
And other distances
only get smaller.

1164
01:04:38,800 --> 01:04:41,610
So if I have to make a
2x distance from there,

1165
01:04:41,610 --> 01:04:43,770
this is my best
chance for doing it.

1166
01:04:43,770 --> 01:04:45,200
And that gives
you a lower bound.

1167
01:04:45,200 --> 01:04:46,533
Doesn't mean it can be achieved.

1168
01:04:46,533 --> 01:04:49,260
But this shows you that
you can achieve it.

1169
01:04:49,260 --> 01:04:53,550
This is a result by Catalano,
Johnson, and Lobe in 2001.

1170
01:04:53,550 --> 01:04:56,341
It's like the only optimality
result we have for folding 3D

1171
01:04:56,341 --> 01:04:56,840
shapes.

1172
01:04:56,840 --> 01:04:58,110
That's why I mention it.

1173
01:04:58,110 --> 01:05:00,379
Tons of fun open
problems like you

1174
01:05:00,379 --> 01:05:01,920
don't want to make
a square-- a cube.

1175
01:05:01,920 --> 01:05:06,190
Maybe you want to
make a triangle.

1176
01:05:06,190 --> 01:05:08,240
If you want to cover a
triangle on both sides,

1177
01:05:08,240 --> 01:05:09,280
that's probably open.

1178
01:05:09,280 --> 01:05:10,946
If you want to make
regular tetrahedron,

1179
01:05:10,946 --> 01:05:12,050
that's probably open.

1180
01:05:12,050 --> 01:05:14,030
Pretty much any problem
you pose here is open.

1181
01:05:14,030 --> 01:05:16,385
It would make fun project.

1182
01:05:16,385 --> 01:05:17,760
You can also think
about, instead

1183
01:05:17,760 --> 01:05:19,176
of starting from
square, you start

1184
01:05:19,176 --> 01:05:22,000
with a rectangle of
some given aspect ratio.

1185
01:05:22,000 --> 01:05:23,696
What's the biggest
cube you can make?

1186
01:05:23,696 --> 01:05:25,320
That's kind of fun
because in the limit

1187
01:05:25,320 --> 01:05:28,800
for a super long rectangle,
you should do strip wrapping.

1188
01:05:28,800 --> 01:05:30,520
For a square, we have
the right answer.

1189
01:05:30,520 --> 01:05:32,760
What's the right
answer in between?

1190
01:05:32,760 --> 01:05:33,260
Who knows.

1191
01:05:37,550 --> 01:05:42,070
The next thing I wanted to
talk about where there's

1192
01:05:42,070 --> 01:05:45,080
been some recent progress
is checkerboard folding.

1193
01:05:53,630 --> 01:05:57,190
In lecture one, I showed
you this model which I never

1194
01:05:57,190 --> 01:05:59,950
go anywhere without,
the four by four

1195
01:05:59,950 --> 01:06:06,460
checkerboard folded from one
square paper, white on one side

1196
01:06:06,460 --> 01:06:08,020
and red on the other.

1197
01:06:08,020 --> 01:06:11,260
And so I think this is
probably the most efficient way

1198
01:06:11,260 --> 01:06:13,310
to fold a four by
four checkerboard.

1199
01:06:13,310 --> 01:06:14,835
You start with a
square of one size,

1200
01:06:14,835 --> 01:06:16,874
and you shrink both
dimensions by two.

1201
01:06:16,874 --> 01:06:18,540
And you get a four
by four checkerboard.

1202
01:06:18,540 --> 01:06:20,081
But we don't know
if this is the best

1203
01:06:20,081 --> 01:06:21,310
way to fold a checkerboard.

1204
01:06:21,310 --> 01:06:22,940
Be nice to know.

1205
01:06:22,940 --> 01:06:28,610
And this has been
studied for a while.

1206
01:06:28,610 --> 01:06:31,110
And this is not
the standard method

1207
01:06:31,110 --> 01:06:32,500
for folding a checkerboard.

1208
01:06:32,500 --> 01:06:34,010
But it's actually
pretty efficient

1209
01:06:34,010 --> 01:06:37,180
which is kind of crazy.

1210
01:06:37,180 --> 01:06:39,000
So you take a
square, white on one

1211
01:06:39,000 --> 01:06:40,620
side and brown on the other.

1212
01:06:40,620 --> 01:06:42,310
You do this accordion pleat.

1213
01:06:42,310 --> 01:06:46,420
You get a bunch of nice color
reversals, bunch of squares.

1214
01:06:46,420 --> 01:06:50,910
And then, you just need to make
a square of squares from that.

1215
01:06:50,910 --> 01:06:53,870
So general problem is, I want
to fold an n by n checkerboard

1216
01:06:53,870 --> 01:06:55,360
from the smallest
possible square.

1217
01:06:55,360 --> 01:06:58,340
How big does it have to
be as a function of n?

1218
01:06:58,340 --> 01:07:01,670
And the standard
approach is-- well,

1219
01:07:01,670 --> 01:07:03,700
this is the first
method that does it

1220
01:07:03,700 --> 01:07:06,590
for all n in a
general simple way.

1221
01:07:06,590 --> 01:07:08,120
But the practical
foldings people

1222
01:07:08,120 --> 01:07:09,955
have designed, like four by four
and there are a bunch of eight

1223
01:07:09,955 --> 01:07:12,850
by eights out there, are little
more efficient than this.

1224
01:07:12,850 --> 01:07:14,410
But they have the
same asymptotics

1225
01:07:14,410 --> 01:07:17,060
which is the perimeter
of the square you start

1226
01:07:17,060 --> 01:07:20,580
with has to be about
twice n squared

1227
01:07:20,580 --> 01:07:22,600
to make an n by n checkerboard.

1228
01:07:22,600 --> 01:07:24,800
And the reason
that is, is if you

1229
01:07:24,800 --> 01:07:27,010
look at the
checkerboard pattern,

1230
01:07:27,010 --> 01:07:29,700
we're trying to get
color reversals along all

1231
01:07:29,700 --> 01:07:35,370
of these lines between the red
and the white, brown and white.

1232
01:07:35,370 --> 01:07:37,120
And the way we're doing
that here is we're

1233
01:07:37,120 --> 01:07:39,020
taking the boundary
of the paper,

1234
01:07:39,020 --> 01:07:42,710
and we're mapping it along
all the color reversals.

1235
01:07:42,710 --> 01:07:45,320
And if you work out how
much color reversal is there

1236
01:07:45,320 --> 01:07:49,200
in an n by n thing, it's
about twice n squared.

1237
01:07:49,200 --> 01:07:52,100
And so either your perimeter
has to be at least that large

1238
01:07:52,100 --> 01:07:54,960
if you're going to cover the
color reversals with perimeter.

1239
01:07:54,960 --> 01:07:58,130
And for a long time, we thought
that was the best we could do

1240
01:07:58,130 --> 01:08:01,090
was to cover color reversals
with a perimeter of paper.

1241
01:08:01,090 --> 01:08:03,620
Of course, know that you
can take a square of paper

1242
01:08:03,620 --> 01:08:05,630
make the perimeter
arbitrarily large.

1243
01:08:05,630 --> 01:08:08,770
So this was never
really a lower bound.

1244
01:08:08,770 --> 01:08:12,890
We never really knew that
you needed 2n squared.

1245
01:08:12,890 --> 01:08:15,440
The four by four
achieves 2n squared.

1246
01:08:15,440 --> 01:08:18,720
We think it's the
best for four by four,

1247
01:08:18,720 --> 01:08:20,939
but we proved last
year-- this is

1248
01:08:20,939 --> 01:08:25,359
with Marty and [INAUDIBLE]
and Robert Lang--

1249
01:08:25,359 --> 01:08:28,735
that you can do better and
get perimeter about n squared.

1250
01:08:28,735 --> 01:08:31,359
Now, there are some lower order
terms there, the order n parts.

1251
01:08:31,359 --> 01:08:33,880
So this is really only
practical for large n.

1252
01:08:33,880 --> 01:08:38,850
I think-- I'll elaborate on
that a little more in a moment.

1253
01:08:38,850 --> 01:08:39,689
But here's the idea.

1254
01:08:39,689 --> 01:08:43,609
Instead of visiting
all the boundaries

1255
01:08:43,609 --> 01:08:45,319
between red and
white squares, I just

1256
01:08:45,319 --> 01:08:47,620
want to visit the
squares themselves.

1257
01:08:47,620 --> 01:08:50,439
So if I could fold
a, in this case

1258
01:08:50,439 --> 01:08:52,830
a rectangle paper
into this shape

1259
01:08:52,830 --> 01:08:55,529
which has slits down
the sides, and it

1260
01:08:55,529 --> 01:08:57,250
has these flaps hanging out.

1261
01:08:57,250 --> 01:08:59,620
Now, you've seen how to make
flaps super efficiently.

1262
01:08:59,620 --> 01:09:01,550
You really don't need
to shrink the paper

1263
01:09:01,550 --> 01:09:04,370
by very much to
make this pattern.

1264
01:09:04,370 --> 01:09:08,779
Then, you take these guys-- and
everything is white side up.

1265
01:09:08,779 --> 01:09:13,090
You take these flaps,
fold them over.

1266
01:09:13,090 --> 01:09:15,189
They become brown.

1267
01:09:15,189 --> 01:09:17,229
And these guys fall over.

1268
01:09:17,229 --> 01:09:18,470
These fall down.

1269
01:09:18,470 --> 01:09:19,289
These guys fall up.

1270
01:09:19,289 --> 01:09:21,330
You can actually make any
two color pixel pattern

1271
01:09:21,330 --> 01:09:22,770
from this idea.

1272
01:09:22,770 --> 01:09:26,710
And it will make white squares
on top of the brown surface

1273
01:09:26,710 --> 01:09:28,420
that you folded.

1274
01:09:28,420 --> 01:09:29,870
So this is the starting point.

1275
01:09:29,870 --> 01:09:31,140
You just fold everything over.

1276
01:09:31,140 --> 01:09:33,350
And you get your checkerboard.

1277
01:09:33,350 --> 01:09:36,760
And now, essentially, you're
visiting each square only once

1278
01:09:36,760 --> 01:09:38,850
instead of the boundary
edge for all the squares.

1279
01:09:38,850 --> 01:09:40,439
And so you end up
using only n squared

1280
01:09:40,439 --> 01:09:41,696
instead of twice n squared.

1281
01:09:41,696 --> 01:09:43,779
And you can do it if you
start from a square also.

1282
01:09:43,779 --> 01:09:47,260
You just need more flaps.

1283
01:09:47,260 --> 01:09:50,920
And there's a bunch of
tabs sticking up here,

1284
01:09:50,920 --> 01:09:52,800
and a bunch of tabs
sticking up there.

1285
01:09:52,800 --> 01:09:54,550
You can fold this again
super efficiently,

1286
01:09:54,550 --> 01:09:57,590
using all these
standard techniques.

1287
01:09:57,590 --> 01:09:59,640
And then, you make a
checkerboard twice as

1288
01:09:59,640 --> 01:10:04,370
efficient for large n as we've
previously thought possible.

1289
01:10:04,370 --> 01:10:07,200
Now, we still don't know
whether this is optimal.

1290
01:10:07,200 --> 01:10:08,330
We think it is.

1291
01:10:08,330 --> 01:10:11,370
But we thought so before also.

1292
01:10:14,340 --> 01:10:16,640
Who knows?

1293
01:10:16,640 --> 01:10:19,410
So big open problem is
[INAUDIBLE] for anything.

1294
01:10:19,410 --> 01:10:25,270
In terms of actual values
of n, for n bigger than 16,

1295
01:10:25,270 --> 01:10:31,120
this method is better than
the standard approach.

1296
01:10:31,120 --> 01:10:36,756
Although if you look just
at seamless-- so seamless,

1297
01:10:36,756 --> 01:10:38,130
I didn't mention,
but we're going

1298
01:10:38,130 --> 01:10:39,820
to talk about it
more in a moment.

1299
01:10:39,820 --> 01:10:41,870
When I make a square
of a checkerboard,

1300
01:10:41,870 --> 01:10:45,160
I'd really like this to
be a single panel of paper

1301
01:10:45,160 --> 01:10:47,200
not divided into little panels.

1302
01:10:47,200 --> 01:10:50,030
And like in this checkerboard,
this white square

1303
01:10:50,030 --> 01:10:51,450
has a bunch of seems on it.

1304
01:10:51,450 --> 01:10:54,080
It's made out of three
smaller triangles.

1305
01:10:54,080 --> 01:10:55,710
And that's not so nice.

1306
01:10:55,710 --> 01:10:57,240
This method is seamless.

1307
01:10:57,240 --> 01:11:00,230
You get whole panels making
each of your squares,

1308
01:11:00,230 --> 01:11:01,930
so it looks a little prettier.

1309
01:11:01,930 --> 01:11:04,450
If you look at the best eight
by eight seamless folding,

1310
01:11:04,450 --> 01:11:07,550
this beats the best seamless
eight by eight folding.

1311
01:11:07,550 --> 01:11:09,700
Although it's rather
difficult to fold.

1312
01:11:09,700 --> 01:11:11,214
Hasn't yet been folded.

1313
01:11:11,214 --> 01:11:12,630
That would be a
good project also.

1314
01:11:16,070 --> 01:11:18,010
Build an actual checkerboard
with this method.

1315
01:11:21,450 --> 01:11:24,040
Questions?

1316
01:11:24,040 --> 01:11:25,810
Now, I want to move
to the general case.

1317
01:11:25,810 --> 01:11:29,100
So I talked a little bit about
checkerboards and about cubes.

1318
01:11:29,100 --> 01:11:31,320
Let's think about
arbitrary polyhedra.

1319
01:11:31,320 --> 01:11:32,660
And this is the Origamizer.

1320
01:11:54,340 --> 01:11:56,880
So Origamizer's
actually two things.

1321
01:11:56,880 --> 01:12:00,460
It's a computer program for
Windows that you can download,

1322
01:12:00,460 --> 01:12:02,880
and it's an algorithm.

1323
01:12:02,880 --> 01:12:04,730
And they're not quite the same.

1324
01:12:04,730 --> 01:12:06,990
So there's original
computer program

1325
01:12:06,990 --> 01:12:09,750
and Tomohiro Tachi wrote
a couple papers about it.

1326
01:12:09,750 --> 01:12:13,020
That program does
not always work.

1327
01:12:13,020 --> 01:12:15,140
Doesn't make every polyhedron.

1328
01:12:15,140 --> 01:12:19,990
It need some finesse to make it
work, but it's super efficient.

1329
01:12:19,990 --> 01:12:20,790
And it's practical.

1330
01:12:20,790 --> 01:12:23,060
He's made lots of models
with it like the bunny

1331
01:12:23,060 --> 01:12:25,450
you've seen on the poster.

1332
01:12:25,450 --> 01:12:27,980
There's the algorithm,
which we developed together.

1333
01:12:27,980 --> 01:12:30,520
And we know it's similar.

1334
01:12:30,520 --> 01:12:32,630
And we know it always works.

1335
01:12:32,630 --> 01:12:35,910
But it's a little
bit less practical.

1336
01:12:35,910 --> 01:12:39,960
So it's-- theory's always
a little behind practice,

1337
01:12:39,960 --> 01:12:40,460
let's say.

1338
01:12:40,460 --> 01:12:42,046
So there's a
practical thing here.

1339
01:12:42,046 --> 01:12:44,170
There's also a theoretically
guaranteed thing here.

1340
01:12:44,170 --> 01:12:46,399
They're not quite the same,
but they're very similar.

1341
01:12:46,399 --> 01:12:47,690
I'm going to tell you a little.

1342
01:12:47,690 --> 01:12:50,640
I'll show you both, basically.

1343
01:12:50,640 --> 01:12:56,160
But the idea is, a
practical algorithm

1344
01:12:56,160 --> 01:12:58,215
to fold any polyhedron.

1345
01:13:11,710 --> 01:13:14,360
And practical here
is a bit vague.

1346
01:13:14,360 --> 01:13:16,920
We don't-- that's not a theorem.

1347
01:13:16,920 --> 01:13:20,791
We don't know how to define
practical in mathematics

1348
01:13:20,791 --> 01:13:21,290
anyway.

1349
01:13:24,550 --> 01:13:27,200
It has some fun features,
though, mathematically.

1350
01:13:27,200 --> 01:13:29,050
One is that it's seamless.

1351
01:13:29,050 --> 01:13:34,076
So for it to be seamless, I
need to assume convex faces.

1352
01:13:36,690 --> 01:13:39,130
So faces are the sides
of the polyhedron.

1353
01:13:39,130 --> 01:13:41,890
So like in a cube,
every face is a square.

1354
01:13:41,890 --> 01:13:43,510
Those are convex.

1355
01:13:43,510 --> 01:13:45,930
And provided all the faces
are convex, if they're not,

1356
01:13:45,930 --> 01:13:48,920
you have to cut them
up into convex pieces.

1357
01:13:48,920 --> 01:13:51,320
My folding will be
seamless in that

1358
01:13:51,320 --> 01:13:53,740
it will be covered by an
entire piece of paper.

1359
01:13:53,740 --> 01:13:56,090
There maybe other things
hidden underneath.

1360
01:13:56,090 --> 01:14:01,300
But there won't be any
visible seems on the top side.

1361
01:14:01,300 --> 01:14:02,290
So that's nice.

1362
01:14:04,870 --> 01:14:07,055
It's also water tight.

1363
01:14:13,290 --> 01:14:15,700
And for this, I have
an illustration.

1364
01:14:15,700 --> 01:14:17,854
This is a feature missed
by the strip method.

1365
01:14:17,854 --> 01:14:19,520
And if you've always
felt like the strip

1366
01:14:19,520 --> 01:14:21,270
method of making
anything is cheating,

1367
01:14:21,270 --> 01:14:25,220
here's a nice formal sense
in which it's cheating.

1368
01:14:25,220 --> 01:14:28,560
We didn't realize it until we
start talking about Origamizer

1369
01:14:28,560 --> 01:14:31,820
which does not have
this cheating sense.

1370
01:14:31,820 --> 01:14:35,140
So here I'm trying to
make a 3D surface, looks

1371
01:14:35,140 --> 01:14:38,590
like a saddle
surface, by a strip.

1372
01:14:38,590 --> 01:14:41,835
If I just visited the guys in
this nice zigzag order, which

1373
01:14:41,835 --> 01:14:46,020
I know is possible, I get all
these slits down the sides.

1374
01:14:46,020 --> 01:14:47,550
This thing would not hold water.

1375
01:14:47,550 --> 01:14:49,008
If you poured water
on it, it would

1376
01:14:49,008 --> 01:14:51,270
fall through all the cracks.

1377
01:14:51,270 --> 01:14:55,110
And if I fold it right,
like in this picture,

1378
01:14:55,110 --> 01:14:56,690
there should be
no seems in here.

1379
01:14:56,690 --> 01:15:00,450
The square, the boundary of the
squares is what's drawn in red.

1380
01:15:00,450 --> 01:15:02,740
So here the boundary
of your piece of paper

1381
01:15:02,740 --> 01:15:04,160
gets mapped all over the place.

1382
01:15:04,160 --> 01:15:05,600
So it's lots of holes.

1383
01:15:05,600 --> 01:15:07,480
Here, I want the
boundary of the paper

1384
01:15:07,480 --> 01:15:11,600
to be the same as the
boundary of the surface.

1385
01:15:11,600 --> 01:15:14,980
So the only place the water
to run off is at the edge.

1386
01:15:14,980 --> 01:15:18,690
I mean, obviously, this
thing is not a closed solid.

1387
01:15:18,690 --> 01:15:20,469
But if you actually
made a cube, you're

1388
01:15:20,469 --> 01:15:22,510
still going to get some
edge because the boundary

1389
01:15:22,510 --> 01:15:24,090
paper has to go somewhere.

1390
01:15:24,090 --> 01:15:26,440
But if you then
sewed up the edge,

1391
01:15:26,440 --> 01:15:30,220
it would totally hold water.

1392
01:15:30,220 --> 01:15:33,390
So that is the informal
version of watertight.

1393
01:15:33,390 --> 01:15:36,470
The formal version
is the boundary

1394
01:15:36,470 --> 01:15:47,890
of the paper maps within
some tiny distance

1395
01:15:47,890 --> 01:15:51,245
epsilon of the boundary
of the surface, boundary

1396
01:15:51,245 --> 01:15:52,000
of the polyhedron.

1397
01:15:58,470 --> 01:16:02,730
And here, when I say
polyhedron, I really

1398
01:16:02,730 --> 01:16:05,875
means something that's
topologically a disk.

1399
01:16:12,770 --> 01:16:15,410
Brief topology.

1400
01:16:15,410 --> 01:16:18,050
This is a disk.

1401
01:16:18,050 --> 01:16:20,730
This is a disk.

1402
01:16:20,730 --> 01:16:23,995
This is not a disk.

1403
01:16:28,520 --> 01:16:30,586
Cube is not a disk.

1404
01:16:30,586 --> 01:16:31,450
It's a sphere.

1405
01:16:34,590 --> 01:16:36,010
This is a disk.

1406
01:16:36,010 --> 01:16:37,140
A piece of paper is a disk.

1407
01:16:37,140 --> 01:16:39,390
So really the only things
you could fold topologically

1408
01:16:39,390 --> 01:16:43,270
in a pure sense in a water
tight sense are disks.

1409
01:16:43,270 --> 01:16:44,900
You can't glue things together.

1410
01:16:44,900 --> 01:16:46,760
That's not in the rules.

1411
01:16:46,760 --> 01:16:49,550
So I can't make--
I could fold this.

1412
01:16:49,550 --> 01:16:51,910
But I'd have to have
an extra seem somewhere

1413
01:16:51,910 --> 01:16:54,730
in order to make this
thing just be a disk.

1414
01:16:54,730 --> 01:16:56,160
I could fold a cube.

1415
01:16:56,160 --> 01:16:58,950
But I have to have
some seem somewhere.

1416
01:16:58,950 --> 01:17:02,840
Here-- the top-- a square
gets cut into four pieces

1417
01:17:02,840 --> 01:17:04,280
in order to make it into a disk.

1418
01:17:04,280 --> 01:17:08,590
Any higher topology can be
cut down and made a disk.

1419
01:17:08,590 --> 01:17:09,770
So this is still universal.

1420
01:17:09,770 --> 01:17:11,270
But in terms of the
water tightness,

1421
01:17:11,270 --> 01:17:14,270
you have to think
about the disk version.

1422
01:17:14,270 --> 01:17:15,035
AUDIENCE: Erik?

1423
01:17:15,035 --> 01:17:15,660
PROFESSOR: Yes.

1424
01:17:15,660 --> 01:17:18,588
AUDIENCE: Even when you go
back and forth with the script

1425
01:17:18,588 --> 01:17:21,887
method, you could argue that
that was topologically a disk.

1426
01:17:21,887 --> 01:17:22,720
PROFESSOR: Oh, yeah.

1427
01:17:22,720 --> 01:17:25,650
Anything, any folding you make
is still topologically a disk.

1428
01:17:25,650 --> 01:17:26,940
This is making a disk.

1429
01:17:26,940 --> 01:17:29,060
But it's doesn't
preserve the boundary.

1430
01:17:29,060 --> 01:17:29,820
AUDIENCE: It what?

1431
01:17:29,820 --> 01:17:30,980
PROFESSOR: It's not
preserving the boundary.

1432
01:17:30,980 --> 01:17:31,480
So yeah.

1433
01:17:31,480 --> 01:17:33,620
Any origami, you're
still disk like,

1434
01:17:33,620 --> 01:17:36,300
and you're watertight
for some disk surface.

1435
01:17:36,300 --> 01:17:40,520
But I want to make this disk
surface with that boundary.

1436
01:17:40,520 --> 01:17:44,310
And watertightness is supposed
to match the given boundary.

1437
01:17:44,310 --> 01:17:46,270
But that boundary
must form a disk.

1438
01:17:46,270 --> 01:17:46,980
That's the point.

1439
01:17:46,980 --> 01:17:48,950
I can't say, oh, there's
no boundary in a cube.

1440
01:17:48,950 --> 01:17:51,490
So you have-- so the boundary
of paper goes nowhere.

1441
01:17:51,490 --> 01:17:53,530
That's not allowed.

1442
01:17:53,530 --> 01:17:57,892
So you get to specify it,
but it has to be a disk.

1443
01:17:57,892 --> 01:17:59,724
I'm going to wave
my hand some more.

1444
01:17:59,724 --> 01:18:02,140
There's another feature which
you can see in this picture.

1445
01:18:02,140 --> 01:18:05,860
This is a schematic of what
Origamizer would produce which

1446
01:18:05,860 --> 01:18:09,510
is that there's some
extra stuff underneath.

1447
01:18:09,510 --> 01:18:13,020
It's slightly lighter because
it's on the bottom side there.

1448
01:18:13,020 --> 01:18:14,850
But you can see
along every edge,

1449
01:18:14,850 --> 01:18:16,770
these are the edges of
the actual polyhedron.

1450
01:18:16,770 --> 01:18:18,650
And then, there's these
extra little tabs,

1451
01:18:18,650 --> 01:18:21,190
extra flaps on the underside.

1452
01:18:21,190 --> 01:18:22,440
This is actually necessary.

1453
01:18:22,440 --> 01:18:24,600
If you want
watertightness, you can't

1454
01:18:24,600 --> 01:18:27,100
fold exactly that polyhedron.

1455
01:18:27,100 --> 01:18:29,720
You fold a slightly
thickened version.

1456
01:18:29,720 --> 01:18:31,980
But you can keep all
those flaps on one side.

1457
01:18:31,980 --> 01:18:34,127
So if you're making
something like a cube,

1458
01:18:34,127 --> 01:18:35,835
you can put all the
garbage on the inside

1459
01:18:35,835 --> 01:18:38,320
where no one can see it.

1460
01:18:38,320 --> 01:18:42,455
So there's another feature
here is we get a little extra.

1461
01:18:48,200 --> 01:18:50,869
And that's necessary if
you want watertightness.

1462
01:18:50,869 --> 01:18:52,410
And it's sort of
the trick that makes

1463
01:18:52,410 --> 01:18:54,640
all of this possible
and efficient and so on.

1464
01:18:57,570 --> 01:19:00,455
So the high level
idea of Origamizer

1465
01:19:00,455 --> 01:19:03,030
is we're going to say,
there's all these faces

1466
01:19:03,030 --> 01:19:05,780
that we need to make.

1467
01:19:05,780 --> 01:19:09,010
So just plop them down on
the piece of paper somewhere.

1468
01:19:09,010 --> 01:19:11,390
And then, fold away the excess.

1469
01:19:11,390 --> 01:19:13,850
Get rid of it by tucking.

1470
01:19:13,850 --> 01:19:15,700
And that excess
paper is going to get

1471
01:19:15,700 --> 01:19:19,480
mapped into these
little chunks here.

1472
01:19:19,480 --> 01:19:21,980
And maybe I'll show you a demo.

1473
01:19:28,110 --> 01:19:31,600
So you takes
something like-- this

1474
01:19:31,600 --> 01:19:34,510
is similar to the
thing I was showing.

1475
01:19:34,510 --> 01:19:36,335
And I forgot a mouse.

1476
01:19:40,610 --> 01:19:42,660
There are all the
faces in the plane.

1477
01:19:42,660 --> 01:19:45,490
And they've conveniently
already been arranged.

1478
01:19:45,490 --> 01:19:48,720
I can't zoom out because
I lack scroll wheel.

1479
01:19:48,720 --> 01:19:55,450
But there's a square
that makes the-- yeah,

1480
01:19:55,450 --> 01:19:57,460
or a multi-touch trackpad.

1481
01:19:57,460 --> 01:19:59,300
Sorry.

1482
01:19:59,300 --> 01:20:00,900
And all the faces
are just there.

1483
01:20:00,900 --> 01:20:02,350
And then there's
this extra stuff.

1484
01:20:02,350 --> 01:20:06,760
And now I say-- I should
probably do this, too.

1485
01:20:06,760 --> 01:20:08,130
Maybe not.

1486
01:20:08,130 --> 01:20:09,520
Well, all right.

1487
01:20:09,520 --> 01:20:15,010
And then I say, crease pattern.

1488
01:20:15,010 --> 01:20:15,510
Boom.

1489
01:20:15,510 --> 01:20:16,810
That folds away the excess.

1490
01:20:16,810 --> 01:20:18,930
And then, just the
white faces, these guys

1491
01:20:18,930 --> 01:20:24,520
which correspond faces
over there, are used.

1492
01:20:24,520 --> 01:20:27,440
And then, you just fold
away the extra junk.

1493
01:20:27,440 --> 01:20:29,940
Easy.

1494
01:20:29,940 --> 01:20:31,170
You want to make a bunny.

1495
01:20:35,280 --> 01:20:36,770
This is actually
an example where

1496
01:20:36,770 --> 01:20:40,985
it will not work by itself.

1497
01:20:40,985 --> 01:20:42,360
Because as I said,
this algorithm

1498
01:20:42,360 --> 01:20:45,550
is not quite guaranteed to work.

1499
01:20:45,550 --> 01:20:49,780
So I'm going to change
the boundary little bit

1500
01:20:49,780 --> 01:20:55,180
by cutting to the ears.

1501
01:20:55,180 --> 01:20:57,360
And so this is still--
it was a disk before.

1502
01:20:57,360 --> 01:21:00,810
It's a disk after because
there's this boundary here.

1503
01:21:00,810 --> 01:21:02,520
But it turns out,
now the algorithm

1504
01:21:02,520 --> 01:21:04,900
will work, assuming
I didn't mess up.

1505
01:21:09,070 --> 01:21:11,760
It's bouncing
around a little bit.

1506
01:21:11,760 --> 01:21:13,480
You can see it's
pretty efficient here.

1507
01:21:13,480 --> 01:21:17,010
There's very tiny gaps
between the triangles.

1508
01:21:17,010 --> 01:21:19,090
There's actually a little
bit a violation here.

1509
01:21:19,090 --> 01:21:23,350
What's happening, which you
can see here, is, on the inside

1510
01:21:23,350 --> 01:21:26,160
are all this extra
structure, these flaps.

1511
01:21:26,160 --> 01:21:28,340
And sometimes they're so
big that they actually

1512
01:21:28,340 --> 01:21:29,780
penetrate the surface.

1513
01:21:29,780 --> 01:21:34,480
But that can be dealt with by
just a little bit of cutting,

1514
01:21:34,480 --> 01:21:36,730
maybe a little more cutting.

1515
01:21:36,730 --> 01:21:38,800
Not cutting in
the literal sense.

1516
01:21:38,800 --> 01:21:40,600
But we just subdivided
these panels

1517
01:21:40,600 --> 01:21:42,260
into lots of smaller panels.

1518
01:21:42,260 --> 01:21:45,450
And now, it is valid.

1519
01:21:45,450 --> 01:21:48,310
This not the design that
you've seen on the poster.

1520
01:21:48,310 --> 01:21:50,390
The design on poster's
little more efficient.

1521
01:21:50,390 --> 01:21:52,020
I'm not so expert.

1522
01:21:52,020 --> 01:21:54,820
I'm not so pro that I can
make exactly that design.

1523
01:21:54,820 --> 01:21:57,020
So it's a little inefficient
on the sides here.

1524
01:21:57,020 --> 01:21:59,840
But you can use this tool
to make super complicated 3D

1525
01:21:59,840 --> 01:22:01,120
models.

1526
01:22:01,120 --> 01:22:06,110
Let me quickly tell you what
goes into the algorithm.

1527
01:22:06,110 --> 01:22:08,560
So the first part
is to figure out

1528
01:22:08,560 --> 01:22:10,820
where all these tucks
are going to go.

1529
01:22:10,820 --> 01:22:13,970
They lie essentially
along angular bisectors

1530
01:22:13,970 --> 01:22:16,410
on one side of every edge.

1531
01:22:16,410 --> 01:22:19,820
But at the vertices, things
are super complicated.

1532
01:22:19,820 --> 01:22:22,880
And in general, if you have a
non-convex surface with tons

1533
01:22:22,880 --> 01:22:25,770
of material coming together,
what I'd like to do

1534
01:22:25,770 --> 01:22:28,570
is add lots of little
flaps on the side.

1535
01:22:28,570 --> 01:22:31,450
So that when I
open it up-- so let

1536
01:22:31,450 --> 01:22:33,700
me draw you a generic picture.

1537
01:22:33,700 --> 01:22:36,350
So we have two faces
coming together.

1538
01:22:36,350 --> 01:22:40,680
What I'd like to do
is add a flap here

1539
01:22:40,680 --> 01:22:43,900
and a corresponding
one just behind it.

1540
01:22:43,900 --> 01:22:45,105
So that's sort of a tab.

1541
01:22:49,110 --> 01:22:52,220
And I can unfold that and
think of some other surface

1542
01:22:52,220 --> 01:22:53,470
that I'm trying to fold.

1543
01:22:53,470 --> 01:22:56,450
So I really wanted just
those two polygons.

1544
01:22:56,450 --> 01:22:59,610
But now, I've made some other
thing which is still a disk.

1545
01:22:59,610 --> 01:23:02,980
You can add those faces in
such a way that you will have,

1546
01:23:02,980 --> 01:23:05,690
at most, 360 degrees
of material everywhere.

1547
01:23:05,690 --> 01:23:07,760
So even though in
the original thing,

1548
01:23:07,760 --> 01:23:10,190
you had maybe tons of material
coming together which you

1549
01:23:10,190 --> 01:23:13,360
cannot make with real paper,
you add in a bunch of tabs along

1550
01:23:13,360 --> 01:23:15,480
the edges and a few
more at the vertices.

1551
01:23:15,480 --> 01:23:18,660
You can fix things so
that the thing is actually

1552
01:23:18,660 --> 01:23:20,940
makeable from one
sheet of paper.

1553
01:23:20,940 --> 01:23:22,852
That's the high level idea.

1554
01:23:22,852 --> 01:23:24,060
Doing that is a bit detailed.

1555
01:23:24,060 --> 01:23:25,351
The paper isn't even published.

1556
01:23:25,351 --> 01:23:26,784
These are some figures from it.

1557
01:23:26,784 --> 01:23:28,200
But this is some--
way this is how

1558
01:23:28,200 --> 01:23:31,050
you resolve a vertex with
some triangulation stuff.

1559
01:23:31,050 --> 01:23:35,180
Each of these corresponds to
a flap in the original thing.

1560
01:23:35,180 --> 01:23:40,216
And then, this is where
we're a little impractical.

1561
01:23:40,216 --> 01:23:42,030
It doesn't quite
match what we do

1562
01:23:42,030 --> 01:23:44,900
in practice in the
computer program.

1563
01:23:44,900 --> 01:23:47,300
But the idea is, you imagine,
you have your faces which

1564
01:23:47,300 --> 01:23:50,430
you want to bring
together somehow.

1565
01:23:50,430 --> 01:23:54,690
They're distributed in the
piece of paper somewhere.

1566
01:23:54,690 --> 01:23:56,810
But you'd really like
to connect them together

1567
01:23:56,810 --> 01:23:58,270
when they have matching edges.

1568
01:23:58,270 --> 01:24:00,570
So this edge might
be glued to some edge

1569
01:24:00,570 --> 01:24:02,700
of some other triangle.

1570
01:24:02,700 --> 01:24:07,775
And I need to just navigate
these little river-like strips

1571
01:24:07,775 --> 01:24:09,150
to visit one edge
from the other.

1572
01:24:09,150 --> 01:24:12,050
And we proved that if these
guys are sufficiently tiny,

1573
01:24:12,050 --> 01:24:13,270
you can always do that.

1574
01:24:13,270 --> 01:24:15,659
And of course, in reality,
you want to arrange things,

1575
01:24:15,659 --> 01:24:17,700
so you can do it efficiently
with little wastage.

1576
01:24:17,700 --> 01:24:19,825
But we proved at least it's
possible with this kind

1577
01:24:19,825 --> 01:24:21,005
of wiggly path stuff.

1578
01:24:24,530 --> 01:24:26,850
So now our picture, the
thing we're trying to make

1579
01:24:26,850 --> 01:24:28,190
looks something like that.

1580
01:24:28,190 --> 01:24:30,400
Where we have-- originally
it was four triangles,

1581
01:24:30,400 --> 01:24:31,480
the gray triangles.

1582
01:24:31,480 --> 01:24:34,350
We added in these extra
flaps so that it's

1583
01:24:34,350 --> 01:24:36,720
nice and well-behaved.

1584
01:24:36,720 --> 01:24:40,780
And each of those flaps we're
covering from both sides.

1585
01:24:40,780 --> 01:24:48,610
And if you think the red
diagram-- I'll get this right.

1586
01:24:48,610 --> 01:24:50,190
The red diagram
corresponds to that.

1587
01:24:50,190 --> 01:24:52,060
It's just been kind of squashed.

1588
01:24:52,060 --> 01:24:53,620
So there are four
triangles, which

1589
01:24:53,620 --> 01:24:55,530
correspond to the
four top flaps.

1590
01:24:55,530 --> 01:25:02,010
And there's this outer chunk
which corresponds to that flap.

1591
01:25:02,010 --> 01:25:05,950
And then, you look at the dual
which is the blue diagram.

1592
01:25:05,950 --> 01:25:07,820
You take that
picture, and that's

1593
01:25:07,820 --> 01:25:11,390
how you set up the crease
pattern essentially.

1594
01:25:11,390 --> 01:25:13,430
So these are the
original four triangles.

1595
01:25:13,430 --> 01:25:13,930
And then.

1596
01:25:13,930 --> 01:25:16,190
There's all this
stuff that represents

1597
01:25:16,190 --> 01:25:19,230
the structure of that
thing that we want to make.

1598
01:25:19,230 --> 01:25:22,220
And then, you just have to fill
in the creases in the middle.

1599
01:25:22,220 --> 01:25:24,900
And you do that just
with something--

1600
01:25:24,900 --> 01:25:26,900
this is how you do it
guaranteed correct.

1601
01:25:26,900 --> 01:25:29,570
And we saw I had to do
lots of subdivision here.

1602
01:25:29,570 --> 01:25:32,030
What I called,
accidentally, cutting.

1603
01:25:32,030 --> 01:25:34,660
But just lots of pleats there.

1604
01:25:34,660 --> 01:25:37,350
Because, essentially,
this is the edge.

1605
01:25:37,350 --> 01:25:40,091
And we want that edge to
lie in a tiny little tab.

1606
01:25:40,091 --> 01:25:41,965
So it's got to go up
and down and up and down

1607
01:25:41,965 --> 01:25:43,970
and up and down,
accordion style.

1608
01:25:43,970 --> 01:25:46,110
And if you do it
right, all those things

1609
01:25:46,110 --> 01:25:49,400
will collapse down to
a little tab attached

1610
01:25:49,400 --> 01:25:51,530
to that edge which is
also attach that edge.

1611
01:25:51,530 --> 01:25:52,771
And they will get joined up.

1612
01:25:52,771 --> 01:25:55,270
Then, you've got to get rid of
all this stuff in the middle.

1613
01:25:55,270 --> 01:25:57,980
And rough idea is,
if you pack it with--

1614
01:25:57,980 --> 01:26:00,230
or I guess you
cover it with disks

1615
01:26:00,230 --> 01:26:02,730
so that everything
is again very tiny.

1616
01:26:02,730 --> 01:26:06,430
And you fold what's called the
[INAUDIBLE] with those points.

1617
01:26:06,430 --> 01:26:09,020
And it works.

1618
01:26:09,020 --> 01:26:11,850
It's a complicated
but very cool.

1619
01:26:11,850 --> 01:26:16,650
And the paper hopefully will
be released later this year

1620
01:26:16,650 --> 01:26:17,560
finally.

1621
01:26:17,560 --> 01:26:19,560
And that's Origamizer.

1622
01:26:19,560 --> 01:26:22,030
And that's efficient
origami design.