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PROFESSOR: Let's get started.

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Today we're really going
to take on folding polygons

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into polyhedra in a big way.

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We got started last time.

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Today we're going
to do the real thing

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with some tape and scissors.

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Folding polyhedra.

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It's pretty much the
last kind of folding

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that we're going to be
talking about in this class.

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We see linkage folding, paper
folding, unfolding polyhedra,

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and finally, folding polyhedra.

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So this is a, sort of, problem.

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We're given some polygon.

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We're not told where to
folder or how to glue it up,

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necessarily.

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We'd like to find,
is there some way

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to glue this
boundary to itself--

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I've drawn this picture
before, something like this--

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in order to make a
convex polyhedron?

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There's several versions
of this problem.

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One is the decision version,
which is, I give you a polygon.

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And I want to know, does it
make any convex polyhedron?

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For the cross,
the answer is yes.

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We will see examples later
today where the answer is no.

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In fact, most of the
time the answer is no.

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So given a polygon, can its
boundary be glued to itself?

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And this is the
distinguishing feature

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from origami is that's
all you get to do

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and no multiple coverage.

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You want exactly
that surface to be

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folded into convex polyhedron.

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And today's lecture is going
to be mainly about mathematics

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for solving this problem.

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The next class will be
algorithms for solving

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this problem.

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So we need some mathematical
tools, some cool geometry

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stuff by a Russian
geometer called Alexandrov,

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from the '40s.

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And that's what we'll
be talking about today.

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Then we'll see how that turns
into an algorithm next class.

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Decision is nice.

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But of course, we'd like to
actually find these gluings.

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I want to actually know that
this guy folds into a cube.

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So there's the
enumeration problem,

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which is list all the
gluings and all the foldings.

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And there's also the more
mathematical combinatorial

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problem, which is,
how many can there be?

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And we'll start
addressing that today

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by showing that there
can be infinitely many.

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There can be continua.

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For example, this
polygon, this rectangle,

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can fold into infinitely
many polyhedra.

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We'll prove that today.

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We won't do all of them.

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But we'll one of them, I guess.

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So that's makes this listing
all of them a little bit

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difficult, when there's
infinitely many.

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But it turns out they'll
fall into some nice classes.

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We'll be able to
characterize that

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at the end of today's class.

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And so this problem is
actually well defined.

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They're a finite number
of distinct classes.

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And you could list them all.

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There can actually
be exponentially

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many of those classes.

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But most the time,
it's polynomial.

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That'll be more next class.

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These are the sorts of problems
we're interested in folding.

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Now, one natural
question is, why am I

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restricting myself to convex?

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And one answer is, well,
that's where most of the work

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has been done.

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But there is some nice
work on the nonconvex case.

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So let me tell you about that.

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So for nonconvex-- or
where you don't care about

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whether the result
is convex-- you just

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want it to make some polyhedron.

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You can show it's
always possible.

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What this means is if
I'm given some polygon,

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and I glue up the
boundary somehow--

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and this result
even holds if you

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don't glue up all
of the boundary.

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Because then we're no longer
trying to make a convex thing.

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For example, I could
glue up all the boundary,

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as long as the way that I glue
things is orientable-- I don't

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put twists in an unresolvable
way, in these gluings--

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then you're golden.

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So you need that the surface
is orientable or has boundary.

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So this is posed as an
open problem in our book.

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And then we
discovered, this year,

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that it was already solved
in 1960, but in Russian.

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And then, it was only
translated in 1996.

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We still should have
known about it in '96.

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But we finally
realized that they

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didn't talk about
folding or gluing.

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They were talking about
realizing polyhedral metrics,

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which is the language we're
going to translate to soon.

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But our bad.

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And I think this is a
pretty complicated result.

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So we're not going to
talk about it here.

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But it's interesting.

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I'd like to learn more about it.

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So it's by Burago
and Zalgaller, I've

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used others of their theorems.

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So it's funny I
didn't know this one.

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In some sense, the
decision question

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isn't interesting for
non-convex polyhedra.

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So by restricting
the convex, saying,

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I want to make
convex shapes, this

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becomes an interesting problem.

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And it leads to lots of cool
mathematics and algorithms.

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And that's what we're
going to talk about.

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So I want to talk about
this guy, Alexandrov.

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Not really the guy, but the
mathematics that he did.

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And we're going to appropriate
his name a little bit

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and talk about
Alexandrov gluings.

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This is a term that we made
up, obviously he didn't.

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You don't usually make up your
own terms named after yourself.

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We'd like to know which
gluings are valid.

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We're not going to think about
where the thing is creased,

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how to fold.

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We're just going to focus
on the gluing for now.

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What makes a gluing valid?

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Well, one thing that we
talked about last time

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is that once you have a
gluing, in this picture,

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you can compute shortest paths.

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You can compute the shortest
path from this vertex

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to this vertex.

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But here, it's a straight line.

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There's probably
another path that

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goes around a different way.

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Last time we did the shortest
path from here to here.

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And it was some
complicated thing.

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I won't try to do it again.

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But you can compute
shortest paths

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once you know what's
connected to what.

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So this defines what
we call a metric.

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We're going to learn
something about that.

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Polygon plus gluing
induce a metric.

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Metric just means it tells you
how far different points are

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from each other.

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And we can compute that metric
by computing shortest paths.

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I'm going to start
abstracting a little bit away

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from polygon plus gluing
and just think about,

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there's some surface
that, in your mind,

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it's not really
geometric thing entirely.

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But locally, you
understand the geometry.

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If you're at some
corner, you can

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understand what other
vertices are near to you

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and how to walk around
on that surface.

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So locally, you understand the
metric and how it's laid out.

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When you're at this
corner, you know--

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and let's say I glued this to
this-- that your curvature is

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90 degrees, that there's 270
degrees of material there.

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And you could go any one of
those 270 degrees of directions

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from there.

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So that's the sense in
which we have a metric.

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Now, there are three things
that this metric should satisfy

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if we want to make
something that's convex.

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Any suggestions?

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AUDIENCE: Less than
or equal to 360.

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PROFESSOR: Less than or equal
to 360 degrees of material

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an any point.

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Yep, that is what we
would call convexity.

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It's really a local convexity.

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All points have 0 or
positive curvature.

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That's the way I would say it.

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Remember, curvature
was 360 minus some

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of the angles at the vertex.

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So this is the same as saying
you have, at most, 360 degrees

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of material any point.

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This is obviously necessary.

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And the point is, you can
evaluate this just knowing

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how things are glued together.

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You don't need to
understand 3D geometry.

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It's a local property.

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And certainly necessary
if we're going

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to try to make a
convex polyhedron.

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Anything else?

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They get less obvious
from here, I guess.

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Any topologists?

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AUDIENCE: You
shouldn't have holes.

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PROFESSOR: Shouldn't have holes.

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Right.

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So I should definitely
glue everything together.

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It's sort of implicit
in this statement

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but not totally explicit.

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And there's a little
more than that.

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Not only should everything
to glued together,

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but for example, if I glued
this edge to this edge

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and I glued this edge to this
edge-- something like that.

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That's probably not
going to work out.

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Crossings are bad.

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And what's really going
wrong there is the topology.

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If we're trying to make
something that's convex,

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it better be
topologically a sphere.

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So that's the third condition.

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And this is, again,
something you

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can figure out just by
looking at the local metric.

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This is a property
about metrics.

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In their case of
polygons plus gluings,

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it's really easy
to figure out when

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this is the case, which is that
the gluing should be complete,

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as you were saying.

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You shouldn't leave
anything blank.

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00:12:14,170 --> 00:12:16,490
Everything gets
glued to something.

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00:12:16,490 --> 00:12:20,460
And it has no
crossings that will

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be equivalent to being
sphere like, in our case.

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00:12:23,952 --> 00:12:25,660
The other property
actually always holds.

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00:12:25,660 --> 00:12:29,560
So it's no surprise that
you didn't think of it.

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00:12:32,430 --> 00:12:34,750
But if you want to think
a little more generally,

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00:12:34,750 --> 00:12:38,550
this a key property we're
using that we started

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00:12:38,550 --> 00:12:43,740
from a polygon, not some
smooth curve thing, which

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00:12:43,740 --> 00:12:50,380
is that only a finite number of
points have nonzero curvature.

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00:12:55,900 --> 00:12:58,150
If we want to make a polyhedron
with a finite a number

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00:12:58,150 --> 00:12:59,800
vertices, well,
these are going to be

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00:12:59,800 --> 00:13:03,490
the vertices, all the points
with nonzero curvature.

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00:13:03,490 --> 00:13:05,730
So there had better be a
finite number of those.

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00:13:05,730 --> 00:13:07,970
Now, when we start with
a polygon and just glue

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some finite set of
segments to each other,

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00:13:10,220 --> 00:13:12,280
that will happen automatically.

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00:13:12,280 --> 00:13:15,790
But it's a property we're using.

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00:13:15,790 --> 00:13:19,020
So these three
properties together

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are what I'm going to
call Alexandrov gluing.

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00:13:22,340 --> 00:13:28,200
If you satisfy all three of
these, you are Alexandrov.

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00:13:28,200 --> 00:13:31,950
And now, we have a cool
theorem about Alexandrov

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00:13:31,950 --> 00:13:34,100
gluings is called
Alexandrov's Theorem.

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00:13:37,160 --> 00:13:39,650
He had a lot of gluings,
but this the one

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00:13:39,650 --> 00:13:42,320
we use a lot in this field.

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00:13:50,990 --> 00:13:54,810
Originally proved
in 1941, I believe.

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00:13:54,810 --> 00:13:59,370
Then finally, his big
book about convex surfaces

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was translated to English
in 2005, pretty recent.

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00:14:04,120 --> 00:14:07,120
But we've been reading other
translations, and so on,

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00:14:07,120 --> 00:14:09,520
to understand this theorem.

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00:14:23,900 --> 00:14:26,010
I guess I'm not using
Alexandrov gluing here.

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00:14:26,010 --> 00:14:28,260
But we have a
convex, polyhedral,

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00:14:28,260 --> 00:14:29,770
and topologically sphere.

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00:14:42,160 --> 00:14:44,490
Take any one of those things.

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00:14:44,490 --> 00:14:50,840
Then it's realized by a
unique convex polyhedron.

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00:14:58,900 --> 00:15:01,490
So this is basically the answer.

241
00:15:01,490 --> 00:15:04,150
And from our perspective, it's
like , yeah, yeah, polyhedral.

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00:15:04,150 --> 00:15:05,094
Yeah, no crossings.

243
00:15:05,094 --> 00:15:06,510
But really, the
central constraint

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00:15:06,510 --> 00:15:09,010
that we have to worry
about is convexity,

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00:15:09,010 --> 00:15:11,810
that we never glue
together too much material.

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00:15:11,810 --> 00:15:14,640
As long as you don't glue
together more than 360 degrees

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00:15:14,640 --> 00:15:17,630
material at any
point, then that thing

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00:15:17,630 --> 00:15:19,555
will realize a unique
convex polyhedron.

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00:15:22,710 --> 00:15:24,944
It's pretty cool but surprising.

250
00:15:24,944 --> 00:15:26,610
The uniqueness should
not be surprising,

251
00:15:26,610 --> 00:15:29,900
because we proved
that last class.

252
00:15:29,900 --> 00:15:31,570
I can remind you briefly.

253
00:15:31,570 --> 00:15:33,070
We proved Cauchy's
Rigidity Theorem,

254
00:15:33,070 --> 00:15:34,570
which is not quite the same.

255
00:15:34,570 --> 00:15:37,824
And then we said, oh, yeah, if
you have some convex polyhedron

256
00:15:37,824 --> 00:15:39,240
metric, you don't
know necessarily

257
00:15:39,240 --> 00:15:40,490
know what it looks like in 3D.

258
00:15:40,490 --> 00:15:42,090
But you can compute
all shortest paths

259
00:15:42,090 --> 00:15:45,070
from every vertex to
every other vertex.

260
00:15:45,070 --> 00:15:48,710
And then however you
try to fold that thing,

261
00:15:48,710 --> 00:15:50,310
every edge is a shortest path.

262
00:15:50,310 --> 00:15:53,630
That's just a fact
about polyhedra.

263
00:15:53,630 --> 00:15:57,490
The edges must be some subset
of these shortest paths.

264
00:15:57,490 --> 00:16:01,344
So you think of this
as a convex polyhedron.

265
00:16:01,344 --> 00:16:02,760
And then Cauchy's
Rigidity Theorem

266
00:16:02,760 --> 00:16:05,470
says, there's only
one convex realization

267
00:16:05,470 --> 00:16:08,000
of that thing with those edges.

268
00:16:08,000 --> 00:16:11,690
And Cauchy allows some of the
edges to be flat like this.

269
00:16:11,690 --> 00:16:14,310
So there will only be one
way to embed that in 3D--

270
00:16:14,310 --> 00:16:15,699
at most one way.

271
00:16:15,699 --> 00:16:17,490
The surprising thing
is that there actually

272
00:16:17,490 --> 00:16:21,740
is one way that you can
realize-- I'm just thinking

273
00:16:21,740 --> 00:16:25,330
totally locally in
this kind of gluing--

274
00:16:25,330 --> 00:16:27,590
and yet, as long as
I don't glue together

275
00:16:27,590 --> 00:16:31,600
too much material at any point,
I get a convex polyhedron.

276
00:16:31,600 --> 00:16:34,560
With the usual gluing here,
without these crossings,

277
00:16:34,560 --> 00:16:35,945
I get a cube.

278
00:16:35,945 --> 00:16:38,670
No surprise, and it's unique.

279
00:16:38,670 --> 00:16:40,920
But there are other ways
to glue this polygon.

280
00:16:40,920 --> 00:16:42,670
We saw them, actually,
in Lecture 1--

281
00:16:42,670 --> 00:16:44,439
I'll show them
again next class--

282
00:16:44,439 --> 00:16:45,980
that can make other
convex polyhedra.

283
00:16:50,910 --> 00:16:56,180
Let me give you a little
idea of the proof.

284
00:16:56,180 --> 00:16:59,460
This is going to be a
pretty high-level sketch.

285
00:16:59,460 --> 00:17:02,010
But I'll tell you the
combinatorial part.

286
00:17:02,010 --> 00:17:04,210
And then there's the
topology analysis part

287
00:17:04,210 --> 00:17:06,710
that's a little less obvious.

288
00:17:15,680 --> 00:17:17,280
We've already proved uniqueness.

289
00:17:17,280 --> 00:17:19,170
I want to look at existence.

290
00:17:19,170 --> 00:17:24,130
And we're going to use induction
on the number of vertices.

291
00:17:27,910 --> 00:17:30,220
And remember, vertices
are well defined.

292
00:17:30,220 --> 00:17:32,640
When I say vertex
here, I really mean

293
00:17:32,640 --> 00:17:35,907
point of nonzero curvature.

294
00:17:35,907 --> 00:17:37,490
We know there's
finitely many of them.

295
00:17:37,490 --> 00:17:39,230
Call the number n.

296
00:17:39,230 --> 00:17:41,350
And we're going to try to
simplify our polyhedron

297
00:17:41,350 --> 00:17:44,020
and reduce the number vertices.

298
00:17:44,020 --> 00:17:47,630
But there's lots of ways
to do that with polyhedra.

299
00:17:47,630 --> 00:17:49,460
But here, we don't
have a polyhedron.

300
00:17:49,460 --> 00:17:52,240
We have this kind of abstract
polyhedron in Never Never Land.

301
00:17:52,240 --> 00:17:54,820
We know how to walk around
on the surface, locally.

302
00:17:54,820 --> 00:17:59,180
But we don't really
know any 3D geometry.

303
00:17:59,180 --> 00:18:01,010
So is this a crazy idea.

304
00:18:01,010 --> 00:18:03,030
Before we get to
the crazy idea, I

305
00:18:03,030 --> 00:18:14,010
need an important fact, which
is total curvature, if you

306
00:18:14,010 --> 00:18:22,690
sum over all vertices,
is 720 degrees.

307
00:18:25,560 --> 00:18:27,870
This is actually true, even
of nonconvex polyhedra,

308
00:18:27,870 --> 00:18:30,420
as long as you have something
homeomorphic to a sphere,

309
00:18:30,420 --> 00:18:32,080
topologically a sphere.

310
00:18:32,080 --> 00:18:35,107
Total curvature is 720.

311
00:18:35,107 --> 00:18:37,440
This is actually why it's
useful to talk about curvature

312
00:18:37,440 --> 00:18:41,310
and not how much angle is
coming together at a point.

313
00:18:41,310 --> 00:18:43,940
So obviously, we've got zillions
of points with a curvature 0.

314
00:18:43,940 --> 00:18:46,330
Those don't count for anything.

315
00:18:46,330 --> 00:18:49,230
But then, somehow this thing
has to wrap around a sphere.

316
00:18:49,230 --> 00:18:52,660
And you can compute the
curvature of a sphere at 720.

317
00:18:52,660 --> 00:18:54,652
And this has to match that.

318
00:18:54,652 --> 00:18:58,286
I can't give a lot of intuition
for this, but it's true.

319
00:18:58,286 --> 00:19:00,415
And it's kind of powerful.

320
00:19:03,590 --> 00:19:05,590
It says you can't have
very many sharp vertices.

321
00:19:05,590 --> 00:19:10,120
If I have a super-sharp point
in my convex polyhedron,

322
00:19:10,120 --> 00:19:12,120
it has a curvature close to 360.

323
00:19:12,120 --> 00:19:15,000
It's going to be
260 minus almost 0.

324
00:19:15,000 --> 00:19:19,420
I can only have two of
those in any polyhedron.

325
00:19:19,420 --> 00:19:21,050
Now, with a
nonconvex polyhedron,

326
00:19:21,050 --> 00:19:22,940
I can have negative
curvature to compensate.

327
00:19:22,940 --> 00:19:25,740
And so then, I can have lots
of spikes, like an our witch's

328
00:19:25,740 --> 00:19:28,310
hats, where we had four spikes.

329
00:19:28,310 --> 00:19:31,114
But for convex, I don't
have any negative numbers

330
00:19:31,114 --> 00:19:33,030
to compensate with these
big positive numbers.

331
00:19:33,030 --> 00:19:36,190
So I can only have two spikes.

332
00:19:36,190 --> 00:19:38,890
We're going to use that
a lot today, this fact.

333
00:19:38,890 --> 00:19:41,077
It's called Descartes Theorem.

334
00:19:41,077 --> 00:19:42,410
You may have heard of Descartes.

335
00:19:44,929 --> 00:19:46,470
Nowadays, you prove
it in other ways.

336
00:19:46,470 --> 00:19:48,094
But we're not going
to look at a proof.

337
00:19:48,094 --> 00:19:49,842
We're just going to use it.

338
00:19:49,842 --> 00:19:51,800
I'm also going to assume
that my polyhedron has

339
00:19:51,800 --> 00:19:52,930
at least five vertices.

340
00:19:52,930 --> 00:19:57,830
Or really, my polyhedral metric
has at least five vertices.

341
00:19:57,830 --> 00:19:59,300
If it has at least
five vertices,

342
00:19:59,300 --> 00:20:01,380
in other words,
more than four, then

343
00:20:01,380 --> 00:20:04,380
there will be some vertex
that has less than a quarter

344
00:20:04,380 --> 00:20:08,630
of 720 curvature, in
fact, two of them.

345
00:20:14,200 --> 00:20:23,000
Two vertices-- and I call
them x and y-- have curvature.

346
00:20:27,188 --> 00:20:31,900
Call them alpha and
beta, less than 180,

347
00:20:31,900 --> 00:20:35,740
which is one quarter of 720.

348
00:20:35,740 --> 00:20:38,050
The average is obviously
going down as n goes up.

349
00:20:38,050 --> 00:20:42,940
So when n is 5, I can get two
reasonably small curvature

350
00:20:42,940 --> 00:20:44,977
vertices.

351
00:20:44,977 --> 00:20:47,060
This is like saying, there's
more than 180 degrees

352
00:20:47,060 --> 00:20:51,740
of material glued at
these two points So what?

353
00:20:51,740 --> 00:20:55,580
Well, now I'm going
to do some surgery,

354
00:20:55,580 --> 00:20:57,610
as topologists call it.

355
00:20:57,610 --> 00:21:05,024
I have these vertices, x and
y, in this sphere-like thing.

356
00:21:05,024 --> 00:21:06,690
There's some shortest
path between them.

357
00:21:06,690 --> 00:21:08,325
I can compute that.

358
00:21:08,325 --> 00:21:09,700
And now we're
saying that there's

359
00:21:09,700 --> 00:21:12,050
a fair amount of material here
and a fair amount of material

360
00:21:12,050 --> 00:21:12,580
here.

361
00:21:12,580 --> 00:21:17,520
What I'd like to do is
slice the surface open along

362
00:21:17,520 --> 00:21:18,770
that shortest path.

363
00:21:18,770 --> 00:21:21,600
I've drawn it twice,
because I've cut it open.

364
00:21:21,600 --> 00:21:24,370
There's surface here, surface
here, and a hole there.

365
00:21:24,370 --> 00:21:28,110
And I'm going to glue
in some more material.

366
00:21:28,110 --> 00:21:33,010
And I'm going to
glue in a triangle--

367
00:21:33,010 --> 00:21:36,190
actually, two triangles
right on top of each other.

368
00:21:36,190 --> 00:21:38,470
So that here, the edges
are glued together.

369
00:21:38,470 --> 00:21:42,760
Back here, the edges are glued
together, here, they're not.

370
00:21:42,760 --> 00:21:45,460
So you can see, this is
still topologically a sphere.

371
00:21:45,460 --> 00:21:49,920
As you walk on to this edge,
you end up on the top triangle.

372
00:21:49,920 --> 00:21:52,670
I can go around the back
to the bottom triangle.

373
00:21:52,670 --> 00:21:55,410
And eventually, I will
get to the back edge.

374
00:21:55,410 --> 00:21:57,186
And then I can walk like that.

375
00:21:57,186 --> 00:21:58,560
Shortest paths
are not preserved.

376
00:21:58,560 --> 00:22:00,730
But it's at least still
topologically a sphere.

377
00:22:00,730 --> 00:22:05,040
I've slipped this open,
wedged in two triangles.

378
00:22:05,040 --> 00:22:06,980
I get a new metric.

379
00:22:06,980 --> 00:22:12,820
Now, I get to choose
what triangles I glue in.

380
00:22:12,820 --> 00:22:17,730
And I'm going to
set these angles.

381
00:22:17,730 --> 00:22:20,770
Ultimately, I'd like
this angle to be

382
00:22:20,770 --> 00:22:26,840
alpha over 2 and this
angle to be beta over 2.

383
00:22:26,840 --> 00:22:30,950
Remember, curvature is 360 minus
the sum of the angles there.

384
00:22:30,950 --> 00:22:33,440
That means that's how
much extra material

385
00:22:33,440 --> 00:22:37,690
you need to stick in
until that vertex is flat.

386
00:22:37,690 --> 00:22:40,260
So if I stick in an
angle of alpha over 2

387
00:22:40,260 --> 00:22:43,420
here, at this point, and
there's two copies of it.

388
00:22:43,420 --> 00:22:46,080
Because there's the front
triangle and the back triangle,

389
00:22:46,080 --> 00:22:47,580
Then I'll be wedging
in alpha, which

390
00:22:47,580 --> 00:22:51,010
is exactly what it
takes to make x flat.

391
00:22:51,010 --> 00:22:54,360
And this is exactly what
it takes to make y flat.

392
00:22:54,360 --> 00:22:56,221
The result will be
I'd obliterate x.

393
00:22:56,221 --> 00:22:57,220
It's no longer a vertex.

394
00:22:57,220 --> 00:22:57,845
I obliterate y.

395
00:22:57,845 --> 00:22:58,870
It's no longer a vertex.

396
00:22:58,870 --> 00:23:01,020
But I add this new vertex, z.

397
00:23:04,240 --> 00:23:06,670
In that way, I remove
two vertices, add one,

398
00:23:06,670 --> 00:23:08,190
which decreases n.

399
00:23:08,190 --> 00:23:10,210
And then I can use induction.

400
00:23:10,210 --> 00:23:13,350
Now, there's lots of things
to say why this works.

401
00:23:13,350 --> 00:23:15,140
But this is the crazy idea.

402
00:23:15,140 --> 00:23:18,464
It's the Alexandrov
induction, and it's useful.

403
00:23:18,464 --> 00:23:19,880
If you want to
prove, for example,

404
00:23:19,880 --> 00:23:21,800
that the star unfolding
doesn't overlap,

405
00:23:21,800 --> 00:23:23,035
you use this induction also.

406
00:23:23,035 --> 00:23:24,441
It's the only
proof that's known.

407
00:23:27,590 --> 00:23:31,540
So why did we need
these properties?

408
00:23:31,540 --> 00:23:34,176
Well, the biggest
triangle you could make,

409
00:23:34,176 --> 00:23:35,550
something like
this, like we were

410
00:23:35,550 --> 00:23:37,800
doing in witch's hats,
where this is almost 90.

411
00:23:37,800 --> 00:23:39,470
This is almost 90.

412
00:23:39,470 --> 00:23:41,430
This is x and y.

413
00:23:44,670 --> 00:23:47,190
The biggest we can make
alpha over 2 is almost 90.

414
00:23:47,190 --> 00:23:49,900
And the biggest we can make
beta over 2 is almost 90.

415
00:23:49,900 --> 00:23:53,440
So we'd really like alpha
and beta to be less than 180.

416
00:23:53,440 --> 00:23:55,660
But then there is a valid
triangle that does this.

417
00:23:55,660 --> 00:24:00,130
Once you fix two angles, it of
course determines third one.

418
00:24:00,130 --> 00:24:02,070
It might be very close to 0.

419
00:24:02,070 --> 00:24:05,260
So z might have very large
curvature, but that's OK.

420
00:24:08,040 --> 00:24:10,580
What else?

421
00:24:10,580 --> 00:24:13,480
The other thing is, in what
sense could you induct?

422
00:24:13,480 --> 00:24:15,700
And this is where I'm going
to be a little hand wavy.

423
00:24:15,700 --> 00:24:19,230
Yeah, I've found some
other polyhedral metric.

424
00:24:19,230 --> 00:24:21,880
It has one fewer vertex.

425
00:24:21,880 --> 00:24:24,960
So by induction I know that
it's realized by a unique convex

426
00:24:24,960 --> 00:24:26,040
polyhedron.

427
00:24:26,040 --> 00:24:29,000
The how in the world
would I convert that

428
00:24:29,000 --> 00:24:32,860
into a realization of
the original polyhedron?

429
00:24:32,860 --> 00:24:37,260
And the answer is
continuity and magic.

430
00:24:37,260 --> 00:24:39,819
The continuity is easy to see.

431
00:24:39,819 --> 00:24:41,360
When I glue in this
triangle, I don't

432
00:24:41,360 --> 00:24:43,160
have to just glue
in the triangle

433
00:24:43,160 --> 00:24:44,810
with those final angles.

434
00:24:44,810 --> 00:24:47,970
I could actually glue in a
very tiny triangle initially.

435
00:24:47,970 --> 00:24:50,616
And I could change these
angles continuously.

436
00:24:50,616 --> 00:24:52,240
That makes me feel
a little bit better.

437
00:24:52,240 --> 00:24:53,650
I start with something
that's very, very

438
00:24:53,650 --> 00:24:54,608
close to my polyhedron.

439
00:24:54,608 --> 00:24:56,220
Obviously, If I
could realize that,

440
00:24:56,220 --> 00:24:58,167
I could still realize
the original, right?

441
00:24:58,167 --> 00:25:00,416
It's almost the same, if I
glue in a tiny, tiny sliver

442
00:25:00,416 --> 00:25:01,850
of a triangle.

443
00:25:01,850 --> 00:25:04,030
That's not obvious,
but it's true.

444
00:25:04,030 --> 00:25:07,490
And you vary it until the
point, where you actually

445
00:25:07,490 --> 00:25:09,070
get something with
one fewer vertex.

446
00:25:09,070 --> 00:25:10,695
Then you actually do
get a realization.

447
00:25:10,695 --> 00:25:15,070
And then you continuously
morph it back to your original.

448
00:25:15,070 --> 00:25:16,804
That's the idea.

449
00:25:16,804 --> 00:25:18,220
I'm not going to
detail that idea.

450
00:25:18,220 --> 00:25:23,510
It uses something called the
inverse function theorem, which

451
00:25:23,510 --> 00:25:28,250
is a cool theorem
in real analysis.

452
00:25:28,250 --> 00:25:30,690
But I think most of you
haven't had real analysis.

453
00:25:30,690 --> 00:25:33,860
So you can read about
it if you're interested.

454
00:25:33,860 --> 00:25:35,420
It's cool.

455
00:25:35,420 --> 00:25:37,350
The main consequence
for us algorithms

456
00:25:37,350 --> 00:25:40,140
people is that it's
non constructive.

457
00:25:40,140 --> 00:25:41,984
It doesn't actually
tell you how to do this.

458
00:25:41,984 --> 00:25:43,400
It just says, well,
by continuity,

459
00:25:43,400 --> 00:25:46,084
it really ought to
work out somehow.

460
00:25:46,084 --> 00:25:47,000
But we don't know how.

461
00:25:47,000 --> 00:25:51,110
We've tried to make it
work out and failed.

462
00:25:51,110 --> 00:25:54,791
While this does sort of prove
the theorem by continuity,

463
00:25:54,791 --> 00:25:56,290
it doesn't actually
give us any idea

464
00:25:56,290 --> 00:25:59,610
of how to find what
polyhedron we are realizing,

465
00:25:59,610 --> 00:26:02,310
which is a bit frustrating.

466
00:26:02,310 --> 00:26:05,780
And that was the state of
the art until four years ago.

467
00:26:14,850 --> 00:26:21,890
Four years ago, these guys,
Bobenko and Izmestiev,

468
00:26:21,890 --> 00:26:25,200
came up with a
constructive version

469
00:26:25,200 --> 00:26:26,360
of Alexandrov's Theorem.

470
00:26:30,020 --> 00:26:33,860
And I just want to sketch
an idea of how this works.

471
00:26:33,860 --> 00:26:41,410
But the end consequence, which
is done in this class, I think,

472
00:26:41,410 --> 00:26:43,650
as a class project
three years ago,

473
00:26:43,650 --> 00:26:46,180
is that you get a pseudo
polynomial algorithm

474
00:26:46,180 --> 00:26:49,160
for solving this problem.

475
00:26:49,160 --> 00:26:51,020
So I'll jump ahead a little bit.

476
00:26:55,030 --> 00:26:57,340
Running time, we
get-- this, again,

477
00:26:57,340 --> 00:27:05,995
for amusement value-- 1891.

478
00:27:05,995 --> 00:27:06,960
It's a good year.

479
00:27:16,290 --> 00:27:20,330
I think I said, last time, that
n to the 125 times r to the 41

480
00:27:20,330 --> 00:27:21,060
was my record.

481
00:27:21,060 --> 00:27:23,460
Well, it's been surpassed.

482
00:27:23,460 --> 00:27:24,900
I just forgot about it.

483
00:27:24,900 --> 00:27:29,850
n to the 456.5 times r to the
1891, over epsilon epsilon.

484
00:27:29,850 --> 00:27:33,650
Here is the desired error.

485
00:27:33,650 --> 00:27:35,740
So you can make the error
as small as you want.

486
00:27:35,740 --> 00:27:37,660
We can't do it perfectly.

487
00:27:37,660 --> 00:27:41,470
But your running time increases
as the error goes down.

488
00:27:41,470 --> 00:27:44,230
This is our usual feature size.

489
00:27:44,230 --> 00:27:51,390
It's the longest length in
your input metric, let's say,

490
00:27:51,390 --> 00:27:54,260
divided by the shortest length.

491
00:27:54,260 --> 00:27:57,010
So that's a measure of how
tight the features are.

492
00:27:57,010 --> 00:28:00,230
We had that before with
carpenter's rule stuff.

493
00:28:00,230 --> 00:28:04,170
And n is the number vertices.

494
00:28:04,170 --> 00:28:06,764
Obviously, this is
a very bad bound.

495
00:28:06,764 --> 00:28:08,430
But the point is it's
pseudo polynomial.

496
00:28:08,430 --> 00:28:10,310
It's a bad bound
for the same reason

497
00:28:10,310 --> 00:28:12,640
that our last giant ginormous
bound was ginormous.

498
00:28:12,640 --> 00:28:16,300
Because differential flow is
hard to get good bounds on.

499
00:28:16,300 --> 00:28:19,264
See it's possible to
get some kind of bound.

500
00:28:19,264 --> 00:28:21,930
But they don't usually represent
how fast the algorithm actually

501
00:28:21,930 --> 00:28:22,430
is.

502
00:28:22,430 --> 00:28:24,764
Again, this algorithm
is pretty practical.

503
00:28:24,764 --> 00:28:25,930
There's an implementational.

504
00:28:25,930 --> 00:28:27,690
I'll show it to you in a moment.

505
00:28:27,690 --> 00:28:29,265
Maybe I'll show it
to you right now.

506
00:28:29,265 --> 00:28:32,530
It'll be more fun.

507
00:28:32,530 --> 00:28:35,630
It is available on
the web for free.

508
00:28:35,630 --> 00:28:37,510
Try it out.

509
00:28:37,510 --> 00:28:41,731
I assume this is a student
of Bobenko and Izmestiev

510
00:28:41,731 --> 00:28:47,400
in Berlin, and he implemented
this program, Java.

511
00:28:47,400 --> 00:28:50,500
So you specify a
polyhedral metric.

512
00:28:50,500 --> 00:28:52,326
Now, that's the tricky part.

513
00:28:52,326 --> 00:28:53,700
And especially,
it's tricky here,

514
00:28:53,700 --> 00:28:56,230
because it has to
be triangulated.

515
00:28:56,230 --> 00:28:58,930
But we would normally
do it by specifying

516
00:28:58,930 --> 00:29:00,662
a polygon plus a gluing.

517
00:29:00,662 --> 00:29:02,620
But of course, by computing
all shortest paths,

518
00:29:02,620 --> 00:29:05,570
you can eventually make that
a nice triangulated thing.

519
00:29:05,570 --> 00:29:09,250
So here I have a very simple
triangulated thing, namely,

520
00:29:09,250 --> 00:29:11,660
a complete graph on
four vertices, which

521
00:29:11,660 --> 00:29:13,280
makes a regular tetrahedron.

522
00:29:13,280 --> 00:29:15,580
It doesn't look so
regular, because I

523
00:29:15,580 --> 00:29:19,230
have a lot of perspective in
a small, narrow angle of view.

524
00:29:19,230 --> 00:29:22,840
But that is a regular
tetrahedron, believe me.

525
00:29:22,840 --> 00:29:24,460
Then I can change
the edge lengths.

526
00:29:24,460 --> 00:29:27,070
Right now, they're
all one, so I should

527
00:29:27,070 --> 00:29:33,470
be able to make one of
the edges longer, like 2.

528
00:29:33,470 --> 00:29:34,370
I haven't tried this.

529
00:29:34,370 --> 00:29:35,328
We'll see what happens.

530
00:29:35,328 --> 00:29:41,400
I say, Compute, and it says,
Starting Calculations--

531
00:29:41,400 --> 00:29:45,880
not my intended answer.

532
00:29:45,880 --> 00:29:47,380
I like it to finish
the calculation.

533
00:29:49,874 --> 00:29:51,665
That's why it prepared
some other examples.

534
00:29:55,210 --> 00:29:56,950
I suspect it's not
the program failing,

535
00:29:56,950 --> 00:30:00,305
but I've had some trouble
running it on this computer.

536
00:30:07,210 --> 00:30:09,160
I think we have demo effect.

537
00:30:09,160 --> 00:30:10,990
Finally, something
failed in this class.

538
00:30:10,990 --> 00:30:12,892
It took a long time.

539
00:30:12,892 --> 00:30:14,100
You'll have to try it online.

540
00:30:14,100 --> 00:30:17,140
You can change the edge lengths
to all sorts of crazy things.

541
00:30:17,140 --> 00:30:19,360
If you have very close
to degenerate situations,

542
00:30:19,360 --> 00:30:20,744
it does tend to fail.

543
00:30:20,744 --> 00:30:22,160
Because there's
some tricky issues

544
00:30:22,160 --> 00:30:25,374
to deal with and
this floating point.

545
00:30:25,374 --> 00:30:26,790
Oops, it might be
doing something.

546
00:30:32,820 --> 00:30:35,190
Dare I try to open a file?

547
00:30:38,760 --> 00:30:41,030
Here, you get to see my
organizational structure

548
00:30:41,030 --> 00:30:42,710
or lack thereof.

549
00:30:42,710 --> 00:30:52,070
Courses, Slides, 15,
and let's take Cube.

550
00:30:52,070 --> 00:30:53,395
Exciting.

551
00:30:53,395 --> 00:30:54,020
This is a cube.

552
00:30:54,020 --> 00:30:57,130
It's not so obvious
that it is a cube.

553
00:30:57,130 --> 00:30:59,050
But you set the edge links.

554
00:30:59,050 --> 00:31:02,910
And a lot of these
are sort of diagonals.

555
00:31:02,910 --> 00:31:06,220
And if I do it right,
maybe add some iterations.

556
00:31:08,564 --> 00:31:10,480
It says, [INAUDIBLE]
successfully constructed.

557
00:31:10,480 --> 00:31:13,540
You can see it, right?

558
00:31:13,540 --> 00:31:15,852
My job is failing, I'm afraid.

559
00:31:15,852 --> 00:31:16,560
Sorry about that.

560
00:31:20,479 --> 00:31:20,979
Go away.

561
00:31:23,850 --> 00:31:25,100
You'll have to try it at home.

562
00:31:25,100 --> 00:31:25,540
AUDIENCE: It was there.

563
00:31:25,540 --> 00:31:26,240
PROFESSOR: It was there?

564
00:31:26,240 --> 00:31:26,740
Oh.

565
00:31:26,740 --> 00:31:29,730
AUDIENCE: [LAUGHTER]

566
00:31:29,730 --> 00:31:32,940
PROFESSOR: Wait, I
don't see a cube.

567
00:31:32,940 --> 00:31:36,190
What did you see?

568
00:31:36,190 --> 00:31:37,170
AUDIENCE: It was there.

569
00:31:37,170 --> 00:31:38,169
PROFESSOR: It was there.

570
00:31:38,169 --> 00:31:40,040
All right, you saw it.

571
00:31:40,040 --> 00:31:40,950
It's like it ghost.

572
00:31:40,950 --> 00:31:43,530
It's the left over
from Halloween.

573
00:31:43,530 --> 00:31:45,620
Let me tell you a little
bit about how this works.

574
00:31:45,620 --> 00:31:47,570
And you'll have to try
the program at home.

575
00:31:54,030 --> 00:31:56,500
A crazy idea is we have
this polyhedron, which

576
00:31:56,500 --> 00:31:58,700
we want to find.

577
00:31:58,700 --> 00:31:59,960
It's a surface.

578
00:31:59,960 --> 00:32:03,080
It's hard to think about
surfaces that live in 3D.

579
00:32:03,080 --> 00:32:06,530
So I want to make it
more three dimensional.

580
00:32:06,530 --> 00:32:07,970
And the way I
going to do that is

581
00:32:07,970 --> 00:32:10,720
by thinking about the
inside of the polyhedron.

582
00:32:10,720 --> 00:32:13,027
And say, well,
imagine it's convex.

583
00:32:13,027 --> 00:32:14,860
So there's some point
inside the polyhedron.

584
00:32:14,860 --> 00:32:16,290
Start with one point.

585
00:32:16,290 --> 00:32:18,930
And then, I'd like to fill
the interior in between.

586
00:32:18,930 --> 00:32:22,137
So the surface, as you saw--
we're assuming that everything

587
00:32:22,137 --> 00:32:23,220
is initially triangulated.

588
00:32:23,220 --> 00:32:25,832
We can do that by
adding shortest paths.

589
00:32:25,832 --> 00:32:27,540
Well, if this triangle
is on the surface,

590
00:32:27,540 --> 00:32:29,960
and this point is in the
center, the natural thing to do

591
00:32:29,960 --> 00:32:33,860
is to make tetrahedra inside.

592
00:32:37,180 --> 00:32:39,760
If I could find
the 3D triangles,

593
00:32:39,760 --> 00:32:41,930
I could also find
the 3D tetrahedra.

594
00:32:41,930 --> 00:32:43,550
So we're going to
do a harder job

595
00:32:43,550 --> 00:32:44,777
of finding the 3D tetrahedra.

596
00:32:44,777 --> 00:32:46,610
From that, we will
derive what the triangles

597
00:32:46,610 --> 00:32:48,820
are-- easy enough idea.

598
00:32:48,820 --> 00:32:50,150
Make your problem harder.

599
00:32:50,150 --> 00:32:51,500
That often makes things easier.

600
00:32:55,690 --> 00:32:56,850
How do we do that?

601
00:32:56,850 --> 00:32:59,440
Well, for every
vertex up here-- call

602
00:32:59,440 --> 00:33:02,000
it vi-- there's some distance.

603
00:33:02,000 --> 00:33:04,740
This dashed line-- we're
going to call that ri.

604
00:33:04,740 --> 00:33:06,850
That's really what we'd
like to figure out.

605
00:33:06,850 --> 00:33:10,530
We already know the triangles--
we know their geometry.

606
00:33:10,530 --> 00:33:12,870
That's given to us as
part of the metric.

607
00:33:12,870 --> 00:33:14,662
We don't know how the
triangles are hinged,

608
00:33:14,662 --> 00:33:15,786
with respect to each other.

609
00:33:15,786 --> 00:33:17,040
That's the dihedral angles.

610
00:33:17,040 --> 00:33:18,320
That's the tricky part.

611
00:33:18,320 --> 00:33:20,361
But we're not going to
worry about that directly.

612
00:33:20,361 --> 00:33:23,530
We're going to think about
these edge lengths, ri.

613
00:33:23,530 --> 00:33:27,570
Because if I knew
all the ri lengths--

614
00:33:27,570 --> 00:33:30,971
they're the sort of radii
from the center point-- then

615
00:33:30,971 --> 00:33:32,970
I'd know all the edge
lengths of the tetrahedron

616
00:33:32,970 --> 00:33:34,689
that uniquely determined
the tetrahedron.

617
00:33:34,689 --> 00:33:36,230
If I do that for
all the tetrahedron,

618
00:33:36,230 --> 00:33:37,820
there would be a unique
way to glue them together.

619
00:33:37,820 --> 00:33:39,050
And I would get the answer.

620
00:33:39,050 --> 00:33:40,500
So I really just
need these ri's.

621
00:33:43,110 --> 00:33:44,660
What I'd like to
do is characterize,

622
00:33:44,660 --> 00:33:48,130
when are the ri's valid?

623
00:33:48,130 --> 00:33:53,230
And there's a few conditions
to make them reasonable.

624
00:33:53,230 --> 00:33:56,610
The first thing, I guess I
didn't even write in the notes.

625
00:33:56,610 --> 00:33:59,550
You might call it
sanity, which is

626
00:33:59,550 --> 00:34:05,145
that these are
real 3D tetrahedra.

627
00:34:08,139 --> 00:34:10,560
I'm not going to try to
characterize when that's true.

628
00:34:10,560 --> 00:34:13,630
It's some function on the ri
lengths and existing lengths.

629
00:34:13,630 --> 00:34:16,260
But I always want these to
correspond actual tetrahedra.

630
00:34:16,260 --> 00:34:18,310
The trouble is, they might
not fit together right

631
00:34:18,310 --> 00:34:19,710
to make 3D polyhedron.

632
00:34:19,710 --> 00:34:22,630
So I won't really have to
worry about this condition.

633
00:34:22,630 --> 00:34:28,254
The next condition-- what
do I call it-- is convexity.

634
00:34:32,830 --> 00:34:34,550
This will also be kind of easy.

635
00:34:34,550 --> 00:34:38,770
What I want is that the
outside surface is convex.

636
00:34:38,770 --> 00:34:42,620
Now, if you look at the outside
surface, what we really want

637
00:34:42,620 --> 00:34:44,810
is that one of these edges,
on the outside surface,

638
00:34:44,810 --> 00:34:47,170
bends the correct way.

639
00:34:47,170 --> 00:34:48,337
So when will that be true?

640
00:34:48,337 --> 00:34:50,170
Well, I'm going to look
at the two triangles

641
00:34:50,170 --> 00:34:51,550
next to that edge.

642
00:34:51,550 --> 00:34:58,874
That defines two tetrahedra
that share a face.

643
00:34:58,874 --> 00:35:00,290
And if I knew what
the ri's where,

644
00:35:00,290 --> 00:35:02,020
I'd know what those
tetrahedra are.

645
00:35:02,020 --> 00:35:05,290
And what I really need is
that if I look at the angle--

646
00:35:05,290 --> 00:35:07,290
I'm going to draw this
a little bit thicker--

647
00:35:07,290 --> 00:35:12,160
this dihedral angle
between-- it's hard to draw.

648
00:35:12,160 --> 00:35:13,840
If you look at
this edge, there's

649
00:35:13,840 --> 00:35:15,690
two tetrahedra glued there.

650
00:35:15,690 --> 00:35:18,210
So there's some
solid dihedral angle

651
00:35:18,210 --> 00:35:20,760
of tetrahedron glued at that
edge from this tetrahedron.

652
00:35:20,760 --> 00:35:22,620
And there's some
angle over here that's

653
00:35:22,620 --> 00:35:24,650
glued to this edge
from that tetrahedron.

654
00:35:24,650 --> 00:35:29,750
I want the sum of those
two dihedral angles,

655
00:35:29,750 --> 00:35:35,100
the total material there, to
be less than or equal to 180.

656
00:35:35,100 --> 00:35:37,170
That is, the sum of
those angles is really

657
00:35:37,170 --> 00:35:39,180
the dihedral angle
along this edge.

658
00:35:39,180 --> 00:35:41,870
If it's less than
180, it'll be convex.

659
00:35:41,870 --> 00:35:45,940
It's hard to draw,
but very intuitive.

660
00:35:45,940 --> 00:35:48,860
This is also pretty easy to get.

661
00:35:48,860 --> 00:35:50,780
You can set the ri
so that that's true.

662
00:35:50,780 --> 00:35:54,700
The hard part is
looking at a vertex.

663
00:35:54,700 --> 00:35:55,970
You think about a vertex.

664
00:35:55,970 --> 00:35:57,940
It has a whole bunch
of triangles around it.

665
00:36:01,080 --> 00:36:06,650
From this vertex to the central
point, which we usually call p,

666
00:36:06,650 --> 00:36:10,350
if you look at that edge-- I'm
not going to try to draw it--

667
00:36:10,350 --> 00:36:12,360
there's, in this
case, five tetrahedra

668
00:36:12,360 --> 00:36:15,530
glued around that edge.

669
00:36:15,530 --> 00:36:19,300
The total dihedral angle
of those five tetrahedra

670
00:36:19,300 --> 00:36:21,620
ought to be exactly 360.

671
00:36:21,620 --> 00:36:23,870
If these tetrahedra are going
to fit together and make

672
00:36:23,870 --> 00:36:29,545
a solid thing-- and this we
call the reality constraint.

673
00:36:32,050 --> 00:36:34,170
I'm not usually bound by
the reality constraint.

674
00:36:34,170 --> 00:36:36,040
In fact, we will
relax it a little bit.

675
00:36:36,040 --> 00:36:42,850
But we would like that the
sum of the dihedral angles

676
00:36:42,850 --> 00:36:52,110
around some edge from
P to vi equals 360.

677
00:36:52,110 --> 00:36:54,270
This is the hard
constraint to get.

678
00:36:54,270 --> 00:36:56,340
Reality is tough.

679
00:36:56,340 --> 00:36:57,740
Sanity, convexity, easy.

680
00:37:02,240 --> 00:37:06,740
I guess I shouldn't
kill this yet.

681
00:37:06,740 --> 00:37:10,290
So let me tell you a little
bit more about how we get this.

682
00:37:36,210 --> 00:37:38,380
I'll just tell you
some of this stuff.

683
00:37:38,380 --> 00:37:41,330
The first thing we do is
get sanity and convexity

684
00:37:41,330 --> 00:37:43,340
and forget about reality.

685
00:37:43,340 --> 00:37:44,215
Reality is overrated.

686
00:37:46,870 --> 00:37:48,600
To do that, we actually
need two things.

687
00:37:48,600 --> 00:37:51,220
One, is we need a
triangulation of the surface.

688
00:37:51,220 --> 00:37:52,970
We're presumably given
some triangulation,

689
00:37:52,970 --> 00:37:56,000
but that one might
be hard to work with.

690
00:37:56,000 --> 00:37:59,790
And the answer is,
to start out, it's

691
00:37:59,790 --> 00:38:03,630
nice to use Delaunay
triangulation.

692
00:38:03,630 --> 00:38:06,330
How many people know about
the Delaunay triangulations?

693
00:38:06,330 --> 00:38:09,440
They're nice triangulations.

694
00:38:09,440 --> 00:38:11,340
And you can define
Delaunay triangulation

695
00:38:11,340 --> 00:38:13,390
in an implicit
surface like this.

696
00:38:13,390 --> 00:38:15,510
I won't do that here.

697
00:38:15,510 --> 00:38:17,980
But it can be done,
and it can be computed.

698
00:38:17,980 --> 00:38:20,380
One of the innovations
in this paper

699
00:38:20,380 --> 00:38:22,120
was to actually show
how to compute it

700
00:38:22,120 --> 00:38:23,680
in pseudo polynomial time.

701
00:38:23,680 --> 00:38:26,580
It's not too hard.

702
00:38:26,580 --> 00:38:28,960
It's just shortest paths.

703
00:38:28,960 --> 00:38:32,140
Then, it turns out, if you
start with that triangulation,

704
00:38:32,140 --> 00:38:35,020
and you let all the ri's be
really, really big, close

705
00:38:35,020 --> 00:38:39,440
to infinity, then sanity
and convexity hold.

706
00:38:39,440 --> 00:38:41,859
So I guess intuition may
be like witch's hats.

707
00:38:41,859 --> 00:38:42,400
I don't know.

708
00:38:42,400 --> 00:38:45,690
You're making these super-sharp
triangles, tetrahedra.

709
00:38:45,690 --> 00:38:46,450
And it will work.

710
00:38:46,450 --> 00:38:50,480
Now, it'll make
everything big and equal.

711
00:38:50,480 --> 00:38:54,210
And so reality will be violated.

712
00:38:54,210 --> 00:38:59,330
But sanity and convexity
are OK, and that

713
00:38:59,330 --> 00:39:01,830
will be your starting point.

714
00:39:01,830 --> 00:39:04,460
So now the crazy
idea is to fix it.

715
00:39:11,920 --> 00:39:14,720
I'm going to define
a kind of curvature

716
00:39:14,720 --> 00:39:25,430
around each edge, around that
center line from p to vi.

717
00:39:25,430 --> 00:39:28,690
That's the sum that we
wanted to be equal to 360.

718
00:39:28,690 --> 00:39:30,570
And now I'm just
taking 360 minus that.

719
00:39:30,570 --> 00:39:31,470
That's the deficit.

720
00:39:31,470 --> 00:39:33,810
This is how much we're off by.

721
00:39:33,810 --> 00:39:35,330
So we have some starting point.

722
00:39:35,330 --> 00:39:40,250
And it turns out the Ki
will be positive, I believe.

723
00:39:40,250 --> 00:39:52,175
And we want to make Ki
evolve with time, so that it

724
00:39:52,175 --> 00:39:55,170
is 1 minus t, times
the initial Ki.

725
00:39:57,750 --> 00:40:00,000
So we start at times
0, with the solution

726
00:40:00,000 --> 00:40:01,439
where all the ri's are large.

727
00:40:01,439 --> 00:40:02,980
So we're staying in
convex, and we're

728
00:40:02,980 --> 00:40:05,510
going to be staying in
convex from then on.

729
00:40:05,510 --> 00:40:09,350
And then, if we could do this,
we would start at t equals 0.

730
00:40:09,350 --> 00:40:12,520
This is 1 times the original
curvature at every edge,

731
00:40:12,520 --> 00:40:14,140
so it's a problem.

732
00:40:14,140 --> 00:40:17,360
But at t equals 1, at time
1, when this is finished,

733
00:40:17,360 --> 00:40:20,140
we will get 0 times
whatever we had.

734
00:40:20,140 --> 00:40:22,310
So all of these numbers
will magically go to 0.

735
00:40:22,310 --> 00:40:25,160
And then it adds up to 360.

736
00:40:25,160 --> 00:40:28,070
It's not obvious that
this works, but it does.

737
00:40:28,070 --> 00:40:29,610
It defines a differential flow.

738
00:40:29,610 --> 00:40:34,670
And those who know
differential equations,

739
00:40:34,670 --> 00:40:38,127
this will look pretty.

740
00:40:38,127 --> 00:40:39,710
This is partial
differential equations

741
00:40:39,710 --> 00:40:41,890
to make it a little
more exciting.

742
00:40:41,890 --> 00:40:49,560
This is a kappa and
inverse and phi.

743
00:40:49,560 --> 00:40:51,270
What was phi?

744
00:40:51,270 --> 00:40:55,690
I'm forgetting my own
notation, some initial value.

745
00:40:58,610 --> 00:41:00,806
It should probably be r.

746
00:41:00,806 --> 00:41:01,306
r?

747
00:41:01,306 --> 00:41:04,590
We'll see.

748
00:41:04,590 --> 00:41:08,500
All this is saying
is what I really

749
00:41:08,500 --> 00:41:11,007
want is to evolve
kappas to go smaller.

750
00:41:11,007 --> 00:41:13,465
I don't know how to do that,
because all I can do is change

751
00:41:13,465 --> 00:41:14,147
the ri's.

752
00:41:14,147 --> 00:41:15,480
That's the only thing I control.

753
00:41:15,480 --> 00:41:18,220
The kappas are computed
as a function of the ri's.

754
00:41:18,220 --> 00:41:21,240
There's a standard trick in
analysis-- or I guess, whatever

755
00:41:21,240 --> 00:41:27,300
that is, calculus--
to rewrite things.

756
00:41:27,300 --> 00:41:28,680
You get to control the ri's.

757
00:41:28,680 --> 00:41:31,950
You want to control the kappas.

758
00:41:31,950 --> 00:41:36,500
So you use this thing, which
is called the Jacobian.

759
00:41:39,140 --> 00:41:40,650
It's a matrix.

760
00:41:40,650 --> 00:41:43,600
But really, it says,
if I change ri,

761
00:41:43,600 --> 00:41:45,230
how does that change kappa j?

762
00:41:45,230 --> 00:41:46,732
It says that for all i and j.

763
00:41:46,732 --> 00:41:48,690
So it's just a whole
bunch of different values.

764
00:41:48,690 --> 00:41:50,550
They're really functions.

765
00:41:50,550 --> 00:41:51,980
You invert that
because you really

766
00:41:51,980 --> 00:41:55,640
want to know how--
I want to change

767
00:41:55,640 --> 00:41:58,350
the ri's in order to
affect the kappas.

768
00:41:58,350 --> 00:42:01,590
So you write this
differential equation.

769
00:42:01,590 --> 00:42:07,590
And you just follow your nose,
and it tells you what to do.

770
00:42:07,590 --> 00:42:10,370
I'm going to slide a whole
bunch of details under the rug.

771
00:42:10,370 --> 00:42:13,520
The hard part is proving that
this relation is actually

772
00:42:13,520 --> 00:42:15,810
feasible, that you can
change the ri's in order

773
00:42:15,810 --> 00:42:18,190
to change the kappas,
no matter how you want.

774
00:42:18,190 --> 00:42:22,010
This is equivalent to saying
that the Jacobian is never 0,

775
00:42:22,010 --> 00:42:24,709
and it always has an inverse.

776
00:42:24,709 --> 00:42:25,750
And that's hard to prove.

777
00:42:25,750 --> 00:42:28,570
And in fact, they prove it using
the Inverse Function Theorem,

778
00:42:28,570 --> 00:42:30,490
which is what we
were trying to avoid.

779
00:42:30,490 --> 00:42:33,610
But hey, it's a
construction of sorts,

780
00:42:33,610 --> 00:42:35,790
and it leads to
that implementation.

781
00:42:35,790 --> 00:42:38,990
This, in some sense,
gives you an algorithm.

782
00:42:38,990 --> 00:42:41,040
To turn it into an
actual algorithm--

783
00:42:41,040 --> 00:42:47,210
as we did in this paper, which
appeared last year-- you need

784
00:42:47,210 --> 00:42:49,800
to make this proof a little
bit more constructive,

785
00:42:49,800 --> 00:42:52,700
a little bit more
algorithmic, let's say.

786
00:42:52,700 --> 00:42:55,342
You don't just want to prove
that this Jacobian is never

787
00:42:55,342 --> 00:42:57,050
zero 0 and that it
always has an inverse.

788
00:42:57,050 --> 00:42:58,570
You need to prove
that it's actually

789
00:42:58,570 --> 00:43:00,300
always at least some epsilon.

790
00:43:00,300 --> 00:43:02,960
It's bounded away from 0,
and same for the inverse.

791
00:43:05,812 --> 00:43:07,520
Instead of using
inverse functions there,

792
00:43:07,520 --> 00:43:08,850
and we're totally
constructive, we just

793
00:43:08,850 --> 00:43:10,730
look at second derivatives
and bound those.

794
00:43:10,730 --> 00:43:14,290
We get a really awful
bound, which is this one.

795
00:43:14,290 --> 00:43:17,830
But we do get an explicit
bound on how long, essentially,

796
00:43:17,830 --> 00:43:23,810
the algorithm is implemented
and the applet takes.

797
00:43:23,810 --> 00:43:26,420
So, a high-level sketch,
but it gives you an idea.

798
00:43:26,420 --> 00:43:30,961
This is pretty new stuff.

799
00:43:30,961 --> 00:43:31,460
Questions?

800
00:43:34,690 --> 00:43:42,639
I want to move on to more
fun topics, let's technical,

801
00:43:42,639 --> 00:43:43,180
I should say.

802
00:43:46,270 --> 00:43:48,340
So that's Alexandrov's
Theorem algorithms.

803
00:43:48,340 --> 00:43:50,550
In theory, there's
now a computer program

804
00:43:50,550 --> 00:43:56,150
that will tell you
how to fold something,

805
00:43:56,150 --> 00:43:58,530
as long as we have
an Alexandrov gluing.

806
00:43:58,530 --> 00:43:59,710
So it has to be polyhedral.

807
00:43:59,710 --> 00:44:00,876
That's pretty much for free.

808
00:44:00,876 --> 00:44:01,724
It has to be convex.

809
00:44:01,724 --> 00:44:02,890
That's the interesting part.

810
00:44:02,890 --> 00:44:04,810
It has to be a
topological sphere that

811
00:44:04,810 --> 00:44:06,850
is just non-crossing.

812
00:44:06,850 --> 00:44:09,090
So let's think about gluings.

813
00:44:09,090 --> 00:44:11,160
We know that as long as
we find a good gluing,

814
00:44:11,160 --> 00:44:13,180
it corresponds to something.

815
00:44:13,180 --> 00:44:15,230
We know that something's
a little hard to find.

816
00:44:15,230 --> 00:44:17,435
But we'll see how we
find them in practice.

817
00:44:17,435 --> 00:44:18,170
It's kind of fun.

818
00:44:21,810 --> 00:44:24,300
We talked about the
decision problem.

819
00:44:24,300 --> 00:44:27,720
Is it possible at all
for a given polygon?

820
00:44:27,720 --> 00:44:31,212
So here's a polygon where
it's not possible at all.

821
00:44:31,212 --> 00:44:32,420
It's kind of fun-- ungluable.

822
00:44:38,740 --> 00:44:39,865
I should make that sharper.

823
00:44:43,970 --> 00:44:45,841
So this is a polygon.

824
00:44:45,841 --> 00:44:46,715
This is the material.

825
00:44:50,100 --> 00:44:52,220
I want to glue it
into a convex shape.

826
00:44:52,220 --> 00:44:55,240
I claim it's not possible.

827
00:44:55,240 --> 00:44:58,620
The reason is, think
about this vertex.

828
00:44:58,620 --> 00:45:02,518
What could get glued
to that vertex?

829
00:45:02,518 --> 00:45:04,031
It has to be less than 360.

830
00:45:04,031 --> 00:45:05,280
Right now, it's less than 360.

831
00:45:05,280 --> 00:45:07,640
It's that much material.

832
00:45:07,640 --> 00:45:11,410
Could I glue any other
vertex to that point?

833
00:45:11,410 --> 00:45:11,910
No.

834
00:45:11,910 --> 00:45:14,450
Because all the other vertices
have at least 90 degrees

835
00:45:14,450 --> 00:45:15,630
of material.

836
00:45:15,630 --> 00:45:17,500
This one has even more.

837
00:45:17,500 --> 00:45:18,750
But all them have at least 90.

838
00:45:18,750 --> 00:45:20,820
And there's less
than 90 remaining.

839
00:45:20,820 --> 00:45:22,680
So no other vertex
could be glued there.

840
00:45:22,680 --> 00:45:24,930
The other possibility, it
doesn't have to be a vertex.

841
00:45:24,930 --> 00:45:26,910
It could be a point in
the middle of an edge.

842
00:45:26,910 --> 00:45:28,330
But that has 180 degrees.

843
00:45:28,330 --> 00:45:31,610
That's way too much,
so nothing fits there.

844
00:45:31,610 --> 00:45:33,620
Therefore, I have
glue like this.

845
00:45:33,620 --> 00:45:36,830
This is what we call zipping.

846
00:45:36,830 --> 00:45:39,300
I zip those two
edges to each other.

847
00:45:39,300 --> 00:45:41,490
Now, they happen to be
exactly the same length.

848
00:45:41,490 --> 00:45:45,310
Therefore, this vertex
gets glued to that vertex.

849
00:45:45,310 --> 00:45:47,640
But that's not allowed,
because there's more than 180

850
00:45:47,640 --> 00:45:50,640
and more than 180
here-- contradiction.

851
00:45:50,640 --> 00:45:52,568
So that's it.

852
00:45:52,568 --> 00:45:56,345
This can actually happen a lot.

853
00:45:56,345 --> 00:45:59,580
Let me formalize that.

854
00:46:06,850 --> 00:46:09,515
So we could look at what you
might call the generic case.

855
00:46:13,520 --> 00:46:15,990
And I won't call it
the generic case.

856
00:46:15,990 --> 00:46:20,400
But there's a back reference
there, all the linkage stuff.

857
00:46:20,400 --> 00:46:22,480
I can explicitly think
about the random case.

858
00:46:22,480 --> 00:46:24,520
Suppose you choose all
the lengths of your edges

859
00:46:24,520 --> 00:46:27,890
and all you angles at
random in your polygon.

860
00:46:27,890 --> 00:46:29,760
You get some messy thing.

861
00:46:29,760 --> 00:46:36,580
But the point is,
around half the vertices

862
00:46:36,580 --> 00:46:42,320
will be reflex
angle, more than 180.

863
00:46:42,320 --> 00:46:43,900
And those are kind of troubled.

864
00:46:43,900 --> 00:46:47,850
Because if you have a
reflex vertex like this,

865
00:46:47,850 --> 00:46:51,150
in particular, you can't
glue the middle of an edge.

866
00:46:51,150 --> 00:46:53,590
Because that has 180.

867
00:46:53,590 --> 00:46:55,800
You can only blue
convex vertices

868
00:46:55,800 --> 00:46:59,400
that have some smaller
angle into there.

869
00:46:59,400 --> 00:47:04,510
Otherwise, use zip, and
zipping would be like this.

870
00:47:04,510 --> 00:47:08,490
So, what do I want to say?

871
00:47:08,490 --> 00:47:09,100
Yeah, I see.

872
00:47:09,100 --> 00:47:10,760
Right, cool.

873
00:47:10,760 --> 00:47:14,071
Before I get to zipping,
maybe I glue some convex angle

874
00:47:14,071 --> 00:47:14,570
in there.

875
00:47:14,570 --> 00:47:15,930
It's the same kind of analysis.

876
00:47:15,930 --> 00:47:17,221
Maybe I glue some convex angle.

877
00:47:17,221 --> 00:47:19,312
But if the angles
are chosen at random,

878
00:47:19,312 --> 00:47:21,520
I'm not going to get lucky
and completely flatten out

879
00:47:21,520 --> 00:47:22,019
the vertex.

880
00:47:22,019 --> 00:47:22,890
It won't disappear.

881
00:47:22,890 --> 00:47:24,860
That would be too
much to hope for.

882
00:47:24,860 --> 00:47:26,460
So most the time,
I'm going to glue

883
00:47:26,460 --> 00:47:28,290
in-- maybe I glue
in some material.

884
00:47:28,290 --> 00:47:29,436
Then I glue in some more.

885
00:47:29,436 --> 00:47:30,560
At some point, I get stuck.

886
00:47:30,560 --> 00:47:33,690
I can't glue anymore in, or
I decide I don't want to.

887
00:47:33,690 --> 00:47:35,950
At that point, I must zip.

888
00:47:35,950 --> 00:47:38,180
This is a rather
ambiguous notation.

889
00:47:38,180 --> 00:47:41,870
But zipping, I want to glue
this edge to that edge.

890
00:47:41,870 --> 00:47:44,370
Well, how long can
I zip them for?

891
00:47:47,660 --> 00:47:49,731
Well, let's say I zip them.

892
00:47:52,350 --> 00:47:54,430
At any moment here,
I'm gluing a 180 degree

893
00:47:54,430 --> 00:47:56,010
angle to another
180 degree angle.

894
00:47:56,010 --> 00:47:57,742
I can't stop zipping
once I start.

895
00:47:57,742 --> 00:47:59,950
Because there's nothing else
that could fit in there.

896
00:47:59,950 --> 00:48:05,210
So I've got to zip
until I get to a corner.

897
00:48:05,210 --> 00:48:08,000
There's going to be some
corner, some vertex, that's

898
00:48:08,000 --> 00:48:10,662
glued to some point.

899
00:48:10,662 --> 00:48:12,370
Now, is it going to
be glued to a vertex?

900
00:48:12,370 --> 00:48:13,710
Well, that would
be awfully lucky.

901
00:48:13,710 --> 00:48:14,920
That would mean
that these two edges

902
00:48:14,920 --> 00:48:16,128
have exactly the same length.

903
00:48:16,128 --> 00:48:18,480
That's not going
to happen randomly.

904
00:48:18,480 --> 00:48:21,150
One of these is going to
be larger than the other.

905
00:48:21,150 --> 00:48:25,620
Now I'm gluing some angle
to a 180 degree angle.

906
00:48:25,620 --> 00:48:28,770
The result is another
reflex vertex.

907
00:48:28,770 --> 00:48:30,730
This will keep going.

908
00:48:30,730 --> 00:48:33,890
And you'll never finish.

909
00:48:33,890 --> 00:48:35,410
So this is a bit hand wavy.

910
00:48:35,410 --> 00:48:39,000
But you can see the textbook
for more formal argument.

911
00:48:43,050 --> 00:48:43,690
I have it here.

912
00:48:43,690 --> 00:48:47,210
I'll finish it for you.

913
00:48:47,210 --> 00:48:49,920
This would be fine if
this vertex is convex.

914
00:48:49,920 --> 00:48:51,610
This is less than 180.

915
00:48:51,610 --> 00:48:54,596
But actually, with
probability 1/2,

916
00:48:54,596 --> 00:48:55,970
it's going to be
a reflex vertex.

917
00:48:55,970 --> 00:48:58,830
And then you're screwed, because
now you have more than 180

918
00:48:58,830 --> 00:48:59,790
gluing to 180.

919
00:48:59,790 --> 00:49:01,450
That's more than 360.

920
00:49:01,450 --> 00:49:03,850
So you don't even have to
go all the way to the end.

921
00:49:03,850 --> 00:49:06,270
Immediately, after
the very first zip,

922
00:49:06,270 --> 00:49:08,400
with probability
1/2, you'll fail.

923
00:49:08,400 --> 00:49:12,710
I you have n vertices, and
half of them are reflex,

924
00:49:12,710 --> 00:49:14,700
a quarter of them
are going to fail.

925
00:49:14,700 --> 00:49:18,108
So that's plenty, as long
as n is bigger than 4.

926
00:49:18,108 --> 00:49:21,160
For n large, you're going to get
exponentially high probability

927
00:49:21,160 --> 00:49:24,300
that this fails.

928
00:49:24,300 --> 00:49:27,170
Most of the time, if you're
choosing random polyhedra,

929
00:49:27,170 --> 00:49:30,100
random polygons,
you can't glue them.

930
00:49:35,110 --> 00:49:40,390
So you've got to be clever,
choose some nice polygons.

931
00:49:40,390 --> 00:49:41,890
It's actually not that hard.

932
00:49:41,890 --> 00:49:45,130
Like the cross, if you
just glued together

933
00:49:45,130 --> 00:49:47,750
a bunch of unit squares,
it'll go into something nice

934
00:49:47,750 --> 00:49:48,710
probably.

935
00:49:48,710 --> 00:49:51,680
Or here's another great, class.

936
00:49:51,680 --> 00:49:52,785
Convex polygons.

937
00:49:56,670 --> 00:50:01,660
Convex polygons do
always glue to something

938
00:50:01,660 --> 00:50:04,490
using something called
perimeter having.

939
00:50:10,170 --> 00:50:13,820
What to do, basically,
by having the perimeter.

940
00:50:13,820 --> 00:50:17,090
I have some convex polygon.

941
00:50:17,090 --> 00:50:20,200
I'm going to choose any
point on the boundary.

942
00:50:20,200 --> 00:50:21,880
And then have the
perimeter from there.

943
00:50:21,880 --> 00:50:30,560
So I walk equal distance,
all around this thing-- equal

944
00:50:30,560 --> 00:50:31,410
speed, I should say.

945
00:50:38,000 --> 00:50:38,940
Something like that.

946
00:50:45,890 --> 00:50:47,510
This is x.

947
00:50:47,510 --> 00:50:53,070
This is the antipodal point, in
terms of measuring perimeter.

948
00:50:53,070 --> 00:50:57,080
Now I zip.

949
00:50:57,080 --> 00:50:59,110
So I'm going to
glue this to this.

950
00:50:59,110 --> 00:51:00,880
I'm going to glue this to this.

951
00:51:00,880 --> 00:51:02,396
I drew all those marks.

952
00:51:02,396 --> 00:51:03,270
I was planning ahead.

953
00:51:03,270 --> 00:51:04,870
Can you tell?

954
00:51:04,870 --> 00:51:07,810
I was trying to make all the
segments the same length,

955
00:51:07,810 --> 00:51:10,252
so that I could properly glue.

956
00:51:10,252 --> 00:51:12,210
Something I've never
really said about gluings,

957
00:51:12,210 --> 00:51:16,400
but you should always glue same
length things to other things.

958
00:51:16,400 --> 00:51:16,900
Segments.

959
00:51:20,004 --> 00:51:20,670
OK, there we go.

960
00:51:20,670 --> 00:51:21,378
There's a gluing.

961
00:51:21,378 --> 00:51:23,920
There's no crossings
in that glowing.

962
00:51:23,920 --> 00:51:24,730
You can check.

963
00:51:24,730 --> 00:51:25,980
Its Alexandrov.

964
00:51:25,980 --> 00:51:30,260
It is convex, because
what do I glue to what?

965
00:51:30,260 --> 00:51:32,895
Well, here, maybe
I'm going this vertex

966
00:51:32,895 --> 00:51:34,020
to the middle of this edge.

967
00:51:34,020 --> 00:51:35,220
But it's a convex vertex.

968
00:51:35,220 --> 00:51:35,820
So that's OK.

969
00:51:38,075 --> 00:51:39,200
Everything looks like that.

970
00:51:39,200 --> 00:51:42,042
You're always gluing some
vertex to some other edge.

971
00:51:42,042 --> 00:51:43,375
And all the vertices are convex.

972
00:51:43,375 --> 00:51:45,670
So it will be less than 360.

973
00:51:45,670 --> 00:51:48,730
Maybe, if you got lucky, some
edge will glue to some edge.

974
00:51:48,730 --> 00:51:51,240
But that's still barely OK.

975
00:51:51,240 --> 00:51:53,850
So for convex polygons,
it's always going to work.

976
00:51:53,850 --> 00:51:56,650
Let's try it out, shall we?

977
00:51:56,650 --> 00:51:59,150
I don't want to use this page,
because I'm not done with it.

978
00:51:59,150 --> 00:52:02,470
We'll take page one,
make a convex polygon.

979
00:52:11,140 --> 00:52:13,790
I don't want to make
it too complicated.

980
00:52:13,790 --> 00:52:16,731
I'll spend a lot of time taping.

981
00:52:16,731 --> 00:52:18,980
You an try this at home with
your own, favorite convex

982
00:52:18,980 --> 00:52:19,479
polygon.

983
00:52:19,479 --> 00:52:21,360
Here's a convex polygon.

984
00:52:21,360 --> 00:52:23,590
And now, we're going
to pick a point.

985
00:52:28,430 --> 00:52:31,670
Do I have a good
point written here?

986
00:52:31,670 --> 00:52:34,750
Let's try here.

987
00:52:37,650 --> 00:52:41,090
See that point, purple?

988
00:52:41,090 --> 00:52:43,480
Now we are going to zip.

989
00:52:43,480 --> 00:52:46,800
So take this.

990
00:52:46,800 --> 00:52:49,110
I glue those edges together.

991
00:52:49,110 --> 00:52:51,636
I don't have glue, or
glue takes a while to dry.

992
00:52:51,636 --> 00:52:52,760
So we're going to use tape.

993
00:52:55,520 --> 00:52:56,020
Ta-da.

994
00:52:58,560 --> 00:53:01,310
Now, we have to
glue a little more.

995
00:53:01,310 --> 00:53:04,796
I'm going to pretend
those just met.

996
00:53:04,796 --> 00:53:08,130
Eh, it's a little ugly.

997
00:53:08,130 --> 00:53:09,680
Man, couldn't I have
chosen something

998
00:53:09,680 --> 00:53:10,721
a little more degenerate?

999
00:53:14,225 --> 00:53:16,100
Here, it really does
work in the random case,

1000
00:53:16,100 --> 00:53:18,605
which is fine, as long as
I was restricted to convex.

1001
00:53:21,636 --> 00:53:23,851
Now, we're going to
If you brought tape,

1002
00:53:23,851 --> 00:53:24,725
you can try yourself.

1003
00:53:28,898 --> 00:53:30,190
Tape a little more.

1004
00:53:34,190 --> 00:53:36,680
See, it's already making
a convex polyhedron.

1005
00:53:36,680 --> 00:53:39,780
It's like magic.

1006
00:53:39,780 --> 00:53:42,230
For a long time, we've
tried to turn this process

1007
00:53:42,230 --> 00:53:43,420
into an algorithm.

1008
00:53:43,420 --> 00:53:46,240
Because it looks
pretty good, right?

1009
00:53:46,240 --> 00:53:49,925
But we have not been
able to formalize this.

1010
00:53:49,925 --> 00:53:52,425
This differential flow does not
really follow our intuition.

1011
00:53:56,390 --> 00:53:58,390
Now, I'm going to glue
these are edges together.

1012
00:54:05,537 --> 00:54:07,120
This is when I wish
I had three hands.

1013
00:54:20,498 --> 00:54:21,310
A little more.

1014
00:54:40,720 --> 00:54:44,050
And there, we've computed
the antipodal point.

1015
00:54:44,050 --> 00:54:45,420
In theory, you would zip that.

1016
00:54:45,420 --> 00:54:47,410
But I'm going to leave
it open to give myself

1017
00:54:47,410 --> 00:54:49,570
some maneuverability here.

1018
00:54:49,570 --> 00:54:51,070
So you puff it out a little.

1019
00:54:51,070 --> 00:54:54,630
It will be kind of
round and convex.

1020
00:54:54,630 --> 00:54:56,090
Now, that's not
quite a polyhedron.

1021
00:54:56,090 --> 00:54:58,460
But this is a polyhedral metric.

1022
00:54:58,460 --> 00:55:00,570
And Alexandrov's
Theorem says this

1023
00:55:00,570 --> 00:55:02,870
should make a unique
convex polyhedron.

1024
00:55:02,870 --> 00:55:06,350
And with a little bit of
practice and thinking about it,

1025
00:55:06,350 --> 00:55:09,310
you can derive that actually
this must be an edge here.

1026
00:55:12,740 --> 00:55:15,660
It's more of an intuition thing.

1027
00:55:15,660 --> 00:55:17,890
See, it looks like this
really wants to bend there.

1028
00:55:17,890 --> 00:55:20,430
So that must be an edge.

1029
00:55:20,430 --> 00:55:23,430
This one looks like it
must be an edge here.

1030
00:55:23,430 --> 00:55:24,680
This gets a little hard to do.

1031
00:55:27,860 --> 00:55:30,220
But this is how we
used to do them.

1032
00:55:30,220 --> 00:55:34,020
Wow, this probably
still is how we do them.

1033
00:55:34,020 --> 00:55:37,557
I haven't done a lot
of these in a while.

1034
00:55:37,557 --> 00:55:39,015
It looks like
there's an edge here.

1035
00:55:41,725 --> 00:55:47,466
Oh boy, something like that.

1036
00:55:47,466 --> 00:55:51,320
Here there already is and
edge, kind of convenient.

1037
00:55:51,320 --> 00:55:52,320
Here there's an edge.

1038
00:55:57,450 --> 00:56:00,715
I'm just folding point to point.

1039
00:56:00,715 --> 00:56:02,750
I'm almost triangulated.

1040
00:56:02,750 --> 00:56:04,625
I've got a nice line there.

1041
00:56:09,784 --> 00:56:11,450
It's starting to look
like a polyhedron.

1042
00:56:11,450 --> 00:56:12,491
There's a couple missing.

1043
00:56:12,491 --> 00:56:15,620
This guy probably has
a diagonal somewhere.

1044
00:56:15,620 --> 00:56:17,560
I would guess here.

1045
00:56:17,560 --> 00:56:20,970
And on the bottom, there's
a diagonal missing,

1046
00:56:20,970 --> 00:56:24,887
I would guess, here.

1047
00:56:24,887 --> 00:56:26,470
One of them is going
to make it reflex

1048
00:56:26,470 --> 00:56:28,250
when the other one
will make it convex.

1049
00:56:28,250 --> 00:56:29,410
You can experiment.

1050
00:56:29,410 --> 00:56:33,090
Then you unfold, ideally,
and make a better version

1051
00:56:33,090 --> 00:56:36,670
of this with more properly
drawn lines and then refold it.

1052
00:56:36,670 --> 00:56:42,430
And I bet you it makes a
beautiful convex polyhedron.

1053
00:56:42,430 --> 00:56:46,560
So this is the
experimental approach.

1054
00:56:46,560 --> 00:56:51,040
And this gives you a hands-on
feel for Alexandrov's Theorem.

1055
00:56:57,844 --> 00:56:58,816
Like that.

1056
00:57:03,690 --> 00:57:06,130
Yeah, almost.

1057
00:57:06,130 --> 00:57:09,350
Believe me, it's a
triangulated convex polyhedron.

1058
00:57:09,350 --> 00:57:09,850
Gorgeous.

1059
00:57:13,660 --> 00:57:14,874
That's how it works.

1060
00:57:14,874 --> 00:57:16,790
And you can try for your
own convex polyhedron

1061
00:57:16,790 --> 00:57:19,070
where you're
guaranteed not to fail.

1062
00:57:19,070 --> 00:57:22,210
Now, you prove some nice things
about these perimeter havings.

1063
00:57:22,210 --> 00:57:25,450
In particular, I had this
flexibility of where I started

1064
00:57:25,450 --> 00:57:26,950
and where I ended.

1065
00:57:26,950 --> 00:57:29,810
And I could move those
points continuously.

1066
00:57:29,810 --> 00:57:33,190
There's infinitely many
choices for x and y.

1067
00:57:33,190 --> 00:57:36,480
And certainly, there's
infinitely many gluings.

1068
00:57:36,480 --> 00:57:39,300
I can just keep moving x and
moving y correspondingly.

1069
00:57:39,300 --> 00:57:41,650
There's a whole
continuum of gluings.

1070
00:57:41,650 --> 00:57:43,490
They're all different.

1071
00:57:43,490 --> 00:57:47,000
They all generate
different polyhedra,

1072
00:57:47,000 --> 00:57:48,190
if you think about it.

1073
00:57:48,190 --> 00:57:55,127
Because if you think
of x being very close.

1074
00:57:55,127 --> 00:57:57,460
x is definitely going to be
a vertex of your polyhedron.

1075
00:57:57,460 --> 00:58:01,359
Because you end up gluing
this part of the edge

1076
00:58:01,359 --> 00:58:02,400
to this part of the edge.

1077
00:58:02,400 --> 00:58:04,710
So you have 180 degrees
of material there.

1078
00:58:04,710 --> 00:58:09,279
That's a vertex, less than 360.

1079
00:58:09,279 --> 00:58:10,820
And this is also a
vertex, let's say.

1080
00:58:10,820 --> 00:58:13,660
Look at the nearest
neighbor to x.

1081
00:58:13,660 --> 00:58:16,200
Here is a distance
between two vertices.

1082
00:58:16,200 --> 00:58:17,216
It's the shortest path.

1083
00:58:17,216 --> 00:58:18,840
If you make x very
close to the vertex,

1084
00:58:18,840 --> 00:58:21,180
it will be a shortest path.

1085
00:58:21,180 --> 00:58:22,850
And you're setting
that to infinitely

1086
00:58:22,850 --> 00:58:26,650
many different values, a whole
continuum, very close to 0.

1087
00:58:26,650 --> 00:58:29,090
If you look at some polyhedron
that this thing realizes,

1088
00:58:29,090 --> 00:58:30,877
it has whatever, n vertices.

1089
00:58:30,877 --> 00:58:32,960
You look at the n squared
different shortest paths

1090
00:58:32,960 --> 00:58:33,501
between them.

1091
00:58:33,501 --> 00:58:35,126
That's some finite
set of real numbers.

1092
00:58:35,126 --> 00:58:37,542
And here, I'm supposed to get
a continuum of real numbers.

1093
00:58:37,542 --> 00:58:39,190
Therefore, the set
of all polyhedra

1094
00:58:39,190 --> 00:58:42,044
that you are making in this
process is a continuum.

1095
00:58:42,044 --> 00:58:42,960
There's a lot of them.

1096
00:58:45,892 --> 00:58:47,600
Most of the things
you'll get out of this

1097
00:58:47,600 --> 00:58:48,927
are different polyhedra.

1098
00:58:48,927 --> 00:58:50,510
If I started with a
different x and y,

1099
00:58:50,510 --> 00:58:52,176
I'd get a slightly
different polyhedron.

1100
00:58:52,176 --> 00:58:57,921
It will change continuously,
but it will be different.

1101
00:58:57,921 --> 00:58:59,920
So this is the sense in
which, at the beginning,

1102
00:58:59,920 --> 00:59:02,255
I said there infinitely
many gluings.

1103
00:59:05,549 --> 00:59:07,340
Really, the only time
that they're infinite

1104
00:59:07,340 --> 00:59:09,790
is when you have a
convex polygon somewhere

1105
00:59:09,790 --> 00:59:11,755
embedded inside your polygon.

1106
00:59:11,755 --> 00:59:14,880
It doesn't have
to be at the top.

1107
00:59:14,880 --> 00:59:16,798
All right, where
do I want to go?

1108
00:59:16,798 --> 00:59:17,298
Here.

1109
00:59:43,670 --> 00:59:48,510
One way to think about
gluings is all these arrows

1110
00:59:48,510 --> 00:59:49,469
that are not crossing.

1111
00:59:49,469 --> 00:59:51,260
But it's a little bit
awkward to work with.

1112
00:59:51,260 --> 00:59:54,080
And next class we're
going to really find

1113
00:59:54,080 --> 00:59:56,070
algorithms to compute
these gluings.

1114
00:59:56,070 --> 00:59:58,320
And it's useful to have
another structure, another way

1115
00:59:58,320 --> 01:00:02,890
of looking and gluings,
called the gluing tree.

1116
01:00:02,890 --> 01:00:06,210
And the main idea is to
take your polygon, which

1117
01:00:06,210 --> 01:00:08,340
I'm going to explicitly shade.

1118
01:00:08,340 --> 01:00:09,820
Because you're
soon going to lose

1119
01:00:09,820 --> 01:00:13,680
track of where it is
if I don't shade it.

1120
01:00:13,680 --> 01:00:15,180
So that's the inside
of the polygon,

1121
01:00:15,180 --> 01:00:16,530
and there's outside space.

1122
01:00:16,530 --> 01:00:19,520
I want to turn this
whole picture inside out.

1123
01:00:19,520 --> 01:00:21,641
Think of this as
living on a sphere.

1124
01:00:21,641 --> 01:00:23,390
On the sphere, there's
some shaded region.

1125
01:00:23,390 --> 01:00:24,760
That's the material.

1126
01:00:24,760 --> 01:00:29,390
And then there's the rest,
the black of the blackboard.

1127
01:00:29,390 --> 01:00:31,510
If I look on the other
side of the sphere,

1128
01:00:31,510 --> 01:00:35,760
my perspective will be, I
have some convex polygon.

1129
01:00:35,760 --> 01:00:38,080
And the interior of the
polygon is on the outside.

1130
01:00:41,040 --> 01:00:44,210
This is a useful way of
thinking about things.

1131
01:00:44,210 --> 01:00:46,820
So there's my
polygons out there.

1132
01:00:46,820 --> 01:00:52,210
Now in fact, in a convex case,
this would be less than 180.

1133
01:00:52,210 --> 01:00:53,840
It's hard to see
that in this picture.

1134
01:00:53,840 --> 01:00:55,330
But just think of
it topologically.

1135
01:00:55,330 --> 01:00:57,184
Don't worry so much
about the geometry.

1136
01:00:57,184 --> 01:00:59,600
Now, what I'm trying to do--
it's a little easier to think

1137
01:00:59,600 --> 01:01:02,450
about, oh, I need to glue
this edge to this edge

1138
01:01:02,450 --> 01:01:06,410
and then maybe glue
up to whatever.

1139
01:01:06,410 --> 01:01:08,070
Maybe these were
the full length.

1140
01:01:08,070 --> 01:01:11,580
Then I'll glue this edge to
this portion of that edge.

1141
01:01:11,580 --> 01:01:21,275
So I still can draw my
arrows, something like that.

1142
01:01:21,275 --> 01:01:24,067
This would be a
parameter having gluing.

1143
01:01:24,067 --> 01:01:25,650
I can still draw
them, but now they're

1144
01:01:25,650 --> 01:01:27,200
on the inside instead
of the outside.

1145
01:01:27,200 --> 01:01:28,860
It's easier to see them.

1146
01:01:28,860 --> 01:01:30,520
And in particular,
I could actually

1147
01:01:30,520 --> 01:01:33,103
say, well, if I want to go that
edge to that edge, why don't I

1148
01:01:33,103 --> 01:01:35,370
draw them next to each other?

1149
01:01:35,370 --> 01:01:39,510
So it would look like this.

1150
01:01:39,510 --> 01:01:43,700
Then have these edges when I
draw them next to each other.

1151
01:01:43,700 --> 01:01:49,500
And then I have those edges
and then that edge and then

1152
01:01:49,500 --> 01:01:54,620
the little edges, except these
are actually the same point.

1153
01:01:54,620 --> 01:01:57,920
So it would look like that.

1154
01:01:57,920 --> 01:02:01,480
I think of these as almost
touching and then the material

1155
01:02:01,480 --> 01:02:04,000
still on the outside.

1156
01:02:04,000 --> 01:02:08,880
It's not looking so white
anymore, but you get the idea.

1157
01:02:08,880 --> 01:02:10,290
Well, that's a path.

1158
01:02:10,290 --> 01:02:12,990
In general, you get a tree.

1159
01:02:12,990 --> 01:02:18,690
So a gluing tree will look
something like this, maybe.

1160
01:02:18,690 --> 01:02:19,760
It's a double tree.

1161
01:02:19,760 --> 01:02:23,440
There's an underlying tree here,
and then you walk around it.

1162
01:02:23,440 --> 01:02:25,080
And the material
is all out here.

1163
01:02:29,110 --> 01:02:32,020
It's like that's
where you're gluing.

1164
01:02:32,020 --> 01:02:34,340
Now, this is actually
really intuitive to us.

1165
01:02:34,340 --> 01:02:36,730
It could be.

1166
01:02:36,730 --> 01:02:39,290
When we were thinking
about where to cut along

1167
01:02:39,290 --> 01:02:41,440
the surface of a
convex polyhedron,

1168
01:02:41,440 --> 01:02:46,030
at some point we said, well,
in a convex polyhedron,

1169
01:02:46,030 --> 01:02:48,050
we had all these properties
that, in the end,

1170
01:02:48,050 --> 01:02:51,000
said the cuts must be a
spanning tree of the vertices.

1171
01:02:51,000 --> 01:02:52,631
Every vertex must get cut too.

1172
01:02:52,631 --> 01:02:54,380
And you can't have any
cycles in the cuts,

1173
01:02:54,380 --> 01:02:56,680
because that would
disconnect the surface.

1174
01:02:56,680 --> 01:02:59,140
So where you cut was
a set of was a tree,

1175
01:02:59,140 --> 01:03:01,140
a set of edges forming a tree.

1176
01:03:01,140 --> 01:03:06,570
That's the cut tree, just
viewed from the other side.

1177
01:03:06,570 --> 01:03:07,904
That's where we want to glue.

1178
01:03:07,904 --> 01:03:09,820
Folding is supposed to
be exactly the opposite

1179
01:03:09,820 --> 01:03:12,120
of unfolding.

1180
01:03:12,120 --> 01:03:14,570
That's why the gluing
tree is a tree.

1181
01:03:21,410 --> 01:03:23,610
You could think about
some interesting things

1182
01:03:23,610 --> 01:03:28,940
like where are the vertices
of my polygon in that picture?

1183
01:03:28,940 --> 01:03:31,577
So here, I made all these
notches, but not all of them

1184
01:03:31,577 --> 01:03:32,160
were vertices.

1185
01:03:32,160 --> 01:03:34,870
Some of them were
the middles of edges.

1186
01:03:34,870 --> 01:03:37,550
So this one might be 180,
meaning it wasn't really

1187
01:03:37,550 --> 01:03:39,140
a vertex of the polygon.

1188
01:03:39,140 --> 01:03:43,110
It was the middle of an edge.

1189
01:03:43,110 --> 01:03:45,690
Obviously, if I
look at some vertex

1190
01:03:45,690 --> 01:03:52,570
where three corners are coming
together, if one of them

1191
01:03:52,570 --> 01:03:55,930
was 180 and another
one was 180, we

1192
01:03:55,930 --> 01:03:59,200
would not have room for a third.

1193
01:03:59,200 --> 01:04:01,510
At a degree 3 vertex
or more, it could have,

1194
01:04:01,510 --> 01:04:03,410
at most, one middle of an edge.

1195
01:04:03,410 --> 01:04:05,890
These guys, the
other remaining guys,

1196
01:04:05,890 --> 01:04:09,080
must be vertices of the polygon.

1197
01:04:09,080 --> 01:04:12,600
The middle of an edge
could be two 180s.

1198
01:04:12,600 --> 01:04:15,360
That's actually
the typical case.

1199
01:04:15,360 --> 01:04:20,245
If you look at one of these
gluings, most of the points,

1200
01:04:20,245 --> 01:04:21,620
it's the middle
of an edge gluing

1201
01:04:21,620 --> 01:04:23,200
to another middle of an edge.

1202
01:04:23,200 --> 01:04:26,080
That's what's happening
almost everywhere.

1203
01:04:26,080 --> 01:04:31,210
But you could also have some
180 gluing to some vertex.

1204
01:04:31,210 --> 01:04:36,727
But then it had better be a
convex vertex, less than 180.

1205
01:04:36,727 --> 01:04:38,310
You can go around
and think about what

1206
01:04:38,310 --> 01:04:41,637
all these situations are.

1207
01:04:41,637 --> 01:04:43,220
When there's a 180,
the remaining guys

1208
01:04:43,220 --> 01:04:44,160
had better be convex.

1209
01:04:44,160 --> 01:04:48,949
And their total angle had
better be less than 180.

1210
01:04:48,949 --> 01:04:49,865
What about the leaves?

1211
01:04:57,740 --> 01:04:58,855
This is a zipping place.

1212
01:04:58,855 --> 01:05:01,650
I was talking about zips before,
were you just locally join

1213
01:05:01,650 --> 01:05:04,730
the two incident
edges to themselves.

1214
01:05:04,730 --> 01:05:06,450
The leaves are
where zips happen.

1215
01:05:06,450 --> 01:05:09,110
Trees have leaves,
so zips happen.

1216
01:05:12,020 --> 01:05:13,536
Zip happens.

1217
01:05:13,536 --> 01:05:15,366
[LAUGHTER]

1218
01:05:15,366 --> 01:05:16,740
PROFESSOR: This
angle, obviously,

1219
01:05:16,740 --> 01:05:18,340
will be less than 360.

1220
01:05:18,340 --> 01:05:21,080
It could be a vertex,
and then at some angle,

1221
01:05:21,080 --> 01:05:23,330
or it could be the
middle of an edge.

1222
01:05:23,330 --> 01:05:25,270
And then it will be 180.

1223
01:05:25,270 --> 01:05:27,360
That's what we had over here.

1224
01:05:33,900 --> 01:05:37,712
I chose one and
x to be a vertex.

1225
01:05:37,712 --> 01:05:39,420
But then, generically,
on the other side,

1226
01:05:39,420 --> 01:05:41,170
you're going to get
the middle of an edge.

1227
01:05:41,170 --> 01:05:44,470
And the example I did
in reality, both of them

1228
01:05:44,470 --> 01:05:46,390
were in the middle of an edge.

1229
01:05:46,390 --> 01:05:50,040
I'm going to make
special note of the ones

1230
01:05:50,040 --> 01:05:55,820
where we have
material like this,

1231
01:05:55,820 --> 01:05:58,390
and you end up
zipping like that,

1232
01:05:58,390 --> 01:06:02,490
we call this a fold point.

1233
01:06:02,490 --> 01:06:05,430
I don't know why,
but you fold at it.

1234
01:06:05,430 --> 01:06:08,190
So it's some shorthand.

1235
01:06:08,190 --> 01:06:11,210
So these are fold
points, the leaves that

1236
01:06:11,210 --> 01:06:14,630
are at the middle
of an edge, they're

1237
01:06:14,630 --> 01:06:19,170
a little special or
a little restricted.

1238
01:06:19,170 --> 01:06:22,580
Because how much curvature
is at that point?

1239
01:06:22,580 --> 01:06:25,050
180.

1240
01:06:25,050 --> 01:06:32,590
Fold points have curvature 180.

1241
01:06:32,590 --> 01:06:35,000
The total curvature is 720.

1242
01:06:35,000 --> 01:06:37,705
So I can't have more
than four fold points.

1243
01:06:46,980 --> 01:06:50,030
They're actually quite
rare, but they can happen.

1244
01:06:50,030 --> 01:06:52,185
Obviously, we had two of
them in these examples.

1245
01:06:56,540 --> 01:06:57,890
Can actually have four?

1246
01:07:01,220 --> 01:07:02,631
Yeah I think.

1247
01:07:02,631 --> 01:07:04,172
AUDIENCE: Could you
have [INAUDIBLE]?

1248
01:07:12,040 --> 01:07:14,010
PROFESSOR: I'm going to
draw an actual folding,

1249
01:07:14,010 --> 01:07:15,940
with creases and everything.

1250
01:07:15,940 --> 01:07:19,260
This is a blintz space,
for the origamists.

1251
01:07:19,260 --> 01:07:20,230
You fold like that.

1252
01:07:20,230 --> 01:07:24,020
You get a doubly covered square.

1253
01:07:24,020 --> 01:07:28,225
The gluing tree is
going to look like this.

1254
01:07:30,869 --> 01:07:31,410
You see that?

1255
01:07:31,410 --> 01:07:35,310
And it will be drawn
right on top of that,

1256
01:07:35,310 --> 01:07:38,440
and the actual
material is back here.

1257
01:07:38,440 --> 01:07:40,600
So this will be exactly
four fold points.

1258
01:07:40,600 --> 01:07:43,040
And just by luck, you have
these four 90 degree angles.

1259
01:07:43,040 --> 01:07:44,649
So there's no more
curvature there.

1260
01:07:44,649 --> 01:07:46,690
So yeah, you can definitely
get four fold points.

1261
01:07:46,690 --> 01:07:48,455
I don't know if you can
get a large examples four

1262
01:07:48,455 --> 01:07:49,200
fold points.

1263
01:07:49,200 --> 01:07:52,180
AUDIENCE: I think four fold
points means it'll be flat.

1264
01:07:52,180 --> 01:07:54,180
PROFESSOR: Four fold
points means it'll be flat.

1265
01:07:54,180 --> 01:07:57,235
AUDIENCE: So that's going
to be 0 curvature at that.

1266
01:07:57,235 --> 01:07:58,610
PROFESSOR: You're
going to have 0

1267
01:07:58,610 --> 01:08:00,700
curvature at every other vertex.

1268
01:08:00,700 --> 01:08:03,230
It's certainly implies
that you are a tetrahedron.

1269
01:08:03,230 --> 01:08:07,250
Because you only have
four vertices at all.

1270
01:08:07,250 --> 01:08:07,751
So yeah.

1271
01:08:07,751 --> 01:08:10,166
You might be able to start
with some complicate a polygon.

1272
01:08:10,166 --> 01:08:11,930
And then, everything
somehow disappears,

1273
01:08:11,930 --> 01:08:15,990
because all the angles match
up in a lucky way, like this.

1274
01:08:15,990 --> 01:08:18,744
But the polyhedron itself will
definitely be a tetrahedron.

1275
01:08:18,744 --> 01:08:20,160
I'm not sure if
it has to be flat,

1276
01:08:20,160 --> 01:08:23,050
but tetrahedron
is pretty simple.

1277
01:08:23,050 --> 01:08:25,180
So it's not so exciting.

1278
01:08:25,180 --> 01:08:26,830
Let me tell you
about rolling belts,

1279
01:08:26,830 --> 01:08:29,760
because they're really cool.

1280
01:08:29,760 --> 01:08:32,280
And then I think we're done.

1281
01:09:00,691 --> 01:09:03,920
A rolling belt is basically
an embedded, convex polygon.

1282
01:09:06,479 --> 01:09:08,870
And we saw that when we
looked at a convex polygon,

1283
01:09:08,870 --> 01:09:13,497
from the perspective of a
gluing tree, it was a path.

1284
01:09:13,497 --> 01:09:14,580
In general, I have a tree.

1285
01:09:14,580 --> 01:09:17,859
And I want to define
when a path in that tree

1286
01:09:17,859 --> 01:09:20,210
is like a convex polygon.

1287
01:09:20,210 --> 01:09:22,319
And like in the sense
that it can roll.

1288
01:09:22,319 --> 01:09:25,582
I could move x continuously
and move y corresponding,

1289
01:09:25,582 --> 01:09:27,290
and I'd always have
an Alexandrov gluing.

1290
01:09:27,290 --> 01:09:29,710
So it's like one of
these infinite families.

1291
01:09:29,710 --> 01:09:34,720
I'd like to detect those things,
and we call them rolling belts.

1292
01:09:34,720 --> 01:09:37,689
They're going to be a
path in the gluing tree.

1293
01:09:37,689 --> 01:09:40,180
Here are the
conditions you need.

1294
01:09:40,180 --> 01:09:52,060
The endpoints are
either fold points

1295
01:09:52,060 --> 01:09:54,184
or convex vertices, no reflex.

1296
01:10:01,020 --> 01:10:03,390
In that case, they
must be leaves.

1297
01:10:03,390 --> 01:10:12,180
And along that path, you
always have less than

1298
01:10:12,180 --> 01:10:19,580
or equal to 180 degrees of
material on either side.

1299
01:10:24,860 --> 01:10:26,630
This is pretty intuitive.

1300
01:10:26,630 --> 01:10:28,210
So you have some tree.

1301
01:10:31,652 --> 01:10:33,610
I'll draw a slightly more
general picture here.

1302
01:10:42,160 --> 01:10:49,082
And then, I look at some path
in that tree, maybe from here

1303
01:10:49,082 --> 01:10:50,380
to here.

1304
01:10:55,632 --> 01:10:57,340
Yeah, this was the
way I glued things up.

1305
01:10:57,340 --> 01:11:00,239
Now, imagine that I recut here.

1306
01:11:00,239 --> 01:11:02,280
Ideally, these are supposed
to be glued together.

1307
01:11:02,280 --> 01:11:05,150
But what if I separated, I
pulled just those parts apart?

1308
01:11:05,150 --> 01:11:07,600
I left these intact, all
the sub trees hanging off I

1309
01:11:07,600 --> 01:11:08,810
didn't touch.

1310
01:11:08,810 --> 01:11:10,540
And then I just
rolled that thing

1311
01:11:10,540 --> 01:11:11,880
like it was a convex polygon.

1312
01:11:11,880 --> 01:11:14,420
Which means, instead
of going here to here,

1313
01:11:14,420 --> 01:11:17,240
I'm going to glue this
point to this point.

1314
01:11:17,240 --> 01:11:21,080
I'm going to slide
this over a little bit.

1315
01:11:21,080 --> 01:11:22,580
Whereas before,
this point was glued

1316
01:11:22,580 --> 01:11:26,790
to itself-- that was the end of
the zip-- the new zip point is

1317
01:11:26,790 --> 01:11:31,250
going to be-- I can't tell,
everything's moving this way.

1318
01:11:31,250 --> 01:11:34,220
This point is going
to move up here.

1319
01:11:34,220 --> 01:11:37,080
And it's going to become
the new zip point.

1320
01:11:37,080 --> 01:11:39,390
Everybody here is moving
in that direction.

1321
01:11:39,390 --> 01:11:41,460
Everybody here is moving
in that direction.

1322
01:11:41,460 --> 01:11:47,080
I just kind of roll that belt
along, and then I reglue.

1323
01:11:47,080 --> 01:11:49,950
And as long as it looks like
a convex polygon-- meaning,

1324
01:11:49,950 --> 01:11:52,680
as you walk around here,
the total amount of material

1325
01:11:52,680 --> 01:11:55,590
getting glued, like, here I
have two portions of material--

1326
01:11:55,590 --> 01:11:59,927
as long as that's, at most,
180 in total, I can do that.

1327
01:11:59,927 --> 01:12:01,760
Because if it's, at
most, 180 on both sides,

1328
01:12:01,760 --> 01:12:03,330
it will be, at
most, 360 in total.

1329
01:12:05,930 --> 01:12:07,060
So that's it.

1330
01:12:07,060 --> 01:12:09,050
And there's a separate
corner condition.

1331
01:12:09,050 --> 01:12:12,707
But all it's saying is that
it's, at most, 180 everywhere.

1332
01:12:12,707 --> 01:12:14,790
And then it's really acting
like a convex polygon,

1333
01:12:14,790 --> 01:12:19,280
but not in total, just in that
little subpath of the tree.

1334
01:12:22,280 --> 01:12:24,060
Turns out, this is
the only way to have

1335
01:12:24,060 --> 01:12:26,470
an infinite set of gluings.

1336
01:12:26,470 --> 01:12:30,590
All infinite goings will
look like rolling belts.

1337
01:12:30,590 --> 01:12:32,850
Now, there can be more
than one rolling belt

1338
01:12:32,850 --> 01:12:35,830
in a single example.

1339
01:12:35,830 --> 01:12:39,011
And this I can even build.

1340
01:12:39,011 --> 01:12:40,080
It's pretty simple.

1341
01:12:40,080 --> 01:12:46,595
I take a rectangle, and I
glue it into a cylinder.

1342
01:12:55,630 --> 01:12:57,535
Beautiful, what an
amazing cylinder.

1343
01:13:06,487 --> 01:13:08,820
But I'm not going to commit
to how the rest of the thing

1344
01:13:08,820 --> 01:13:10,420
is glued.

1345
01:13:10,420 --> 01:13:11,470
There's a cylinder.

1346
01:13:11,470 --> 01:13:12,740
What did I draw?

1347
01:13:12,740 --> 01:13:13,420
The gluing tree.

1348
01:13:13,420 --> 01:13:18,270
So initially, my rectangles
out here, imagine.

1349
01:13:18,270 --> 01:13:20,370
And I've glued--
what did I glue,

1350
01:13:20,370 --> 01:13:23,290
the shortage to the shortage?

1351
01:13:23,290 --> 01:13:25,410
So what I end up with,
in terms of gluing tree,

1352
01:13:25,410 --> 01:13:27,050
is I have these two edges
that are glued to each other.

1353
01:13:27,050 --> 01:13:29,080
And then I have this
big thing, which

1354
01:13:29,080 --> 01:13:31,450
I don't know how I
want to glue that yet.

1355
01:13:31,450 --> 01:13:32,960
I haven't decided.

1356
01:13:32,960 --> 01:13:35,720
So those loops are supposed
to get squashed into a tree.

1357
01:13:35,720 --> 01:13:38,760
And that's this
loop and this loop.

1358
01:13:38,760 --> 01:13:43,186
You can almost see this is a
weird projection of a cylinder.

1359
01:13:43,186 --> 01:13:44,810
A cylinder goes around
on the backside,

1360
01:13:44,810 --> 01:13:46,351
because all the
material is out here.

1361
01:13:48,860 --> 01:13:50,990
I'm totally free, what
I do with the top here.

1362
01:13:50,990 --> 01:13:52,030
It's a rolling belt.

1363
01:13:52,030 --> 01:13:53,071
Look, I can even role it.

1364
01:13:53,071 --> 01:13:53,580
It's great.

1365
01:13:53,580 --> 01:13:59,890
I can slide it around,
slide, slide, slide.

1366
01:13:59,890 --> 01:14:02,640
I can do this forever,
because it's cyclic.

1367
01:14:02,640 --> 01:14:06,700
And then separately, I can
slide this one however I want.

1368
01:14:06,700 --> 01:14:07,750
It doesn't matter.

1369
01:14:07,750 --> 01:14:11,590
At all times, I'm just
gluing 180 to 180.

1370
01:14:11,590 --> 01:14:14,170
And at the ends, I'm getting
to just single 180's.

1371
01:14:14,170 --> 01:14:16,356
Those are fold points.

1372
01:14:16,356 --> 01:14:18,230
Well, this one's not
technically a fold point

1373
01:14:18,230 --> 01:14:22,660
if I did it right at the vertex,
but it acts the same way.

1374
01:14:22,660 --> 01:14:24,320
So the gluing tree
is always going

1375
01:14:24,320 --> 01:14:27,982
to look something like this.

1376
01:14:27,982 --> 01:14:29,190
But each of these could roll.

1377
01:14:29,190 --> 01:14:30,340
So it could look like that.

1378
01:14:30,340 --> 01:14:33,290
It could look like it's even.

1379
01:14:33,290 --> 01:14:36,450
You get to slide this around.

1380
01:14:36,450 --> 01:14:40,870
So it's a double continuum,
a double rainbow.

1381
01:14:43,870 --> 01:14:46,460
In some cases, like this
one, when there are lines,

1382
01:14:46,460 --> 01:14:49,280
you actually get a flat thing,
which is kind of boring.

1383
01:14:49,280 --> 01:14:55,550
In all other cases, you will get
a convex tetrahedron, non-flat,

1384
01:14:55,550 --> 01:14:57,510
for this one example.

1385
01:14:57,510 --> 01:14:58,640
That's two rolling belts.

1386
01:14:58,640 --> 01:15:00,280
Now notice, to have
to rolling belts,

1387
01:15:00,280 --> 01:15:03,040
I basically had to
have four fold points.

1388
01:15:03,040 --> 01:15:04,290
Sounds familiar.

1389
01:15:04,290 --> 01:15:08,050
I can only have
four fold points.

1390
01:15:08,050 --> 01:15:09,880
So it seems like
maybe two rolling

1391
01:15:09,880 --> 01:15:11,830
belts is the most I can have.

1392
01:15:11,830 --> 01:15:15,670
Because each rolling belt--
if you roll a little bit,

1393
01:15:15,670 --> 01:15:17,450
it will have a fold
point on either end.

1394
01:15:19,980 --> 01:15:24,220
Two is kind of the
max, but only kind of.

1395
01:15:24,220 --> 01:15:25,100
You can do three.

1396
01:15:29,036 --> 01:15:29,790
Let's do three.

1397
01:15:38,500 --> 01:15:40,055
Three rolling belts.

1398
01:15:44,880 --> 01:15:48,570
I'm going to take this
example, the blintz space.

1399
01:15:52,100 --> 01:15:54,400
Yeah, let's look at this
example for a little bit.

1400
01:15:54,400 --> 01:15:57,650
I hadn't planned this.

1401
01:15:57,650 --> 01:16:01,630
The tree here, is this nice
four star, four-armed star.

1402
01:16:01,630 --> 01:16:07,780
And there's kind of a rolling
belt from here to here.

1403
01:16:07,780 --> 01:16:10,860
If I ignore the fact that there
are these things hanging off,

1404
01:16:10,860 --> 01:16:14,605
then I have, at most,
180 on both sides.

1405
01:16:14,605 --> 01:16:16,960
It acts just like a
rolling belt Also,

1406
01:16:16,960 --> 01:16:18,360
there's a rolling belt here.

1407
01:16:18,360 --> 01:16:20,090
They cross each other.

1408
01:16:20,090 --> 01:16:21,820
Crazy.

1409
01:16:21,820 --> 01:16:28,480
But in theory, it's
a little tricky.

1410
01:16:28,480 --> 01:16:30,210
They're not independent anymore.

1411
01:16:30,210 --> 01:16:33,480
I can roll one of
them however I want.

1412
01:16:33,480 --> 01:16:37,090
But if I roll that one at all,
I cannot roll the other one.

1413
01:16:37,090 --> 01:16:39,320
Because as soon as
you roll-- let's say

1414
01:16:39,320 --> 01:16:41,100
I roll the horizontal one.

1415
01:16:41,100 --> 01:16:44,060
It'll look like this.

1416
01:16:46,620 --> 01:16:49,880
They'll be if you roll that.

1417
01:16:49,880 --> 01:16:53,690
So these top and bottom
guys are no longer touching.

1418
01:16:53,690 --> 01:16:57,402
And if you try to make
this path a rolling belt,

1419
01:16:57,402 --> 01:17:00,420
it doesn't look so good.

1420
01:17:00,420 --> 01:17:08,210
Because here, this side,
there's more than 180.

1421
01:17:08,210 --> 01:17:12,232
It's going to have a 90 from
here and 180 from down there.

1422
01:17:12,232 --> 01:17:13,690
So that's no longer
a rolling belt.

1423
01:17:13,690 --> 01:17:15,064
Right now it is,
and I could roll

1424
01:17:15,064 --> 01:17:18,040
just the vertical one or the
horizontal one but not both.

1425
01:17:20,430 --> 01:17:22,680
But the point here is, you
can have rolling belts that

1426
01:17:22,680 --> 01:17:26,670
cross or overlap each
other, in general.

1427
01:17:26,670 --> 01:17:35,020
If we do the same thing
with a triangle-- in fact,

1428
01:17:35,020 --> 01:17:38,460
I think I'm going to make a
tetrahedron-- look at that.

1429
01:17:38,460 --> 01:17:39,089
Amazing.

1430
01:17:39,089 --> 01:17:40,505
Bring those three
points together.

1431
01:17:44,320 --> 01:17:47,932
The gluing tree looks like that.

1432
01:17:47,932 --> 01:17:49,390
This is the top of
the tetrahedron.

1433
01:17:49,390 --> 01:17:52,030
These are the other three
vertices of the tetrahedron.

1434
01:17:52,030 --> 01:17:53,571
It's a spanning tree
of the vertices.

1435
01:17:56,450 --> 01:17:58,790
I claim this is a rolling belt.

1436
01:17:58,790 --> 01:18:01,930
And therefore, so is
this and so is this.

1437
01:18:01,930 --> 01:18:04,610
So there are three of them.

1438
01:18:04,610 --> 01:18:07,200
Because at this point, well,
that was like a fold point.

1439
01:18:07,200 --> 01:18:09,220
There's only 180 degrees there.

1440
01:18:09,220 --> 01:18:13,910
As I walk around, here I
have two 60s going together.

1441
01:18:13,910 --> 01:18:16,620
That's less than 180.

1442
01:18:16,620 --> 01:18:18,670
And on the other
side, I just have 160.

1443
01:18:18,670 --> 01:18:21,560
So it's definitely less
than 180 on both sides,

1444
01:18:21,560 --> 01:18:22,910
all the way along this curve.

1445
01:18:22,910 --> 01:18:24,430
And by symmetry,
along all three.

1446
01:18:24,430 --> 01:18:26,390
So I can kind of get
three rolling belts.

1447
01:18:26,390 --> 01:18:29,200
They're not independent,
I don't think.

1448
01:18:29,200 --> 01:18:34,220
If I roll one of them, it
looks like actually maybe I

1449
01:18:34,220 --> 01:18:36,960
can keep rolling the others.

1450
01:18:36,960 --> 01:18:39,420
But this is the only way you
can get three rolling belts.

1451
01:18:39,420 --> 01:18:42,685
It turns out, four rolling
belts are impossible.

1452
01:18:50,470 --> 01:18:53,410
This is actually something
we proved in this class.

1453
01:18:53,410 --> 01:18:56,880
We've proved it six years ago.

1454
01:18:56,880 --> 01:18:58,580
So it's probably
even in the textbook.

1455
01:18:58,580 --> 01:19:00,215
I don't remember,
very simple proof.

1456
01:19:05,080 --> 01:19:08,270
If you have four rolling
belts, even if they overlap,

1457
01:19:08,270 --> 01:19:14,015
you have to have a lot of fold
points, at least four of them.

1458
01:19:14,015 --> 01:19:16,390
There has to be at least one
fold point per rolling belt,

1459
01:19:16,390 --> 01:19:17,764
because they have
different ends.

1460
01:19:17,764 --> 01:19:19,740
At least one end is different.

1461
01:19:19,740 --> 01:19:23,006
Here, we had three fold
points and three belts.

1462
01:19:23,006 --> 01:19:26,890
If we have four fold
points, at least four belts.

1463
01:19:26,890 --> 01:19:30,190
If you have four rolling belts,
at least four fold points.

1464
01:19:30,190 --> 01:19:31,780
Get the right direction.

1465
01:19:31,780 --> 01:19:33,480
We know there's only
four fold points.

1466
01:19:33,480 --> 01:19:35,409
So it's actually exactly
four fold points.

1467
01:19:35,409 --> 01:19:37,700
But as you were saying, when
you have four fold points,

1468
01:19:37,700 --> 01:19:39,616
there's really no more
curvature to go around.

1469
01:19:41,986 --> 01:19:43,110
In fact, this can't happen.

1470
01:19:43,110 --> 01:19:45,620
Because your four fold
points-- all the curvature

1471
01:19:45,620 --> 01:19:47,440
is at those four leaves.

1472
01:19:47,440 --> 01:19:51,130
Every other vertex must be
perfectly balanced, has to have

1473
01:19:51,130 --> 01:19:51,740
0 curvature.

1474
01:19:56,690 --> 01:19:58,606
And then things break.

1475
01:19:58,606 --> 01:20:00,105
It's not a very
convincing argument.

1476
01:20:07,190 --> 01:20:08,460
You imagine these fold points.

1477
01:20:08,460 --> 01:20:15,586
They're somewhere
in your gluing tree.

1478
01:20:15,586 --> 01:20:17,890
Maybe they're here.

1479
01:20:17,890 --> 01:20:22,330
In order to fit four rolling
belts, there's got to be,

1480
01:20:22,330 --> 01:20:26,990
on average, two rolling belts
coming out of every vertex.

1481
01:20:26,990 --> 01:20:30,000
And where they diverge,
it's a problem.

1482
01:20:30,000 --> 01:20:32,749
Because there has to be
basically 0 degrees of material

1483
01:20:32,749 --> 01:20:35,040
out here for them to roll
that way and to role that way

1484
01:20:35,040 --> 01:20:36,760
and have 0 curvature.

1485
01:20:36,760 --> 01:20:38,230
So it fails.

1486
01:20:41,630 --> 01:20:43,130
One rolling belt,
two rolling belts,

1487
01:20:43,130 --> 01:20:45,320
three rolling belts, no more.

1488
01:20:45,320 --> 01:20:47,200
So it's not that
complicated, in terms

1489
01:20:47,200 --> 01:20:48,660
of the infinite structure.

1490
01:20:48,660 --> 01:20:51,530
What's really interesting is
all the very combinatorially

1491
01:20:51,530 --> 01:20:52,459
different trees.

1492
01:20:52,459 --> 01:20:54,000
When I roll a belt,
not much changes.

1493
01:20:54,000 --> 01:20:56,180
It's just changing a
few of the lengths.

1494
01:20:56,180 --> 01:21:00,180
Next class, we're going to look
at how the trees themselves

1495
01:21:00,180 --> 01:21:03,250
can actually change
combinatorially and get bounds

1496
01:21:03,250 --> 01:21:04,980
on how many there or
how to compute them.

1497
01:21:04,980 --> 01:21:07,840
There's good algorithms
for it, all that stuff.

1498
01:21:07,840 --> 01:21:09,640
Still some open problems too.

1499
01:21:09,640 --> 01:21:12,774
That'll be next class,
which is on Monday.

1500
01:21:12,774 --> 01:21:13,690
No class on Wednesday.

1501
01:21:16,240 --> 01:21:17,930
That's it.