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I'm Erik Demaine.

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You can call me Erik.

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This is the web
page for the class.

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You should all go there if you
haven't already, and sign up

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on this sheet if you want
to be on the mailing list.

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Good.

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So, maybe I'll tell you a little
bit what this class about.

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Then I'll tell you about
how the class works,

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and then we'll do more
about what it's about.

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The idea of lecture 1 is
to cover the entire class

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in one lecture.

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Obviously, I will omit
a few of the details

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because we have a
lot to cover, but I

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thought it would
be fun to give you

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a picture what the whole class
is, what the sort of content

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is, so you know whether
you want to be here.

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And my chalk.

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So this class is about geometry.

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It's about folding.

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It's about algorithms.

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In general, we are interested in
the mathematics and algorithms

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behind folding things.

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And also unfolding things,
because that turns out

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to be pretty interesting.

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And the formal term for
things is geometric objects.

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And you think of things
like your arm folds.

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Pieces of paper fold.

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All sorts of things
fold in the world.

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A lot of the-- if there's
any sheet metal objects

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in this room, maybe
some of these parts

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are folded out of sheet.

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I would guess so.

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So folding is everywhere,
and this class

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is all about how that
works mathematically.

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And I'm a theoretical
computer scientist.

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I do algorithms, so my
slant is towards algorithms,

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which is getting computers
to do all this for you.

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How many people here
are computer scientists?

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How many people are-- you've got
to pick one-- mathematicians?

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How many people are
neither of the above?

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Wow, cool.

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It's like not a lot of
mathematicians only,

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but a lot of
computer scientists,

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a lot of everyone else.

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So I'll go around later
maybe and find out

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what your background is.

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You may be interested
less in the mathematics

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and more in the
applications, so let

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me tell you a little
bit about those.

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I have a long list of
applications, some of which

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have been realized,
some of which

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are still in progress
or just ideas.

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Robotics is a big one,
folding robotic arms.

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And I'll show later different
kinds of transforming robots,

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like Transformers or
Terminator 2-style, where

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you want one robot to take on
many different forms, this idea

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of programmable matter,
where you can program

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the geometry of your
object as well as you

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could program the software.

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So next time when you
buy the new iPhone,

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you download the
hardware in addition

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to downloading software.

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That would be the crazy idea.

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That, of course,
doesn't exist yet.

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But it exists in
some simple forms.

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Computer graphics.

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If you're making
Toy Story 4, and you

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want to animate your characters
from one position to another,

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you have some skeleton
which is a foldable object,

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and you'd like to interpolate
between two key frames

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that the animator
draws automatically.

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That's a morphing problem,
and it's a folding morphing

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problem.

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We have mechanics.

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A lot of the early folding
work is in like 17, 1800s,

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and is motivated by
building mechanical linkages

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to do useful things.

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This is back before
electrical computers,

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we had mechanical
computers of sorts.

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Things like you could sign
your name in triplicate

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by just signing once,
and having a linkage that

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made many copies of it.

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I think Jefferson
used such a linkage.

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We have manufacturing, which
is a pretty broad term--

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things like sheet metal bending.

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Nano manufacturing, I think,
is an exciting context,

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where you can-- we're
really good at building

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flat nano-scale
objects like CPUs.

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And if you could
get them to fold up,

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then you could manufacture
3D nano-scale objects.

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I have optics here, Jason.

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Sure.

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George [INAUDIBLE] group
does some optical devices

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through folding here at MIT.

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Medical is a big one.

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Imagine folding a
stent really small,

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so it can fit through
small blood vessels

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until it gets to where
you want in your body

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and expands-- do
non-intrusive heart surgery.

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Drug delivery is another one.

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You have some object you want
to deliver, your cancer drug,

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so it follows through your body
until it detects the cancer,

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and then it releases
the drug, for example.

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Something we just
started looking at.

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Aero-astro.

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You want to deploy something
out in outer space.

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You've got to get it there
first within the space shuttle.

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So how do you fold
it down small so it

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fits in your space
shuttle, and then expands

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when it gets there?

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Biology.

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Big one.

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I'm interested in
protein folding.

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Proteins are the building block
of all life forms we know,

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and we'd like to
know how they fold.

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I'll show you some
pictures of them later.

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We'll talk more
about it, obviously,

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in the rest of the class.

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And the goal is to use
this mathematics of folding

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to try to figure out what
biology might be doing

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in real life, though that's
still obviously very difficult.

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Sculpture, sort
of an obvious one.

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Designing cooler
origami is possible,

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thanks to mathematics
and algorithms.

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Hopefully Jason agrees.

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Jason Ku is, I guess, the top
origami designer currently

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at MIT.

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That's a safe bet.

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You could make
some broader claim.

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And we'll get him to
give a guest lecture

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on the more artistic
side of origami,

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and I'll talk about the
mathematics-- some of which

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is currently used
in origami design

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today, either implicitly or
explicitly, some of which

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is yet to be used, but hopefully
will make cooler sculpture.

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You could also imagine
building interactive sculpture.

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Interactive buildings'
architecture

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is another big one.

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How many people
here are architects?

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Cool.

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A bunch.

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So I think this is really
exciting and underexploited

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at the moment,
reconfigurable buildings.

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Hoberman is one
example of somebody

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exploring this, getting the
building to fold from one shape

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to another, or getting your
shades to fold from one shape

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to another, all sorts of things.

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That's all I have.

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Maybe there are more.

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You're welcome to tell me.

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But this is somehow
why you might

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care about the mathematics
of folding things.

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And what else do I have?

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Yeah, let me tell you a
little bit about the field.

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It's, in some sense, old.

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There are some problems in
this world of geometric folding

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that go back four
or five centuries.

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And some of those problems
are still unsolved,

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but a lot of the
action in the field

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has been in the
last 12 years or so.

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And it's been really exciting.

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A lot of theorems
have been proved.

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In fact, a lot of
theorems have been

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proved in the context
of previous iterations

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of this class.

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And so I always have
to update things,

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because we keep
getting new results.

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And the idea of this class
is to cover the bleeding

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edge, whatever the
frontiers of what's known,

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and also to push
that edge further.

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So there is an open problem
session, which is optional.

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But if you want to come solve
new problems that haven't been

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solved before,
every week probably

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we'll work on a problem related
to what I cover in class.

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So you know what's
known, and then we

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try to prove what's unknown.

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That's the idea.

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And that's worked
pretty well in the past.

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Let's dive a little
bit into the sort

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of mathematical structures.

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So say geometric objects.

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There are three main
things that we typically

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think about, linkages, pieces
of paper, and polyhedra.

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A linkage is something like
your arm or a robotic arm.

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You have one-dimensional
straight links like bones,

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and hinges that
connect them together.

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And you'd like to know
how that thing can fold.

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So this is like in the graphics
world or mechanical linkage

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world.

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The typical way we
think about linkages

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is that they have rigid bars.

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As opposed to a string,
which is really floppy,

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here you have these rigid,
like maybe metal, parts,

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and you can only
fold at the hinges.

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That's typical
thinking on linkages.

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Makes it interesting.

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And sometimes we also
require no crossing.

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I'm trying to take a physical
thing you have intuition for,

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and write some of the
mathematical constraints.

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No crossing means you
can't intersect yourself.

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Sometimes that's important.

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Sometimes it's not.

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In a lot of the
mechanical linkage world

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you can have crossing
bars, but just

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because they're in different
planes in real life,

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if you're making a
two-dimensional linkage.

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Paper.

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This is my drawing
of a piece of paper.

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The rules are you
are not allowed

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to stretch the
material, and you're not

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allowed to tear or cut.

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So all you can do is fold.

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And so intuitively
this is saying

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you can't make paper any longer.

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You can't really
make it shorter.

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The only thing you can do
is change its 3D geometry.

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But if you look really closely,
the geometry is the same.

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It has a fixed
intrinsic geometry.

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All the distances on the
surface are staying the same.

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And usually in modern
origami, you're

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not allowed to cut the
paper, because that

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makes things too
easy, basically.

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And I guess also
no crossing here.

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It's a requirement.

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This is a pretty informal
description of paper.

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You can formalize it.

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We will do that at some point.

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But it's pretty intuitive.

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You've all done it before.

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Polyhedron is something
more three-dimensional.

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So here we had one-dimensional
things we were folding.

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Two-dimensional things
we were folding.

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A three-dimensional thing, or at
least a two-dimensional surface

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in three dimensions,
a typical thing

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you might want to do with
a polyhedron is build it.

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And often you want to
build something out

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of flat material, something
like-- you've probably

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made a cube by building a
cross, and then folding it up

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into that thing.

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That's the folding problem.

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The unfolding problem
is, where do I

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cut along this surface
in order to make

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one of these nice unfoldings?

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I think there's one more
cut here in the back.

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If I did it right, that
will unfold into the cross.

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So if I have some complicated
3D shape, what 2D shape

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do I cut out in order to
bend it into that 3D shape?

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That's the unfolding problem.

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There are actually
lots of different kinds

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of polyhedron folding
problems, but here it

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is you want to cut the surface.

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You want one piece-- it's
a little harder to assemble

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multiple pieces-- and you'd like
no overlap in that unfolding.

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00:12:23,460 --> 00:12:25,390
Because if, when you
unfold, it overlaps,

255
00:12:25,390 --> 00:12:26,848
then you can't
actually make it out

256
00:12:26,848 --> 00:12:30,640
of one piece of sheet material.

257
00:12:30,640 --> 00:12:33,970
So these are the sorts
of things we study.

258
00:12:33,970 --> 00:12:35,520
And I think the
plan in the class

259
00:12:35,520 --> 00:12:39,230
is I will start with paper,
because it's kind of the most

260
00:12:39,230 --> 00:12:42,540
fun, origami design,
computational origami design.

261
00:12:42,540 --> 00:12:44,910
And then we'll talk about
linkages and polyhedra.

262
00:12:44,910 --> 00:12:48,040
And I'll probably jump around
from week to week, just

263
00:12:48,040 --> 00:12:50,540
to keep it exciting, because
they're all interesting.

264
00:12:50,540 --> 00:12:52,890
You might each have
your favorite one.

265
00:12:52,890 --> 00:12:55,050
And so that we can talk
about all of them at once.

266
00:13:01,647 --> 00:13:03,230
So what kind of
mathematical questions

267
00:13:03,230 --> 00:13:06,580
would you ask about
these structures?

268
00:13:06,580 --> 00:13:11,870
Well, there are two main
types I like to characterize.

269
00:13:11,870 --> 00:13:17,330
On the one hand, we have what
I call foldability questions,

270
00:13:17,330 --> 00:13:23,090
where you're given some existing
structure like a linkage,

271
00:13:23,090 --> 00:13:27,159
or a piece of paper with
some creases on it maybe,

272
00:13:27,159 --> 00:13:29,200
and you'd like to know,
how does that thing fold?

273
00:13:36,990 --> 00:13:41,180
So I give you some structure,
and you'd like to know,

274
00:13:41,180 --> 00:13:42,750
does it fold?

275
00:13:42,750 --> 00:13:45,720
Maybe just does it fold at all?

276
00:13:45,720 --> 00:13:49,090
Or, in some particular
way, can you

277
00:13:49,090 --> 00:13:51,750
make it fold into
something interesting?

278
00:13:51,750 --> 00:13:53,220
Or some notion of interesting?

279
00:13:53,220 --> 00:13:55,200
I'm going to be generic
here, because there's

280
00:13:55,200 --> 00:13:57,320
a lot of different
questions obviously.

281
00:13:57,320 --> 00:14:00,730
And this is in contrast
to a design question,

282
00:14:00,730 --> 00:14:04,230
where you're given-- what
you start with is a goal.

283
00:14:04,230 --> 00:14:07,477
Like I want to make a butterfly.

284
00:14:07,477 --> 00:14:09,810
And then the question is, how
do I fold a piece of paper

285
00:14:09,810 --> 00:14:11,110
to make that butterfly?

286
00:14:11,110 --> 00:14:13,140
So here you're given
maybe a crease pattern

287
00:14:13,140 --> 00:14:14,410
and given some structure.

288
00:14:14,410 --> 00:14:17,940
You want to see how it folds.

289
00:14:17,940 --> 00:14:19,842
In design, you're
given the goal and you

290
00:14:19,842 --> 00:14:21,550
want to figure out
what you should build,

291
00:14:21,550 --> 00:14:23,404
what folding structure
should I make

292
00:14:23,404 --> 00:14:24,570
that will achieve this goal?

293
00:14:27,140 --> 00:14:31,150
So what shapes?

294
00:14:31,150 --> 00:14:35,230
Shape is one thing
you might want.

295
00:14:35,230 --> 00:14:39,445
But in general, you have some
property you want to achieve.

296
00:14:45,550 --> 00:14:46,570
Can you fold it?

297
00:14:49,690 --> 00:14:54,470
And of course if you
can, how do you do it?

298
00:14:54,470 --> 00:14:56,205
So this tends-- in
general you could

299
00:14:56,205 --> 00:14:58,330
think of this as a more
mathematical question, this

300
00:14:58,330 --> 00:14:59,510
as a more algorithmic question.

301
00:14:59,510 --> 00:15:00,884
But usually actually
both of them

302
00:15:00,884 --> 00:15:03,460
are addressed with
algorithms to some extent.

303
00:15:06,220 --> 00:15:08,040
Of course design
is somehow cooler,

304
00:15:08,040 --> 00:15:09,960
but often we need to
understand foldability

305
00:15:09,960 --> 00:15:12,140
before we can solve design.

306
00:15:12,140 --> 00:15:13,420
But not always.

307
00:15:13,420 --> 00:15:15,630
It depends.

308
00:15:15,630 --> 00:15:17,940
And just to give
you like-- I mean

309
00:15:17,940 --> 00:15:20,430
if you looked at the entire
set of results, the questions

310
00:15:20,430 --> 00:15:23,430
and answers, that we consider,
and you filter them down,

311
00:15:23,430 --> 00:15:25,140
there would be three
kinds of things

312
00:15:25,140 --> 00:15:27,570
that we prove in this class.

313
00:15:27,570 --> 00:15:29,040
These are the results.

314
00:15:29,040 --> 00:15:31,420
All the results
in three bullets.

315
00:15:31,420 --> 00:15:33,993
It could be either
you get universality.

316
00:15:38,332 --> 00:15:39,790
This is the coolest
kind of result,

317
00:15:39,790 --> 00:15:41,590
and it's surprisingly common.

318
00:15:41,590 --> 00:15:46,077
Everything can be folded, for
some notion of everything.

319
00:15:46,077 --> 00:15:47,160
That's always a challenge.

320
00:15:52,040 --> 00:15:56,080
And you get an
algorithm to do it.

321
00:15:56,080 --> 00:15:58,312
So if you say I want
to fold a butterfly,

322
00:15:58,312 --> 00:16:00,020
you put the butterfly
into the algorithm,

323
00:16:00,020 --> 00:16:02,580
out come your design
for folding a butterfly.

324
00:16:02,580 --> 00:16:09,270
That's the ideal picture when
you get a universality result.

325
00:16:09,270 --> 00:16:11,900
The next best thing
you could hope for

326
00:16:11,900 --> 00:16:14,800
is a sort of decision
result, which

327
00:16:14,800 --> 00:16:18,635
is that there's a fast
algorithm that will tell you

328
00:16:18,635 --> 00:16:19,885
whether something is foldable.

329
00:16:30,080 --> 00:16:32,160
So we say it
decides foldability.

330
00:16:36,166 --> 00:16:37,290
So you give it a butterfly.

331
00:16:37,290 --> 00:16:39,170
It says, oh, that's impossible.

332
00:16:39,170 --> 00:16:39,970
That's not true.

333
00:16:39,970 --> 00:16:41,924
But you could imagine that.

334
00:16:41,924 --> 00:16:43,840
Or maybe you give it a
butterfly, it says yes.

335
00:16:43,840 --> 00:16:46,770
You give it the
skyline of New York,

336
00:16:46,770 --> 00:16:48,770
and it says no, you
can't fold that.

337
00:16:48,770 --> 00:16:49,530
That's not true.

338
00:16:49,530 --> 00:16:54,332
But if it's not the case
that everything is foldable,

339
00:16:54,332 --> 00:16:56,540
the next best thing is you
could at least distinguish

340
00:16:56,540 --> 00:16:58,690
which things are foldable
from which things are not.

341
00:16:58,690 --> 00:17:00,398
And then, of course,
the worst thing that

342
00:17:00,398 --> 00:17:04,650
could happen, in some sense,
is you get a hardness results

343
00:17:04,650 --> 00:17:08,829
which says, there's no good
way to even distinguish

344
00:17:08,829 --> 00:17:10,640
foldable things from
unfoldable things.

345
00:17:19,621 --> 00:17:21,079
So I'm going to
say computationally

346
00:17:21,079 --> 00:17:23,960
intractable to mean generically
that there's no good algorithm

347
00:17:23,960 --> 00:17:25,429
to solve a problem.

348
00:17:25,429 --> 00:17:27,720
We're going to be more formal
about that in the future.

349
00:17:33,160 --> 00:17:35,660
Because different people
have different backgrounds,

350
00:17:35,660 --> 00:17:37,410
you may know about
competition complexity.

351
00:17:37,410 --> 00:17:38,360
Many of you don't.

352
00:17:38,360 --> 00:17:39,410
That's cool.

353
00:17:39,410 --> 00:17:42,960
We're going to talk
about all that here.

354
00:17:42,960 --> 00:17:44,190
All right.

355
00:17:44,190 --> 00:17:45,610
That's the class in a nutshell.

356
00:17:45,610 --> 00:17:46,650
Totally generic.

357
00:17:46,650 --> 00:17:48,830
Any questions so far?

358
00:17:48,830 --> 00:17:49,630
Probably not.

359
00:17:52,300 --> 00:17:55,500
I thought now I would tell you
a little bit about the class

360
00:17:55,500 --> 00:17:57,920
structure.

361
00:17:57,920 --> 00:18:01,340
The main part of the
class are these lectures,

362
00:18:01,340 --> 00:18:04,470
and attendance is
mandatory for classes,

363
00:18:04,470 --> 00:18:07,270
because you're not going to
learn it unless you come here.

364
00:18:07,270 --> 00:18:08,270
You can have exceptions.

365
00:18:08,270 --> 00:18:10,770
Just email me and
watch the videos later.

366
00:18:10,770 --> 00:18:13,700
That's one of the fun
features of the videos,

367
00:18:13,700 --> 00:18:18,120
although the videos are mainly
to reach the world for fun.

368
00:18:18,120 --> 00:18:22,140
There will be a few problem sets
in this class, not too many.

369
00:18:22,140 --> 00:18:26,170
And the other main thing that
you turn in is the project.

370
00:18:26,170 --> 00:18:29,290
So sort of a
project-focused class.

371
00:18:29,290 --> 00:18:31,490
You will hand in some write up.

372
00:18:31,490 --> 00:18:34,040
You will give some presentation
in the last few lectures

373
00:18:34,040 --> 00:18:36,040
of class, if you're taking
the class for credit.

374
00:18:36,040 --> 00:18:37,680
If you're listening, you
don't have to do this.

375
00:18:37,680 --> 00:18:38,420
You can, if you want.

376
00:18:38,420 --> 00:18:39,586
You can do the problem sets.

377
00:18:39,586 --> 00:18:40,910
Some of them will be fun.

378
00:18:40,910 --> 00:18:43,230
Hopefully all of
them will be fun.

379
00:18:43,230 --> 00:18:47,090
None of them will
bee too painful.

380
00:18:47,090 --> 00:18:49,410
But for people who are
taking the class for credit,

381
00:18:49,410 --> 00:18:51,550
you have to do a project
and presentation.

382
00:18:51,550 --> 00:18:53,350
The project could be
about tons of things.

383
00:18:53,350 --> 00:18:54,980
You could build a sculpture.

384
00:18:54,980 --> 00:18:57,660
You could come up with a cool
virtual design of something

385
00:18:57,660 --> 00:19:00,110
amazing that somehow
relates to folding.

386
00:19:00,110 --> 00:19:02,842
It doesn't have to be direct.

387
00:19:02,842 --> 00:19:05,050
If you're a coder, you could
implement some algorithm

388
00:19:05,050 --> 00:19:09,780
we talk about, or make a
beautiful image or animation

389
00:19:09,780 --> 00:19:11,620
or applet or something.

390
00:19:11,620 --> 00:19:15,690
If you're more theory, you
could solve an open problem.

391
00:19:15,690 --> 00:19:17,070
Obviously that's a big win.

392
00:19:17,070 --> 00:19:19,990
But even trying to solve
an open problem is fine.

393
00:19:19,990 --> 00:19:23,250
You can talk about how
you failed to solve it,

394
00:19:23,250 --> 00:19:26,202
in the unhappy
case that happens.

395
00:19:26,202 --> 00:19:27,910
But if you want to
solve an open problem,

396
00:19:27,910 --> 00:19:29,720
I would encourage you to come
to the open problem session,

397
00:19:29,720 --> 00:19:31,350
so we can all solve it together.

398
00:19:31,350 --> 00:19:33,120
Then you can have
a joint project

399
00:19:33,120 --> 00:19:37,450
on what we do, but it's totally
unpredictable, of course.

400
00:19:37,450 --> 00:19:39,410
Even posing an open
problem in this field

401
00:19:39,410 --> 00:19:40,510
is pretty interesting.

402
00:19:40,510 --> 00:19:42,270
I know a lot of the
hard open problems.

403
00:19:42,270 --> 00:19:46,540
I would like to find more of
them, more tractable ones.

404
00:19:46,540 --> 00:19:48,810
And so if you have some
idea, especially related

405
00:19:48,810 --> 00:19:51,620
to some application field
that you know a lot about,

406
00:19:51,620 --> 00:19:53,460
it'd be cool to try
to extract, what

407
00:19:53,460 --> 00:19:58,010
is the mathematical problem
in a lot of these fields?

408
00:19:58,010 --> 00:19:59,300
Or you can write a survey.

409
00:19:59,300 --> 00:20:01,947
That's a typical project.

410
00:20:01,947 --> 00:20:04,530
You should avoid overlap with--
I don't actually have it here,

411
00:20:04,530 --> 00:20:06,200
but the textbook
for this class--

412
00:20:06,200 --> 00:20:09,050
you can imagine it being
here-- is Geometric Folding

413
00:20:09,050 --> 00:20:11,010
Algorithms.

414
00:20:11,010 --> 00:20:11,790
Here it is.

415
00:20:11,790 --> 00:20:12,660
Thank you, Jason.

416
00:20:12,660 --> 00:20:13,160
That's good.

417
00:20:13,160 --> 00:20:14,250
I ran out of copies.

418
00:20:14,250 --> 00:20:15,991
Now I have one.

419
00:20:15,991 --> 00:20:17,710
No, I'll give it back.

420
00:20:17,710 --> 00:20:21,096
So this is by me and Joe
O'Rourke at Smith College.

421
00:20:21,096 --> 00:20:22,595
Any questions about
class structure?

422
00:20:25,150 --> 00:20:25,710
Yeah.

423
00:20:25,710 --> 00:20:28,390
AUDIENCE: Can projects be
individual? [INAUDIBLE]

424
00:20:28,390 --> 00:20:30,545
PROFESSOR: Projects
can be with groups.

425
00:20:30,545 --> 00:20:32,350
I forget whether the
website has a limit

426
00:20:32,350 --> 00:20:33,350
on the number of people.

427
00:20:33,350 --> 00:20:34,849
I don't think so.

428
00:20:34,849 --> 00:20:37,140
But at some point you'll have
to do a project proposal.

429
00:20:37,140 --> 00:20:38,610
I should mention that also.

430
00:20:38,610 --> 00:20:41,950
And then I will vet
your project group.

431
00:20:41,950 --> 00:20:43,447
But I think anything's fine.

432
00:20:43,447 --> 00:20:45,030
Just when you have
more people, you're

433
00:20:45,030 --> 00:20:47,720
expected to do a
little more, naturally.

434
00:20:47,720 --> 00:20:49,250
Other questions?

435
00:20:49,250 --> 00:20:49,880
Yeah.

436
00:20:49,880 --> 00:20:51,460
I love collaboration.

437
00:20:51,460 --> 00:20:53,940
I think I've never written
a paper without a co-author.

438
00:20:53,940 --> 00:20:55,360
It wasn't a survey paper.

439
00:20:55,360 --> 00:20:57,130
So collaboration's good.

440
00:20:59,770 --> 00:21:00,921
All right.

441
00:21:00,921 --> 00:21:01,420
Doing well.

442
00:21:04,860 --> 00:21:08,070
So the next thing I
want to do is actually

443
00:21:08,070 --> 00:21:10,390
dive into actual content.

444
00:21:10,390 --> 00:21:12,012
This was totally generic.

445
00:21:12,012 --> 00:21:12,970
And to me, it's useful.

446
00:21:12,970 --> 00:21:14,637
But maybe to you,
it's less useful.

447
00:21:14,637 --> 00:21:15,970
This is sort of an organization.

448
00:21:15,970 --> 00:21:17,880
As we go through,
everything you'll see

449
00:21:17,880 --> 00:21:20,769
will fit almost always into
one of these three categories.

450
00:21:20,769 --> 00:21:23,060
The question will fit into
one of these two categories,

451
00:21:23,060 --> 00:21:25,590
and the answer will fit into
one of these three categories.

452
00:21:25,590 --> 00:21:28,660
Now let's see what some of
those actual problems, questions

453
00:21:28,660 --> 00:21:29,990
and answers, are.

454
00:21:29,990 --> 00:21:31,600
That's the fun part.

455
00:21:31,600 --> 00:21:33,820
There's so many
cool results here,

456
00:21:33,820 --> 00:21:35,420
and so many cool open problems.

457
00:21:35,420 --> 00:21:40,530
I thought I'd tell you a bunch
of the big open problems, too.

458
00:21:40,530 --> 00:21:43,790
So in today's class, I'm
going to go in order.

459
00:21:43,790 --> 00:21:46,260
We're going to start with
linkages, and then paper,

460
00:21:46,260 --> 00:21:49,120
and then polyhedra.

461
00:21:49,120 --> 00:21:59,290
And I'm going to start with
linkages allowing intersection.

462
00:22:03,130 --> 00:22:07,550
So magically, bars of the
linkage can overlap each other.

463
00:22:07,550 --> 00:22:09,920
And then later I'll
talk about linkages,

464
00:22:09,920 --> 00:22:12,040
which is most interesting
in two dimensions,

465
00:22:12,040 --> 00:22:14,373
because you can actually build
them in three dimensions.

466
00:22:14,373 --> 00:22:18,710
And then we'll go to linkages
that do not have intersection.

467
00:22:18,710 --> 00:22:24,450
So an early
motivation here where

468
00:22:24,450 --> 00:22:26,710
a lot of this linkage folding
came from originally--

469
00:22:26,710 --> 00:22:35,220
this is the 17, 1800s-- is
converting linear motion

470
00:22:35,220 --> 00:22:36,300
into circular motion.

471
00:22:41,290 --> 00:22:45,280
Actually, it's really--
yeah, get it right.

472
00:22:45,280 --> 00:22:47,505
Linear motion to
circular motion.

473
00:22:50,624 --> 00:22:52,540
I don't know what order
to cover these things.

474
00:22:52,540 --> 00:22:54,750
I'll show you a slide.

475
00:22:54,750 --> 00:22:58,130
My rules are essentially
no words on any slide.

476
00:22:58,130 --> 00:23:00,330
This is just for
pretty pictures,

477
00:23:00,330 --> 00:23:02,250
and I'll write
stuff on the board.

478
00:23:02,250 --> 00:23:04,670
There's this fun book called
How to Draw a Straight

479
00:23:04,670 --> 00:23:09,620
Line by Kempe in 1877, which
is all about this problem.

480
00:23:09,620 --> 00:23:11,940
How do I turn a
circular crank and make

481
00:23:11,940 --> 00:23:14,510
a straight line come
out as a result?

482
00:23:14,510 --> 00:23:17,790
And the motivation for this
problem is steam engines.

483
00:23:17,790 --> 00:23:20,280
You have a steam piston,
which is moving something

484
00:23:20,280 --> 00:23:21,980
up and down along
a straight line.

485
00:23:21,980 --> 00:23:25,500
And maybe you're building
a locomotive train,

486
00:23:25,500 --> 00:23:29,609
and you want to turn
a wheel in a circle,

487
00:23:29,609 --> 00:23:30,650
because wheels are round.

488
00:23:30,650 --> 00:23:32,360
So how do you convert
this linear motion

489
00:23:32,360 --> 00:23:33,950
into a circular motion?

490
00:23:33,950 --> 00:23:39,230
That is How to Draw a Straight
Line, affectionately titled.

491
00:23:39,230 --> 00:23:42,960
But you can see that one of
the earliest linkages for this

492
00:23:42,960 --> 00:23:46,270
is called the Watt
parallel motion.

493
00:23:46,270 --> 00:23:48,990
And we have a little
animation here.

494
00:23:48,990 --> 00:23:52,170
So the idea is this vertex in
the top left is pinned down.

495
00:23:52,170 --> 00:23:54,750
This one in the bottom
right is pinned down.

496
00:23:54,750 --> 00:23:58,880
And then, I think if I move
this around the circle,

497
00:23:58,880 --> 00:24:04,050
the green guy moves
along that figure eight.

498
00:24:04,050 --> 00:24:06,950
And there's a limit
to how far it can go.

499
00:24:06,950 --> 00:24:09,730
So and if you draw it
right, the figure eight

500
00:24:09,730 --> 00:24:11,880
is almost a straight line.

501
00:24:11,880 --> 00:24:15,130
So that was mathematics
back in the day.

502
00:24:15,130 --> 00:24:17,282
No, no one thought
that that was perfect,

503
00:24:17,282 --> 00:24:18,240
but it was pretty good.

504
00:24:18,240 --> 00:24:20,780
And that actually led Watt--
you may have heard of Watt.

505
00:24:20,780 --> 00:24:22,440
He's a unit.

506
00:24:22,440 --> 00:24:25,760
And he was very proud
of this invention.

507
00:24:25,760 --> 00:24:29,310
And he made tons of innovations
in the steam engine world,

508
00:24:29,310 --> 00:24:31,160
and this was his favorite.

509
00:24:31,160 --> 00:24:33,100
And, I mean, it changed things.

510
00:24:33,100 --> 00:24:35,660
But later on-- I mean this
is like 100 years later,

511
00:24:35,660 --> 00:24:38,850
1864 versus 1764.

512
00:24:38,850 --> 00:24:40,800
Exactly 100 years later.

513
00:24:40,800 --> 00:24:45,110
Peaucellier a French guy
in the army, I think,

514
00:24:45,110 --> 00:24:46,360
came up with this linkage.

515
00:24:46,360 --> 00:24:49,430
So again, you have two
pinned vertices here.

516
00:24:49,430 --> 00:24:51,120
This one's moving
around a circle.

517
00:24:51,120 --> 00:24:53,130
So you just turn
the crank, and look.

518
00:24:53,130 --> 00:24:56,340
This guy moves
along that red line.

519
00:24:56,340 --> 00:24:57,880
Perfect.

520
00:24:57,880 --> 00:24:58,380
Very cool.

521
00:24:58,380 --> 00:25:02,500
Again it has a limit
how far you can go.

522
00:25:02,500 --> 00:25:04,010
But it's pretty awesome.

523
00:25:04,010 --> 00:25:06,095
I mean you could play
with this forever.

524
00:25:09,820 --> 00:25:10,567
I won't bore you.

525
00:25:10,567 --> 00:25:12,275
There's this other
guy you may have heard

526
00:25:12,275 --> 00:25:14,690
of, Kelvin, Lord Kelvin.

527
00:25:14,690 --> 00:25:16,759
Another unit.

528
00:25:16,759 --> 00:25:18,550
There's a story about
him playing with one.

529
00:25:18,550 --> 00:25:20,299
He wouldn't want to
give it up, because he

530
00:25:20,299 --> 00:25:23,270
was having too much fun just
pushing it back and forth.

531
00:25:23,270 --> 00:25:27,394
It's like, wow, a straight
line out of a circle.

532
00:25:27,394 --> 00:25:28,310
So that's pretty cool.

533
00:25:28,310 --> 00:25:30,560
Making straight lines out
of circles is pretty neat.

534
00:25:30,560 --> 00:25:34,026
In fact, you could think,
well, what else could I make?

535
00:25:34,026 --> 00:25:35,150
I can make a straight line.

536
00:25:35,150 --> 00:25:35,983
That's kind of nice.

537
00:25:35,983 --> 00:25:40,640
But could I make, I don't
know, some other curve?

538
00:25:40,640 --> 00:25:42,770
I mean a straight line is
a special kind of curve.

539
00:25:42,770 --> 00:25:47,420
Maybe I could make this curve.

540
00:25:47,420 --> 00:25:48,393
That's a nice curve.

541
00:25:51,070 --> 00:25:53,380
In general, there's
a universality result

542
00:25:53,380 --> 00:25:56,390
which says there's a
linkage to sign your name.

543
00:25:58,824 --> 00:25:59,865
That's the cute phrasing.

544
00:26:03,240 --> 00:26:05,850
So it's a
two-dimensional linkage.

545
00:26:05,850 --> 00:26:13,510
You turn one circular crank,
and it signs your name.

546
00:26:13,510 --> 00:26:15,950
The mathematical
version is that you

547
00:26:15,950 --> 00:26:21,130
trace a piecewise
polynomial curve.

548
00:26:21,130 --> 00:26:24,259
Or if you're an architect
or graphics person,

549
00:26:24,259 --> 00:26:25,050
call this a spline.

550
00:26:31,340 --> 00:26:32,880
Pick your favorite word.

551
00:26:32,880 --> 00:26:36,110
You can make it all
with one linkage.

552
00:26:36,110 --> 00:26:39,480
It's pretty crazy, and
not super practical.

553
00:26:39,480 --> 00:26:44,370
Building this would probably
require thousands of bars.

554
00:26:44,370 --> 00:26:46,970
But hey, who's counting?

555
00:26:46,970 --> 00:26:48,921
At least you can get
a universality result.

556
00:26:48,921 --> 00:26:50,670
This is actually
something that goes back,

557
00:26:50,670 --> 00:26:52,910
in particular, to the
first time this class was

558
00:26:52,910 --> 00:26:55,220
taught six years ago.

559
00:26:55,220 --> 00:26:56,920
We found some better
ways to do, so it's

560
00:26:56,920 --> 00:26:58,680
known that it could be done.

561
00:26:58,680 --> 00:27:02,690
And I'll talk about
those ways later on.

562
00:27:02,690 --> 00:27:03,620
Cool.

563
00:27:03,620 --> 00:27:04,810
Open problem.

564
00:27:04,810 --> 00:27:08,264
Not everything is known.

565
00:27:08,264 --> 00:27:10,430
There's a lot of open
problems here, but one of them

566
00:27:10,430 --> 00:27:12,850
is, what if you
forbid crossings?

567
00:27:16,500 --> 00:27:20,040
So these linkages, like the
ones I've been showing here,

568
00:27:20,040 --> 00:27:22,560
like the Peaucellier linkage,
there's crossing bars.

569
00:27:22,560 --> 00:27:26,070
And yeah, you could do that, but
what if you forbid crossings?

570
00:27:26,070 --> 00:27:29,150
It'd be really cool to sign
your name with a linkage that

571
00:27:29,150 --> 00:27:30,850
doesn't even cross itself.

572
00:27:30,850 --> 00:27:31,840
That's totally open.

573
00:27:31,840 --> 00:27:34,100
Getting any kind of
result, positive result,

574
00:27:34,100 --> 00:27:35,790
maybe not everything,
but at least

575
00:27:35,790 --> 00:27:37,123
getting some interesting things.

576
00:27:37,123 --> 00:27:39,590
Even drawing a straight
line, I think, is open.

577
00:27:39,590 --> 00:27:42,430
Could be a fun
problem to work on.

578
00:27:42,430 --> 00:27:43,040
All right.

579
00:27:43,040 --> 00:27:45,770
That's one.

580
00:27:45,770 --> 00:27:52,510
Let's go over here.

581
00:27:55,234 --> 00:27:57,400
Let's see, is that the end
of linkages, is crossing?

582
00:27:57,400 --> 00:27:58,030
No.

583
00:27:58,030 --> 00:28:01,840
Next question you might
ask is-- so that's

584
00:28:01,840 --> 00:28:04,180
sort of a design question.

585
00:28:04,180 --> 00:28:07,040
I was given the goal, which
was to make a straight line

586
00:28:07,040 --> 00:28:09,990
or sign my name.

587
00:28:09,990 --> 00:28:12,180
And I wanted to find
the linkage that did it.

588
00:28:12,180 --> 00:28:15,610
The other question, the
foldability question,

589
00:28:15,610 --> 00:28:19,270
one of them is rigidity, which
is, does a linkage fold at all?

590
00:28:23,310 --> 00:28:24,640
I should say a given linkage.

591
00:28:32,520 --> 00:28:36,690
So for example, I'll
give you three examples.

592
00:28:36,690 --> 00:28:37,530
Test you out here.

593
00:28:40,692 --> 00:28:42,400
These are rigid bars
connected by hinges.

594
00:28:53,920 --> 00:28:56,850
OK, rigid or flexible?

595
00:28:56,850 --> 00:28:58,280
Rigid, I agree.

596
00:28:58,280 --> 00:28:59,971
Rigid or flexible.

597
00:28:59,971 --> 00:29:00,470
Rigid.

598
00:29:00,470 --> 00:29:02,260
Any other answers?

599
00:29:02,260 --> 00:29:04,230
Flexible, correct.

600
00:29:04,230 --> 00:29:05,550
Both are correct.

601
00:29:05,550 --> 00:29:10,190
In two dimensions, OK, it
depends where you live?

602
00:29:10,190 --> 00:29:13,456
In two dimensions, this
is rigid no matter what,

603
00:29:13,456 --> 00:29:15,830
in any dimension.

604
00:29:15,830 --> 00:29:20,770
But in three dimensions, this is
flexible and in two dimensions,

605
00:29:20,770 --> 00:29:22,632
it's rigid.

606
00:29:22,632 --> 00:29:24,506
In three dimensions,
you can pick this guy up

607
00:29:24,506 --> 00:29:27,680
and it spins around a circle,
out of the board here.

608
00:29:27,680 --> 00:29:28,930
Or you could pick this guy up.

609
00:29:28,930 --> 00:29:30,450
It spins around a circle.

610
00:29:30,450 --> 00:29:32,385
In two dimensions,
though, it's rigid

611
00:29:32,385 --> 00:29:34,010
because it's really
just two triangles.

612
00:29:34,010 --> 00:29:35,820
Triangles are rigid
in two dimensions.

613
00:29:35,820 --> 00:29:37,010
This is a tetrahedron.

614
00:29:37,010 --> 00:29:39,350
Tetrahedra are rigid
in three dimensions.

615
00:29:39,350 --> 00:29:40,910
Rigid or flexible?

616
00:29:40,910 --> 00:29:42,559
Everyone agrees.

617
00:29:42,559 --> 00:29:43,975
In both dimensions
it is flexible.

618
00:29:46,580 --> 00:29:50,980
OK, pretty intuitive
for four vertices.

619
00:29:50,980 --> 00:29:53,132
But you can ask the
mathematical question

620
00:29:53,132 --> 00:29:54,090
and give you a linkage.

621
00:29:54,090 --> 00:29:55,350
Is it rigid in 2D?

622
00:29:55,350 --> 00:29:57,290
Is it rigid in 3D?

623
00:29:57,290 --> 00:30:00,430
And there are many
versions of this question.

624
00:30:00,430 --> 00:30:05,410
But the short version, a
short answer, let's say,

625
00:30:05,410 --> 00:30:08,060
is that distinguishing rigid
from flexible two-dimensional

626
00:30:08,060 --> 00:30:09,594
linkages is easy.

627
00:30:09,594 --> 00:30:11,010
There's a good
algorithm to do it.

628
00:30:11,010 --> 00:30:12,970
It's very powerful, very useful.

629
00:30:12,970 --> 00:30:15,900
In 3D, we have no idea.

630
00:30:15,900 --> 00:30:18,040
Very open.

631
00:30:18,040 --> 00:30:18,540
Tough.

632
00:30:18,540 --> 00:30:19,200
Very tough problem.

633
00:30:19,200 --> 00:30:20,616
A lot of people
have been thinking

634
00:30:20,616 --> 00:30:24,120
about that for decades.

635
00:30:24,120 --> 00:30:24,950
So that's rigidity.

636
00:30:24,950 --> 00:30:28,740
I'm just going to touch on
lots of topics very briefly.

637
00:30:42,120 --> 00:30:43,490
All right.

638
00:30:43,490 --> 00:30:46,990
Next we go on to, I
guess, one and a half.

639
00:30:46,990 --> 00:30:52,240
This is linkages
forbidding intersection.

640
00:31:04,336 --> 00:31:05,710
And this is more
interesting when

641
00:31:05,710 --> 00:31:07,584
you're talking about 3D
linkages like my arm.

642
00:31:07,584 --> 00:31:09,810
I really don't want it
to penetrate other bars.

643
00:31:09,810 --> 00:31:10,585
It's not possible.

644
00:31:13,900 --> 00:31:18,430
And the first question you
might ask in this world, which

645
00:31:18,430 --> 00:31:22,970
I guess is a
foldability question,

646
00:31:22,970 --> 00:31:25,300
sort of a
reconfiguration question,

647
00:31:25,300 --> 00:31:30,840
let's say I want to fold
my robotic arm from one

648
00:31:30,840 --> 00:31:33,130
configuration that
I know-- call it

649
00:31:33,130 --> 00:31:40,130
configuration A-- to some
other configuration, B.

650
00:31:40,130 --> 00:31:41,880
When is that possible?

651
00:31:41,880 --> 00:31:42,990
Can I go from A to B?

652
00:31:42,990 --> 00:31:46,954
Can I go from some other A
prime to some other B prime?

653
00:31:46,954 --> 00:31:48,120
Sometimes yes, sometimes no.

654
00:31:48,120 --> 00:31:49,320
It depends.

655
00:31:49,320 --> 00:31:51,050
Sometimes it's not
foldable at all,

656
00:31:51,050 --> 00:31:53,644
even when you allow
intersections.

657
00:31:53,644 --> 00:31:55,310
So this is a pretty
open-ended question.

658
00:31:55,310 --> 00:31:57,710
In general, it's
computationally intractable.

659
00:31:57,710 --> 00:32:00,164
If I give you a linkage
and two configurations,

660
00:32:00,164 --> 00:32:01,830
to decide whether you
can go from A to B

661
00:32:01,830 --> 00:32:04,180
is, for the
complexity theorists,

662
00:32:04,180 --> 00:32:05,450
piece space complete.

663
00:32:05,450 --> 00:32:08,020
Talk about what
that means later.

664
00:32:08,020 --> 00:32:09,870
Really, really hard
is the short version.

665
00:32:12,720 --> 00:32:19,250
But a lot of the times you can
think about special linkages.

666
00:32:19,250 --> 00:32:22,660
There are a lot of
interesting special cases.

667
00:32:26,430 --> 00:32:30,550
In particular, we like
to think about chains

668
00:32:30,550 --> 00:32:31,430
like I drew before.

669
00:32:37,160 --> 00:32:39,990
Also polygons.

670
00:32:39,990 --> 00:32:42,880
That's a little
messy, but imagine

671
00:32:42,880 --> 00:32:44,859
those don't self-intersect.

672
00:32:44,859 --> 00:32:46,400
So this is what I
call an open chain,

673
00:32:46,400 --> 00:32:47,500
and this is a closed chain.

674
00:32:47,500 --> 00:32:48,708
In general, these are chains.

675
00:32:51,030 --> 00:32:54,410
And the other thing I might like
to think about, in particular

676
00:32:54,410 --> 00:32:57,145
because proteins look kind
of like this, are trees.

677
00:33:02,330 --> 00:33:05,680
Trees are just linkages
without any cycles in them.

678
00:33:05,680 --> 00:33:07,400
So those are nice and simple.

679
00:33:07,400 --> 00:33:10,780
Here I have no cycles,
no cycles, one cycle.

680
00:33:10,780 --> 00:33:11,890
Easier to think about.

681
00:33:11,890 --> 00:33:14,310
And sometimes you actually
get a universality result

682
00:33:14,310 --> 00:33:16,935
that these linkages can
fold from any configuration

683
00:33:16,935 --> 00:33:18,580
to any configuration.

684
00:33:18,580 --> 00:33:20,340
And that's especially cool.

685
00:33:20,340 --> 00:33:21,710
Let me tell you about them.

686
00:33:28,694 --> 00:33:29,610
Where do I want to go?

687
00:33:38,090 --> 00:33:41,140
So it depends again what
dimension you live in.

688
00:33:43,867 --> 00:33:45,200
I'm very flexible in this class.

689
00:33:45,200 --> 00:33:46,574
You can live in
any dimension you

690
00:33:46,574 --> 00:33:50,600
want to, and even
fictional ones.

691
00:33:50,600 --> 00:33:54,320
And you can think about
chains and trees, let's say.

692
00:33:54,320 --> 00:33:55,800
You could go more
general, but this

693
00:33:55,800 --> 00:33:57,920
is where most things
have been studied.

694
00:33:57,920 --> 00:34:01,960
And the answer is,
for chains in 2D

695
00:34:01,960 --> 00:34:04,220
you get a universality result.

696
00:34:04,220 --> 00:34:06,730
You can fold from
anything into anything.

697
00:34:06,730 --> 00:34:10,975
For trees in 2D, you don't.

698
00:34:10,975 --> 00:34:13,350
There are some trees you can't
get from one configuration

699
00:34:13,350 --> 00:34:14,810
to another.

700
00:34:14,810 --> 00:34:15,889
Which ones?

701
00:34:15,889 --> 00:34:16,580
We don't know.

702
00:34:16,580 --> 00:34:19,989
But at least you don't
get a universality result.

703
00:34:19,989 --> 00:34:23,360
In 3D chains you do not
get a universality result.

704
00:34:23,360 --> 00:34:26,670
And so also for trees,
because that's even harder.

705
00:34:26,670 --> 00:34:30,239
And for 4D, everything's easy.

706
00:34:30,239 --> 00:34:32,780
Also in 5D, any
higher dimension.

707
00:34:32,780 --> 00:34:40,400
Because the intuition here,
at least for this column,

708
00:34:40,400 --> 00:34:42,400
is you think about tying knots.

709
00:34:42,400 --> 00:34:45,027
I have a one-dimensional
linkage here.

710
00:34:45,027 --> 00:34:46,610
Think of it as a
one-dimensional cord.

711
00:34:46,610 --> 00:34:48,190
It's kind of a kinky cord.

712
00:34:48,190 --> 00:34:52,480
It has kinks-- not the
other kind of kinky.

713
00:34:52,480 --> 00:34:54,880
And in two dimensions,
if you draw

714
00:34:54,880 --> 00:34:57,904
a non-self-intersecting
loop, it's never knotted.

715
00:34:57,904 --> 00:34:59,570
You can't draw a knot
in two dimensions.

716
00:34:59,570 --> 00:35:01,040
You can drawn a knot
in three dimensions,

717
00:35:01,040 --> 00:35:03,570
and you cannot draw a knot
in four dimensions or higher.

718
00:35:03,570 --> 00:35:05,410
You may not know that
result, but it's true.

719
00:35:05,410 --> 00:35:09,180
So it matches, but things are
little tricky, even trees.

720
00:35:09,180 --> 00:35:11,430
If this was a piece
of string, you'd

721
00:35:11,430 --> 00:35:15,310
be able to always fold this
piece of a tree-shaped piece

722
00:35:15,310 --> 00:35:16,852
of string into
anything you wanted.

723
00:35:16,852 --> 00:35:18,310
But it's a little
more complicated.

724
00:35:21,460 --> 00:35:26,690
Let me show you a locked
tree, I think, is next.

725
00:35:26,690 --> 00:35:28,950
Yeah.

726
00:35:28,950 --> 00:35:33,090
For a long time, these were
the only known locked trees.

727
00:35:33,090 --> 00:35:37,087
These are configurations of tree
linkages that cannot reach some

728
00:35:37,087 --> 00:35:37,920
other configuration.

729
00:35:37,920 --> 00:35:40,050
In fact, they can
barely move at all.

730
00:35:40,050 --> 00:35:42,550
It's less obvious
for some of them.

731
00:35:42,550 --> 00:35:45,570
But say, in this
top left one, you

732
00:35:45,570 --> 00:35:49,820
have these little sort of petals
tucked into their armpits,

733
00:35:49,820 --> 00:35:52,880
I guess, and you can't
get any of those arms

734
00:35:52,880 --> 00:35:55,799
open unless you had a
lot of room to open it.

735
00:35:55,799 --> 00:35:57,340
And in order to make
room, you'd have

736
00:35:57,340 --> 00:35:58,964
to squeeze all the
others really tight.

737
00:35:58,964 --> 00:36:01,260
And if you draw this
example tight enough,

738
00:36:01,260 --> 00:36:04,530
also none of the arms can
get compressed very much.

739
00:36:04,530 --> 00:36:06,460
And so it's locked.

740
00:36:06,460 --> 00:36:09,030
And this is one of the first
examples actually discovered

741
00:36:09,030 --> 00:36:11,910
in 1998, and publication
took a while.

742
00:36:11,910 --> 00:36:14,332
There are a few others,
which you see here.

743
00:36:14,332 --> 00:36:15,790
This one's kind of
crazy because it

744
00:36:15,790 --> 00:36:19,900
has only one vertex with
three incident bars.

745
00:36:19,900 --> 00:36:21,400
Everybody else is like a chain.

746
00:36:21,400 --> 00:36:23,980
So it's like three chains
joined together at that point,

747
00:36:23,980 --> 00:36:26,257
and still it is locked.

748
00:36:26,257 --> 00:36:28,090
And for a long time,
these were the examples

749
00:36:28,090 --> 00:36:29,298
we would always carry around.

750
00:36:29,298 --> 00:36:31,350
These are the ones that
appear in the textbook.

751
00:36:31,350 --> 00:36:33,230
But I thought it would
be neat to see, well,

752
00:36:33,230 --> 00:36:34,890
are there any simpler examples?

753
00:36:34,890 --> 00:36:37,530
And last time this class
was offered three years ago,

754
00:36:37,530 --> 00:36:40,780
we found what is believed
to be the smallest

755
00:36:40,780 --> 00:36:43,780
locked tree in existence.

756
00:36:43,780 --> 00:36:51,500
It has 1, 2, 3, 4, 5, 6, 7,
8 bars, if I counted right.

757
00:36:51,500 --> 00:36:54,820
And it looks curved here,
just to make it easier to see.

758
00:36:54,820 --> 00:36:56,730
But, in fact, you could
straighten these out

759
00:36:56,730 --> 00:36:57,950
and it's still locked.

760
00:36:57,950 --> 00:37:02,400
If you squeeze these
little regions down,

761
00:37:02,400 --> 00:37:04,550
they'd be very tight.

762
00:37:04,550 --> 00:37:07,290
So that was with a bunch of
students from this class.

763
00:37:10,440 --> 00:37:11,560
Cool.

764
00:37:11,560 --> 00:37:14,380
So that gives you some
idea of this answer.

765
00:37:17,320 --> 00:37:23,930
Maybe I'll draw you
a picture for chains,

766
00:37:23,930 --> 00:37:25,055
because it's really simple.

767
00:37:31,040 --> 00:37:42,160
OK, imagine tying a knot, but
don't actually close the loops.

768
00:37:42,160 --> 00:37:44,690
And make these end lengths
really, really long.

769
00:37:44,690 --> 00:37:46,450
We call this the
knitting needles example,

770
00:37:46,450 --> 00:37:48,991
because the intuition was you
have two long knitting needles,

771
00:37:48,991 --> 00:37:52,270
and then a very short cord
connecting them in a knot.

772
00:37:52,270 --> 00:37:53,740
Mathematically,
this is not a knot,

773
00:37:53,740 --> 00:37:56,590
because if it were string, you
could untie it, no problem.

774
00:37:56,590 --> 00:38:00,130
But because of these really
long bars, you can't untie it.

775
00:38:00,130 --> 00:38:03,910
So that's why 3D is hard.

776
00:38:03,910 --> 00:38:05,690
Or one example of
why 3D is hard.

777
00:38:05,690 --> 00:38:07,290
It's pretty much
our only example.

778
00:38:07,290 --> 00:38:10,260
It is the smallest example, and
we can prove things about it.

779
00:38:10,260 --> 00:38:14,810
But there's a pretty
fascinating open question here,

780
00:38:14,810 --> 00:38:20,880
I would say, which
is, characterize

781
00:38:20,880 --> 00:38:22,450
these bad examples.

782
00:38:22,450 --> 00:38:32,064
Which 3D chains and
which 2D trees have

783
00:38:32,064 --> 00:38:32,980
locked configurations?

784
00:38:38,547 --> 00:38:40,880
And all that means is that
there are two configurations,

785
00:38:40,880 --> 00:38:45,020
A and B, for which you
cannot get from A to B.

786
00:38:45,020 --> 00:38:47,500
So this is an example.

787
00:38:47,500 --> 00:38:48,990
Those locked trees
are an example.

788
00:38:48,990 --> 00:38:52,260
It'd be really fascinating
if you could do this.

789
00:38:52,260 --> 00:38:54,960
I would guess that this is a
hard problem, but I don't know.

790
00:38:54,960 --> 00:38:57,210
It's hard to know
whether it's hard.

791
00:39:04,490 --> 00:39:07,370
It'd be nice to understand
3D chains in particular,

792
00:39:07,370 --> 00:39:10,830
because they relate--
and 3D trees.

793
00:39:10,830 --> 00:39:11,470
Oh, sorry.

794
00:39:11,470 --> 00:39:12,560
I have more animations.

795
00:39:12,560 --> 00:39:13,740
I forgot.

796
00:39:13,740 --> 00:39:16,250
Let me show you some pretty
pictures for this result.

797
00:39:16,250 --> 00:39:20,960
This was actually my PhD
thesis, way back in 2001.

798
00:39:20,960 --> 00:39:24,110
And this is a more
modern algorithm

799
00:39:24,110 --> 00:39:25,840
for solving this problem.

800
00:39:25,840 --> 00:39:27,644
I give you some
complicated polygon.

801
00:39:27,644 --> 00:39:29,310
First thing you want
to know is, can you

802
00:39:29,310 --> 00:39:31,591
unfold it into a
nice convex shape?

803
00:39:31,591 --> 00:39:34,090
Once you get there, you could
refolded into some other shape

804
00:39:34,090 --> 00:39:37,130
by playing one of these
motions backwards.

805
00:39:37,130 --> 00:39:39,750
So that's how you
unfold some teeth.

806
00:39:42,400 --> 00:39:44,970
Here's one of those tree
examples, but doubled.

807
00:39:44,970 --> 00:39:47,560
For a while, people thought
that might still be locked,

808
00:39:47,560 --> 00:39:48,620
but it's not.

809
00:39:48,620 --> 00:39:52,290
Can do this crazy fivefold
rotationally symmetric motion

810
00:39:52,290 --> 00:39:53,530
to unfold that thing.

811
00:39:53,530 --> 00:39:55,759
We're zooming in
and out, so it looks

812
00:39:55,759 --> 00:39:57,550
like things are getting
bigger and smaller.

813
00:39:57,550 --> 00:40:00,008
But in fact, each of these bars
is staying the same length,

814
00:40:00,008 --> 00:40:02,656
and they're never
crossing each other.

815
00:40:02,656 --> 00:40:04,280
Here's a much more
complicated example.

816
00:40:04,280 --> 00:40:05,500
This is the first
algorithm that could

817
00:40:05,500 --> 00:40:07,020
handle examples of this size.

818
00:40:07,020 --> 00:40:10,030
It's pretty fast.

819
00:40:10,030 --> 00:40:14,990
That's like it's
going to come back.

820
00:40:14,990 --> 00:40:17,500
You can do it.

821
00:40:17,500 --> 00:40:20,230
It's like a spider.

822
00:40:20,230 --> 00:40:21,990
Spooky.

823
00:40:21,990 --> 00:40:24,735
I think it has 500
vertices, and it probably

824
00:40:24,735 --> 00:40:26,110
took a couple
minutes to compute.

825
00:40:26,110 --> 00:40:27,692
I'm not computing it live here.

826
00:40:27,692 --> 00:40:29,150
There's an applet
on the web if you

827
00:40:29,150 --> 00:40:30,782
want to pick your
favorite polygon

828
00:40:30,782 --> 00:40:31,990
and run this algorithm on it.

829
00:40:31,990 --> 00:40:35,630
It doesn't make quite
movies like in this style,

830
00:40:35,630 --> 00:40:40,382
but it will show
you how it unfolds.

831
00:40:40,382 --> 00:40:41,465
We call this the tentacle.

832
00:40:45,959 --> 00:40:48,250
Yeah, so we'll talk about
this algorithm, how it works.

833
00:40:48,250 --> 00:40:49,140
There's a couple
other algorithms

834
00:40:49,140 --> 00:40:50,181
for solving this problem.

835
00:40:50,181 --> 00:40:51,830
It's pretty cool,
and you could imagine

836
00:40:51,830 --> 00:40:54,460
using this for planning the
motion of a robotic arm in two

837
00:40:54,460 --> 00:40:55,360
dimensions.

838
00:40:55,360 --> 00:40:57,830
But in three dimensions,
things are a lot harder.

839
00:40:57,830 --> 00:41:00,340
We lack good algorithms.

840
00:41:00,340 --> 00:41:02,374
I would like to study
four dimensions actually.

841
00:41:02,374 --> 00:41:04,040
There's some pretty
neat questions here,

842
00:41:04,040 --> 00:41:05,820
but I'm not supposed
to talk about that.

843
00:41:05,820 --> 00:41:07,060
It's not on my list.

844
00:41:07,060 --> 00:41:10,940
I've got to move quickly, but
we're doing all right on time.

845
00:41:10,940 --> 00:41:13,790
What I wanted to show
you was a protein.

846
00:41:13,790 --> 00:41:16,650
This is a particular
enzyme protein

847
00:41:16,650 --> 00:41:21,740
called hexokinase, whatever.

848
00:41:21,740 --> 00:41:23,030
Embarrassing myself.

849
00:41:23,030 --> 00:41:24,990
It's a particularly
complicated one.

850
00:41:24,990 --> 00:41:28,430
It is one of a few I could
find a nice free image of.

851
00:41:28,430 --> 00:41:32,060
But you can see closely,
if you look closely here,

852
00:41:32,060 --> 00:41:33,360
it's a linkage.

853
00:41:33,360 --> 00:41:35,330
And this one actually
has lots of cycles.

854
00:41:35,330 --> 00:41:38,750
But the backbone of a
protein is like a tree,

855
00:41:38,750 --> 00:41:41,640
and so it fits into
this kind of world.

856
00:41:41,640 --> 00:41:43,770
Unfortunately it
fits in this world

857
00:41:43,770 --> 00:41:46,521
of three-dimensional trees,
which are really hard to fold.

858
00:41:46,521 --> 00:41:48,270
Or three-dimensional
chains, if you really

859
00:41:48,270 --> 00:41:49,714
just look at the backbone.

860
00:41:49,714 --> 00:41:51,880
They're a little bit more
complicated than the sorts

861
00:41:51,880 --> 00:41:53,421
of linkages we're
talking about here,

862
00:41:53,421 --> 00:41:56,550
but it makes it even harder
to fold things like this.

863
00:41:56,550 --> 00:41:58,840
But there's something
special about proteins

864
00:41:58,840 --> 00:42:00,920
that makes them
fold really well.

865
00:42:00,920 --> 00:42:04,530
That enzyme over there is
in every living organism

866
00:42:04,530 --> 00:42:08,310
we have ever tested for the
existence of that thing, which

867
00:42:08,310 --> 00:42:11,050
I assume is everything.

868
00:42:11,050 --> 00:42:13,520
And yet it's folding.

869
00:42:13,520 --> 00:42:15,930
It's produced by this
machine-- the Ribosome,

870
00:42:15,930 --> 00:42:18,990
you probably know about it--
in a sort of straight state,

871
00:42:18,990 --> 00:42:21,440
and then it folds into
this shape pretty reliably,

872
00:42:21,440 --> 00:42:25,584
less than a second, usually
like within nanoseconds.

873
00:42:25,584 --> 00:42:27,500
So it's really hard to
watch what's happening.

874
00:42:27,500 --> 00:42:29,490
It's very hot and jiggly,
so it's a little hard

875
00:42:29,490 --> 00:42:32,400
to see what's actually
going on in the real thing.

876
00:42:32,400 --> 00:42:36,290
But somehow, these kinds
of barriers to foldability

877
00:42:36,290 --> 00:42:37,440
don't happen.

878
00:42:37,440 --> 00:42:41,430
Maybe that's because evolution
found the right things,

879
00:42:41,430 --> 00:42:46,460
or maybe it's because
protein chains don't really

880
00:42:46,460 --> 00:42:48,250
look like this.

881
00:42:48,250 --> 00:42:51,000
They don't have these super
long bars and super short bars.

882
00:42:51,000 --> 00:42:53,590
If you've ever played with a
chemistry set, all of the bars

883
00:42:53,590 --> 00:42:56,534
are within a factor of
like 1.5 of each other.

884
00:42:56,534 --> 00:42:58,450
So there's lots of cool
mathematical questions

885
00:42:58,450 --> 00:42:59,830
that come out of
protein folding,

886
00:42:59,830 --> 00:43:01,770
and we will talk
about the ones I know.

887
00:43:01,770 --> 00:43:03,290
I'd love to find more.

888
00:43:03,290 --> 00:43:05,050
But the ones I know
I will talk about.

889
00:43:05,050 --> 00:43:05,960
And that's linkages.

890
00:43:05,960 --> 00:43:08,910
Let me move on to paper,
unless there are questions.

891
00:43:11,625 --> 00:43:13,080
All right.

892
00:43:13,080 --> 00:43:15,560
Let's go over here.

893
00:43:15,560 --> 00:43:19,510
So we've seen some universality
results, some hardness results.

894
00:43:19,510 --> 00:43:20,892
I didn't go into them.

895
00:43:20,892 --> 00:43:21,850
Folding these linkages.

896
00:43:26,630 --> 00:43:27,745
Let's do that for paper.

897
00:43:36,170 --> 00:43:37,670
I think first up I
have foldability.

898
00:43:40,435 --> 00:43:41,810
And then we'll
talk about design.

899
00:43:44,720 --> 00:43:48,960
So there are lots of
questions in both.

900
00:43:48,960 --> 00:43:51,340
But for foldability, the
sort of first question people

901
00:43:51,340 --> 00:43:54,740
like to ask is, which
crease patterns fold flat?

902
00:44:04,620 --> 00:44:07,390
So a crease pattern
is just a graph

903
00:44:07,390 --> 00:44:08,490
drawn on a piece of paper.

904
00:44:08,490 --> 00:44:11,220
So you have some
collection of lines,

905
00:44:11,220 --> 00:44:14,240
and I think that will fold flat.

906
00:44:14,240 --> 00:44:16,740
But I'm not going to try to
draw something more complicated,

907
00:44:16,740 --> 00:44:18,320
but you could imagine doing it.

908
00:44:22,220 --> 00:44:22,840
I don't know.

909
00:44:22,840 --> 00:44:25,000
That probably is
not flat foldable.

910
00:44:25,000 --> 00:44:25,795
It might be.

911
00:44:25,795 --> 00:44:28,167
It's close.

912
00:44:28,167 --> 00:44:29,500
But that's the sort of question.

913
00:44:29,500 --> 00:44:31,380
If you take some
origami and unfold it,

914
00:44:31,380 --> 00:44:33,710
what kind of
patterns do you get?

915
00:44:33,710 --> 00:44:35,465
Some flat origami,
something that

916
00:44:35,465 --> 00:44:37,230
folds into two
dimensions in the end.

917
00:44:37,230 --> 00:44:39,360
There's some really
nice structure here.

918
00:44:39,360 --> 00:44:42,410
If I drew it right, this
angle plus this angle

919
00:44:42,410 --> 00:44:45,026
should equal 180
degrees, for example.

920
00:44:45,026 --> 00:44:46,150
And that's true everywhere.

921
00:44:46,150 --> 00:44:48,510
And there's all sorts
of cool properties.

922
00:44:48,510 --> 00:44:52,050
But unfortunately,
this is really hard.

923
00:44:52,050 --> 00:44:54,862
This is NP hard problem.

924
00:44:54,862 --> 00:44:57,670
You've probably at least
heard of P versus NP.

925
00:44:57,670 --> 00:44:59,510
Again, I'll define it later.

926
00:44:59,510 --> 00:45:02,350
But it means probably
there is no good algorithm

927
00:45:02,350 --> 00:45:03,520
to solve this problem.

928
00:45:03,520 --> 00:45:05,530
It's really hard
to figure out which

929
00:45:05,530 --> 00:45:07,025
crease patterns fold flat.

930
00:45:07,025 --> 00:45:10,190
It's kind of annoying.

931
00:45:10,190 --> 00:45:12,570
One good news is
that if you just

932
00:45:12,570 --> 00:45:15,400
have an example like the
original thing I drew,

933
00:45:15,400 --> 00:45:19,940
which is like this-- so it
has one vertex and a bunch

934
00:45:19,940 --> 00:45:23,830
of creases emanating out,
that picture we understand.

935
00:45:23,830 --> 00:45:29,700
So it's easy for
a single vertex.

936
00:45:29,700 --> 00:45:33,610
That may seem kind of trivial,
but it's actually really useful

937
00:45:33,610 --> 00:45:36,540
because it lets you understand
the local behavior around one

938
00:45:36,540 --> 00:45:37,052
vertex.

939
00:45:37,052 --> 00:45:38,510
If you check that
for every vertex,

940
00:45:38,510 --> 00:45:40,801
you don't know that the whole
thing folds, but at least

941
00:45:40,801 --> 00:45:43,280
you know it mostly
folds, at least locally.

942
00:45:43,280 --> 00:45:46,770
And you can't tell whether
it globally folds correctly,

943
00:45:46,770 --> 00:45:48,890
because that's NP hard.

944
00:45:48,890 --> 00:45:52,174
There are so many questions here
like, what about two vertices?

945
00:45:52,174 --> 00:45:53,090
No one's studied that.

946
00:45:53,090 --> 00:45:57,055
I think it's polynomial,
but well, it's

947
00:45:57,055 --> 00:46:01,120
certainly polynomial,
but I think

948
00:46:01,120 --> 00:46:02,520
you could do it in linear time.

949
00:46:02,520 --> 00:46:04,103
Anyway, there's lots
of open questions

950
00:46:04,103 --> 00:46:07,070
there I haven't
even listed here.

951
00:46:07,070 --> 00:46:10,340
One of the bigger open
questions is a particular kind

952
00:46:10,340 --> 00:46:14,430
of crease pattern, which you may
have encountered in real life

953
00:46:14,430 --> 00:46:16,792
refolding your roadmaps.

954
00:46:16,792 --> 00:46:18,880
The saying goes, the
easiest way to refold

955
00:46:18,880 --> 00:46:21,190
your road map is differently.

956
00:46:21,190 --> 00:46:26,670
So suppose you have your road
map, and each of these creases

957
00:46:26,670 --> 00:46:28,229
is marked.

958
00:46:28,229 --> 00:46:30,020
I'm not going to mark
all them because it's

959
00:46:30,020 --> 00:46:31,970
a little messy with
black and white chalk.

960
00:46:31,970 --> 00:46:34,170
But some of them are marked
mountain, some of them

961
00:46:34,170 --> 00:46:39,830
are marked valley, meaning
you fold-- this is valley.

962
00:46:39,830 --> 00:46:41,020
This is mountain.

963
00:46:41,020 --> 00:46:43,896
So they just have a relative
orientation to each other.

964
00:46:43,896 --> 00:46:45,520
Suppose you look
carefully at your map.

965
00:46:45,520 --> 00:46:48,230
You can recover which
way it was folded last,

966
00:46:48,230 --> 00:46:49,320
hopefully correctly.

967
00:46:49,320 --> 00:46:52,730
And then you want to find,
does that thing fold flat?

968
00:46:52,730 --> 00:46:54,770
Sometimes it does,
sometimes it doesn't.

969
00:46:54,770 --> 00:46:58,600
For two by n maps, we don't
know whether we can even

970
00:46:58,600 --> 00:47:00,430
detect this with an
efficient algorithm.

971
00:47:00,430 --> 00:47:04,329
Can I decide whether
this thing folds flat

972
00:47:04,329 --> 00:47:06,120
using those creases
and those orientations?

973
00:47:06,120 --> 00:47:07,210
For two by n, it's open.

974
00:47:07,210 --> 00:47:09,210
For one by n, it's easy.

975
00:47:09,210 --> 00:47:10,870
Obviously for bigger,
it's also open.

976
00:47:10,870 --> 00:47:12,400
Two by n is the smallest.

977
00:47:12,400 --> 00:47:14,410
This is a really
annoying problem.

978
00:47:14,410 --> 00:47:15,610
Worked on it many times.

979
00:47:15,610 --> 00:47:18,100
It's very difficult.

980
00:47:18,100 --> 00:47:21,660
All right, so that's a quick
overview of foldability.

981
00:47:21,660 --> 00:47:24,400
It's hard, but there are a lot
of interesting special cases

982
00:47:24,400 --> 00:47:26,750
where we might be
able to solve it.

983
00:47:31,210 --> 00:47:34,300
Let's move on to design,
which is probably

984
00:47:34,300 --> 00:47:38,330
where most of the action
is in the origami world,

985
00:47:38,330 --> 00:47:39,760
in the mathematical
origami world.

986
00:47:46,990 --> 00:47:52,770
So and there are
tons of results,

987
00:47:52,770 --> 00:47:56,190
and I had a much
longer list initially,

988
00:47:56,190 --> 00:47:59,245
but I trimmed just for
brevity down to a few things.

989
00:48:05,935 --> 00:48:08,130
The central question--
at least it's

990
00:48:08,130 --> 00:48:13,240
been considered so far-- in
computational origami design

991
00:48:13,240 --> 00:48:14,485
is, what shapes can you make?

992
00:48:14,485 --> 00:48:16,610
You could imagine other
properties than just shape,

993
00:48:16,610 --> 00:48:18,750
if you want stability
in your folding,

994
00:48:18,750 --> 00:48:21,630
lots of practical things--
not much thickness,

995
00:48:21,630 --> 00:48:23,600
not too many creases, whatever.

996
00:48:23,600 --> 00:48:27,810
But shape has sort of
been the fun centerpiece.

997
00:48:27,810 --> 00:48:34,050
And there's an early result
that says universally,

998
00:48:34,050 --> 00:48:35,760
you can make anything you want.

999
00:48:35,760 --> 00:48:37,530
And in the
mathematical world, you

1000
00:48:37,530 --> 00:48:39,240
might try to make a polygon.

1001
00:48:39,240 --> 00:48:42,250
Try to make a
silhouette or something.

1002
00:48:42,250 --> 00:48:46,630
You could make a
polyhedron in 3D.

1003
00:48:46,630 --> 00:48:50,860
Maybe you want to wrap a box,
so you want to fold a box,

1004
00:48:50,860 --> 00:48:54,480
or you want to fold a
3D model of a dragon.

1005
00:48:54,480 --> 00:48:56,390
That's all possible.

1006
00:48:56,390 --> 00:48:59,810
And for fun, you can even
make any two-color pattern

1007
00:48:59,810 --> 00:49:02,590
on either of those things.

1008
00:49:02,590 --> 00:49:06,440
So I have a tiny example
here, which is-- yeah,

1009
00:49:06,440 --> 00:49:10,170
it's still alive-- a four
by four checkerboard.

1010
00:49:10,170 --> 00:49:13,680
And it's made from
one square paper.

1011
00:49:17,030 --> 00:49:20,600
It's white on one side
and red on the other.

1012
00:49:20,600 --> 00:49:23,479
So you get some idea of
making two-color patterns.

1013
00:49:23,479 --> 00:49:24,770
This one's pretty easy to fold.

1014
00:49:24,770 --> 00:49:28,660
It even folds itself almost.

1015
00:49:28,660 --> 00:49:29,330
That's great.

1016
00:49:29,330 --> 00:49:32,012
Usually when I give a
class like to high school

1017
00:49:32,012 --> 00:49:34,470
students or something, I say,
could you refold this for me?

1018
00:49:34,470 --> 00:49:36,180
And they're like, oh, no!

1019
00:49:36,180 --> 00:49:37,580
But it actually does it itself.

1020
00:49:37,580 --> 00:49:39,110
It's great.

1021
00:49:39,110 --> 00:49:41,200
Like magic.

1022
00:49:41,200 --> 00:49:42,500
So you could try that at home.

1023
00:49:42,500 --> 00:49:43,580
Just take a square paper.

1024
00:49:43,580 --> 00:49:44,080
It works.

1025
00:49:48,740 --> 00:49:51,140
So this is great.

1026
00:49:51,140 --> 00:49:52,220
Super general.

1027
00:49:52,220 --> 00:49:54,810
This is something actually
my dad and I proved

1028
00:49:54,810 --> 00:49:57,970
with Joe Mitchell
back in '99, I think.

1029
00:49:57,970 --> 00:49:59,870
I don't remember exactly.

1030
00:49:59,870 --> 00:50:03,460
So beginning of modern
computational origami design.

1031
00:50:03,460 --> 00:50:05,820
But the way that we do
this, while algorithmic,

1032
00:50:05,820 --> 00:50:07,720
is completely impractical.

1033
00:50:07,720 --> 00:50:09,705
We'd never want to
use the foldings that

1034
00:50:09,705 --> 00:50:12,520
are designed by this algorithm,
unless you're starting out

1035
00:50:12,520 --> 00:50:14,370
with a big ticker tape of paper.

1036
00:50:14,370 --> 00:50:15,629
Then it's a great method.

1037
00:50:15,629 --> 00:50:18,170
But if you're starting with a
square, the first thing it does

1038
00:50:18,170 --> 00:50:20,500
is fold it down to a
tiny little narrow strip,

1039
00:50:20,500 --> 00:50:21,920
and then wraps the shape.

1040
00:50:21,920 --> 00:50:23,170
And we'll see how that's done.

1041
00:50:23,170 --> 00:50:24,878
It's interesting to
do it mathematically,

1042
00:50:24,878 --> 00:50:27,330
but it's not how you want
to do it practically.

1043
00:50:27,330 --> 00:50:30,610
Good news is-- that used to be
sort of the end of the story.

1044
00:50:30,610 --> 00:50:34,320
But now you can even do
it pretty practically--

1045
00:50:34,320 --> 00:50:40,730
practically is a
relative term-- using

1046
00:50:40,730 --> 00:50:43,510
something called Origamizer.

1047
00:50:43,510 --> 00:50:47,389
And if you've seen the poster
of this class, you've seen--

1048
00:50:47,389 --> 00:50:49,180
and if I'm lucky, it's
even the next slide,

1049
00:50:49,180 --> 00:50:54,080
yes it is-- this rabbit--
bunny, I should say.

1050
00:50:54,080 --> 00:50:55,250
This is the Stanford bunny.

1051
00:50:55,250 --> 00:50:57,660
The original Stanford
bunny is in the top right.

1052
00:50:57,660 --> 00:50:59,187
That's a classic model.

1053
00:50:59,187 --> 00:51:01,770
Everybody in computer graphics
does something with that bunny.

1054
00:51:01,770 --> 00:51:02,960
It has a zillion triangles.

1055
00:51:02,960 --> 00:51:04,646
I don't know offhand how many.

1056
00:51:04,646 --> 00:51:06,020
It's been simplified
here to make

1057
00:51:06,020 --> 00:51:11,296
it feasible to fold to
this mesh of triangles.

1058
00:51:11,296 --> 00:51:13,670
So the input to the algorithm
was this mesh of triangles.

1059
00:51:13,670 --> 00:51:15,970
The output of the algorithm--
and it's a real program,

1060
00:51:15,970 --> 00:51:18,500
you can go and download
it right now for free--

1061
00:51:18,500 --> 00:51:20,189
is this crease pattern.

1062
00:51:20,189 --> 00:51:21,980
Now it doesn't look
like a square or paper,

1063
00:51:21,980 --> 00:51:23,646
but if you start with
a square of paper,

1064
00:51:23,646 --> 00:51:25,540
you just fold away
the excess stuff.

1065
00:51:25,540 --> 00:51:27,690
You'll get to a piece
of paper like this.

1066
00:51:27,690 --> 00:51:31,950
And then you just fold along
all those little creases there,

1067
00:51:31,950 --> 00:51:34,240
and eight hours later
you have this bunny.

1068
00:51:34,240 --> 00:51:36,630
This is a real photograph
of a folded bunny

1069
00:51:36,630 --> 00:51:38,970
by Tomohiro Tachi, who
designed this thing.

1070
00:51:38,970 --> 00:51:42,920
He came up with the original
algorithm and computer program.

1071
00:51:42,920 --> 00:51:44,500
In the last few
years, we've been

1072
00:51:44,500 --> 00:51:46,700
proving this algorithm
actually always works.

1073
00:51:46,700 --> 00:51:48,908
We still don't know how to
prove that it's efficient,

1074
00:51:48,908 --> 00:51:51,350
or even what
efficient should mean.

1075
00:51:51,350 --> 00:51:53,780
But in practice it's super good.

1076
00:51:53,780 --> 00:51:55,540
If you look closely,
the white parts

1077
00:51:55,540 --> 00:51:56,660
are the parts of
the paper that you

1078
00:51:56,660 --> 00:51:58,076
need to use in
order to make this.

1079
00:51:58,076 --> 00:51:59,740
They are the triangles
of the surface.

1080
00:51:59,740 --> 00:52:01,560
These grey regions
are the excess.

1081
00:52:01,560 --> 00:52:02,850
And it's pretty small.

1082
00:52:02,850 --> 00:52:06,440
I don't know, maybe 50% of the
area is used in a useful way

1083
00:52:06,440 --> 00:52:08,250
here, which is a lot.

1084
00:52:08,250 --> 00:52:11,330
Whereas this method
over here would use

1085
00:52:11,330 --> 00:52:14,650
like a one millionth
of a percent

1086
00:52:14,650 --> 00:52:15,740
of the paper or something.

1087
00:52:15,740 --> 00:52:16,845
A very tiny amount.

1088
00:52:16,845 --> 00:52:18,262
The Origamizer is
super efficient.

1089
00:52:18,262 --> 00:52:20,470
We don't know how to prove
that it's super efficient,

1090
00:52:20,470 --> 00:52:22,260
but at least we can
prove that it works.

1091
00:52:22,260 --> 00:52:25,380
It will make any 3D
polyhedron you want.

1092
00:52:25,380 --> 00:52:28,420
And in practice, it
seems really good at it.

1093
00:52:28,420 --> 00:52:32,710
Now it's a very different
style from typical origami,

1094
00:52:32,710 --> 00:52:35,770
so this has not yet hit the
sculpture world, let's say.

1095
00:52:35,770 --> 00:52:38,595
But I think it opens the door
to a lot of new possibilities.

1096
00:52:41,320 --> 00:52:44,880
Traditionally
though-- traditionally

1097
00:52:44,880 --> 00:52:48,020
before this Origamizer
or in practice

1098
00:52:48,020 --> 00:52:50,970
today, a much more
commonly used approach

1099
00:52:50,970 --> 00:52:54,230
to computational
origami design is

1100
00:52:54,230 --> 00:52:56,882
something called
the tree method.

1101
00:52:56,882 --> 00:52:58,590
I'm just going to call
it TreeMaker here.

1102
00:52:58,590 --> 00:53:00,256
TreeMaker's the name
of the program that

1103
00:53:00,256 --> 00:53:04,700
implements the tree method,
and it's made by Robert Lang.

1104
00:53:04,700 --> 00:53:13,164
And this is sort of making
stick figures, which

1105
00:53:13,164 --> 00:53:14,080
sounds a little silly.

1106
00:53:14,080 --> 00:53:19,250
But TreeMaker or the
tree method is powerful

1107
00:53:19,250 --> 00:53:25,500
for if I want to make what's
called an origami base, that

1108
00:53:25,500 --> 00:53:31,229
has a lot of limbs in different
places of different lengths--

1109
00:53:31,229 --> 00:53:33,020
maybe you want to make
something like that.

1110
00:53:33,020 --> 00:53:35,960
I don't know.

1111
00:53:35,960 --> 00:53:38,380
The typical origami base might
look something like this.

1112
00:53:38,380 --> 00:53:39,720
Maybe you want to make a lizard.

1113
00:53:39,720 --> 00:53:43,210
It has a head, some forearms,
some really big hind legs,

1114
00:53:43,210 --> 00:53:46,090
a little tail, and some body.

1115
00:53:46,090 --> 00:53:52,270
You can abstract
this into a tree

1116
00:53:52,270 --> 00:53:54,590
and then say, well can
I fold a piece of paper

1117
00:53:54,590 --> 00:53:56,330
into some shape
whose projection is

1118
00:53:56,330 --> 00:53:59,480
that tree, with all
the right edge links?

1119
00:53:59,480 --> 00:54:00,550
And the answer is yes.

1120
00:54:00,550 --> 00:54:02,470
I mean, you can make
anything, of course.

1121
00:54:02,470 --> 00:54:06,420
But what TreeMaker tries to do
is find the most efficient way

1122
00:54:06,420 --> 00:54:08,002
to make that thing.

1123
00:54:08,002 --> 00:54:09,960
Unfortunately, finding
the most efficient way--

1124
00:54:09,960 --> 00:54:13,100
like the smallest square
paper, so the most paper

1125
00:54:13,100 --> 00:54:16,300
usage possible to make
something with that projection--

1126
00:54:16,300 --> 00:54:18,620
finding that is NP complete.

1127
00:54:18,620 --> 00:54:21,710
We just proved that this
year with Robert Lang

1128
00:54:21,710 --> 00:54:23,420
and Sandor Fekete.

1129
00:54:23,420 --> 00:54:26,250
But there's a theory.

1130
00:54:26,250 --> 00:54:28,184
There's a hard problem
you have to solve,

1131
00:54:28,184 --> 00:54:29,600
something related
to disk packing.

1132
00:54:29,600 --> 00:54:31,650
If you solve it, you
find the best way.

1133
00:54:31,650 --> 00:54:34,460
In practice, use some heuristics
which are really good.

1134
00:54:34,460 --> 00:54:36,682
And usually you
find the optimal way

1135
00:54:36,682 --> 00:54:37,890
to make a given stick figure.

1136
00:54:37,890 --> 00:54:41,520
But for super complicated
things, it may not be perfect.

1137
00:54:41,520 --> 00:54:45,710
Now a lot of people use this
method for designing origami.

1138
00:54:45,710 --> 00:54:50,950
I have a slide of cool
origami, just to show you.

1139
00:54:50,950 --> 00:54:52,750
Not all of this uses
the tree method.

1140
00:54:56,360 --> 00:54:58,360
These three designs
by Brian Chan,

1141
00:54:58,360 --> 00:55:02,430
who was an MIT grad student,
graduated last year--

1142
00:55:02,430 --> 00:55:05,440
these are all designed
partly using the tree method.

1143
00:55:05,440 --> 00:55:07,400
I'm sure they're more
complicated than that.

1144
00:55:07,400 --> 00:55:10,860
But to get the initial structure
of the arms in the right place,

1145
00:55:10,860 --> 00:55:13,306
the legs in the right place,
the head, body segments,

1146
00:55:13,306 --> 00:55:14,680
fingers-- some of
these guys have

1147
00:55:14,680 --> 00:55:20,020
fingers-- all that stuff
is done using tree theory,

1148
00:55:20,020 --> 00:55:22,790
almost certainly not with
TreeMaker, the program.

1149
00:55:22,790 --> 00:55:25,960
Most origami designers do it by
hand, in their head, or drawing

1150
00:55:25,960 --> 00:55:29,260
pictures with a drawing
program, to design out the base.

1151
00:55:29,260 --> 00:55:31,410
Then they fold by
hand everything.

1152
00:55:31,410 --> 00:55:35,070
But still, the math is in there.

1153
00:55:35,070 --> 00:55:38,120
These guys are not yet
mathematically analyzed,

1154
00:55:38,120 --> 00:55:38,730
let's say.

1155
00:55:38,730 --> 00:55:45,500
They're experiments by
[? Garn ?] and Joel Cooper.

1156
00:55:45,500 --> 00:55:49,742
[? Garn ?] is a computer
scientist slash mathematician,

1157
00:55:49,742 --> 00:55:51,950
but he doesn't understand
these yet, and sort of he's

1158
00:55:51,950 --> 00:55:52,970
trying to figure it out.

1159
00:55:52,970 --> 00:55:54,860
So there's a lot of
interesting questions,

1160
00:55:54,860 --> 00:55:59,430
and you get some really
cool sculpture as a result,

1161
00:55:59,430 --> 00:56:01,360
out of this world.

1162
00:56:01,360 --> 00:56:04,380
So this is something
we've also been working on

1163
00:56:04,380 --> 00:56:07,790
with my dad and Robert Lang,
proving that this actually

1164
00:56:07,790 --> 00:56:08,770
works.

1165
00:56:08,770 --> 00:56:10,230
It's been around
for a long time,

1166
00:56:10,230 --> 00:56:12,941
but it turns out to
be really complicated

1167
00:56:12,941 --> 00:56:14,190
to prove that it always works.

1168
00:56:14,190 --> 00:56:17,190
But it looks like it
does, so stay tuned.

1169
00:56:17,190 --> 00:56:19,716
Some time we will
publish that paper.

1170
00:56:19,716 --> 00:56:21,860
It's still in process.

1171
00:56:21,860 --> 00:56:23,410
All right.

1172
00:56:23,410 --> 00:56:25,717
I have some more fun things.

1173
00:56:25,717 --> 00:56:27,050
This is just a random selection.

1174
00:56:27,050 --> 00:56:29,990
There's a ton of work here.

1175
00:56:29,990 --> 00:56:32,830
One of my favorites-- this was
my first result in this field--

1176
00:56:32,830 --> 00:56:35,880
is the folding cut
problem, theorem, whatever.

1177
00:56:35,880 --> 00:56:42,490
So you start with a rectangle
of paper, and you fold it flat.

1178
00:56:42,490 --> 00:56:44,060
Now you take your
scissors-- I know

1179
00:56:44,060 --> 00:56:47,130
this is blasphemy
for most origamists.

1180
00:56:47,130 --> 00:56:48,840
But I'm not going to
make a lot of cuts.

1181
00:56:48,840 --> 00:56:52,410
I'm just going to make
one complete straight cut.

1182
00:56:52,410 --> 00:56:55,920
I get two pieces in this case,
and then I unfold the pieces

1183
00:56:55,920 --> 00:56:59,120
and see what I get.

1184
00:56:59,120 --> 00:57:02,540
In this case I
get a little swan.

1185
00:57:02,540 --> 00:57:05,320
And the general theorem is you
can make any polygon you want.

1186
00:57:05,320 --> 00:57:07,310
In fact, you can
make one cut and make

1187
00:57:07,310 --> 00:57:09,360
any collection of
polygons you want.

1188
00:57:09,360 --> 00:57:11,740
So like on my website,
you can make the MIT logo

1189
00:57:11,740 --> 00:57:15,140
with all the little rectangles
with one complete straight cut.

1190
00:57:15,140 --> 00:57:17,430
And the outside
shape will have holes

1191
00:57:17,430 --> 00:57:19,820
exactly where you want them.

1192
00:57:19,820 --> 00:57:21,520
And you can download
this, too, if you

1193
00:57:21,520 --> 00:57:23,550
want to impress
all your friends,

1194
00:57:23,550 --> 00:57:26,630
especially recommended
for, like, kids.

1195
00:57:26,630 --> 00:57:27,877
It's a good magic trick.

1196
00:57:27,877 --> 00:57:29,710
In fact, Harry Houdini
used to do this trick

1197
00:57:29,710 --> 00:57:32,110
in the '20s before he
was an escape artist.

1198
00:57:32,110 --> 00:57:33,580
He did general magic.

1199
00:57:33,580 --> 00:57:34,240
Not this trick.

1200
00:57:34,240 --> 00:57:35,770
He made stars.

1201
00:57:35,770 --> 00:57:39,037
And we thought about it, and we
proved you could make anything.

1202
00:57:39,037 --> 00:57:39,620
So that's fun.

1203
00:57:39,620 --> 00:57:41,320
Another universality result.

1204
00:57:41,320 --> 00:57:42,394
It's what got us started.

1205
00:57:42,394 --> 00:57:44,310
And it relates to things
like the tree theory,

1206
00:57:44,310 --> 00:57:47,960
in fact, because what
you're really solving here

1207
00:57:47,960 --> 00:57:52,120
is, how do I take a polygon
and-- if I fold this guy back

1208
00:57:52,120 --> 00:57:56,460
up-- how do I make all
the edges of that polygon

1209
00:57:56,460 --> 00:57:58,680
lie along-- this
guy's not very happy,

1210
00:57:58,680 --> 00:58:02,270
get rid of that-- lie
along a straight line.

1211
00:58:02,270 --> 00:58:04,729
That actually turns out to be
important for origami design.

1212
00:58:04,729 --> 00:58:06,353
A lot of these things--
an Origamizer--

1213
00:58:06,353 --> 00:58:07,530
it's all over the place.

1214
00:58:07,530 --> 00:58:11,490
And so this structure
is helpful for solving

1215
00:58:11,490 --> 00:58:15,812
sort of pure origami
problems, not with cutting.

1216
00:58:15,812 --> 00:58:16,770
So what else do I have?

1217
00:58:16,770 --> 00:58:20,060
Curved creases.

1218
00:58:20,060 --> 00:58:23,660
You may not know this,
because most origami

1219
00:58:23,660 --> 00:58:24,940
is straight creases.

1220
00:58:24,940 --> 00:58:27,540
But if you take
a piece of paper,

1221
00:58:27,540 --> 00:58:30,840
you can put a curved
crease into it.

1222
00:58:30,840 --> 00:58:34,530
It's a little hard to do
by hand, but there you go.

1223
00:58:34,530 --> 00:58:37,010
Curved crease.

1224
00:58:37,010 --> 00:58:38,290
Is that allowed?

1225
00:58:38,290 --> 00:58:40,980
What can you make like that?

1226
00:58:40,980 --> 00:58:42,190
It's not too well studied.

1227
00:58:42,190 --> 00:58:44,700
Jeannine Mosely has done some
of the curved crease stuff.

1228
00:58:44,700 --> 00:58:46,760
Early stuff was done
by David Huffman,

1229
00:58:46,760 --> 00:58:50,260
you may know from Huffman codes
in every cell phone, whatever.

1230
00:58:50,260 --> 00:58:53,800
And by Ron Resch.

1231
00:58:53,800 --> 00:58:55,200
That's wrong queuing.

1232
00:58:55,200 --> 00:58:57,067
All right, I'm going
to show that later.

1233
00:58:57,067 --> 00:58:58,900
This are some of the
things that we've made,

1234
00:58:58,900 --> 00:59:02,080
this is my dad and I,
out of curved creases.

1235
00:59:02,080 --> 00:59:06,290
This is kind of a crazy-- it's
like a circular piece of paper,

1236
00:59:06,290 --> 00:59:08,990
but it goes around
twice like a ramp.

1237
00:59:08,990 --> 00:59:11,830
So you make two circles
and then join the ends,

1238
00:59:11,830 --> 00:59:13,780
and you pleat along
concentric circles,

1239
00:59:13,780 --> 00:59:15,460
alternating mountain and valley.

1240
00:59:15,460 --> 00:59:17,442
And you change a few
different parameters,

1241
00:59:17,442 --> 00:59:19,150
and you get these
three different shapes.

1242
00:59:19,150 --> 00:59:22,610
These are in the permanent
collection at MOMA in New York.

1243
00:59:22,610 --> 00:59:26,580
So you can make some really
cool sculpture out of this.

1244
00:59:26,580 --> 00:59:28,160
This is a more
recent piece we did.

1245
00:59:28,160 --> 00:59:34,066
And this is taking just a
regular circular band of paper

1246
00:59:34,066 --> 00:59:35,440
with concentric
circular creases,

1247
00:59:35,440 --> 00:59:41,030
and then taking three of them
and joining them together

1248
00:59:41,030 --> 00:59:42,920
at a few points.

1249
00:59:42,920 --> 00:59:44,900
And then you get
some cool structures.

1250
00:59:44,900 --> 00:59:47,790
These are interesting to us
especially because the paper

1251
00:59:47,790 --> 00:59:49,360
actually does fold itself.

1252
00:59:49,360 --> 00:59:51,020
You put in these
circular creases.

1253
00:59:51,020 --> 00:59:52,900
You just squeeze
a little, and it

1254
00:59:52,900 --> 00:59:55,325
will make these
shapes automatically.

1255
00:59:55,325 --> 00:59:57,730
It's called
self-folding origami.

1256
00:59:57,730 --> 00:59:59,640
And in that spirit,
I will show you

1257
00:59:59,640 --> 01:00:01,980
a different kind of
self-folding origami.

1258
01:00:01,980 --> 01:00:05,280
I think I need to
click this button.

1259
01:00:05,280 --> 01:00:11,180
Which is approach to making
paper that folds itself.

1260
01:00:11,180 --> 01:00:15,830
This is a self-folding sheet.

1261
01:00:15,830 --> 01:00:18,190
It's made of rigid panels
connected by little rubber

1262
01:00:18,190 --> 01:00:22,540
hinges, and there's little
muscles, you could say,

1263
01:00:22,540 --> 01:00:25,680
that are folding
those creases shut.

1264
01:00:25,680 --> 01:00:28,290
By turning on some
electrical signal,

1265
01:00:28,290 --> 01:00:29,970
that thing folds
itself into a boat.

1266
01:00:29,970 --> 01:00:33,890
That same sheet, you can send
it a different electrical signal

1267
01:00:33,890 --> 01:00:36,950
and it will just fold
along these two creases.

1268
01:00:36,950 --> 01:00:39,660
And then there's little
magnets to hold it into place,

1269
01:00:39,660 --> 01:00:41,520
so you can turn off
that electricity now.

1270
01:00:41,520 --> 01:00:42,450
No power.

1271
01:00:42,450 --> 01:00:45,140
Then you send it a third signal,
and it will fold those three

1272
01:00:45,140 --> 01:00:49,140
creases, and you get
a paper airplane.

1273
01:00:49,140 --> 01:00:52,070
No origamist required.

1274
01:00:52,070 --> 01:00:57,089
So the idea here, this is the
programmable matter vision,

1275
01:00:57,089 --> 01:00:59,130
where you could download
hardware in the same way

1276
01:00:59,130 --> 01:01:00,370
you could download software.

1277
01:01:00,370 --> 01:01:05,000
You can make one sheet that
has lots of panels in it,

1278
01:01:05,000 --> 01:01:07,139
enough creases to make
anything you want.

1279
01:01:07,139 --> 01:01:09,430
And then you just push a
button and some of the creases

1280
01:01:09,430 --> 01:01:09,910
turn on.

1281
01:01:09,910 --> 01:01:11,640
Then push a button and
some other creases turn on,

1282
01:01:11,640 --> 01:01:13,640
and it folds into some
complicated origami

1283
01:01:13,640 --> 01:01:15,020
all by itself.

1284
01:01:15,020 --> 01:01:16,950
You could imagine making
this building scale.

1285
01:01:16,950 --> 01:01:19,780
You could imagine making it
nano scale, some scale where

1286
01:01:19,780 --> 01:01:22,280
an origamist can't
do it, or you don't

1287
01:01:22,280 --> 01:01:24,080
have an origamist
to do it for you.

1288
01:01:24,080 --> 01:01:27,420
This would be the
self-refolding, transforming

1289
01:01:27,420 --> 01:01:29,630
robot thing.

1290
01:01:29,630 --> 01:01:32,300
To do it, though, we
needed to prove new theory.

1291
01:01:32,300 --> 01:01:36,965
And this is the world of, call
it, universal hinge patterns.

1292
01:01:42,950 --> 01:01:46,125
And this is the topic of a
master's thesis just finished

1293
01:01:46,125 --> 01:01:47,750
last week, I think,
by [? Viva Vadia ?]

1294
01:01:47,750 --> 01:01:51,440
where-- and maybe
we'll get him to talk

1295
01:01:51,440 --> 01:01:54,560
about it at some point--
it's one crease pattern.

1296
01:01:54,560 --> 01:01:58,310
In that case we were using
a crease pattern called

1297
01:01:58,310 --> 01:02:03,080
box pleating, where
you take a square grid,

1298
01:02:03,080 --> 01:02:05,985
and then you put alternating
diagonals in those squares.

1299
01:02:12,710 --> 01:02:14,730
Get my alternation going here.

1300
01:02:14,730 --> 01:02:16,609
So that's the so-called
box pleating pattern.

1301
01:02:16,609 --> 01:02:18,900
We proved that that pattern,
if you make it big enough,

1302
01:02:18,900 --> 01:02:23,790
can fold any 3D shape that's
made out of little cubes.

1303
01:02:23,790 --> 01:02:28,720
I think, yeah, so in fact if
you take a sort of n by n sheet,

1304
01:02:28,720 --> 01:02:30,890
you could make about n cubes.

1305
01:02:30,890 --> 01:02:33,704
And in the worst case,
that's the best you can do.

1306
01:02:33,704 --> 01:02:35,120
So that's cool,
because that tells

1307
01:02:35,120 --> 01:02:36,670
you here's one robot you build.

1308
01:02:36,670 --> 01:02:39,960
You just make this sheet
with those creases,

1309
01:02:39,960 --> 01:02:42,660
those rubber creases with
muscles on each of those edges.

1310
01:02:42,660 --> 01:02:45,920
And if you can turn those
edges on and off programmably,

1311
01:02:45,920 --> 01:02:49,050
you could make anything you
want up to some resolution.

1312
01:02:49,050 --> 01:02:51,720
If you want to make a
bunny, you can sort of

1313
01:02:51,720 --> 01:02:55,380
build that bunny our of
little cubes in simulation,

1314
01:02:55,380 --> 01:02:59,672
and then fold that cubefied,
voxillized form of the bunny.

1315
01:02:59,672 --> 01:03:01,380
And then you want to
make something else,

1316
01:03:01,380 --> 01:03:04,165
you unfold it and refold
it all automatically.

1317
01:03:07,472 --> 01:03:08,918
AUDIENCE: This is
like [INAUDIBLE]

1318
01:03:08,918 --> 01:03:11,330
no width of the material?

1319
01:03:11,330 --> 01:03:14,590
PROFESSOR: Yeah, there's a
lot of open questions here.

1320
01:03:14,590 --> 01:03:16,700
This result assumes 0 thickness.

1321
01:03:16,700 --> 01:03:18,460
In fact, every result
I've talked about

1322
01:03:18,460 --> 01:03:19,294
assumes 0 thickness.

1323
01:03:19,294 --> 01:03:21,376
I think there's one theorem
in the literature that

1324
01:03:21,376 --> 01:03:22,650
looks at thickness of paper.

1325
01:03:22,650 --> 01:03:25,080
That's a great open
question, and you really

1326
01:03:25,080 --> 01:03:27,290
see it in something
like that video.

1327
01:03:27,290 --> 01:03:31,650
That just appeared at PNAS,
and the video's online.

1328
01:03:31,650 --> 01:03:33,770
Because in the
airplane, you have

1329
01:03:33,770 --> 01:03:35,760
to fold through multiple layers.

1330
01:03:35,760 --> 01:03:36,527
And that's tricky.

1331
01:03:36,527 --> 01:03:39,110
We get around that right now by
having the creases be a little

1332
01:03:39,110 --> 01:03:40,810
bit stretchy, some rubber.

1333
01:03:40,810 --> 01:03:44,630
But there's a limit, and it's
related to the thickness.

1334
01:03:44,630 --> 01:03:48,090
So yes, some complicated shapes
are not going to work this way.

1335
01:03:48,090 --> 01:03:50,310
I think one thing that
would work really well-- we

1336
01:03:50,310 --> 01:03:53,840
have another result with Jason
Ku, which is folding a maze.

1337
01:03:53,840 --> 01:03:58,080
This is sort of like folding
a shallow terrain in this way,

1338
01:03:58,080 --> 01:04:01,210
so you could make a rat maze
out of a square-- just tracks

1339
01:04:01,210 --> 01:04:03,420
by 50%, and then
you have arbitrary

1340
01:04:03,420 --> 01:04:05,640
undulation in the middle,
which is pretty cool.

1341
01:04:05,640 --> 01:04:07,723
So some of these things
are going to be practical.

1342
01:04:07,723 --> 01:04:08,360
Some are not.

1343
01:04:08,360 --> 01:04:12,150
So there's still interesting
theory questions to ask about.

1344
01:04:12,150 --> 01:04:14,840
I think thickness is
a really good one.

1345
01:04:14,840 --> 01:04:16,890
Other questions?

1346
01:04:16,890 --> 01:04:17,390
Yeah.

1347
01:04:20,255 --> 01:04:22,880
AUDIENCE: [INAUDIBLE] I thought
I heard something about how you

1348
01:04:22,880 --> 01:04:24,802
can't fold a sheet of
paper more than like--

1349
01:04:24,802 --> 01:04:26,510
PROFESSOR: Yeah, so
that's the one thing.

1350
01:04:26,510 --> 01:04:31,020
The question is, you can't
fold a piece of paper in half 8

1351
01:04:31,020 --> 01:04:35,600
times, 7 times, whatever the
number is everyone quotes.

1352
01:04:35,600 --> 01:04:38,960
The answer was solved by
someone who was a high school

1353
01:04:38,960 --> 01:04:42,140
student a few years ago.

1354
01:04:42,140 --> 01:04:43,660
What's her name?

1355
01:04:43,660 --> 01:04:45,440
Britney Gallivan.

1356
01:04:45,440 --> 01:04:48,860
And so this is the one paper
that's about thickness,

1357
01:04:48,860 --> 01:04:54,160
and she analyzed-- the issue
is when you fold-- if you have

1358
01:04:54,160 --> 01:04:59,250
a folding that is
1,000 layers thick,

1359
01:04:59,250 --> 01:05:01,110
and you want to
fold in the middle,

1360
01:05:01,110 --> 01:05:05,160
you can't just like cut here
and move it over here and then

1361
01:05:05,160 --> 01:05:06,450
be folded.

1362
01:05:06,450 --> 01:05:09,850
Paper has to actually
turn that corner.

1363
01:05:09,850 --> 01:05:14,900
And the thicker that thing
is, the more length of paper

1364
01:05:14,900 --> 01:05:17,306
it takes to turn that corner.

1365
01:05:17,306 --> 01:05:18,680
And you can compute
how much that

1366
01:05:18,680 --> 01:05:22,020
is if you assume sort of
a circular trajectory,

1367
01:05:22,020 --> 01:05:23,400
and it matches reality.

1368
01:05:23,400 --> 01:05:27,080
So she computed how thin does
a piece of paper need to be,

1369
01:05:27,080 --> 01:05:29,060
or how long does
a piece of paper

1370
01:05:29,060 --> 01:05:31,230
have to be-- it's
that ratio-- in order

1371
01:05:31,230 --> 01:05:33,152
to fold it in half 12 times.

1372
01:05:33,152 --> 01:05:35,110
So she took a piece of
paper that was, I think,

1373
01:05:35,110 --> 01:05:38,717
three-quarters of a mile long,
and folded it in half 12 times.

1374
01:05:38,717 --> 01:05:40,800
And it's like this big
monstrosity when it's done.

1375
01:05:40,800 --> 01:05:43,020
And it follows the
theory exactly.

1376
01:05:43,020 --> 01:05:44,870
So that seems to
be the right model,

1377
01:05:44,870 --> 01:05:46,590
but that's the only
paper that analyzes

1378
01:05:46,590 --> 01:05:48,631
paper folding in that
model, because it's so much

1379
01:05:48,631 --> 01:05:50,300
easier to think
about zero thickness.

1380
01:05:50,300 --> 01:05:51,550
But that is what we should do.

1381
01:05:51,550 --> 01:05:53,890
So that particular
question we understand,

1382
01:05:53,890 --> 01:05:56,250
but everything else is open.

1383
01:05:56,250 --> 01:05:59,454
I'll show you some
pictures of that next time.

1384
01:05:59,454 --> 01:06:00,120
Other questions?

1385
01:06:00,120 --> 01:06:01,660
All right.

1386
01:06:01,660 --> 01:06:04,140
That is paper, and
then I was going

1387
01:06:04,140 --> 01:06:08,450
to show you a little bit
about polyhedra and hinged

1388
01:06:08,450 --> 01:06:09,020
dissections.

1389
01:06:09,020 --> 01:06:10,240
All right.

1390
01:06:10,240 --> 01:06:10,910
Let's go.

1391
01:06:31,679 --> 01:06:33,720
So I mentioned there's a
bunch of different kinds

1392
01:06:33,720 --> 01:06:36,740
of polyhedron folding
and unfolding problems.

1393
01:06:36,740 --> 01:06:39,900
Probably the coolest
one-- because it's

1394
01:06:39,900 --> 01:06:44,100
the oldest problem in
this entire field--

1395
01:06:44,100 --> 01:06:50,570
is unfolding convex polyhedron
by cutting along the edges.

1396
01:06:50,570 --> 01:06:59,980
We call this edge
unfolding convex polyhedra.

1397
01:06:59,980 --> 01:07:02,380
At the beginning of
class, I showed you

1398
01:07:02,380 --> 01:07:04,660
a cube which you could
unfold into a cross.

1399
01:07:04,660 --> 01:07:05,470
I want that.

1400
01:07:05,470 --> 01:07:07,690
So I was only cutting
along edges of the cube.

1401
01:07:07,690 --> 01:07:09,380
That's this edge unfolding.

1402
01:07:09,380 --> 01:07:10,800
The cube is a convex shape.

1403
01:07:10,800 --> 01:07:13,950
It doesn't have any dents.

1404
01:07:13,950 --> 01:07:17,200
And I got to unfold it with
one piece without overlap.

1405
01:07:17,200 --> 01:07:18,730
Is that always possible?

1406
01:07:18,730 --> 01:07:21,420
We have no idea.

1407
01:07:21,420 --> 01:07:23,920
Can every convex polyhedron
be made in that way?

1408
01:07:23,920 --> 01:07:24,820
We don't know.

1409
01:07:24,820 --> 01:07:30,630
This problem goes back
to Albrecht Durer,

1410
01:07:30,630 --> 01:07:31,680
this guy on the left.

1411
01:07:31,680 --> 01:07:33,430
This is his self-portrait.

1412
01:07:33,430 --> 01:07:37,590
In 1525, he wrote this book
called The Painter's Manual

1413
01:07:37,590 --> 01:07:38,810
in German.

1414
01:07:38,810 --> 01:07:42,310
And he tried this out for
a whole bunch of polyhedra.

1415
01:07:42,310 --> 01:07:46,970
This is the so-called snub
cube, and unfolded in his book.

1416
01:07:46,970 --> 01:07:50,060
And he has page after page
after page of these unfoldings.

1417
01:07:50,060 --> 01:07:51,519
And they're all edge unfoldings.

1418
01:07:51,519 --> 01:07:52,560
Most of them are correct.

1419
01:07:52,560 --> 01:07:54,715
There's a couple small
errors, probably just

1420
01:07:54,715 --> 01:07:55,590
transcription errors.

1421
01:07:55,590 --> 01:07:56,635
And he did this
because he wanted

1422
01:07:56,635 --> 01:07:57,970
to understand these 3D shapes.

1423
01:07:57,970 --> 01:07:59,070
He had to build them.

1424
01:07:59,070 --> 01:08:01,310
How do you build them?

1425
01:08:01,310 --> 01:08:03,720
So he didn't pose this
mathematical question.

1426
01:08:03,720 --> 01:08:05,470
He was not a mathematician.

1427
01:08:05,470 --> 01:08:08,050
But it's sort of implicit
there, and people

1428
01:08:08,050 --> 01:08:12,420
have been thinking about it for
at least decades if not more.

1429
01:08:12,420 --> 01:08:15,710
At least since '65, I think,
was the first formal posing.

1430
01:08:15,710 --> 01:08:17,660
And it's really hard.

1431
01:08:17,660 --> 01:08:21,050
A lot of people
have worked on it.

1432
01:08:21,050 --> 01:08:23,029
We'll talk about it.

1433
01:08:23,029 --> 01:08:25,854
There are some interesting
things that are known.

1434
01:08:25,854 --> 01:08:27,229
I mean the depressing
thing is we

1435
01:08:27,229 --> 01:08:29,359
have no algorithm which
we think will work.

1436
01:08:29,359 --> 01:08:32,845
Every algorithm we've tried
has a counter example.

1437
01:08:32,845 --> 01:08:34,220
Every counter
example we've tried

1438
01:08:34,220 --> 01:08:36,430
has an algorithm
for which it works,

1439
01:08:36,430 --> 01:08:42,100
but they don't match
up, so it's frustrating.

1440
01:08:42,100 --> 01:08:46,060
If you allow
non-convex polyhedra,

1441
01:08:46,060 --> 01:08:47,170
then the answer is no.

1442
01:08:47,170 --> 01:08:51,439
So that's something I did
with a bunch of people

1443
01:08:51,439 --> 01:08:52,420
back in the day.

1444
01:08:52,420 --> 01:08:54,945
This was actually
done in '98, then

1445
01:08:54,945 --> 01:08:57,770
it took forever for the
journal paper to appear.

1446
01:08:57,770 --> 01:08:59,240
This is a polyhedron.

1447
01:08:59,240 --> 01:09:01,020
All the faces are triangles.

1448
01:09:01,020 --> 01:09:03,160
So it's sort of
topologically convex.

1449
01:09:03,160 --> 01:09:05,200
And then if I made the
spike a little shorter,

1450
01:09:05,200 --> 01:09:07,120
it would be a convex polyhedron.

1451
01:09:07,120 --> 01:09:10,113
But this particular embedding--
these are just two views.

1452
01:09:10,113 --> 01:09:12,029
If you cross your eyes,
it's three dimensions.

1453
01:09:12,029 --> 01:09:13,290
No, I'm just kidding.

1454
01:09:13,290 --> 01:09:14,770
It's not designed for that.

1455
01:09:14,770 --> 01:09:16,279
Just two views of
the same thing.

1456
01:09:16,279 --> 01:09:20,399
And if you cut along the
edges any way you want,

1457
01:09:20,399 --> 01:09:22,682
it will overlap itself
or be multiple pieces.

1458
01:09:22,682 --> 01:09:24,140
And we'll prove
that at some point.

1459
01:09:24,140 --> 01:09:24,890
It's not too hard.

1460
01:09:28,149 --> 01:09:31,850
Yeah, so good.

1461
01:09:31,850 --> 01:09:33,620
That's edge unfolding.

1462
01:09:33,620 --> 01:09:35,460
But there's another
kind of unfolding

1463
01:09:35,460 --> 01:09:38,080
called general unfolding.

1464
01:09:38,080 --> 01:09:41,500
This seems a lot more
interesting to me.

1465
01:09:41,500 --> 01:09:43,279
Edges are sort of artificial.

1466
01:09:43,279 --> 01:09:45,439
What if you let me cut
anywhere on the surface,

1467
01:09:45,439 --> 01:09:47,560
not just at the edges?

1468
01:09:47,560 --> 01:09:52,760
Well, then it turns out,
for convex polyhedra,

1469
01:09:52,760 --> 01:09:54,950
you can do it.

1470
01:09:54,950 --> 01:09:56,295
It's always possible.

1471
01:10:03,420 --> 01:10:05,310
I don't have pictures
of that here,

1472
01:10:05,310 --> 01:10:08,740
but even for some non-convex
polyhedra you could do it.

1473
01:10:08,740 --> 01:10:10,880
For example, this
polyhedron, if you

1474
01:10:10,880 --> 01:10:13,970
let me cut not just at the edges
but anywhere on the surface,

1475
01:10:13,970 --> 01:10:17,460
you can do this crazy thing.

1476
01:10:17,460 --> 01:10:21,530
These are the spikes, and you
cut out a tiny little sliver

1477
01:10:21,530 --> 01:10:23,960
out to the side,
and that lets you

1478
01:10:23,960 --> 01:10:27,390
attach one of those spikes
from one of these witch's hats

1479
01:10:27,390 --> 01:10:29,330
to the wrong side.

1480
01:10:29,330 --> 01:10:33,540
Like you see, this blue guy, the
blue spike-- because this edge

1481
01:10:33,540 --> 01:10:35,260
ends up being
glued over here, we

1482
01:10:35,260 --> 01:10:37,260
can attach the spike with
the little sliver that

1483
01:10:37,260 --> 01:10:40,410
gets opened up here, over
there, and avoid overlap.

1484
01:10:40,410 --> 01:10:42,000
Even if these spikes
are super tall,

1485
01:10:42,000 --> 01:10:44,520
does this go out to
infinity and overlap.

1486
01:10:44,520 --> 01:10:46,290
That's one example.

1487
01:10:46,290 --> 01:10:49,060
Can you do it for every
non-convex polyhedron?

1488
01:10:49,060 --> 01:10:49,850
We don't know.

1489
01:10:49,850 --> 01:10:51,730
I would love to
answer that question.

1490
01:10:51,730 --> 01:10:55,230
The one thing we know or
the big thing that we know

1491
01:10:55,230 --> 01:10:58,210
is so-called
orthogonal polyhedra.

1492
01:10:58,210 --> 01:10:59,210
This is always possible.

1493
01:10:59,210 --> 01:11:01,209
This the same kind of
thing I was talking about.

1494
01:11:01,209 --> 01:11:03,820
You take any 3D shape
made of little cubes--

1495
01:11:03,820 --> 01:11:08,969
so all the faces are horizontal,
vertical, or the other way--

1496
01:11:08,969 --> 01:11:10,260
then it's also always possible.

1497
01:11:13,950 --> 01:11:16,860
The current method takes an
exponential number of cuts.

1498
01:11:16,860 --> 01:11:17,992
It's really impractical.

1499
01:11:17,992 --> 01:11:19,450
I think we could
make it practical.

1500
01:11:19,450 --> 01:11:21,660
I think we could generalize
it to everything.

1501
01:11:21,660 --> 01:11:23,355
But those are all
open questions.

1502
01:11:25,950 --> 01:11:28,220
One more picture I
wanted to show you

1503
01:11:28,220 --> 01:11:31,430
is, so here we really want
to keep the shape connected.

1504
01:11:31,430 --> 01:11:33,650
You have to have a
positive area of connection

1505
01:11:33,650 --> 01:11:35,010
between everything.

1506
01:11:35,010 --> 01:11:38,060
If you let that connection
go down to a single point--

1507
01:11:38,060 --> 01:11:40,690
sort of cheating, this is
called vertex unfolding--

1508
01:11:40,690 --> 01:11:45,890
and then it's possible for any
surface made out of triangles.

1509
01:11:45,890 --> 01:11:47,390
So it could be
non-convex, anything.

1510
01:11:47,390 --> 01:11:48,830
These happen to be
convex polyhedra.

1511
01:11:48,830 --> 01:11:51,205
It's a little hard to see that
they're three-dimensional,

1512
01:11:51,205 --> 01:11:51,790
but they are.

1513
01:11:51,790 --> 01:11:54,600
You cut them up
into little faces.

1514
01:11:54,600 --> 01:11:57,380
Here I'm only
cutting along edges.

1515
01:11:57,380 --> 01:12:00,380
And I string them
out on a line here.

1516
01:12:00,380 --> 01:12:02,080
This thing will not
overlap, but I'm

1517
01:12:02,080 --> 01:12:04,121
kind of cheating because
the connections are only

1518
01:12:04,121 --> 01:12:05,650
at single points.

1519
01:12:05,650 --> 01:12:10,340
But hey, we'll take a positive
result if we can get it.

1520
01:12:10,340 --> 01:12:13,055
A lot of these problems
are really hard.

1521
01:12:13,055 --> 01:12:14,680
OK, the next thing
I want to talk about

1522
01:12:14,680 --> 01:12:16,490
is the reverse
direction folding.

1523
01:12:16,490 --> 01:12:19,065
So we know-- this is
a video I made back

1524
01:12:19,065 --> 01:12:22,760
when I was a grad
student, a long time ago.

1525
01:12:22,760 --> 01:12:25,180
So we know you can take
a cube, and there's

1526
01:12:25,180 --> 01:12:27,780
a whole bunch of different ways
to just cut along its edges

1527
01:12:27,780 --> 01:12:29,820
and unfold it into one piece.

1528
01:12:29,820 --> 01:12:32,970
And the particular
unfolding I want to consider

1529
01:12:32,970 --> 01:12:36,520
is the cross unfolding,
because that's our favorite.

1530
01:12:36,520 --> 01:12:38,860
So you take this
cross unfolding,

1531
01:12:38,860 --> 01:12:41,170
and ideally don't stutter.

1532
01:12:41,170 --> 01:12:43,620
And you say, well what if
I gave you that polygon?

1533
01:12:43,620 --> 01:12:45,740
What 3D shapes
could it fold into?

1534
01:12:45,740 --> 01:12:47,892
One thing is the
cube, but it can also

1535
01:12:47,892 --> 01:12:49,725
make this flat doubly
covered quadrilateral.

1536
01:12:54,090 --> 01:12:57,250
So I'm taking this
as a piece of paper,

1537
01:12:57,250 --> 01:13:00,090
but I can set the
creases however I want.

1538
01:13:00,090 --> 01:13:04,470
I can fold it into this
five-sided, reflectionally

1539
01:13:04,470 --> 01:13:05,870
symmetric polyhedron.

1540
01:13:08,850 --> 01:13:11,490
All of these things are
convex, and they all

1541
01:13:11,490 --> 01:13:13,670
have perfect coverage,
unlike an origami where

1542
01:13:13,670 --> 01:13:15,170
I'm allowed to have
multiple layers.

1543
01:13:15,170 --> 01:13:16,950
Here, one layer everywhere.

1544
01:13:16,950 --> 01:13:18,540
Like sheet metal bending.

1545
01:13:18,540 --> 01:13:20,806
So here, this pocket fits
perfectly into that tab.

1546
01:13:20,806 --> 01:13:21,680
I make a tetrahedron.

1547
01:13:25,350 --> 01:13:27,720
And the last thing you can
make with something called

1548
01:13:27,720 --> 01:13:30,190
edge-to-edge [INAUDIBLE]--
I'll talk about this more.

1549
01:13:30,190 --> 01:13:32,650
There's an algorithm here
which, given a polygon,

1550
01:13:32,650 --> 01:13:36,050
will list all of the
things it can fold into,

1551
01:13:36,050 --> 01:13:38,720
in a reasonable amount of time.

1552
01:13:38,720 --> 01:13:39,990
This is an octahedron.

1553
01:13:39,990 --> 01:13:41,365
Those are the five
things you can

1554
01:13:41,365 --> 01:13:45,610
make by edge-to-edge
[INAUDIBLE] of the cross.

1555
01:13:45,610 --> 01:13:51,037
Let me stop that.

1556
01:13:51,037 --> 01:13:52,120
So that's also on the web.

1557
01:13:52,120 --> 01:13:53,900
You can watch it.

1558
01:13:53,900 --> 01:13:56,910
In fact, all the lecture notes
I use, my handwritten notes,

1559
01:13:56,910 --> 01:13:58,201
are on the class web page.

1560
01:13:58,201 --> 01:14:00,450
The slides that I'm showing
are on the class web page.

1561
01:14:00,450 --> 01:14:02,658
And when there's videos,
there's links to the videos,

1562
01:14:02,658 --> 01:14:07,570
so it's all there for your
perusal if you miss something.

1563
01:14:07,570 --> 01:14:10,450
The last thing I wanted to talk
about, because I've already

1564
01:14:10,450 --> 01:14:13,410
done one, two and three,
there should always

1565
01:14:13,410 --> 01:14:16,140
be one more thing.

1566
01:14:16,140 --> 01:14:17,499
Fourth thing.

1567
01:14:17,499 --> 01:14:19,540
And this is kind of funny,
because I don't really

1568
01:14:19,540 --> 01:14:20,415
know where to put it.

1569
01:14:20,415 --> 01:14:21,780
I'm still figuring it out.

1570
01:14:21,780 --> 01:14:23,267
We have these
debates, because it's

1571
01:14:23,267 --> 01:14:25,100
so easy when you're
thinking about linkages.

1572
01:14:25,100 --> 01:14:26,224
Oh, that's one dimensional.

1573
01:14:26,224 --> 01:14:27,670
Paper, that's two-dimensional.

1574
01:14:27,670 --> 01:14:30,290
Polyhedra, well locally
it's two-dimensional,

1575
01:14:30,290 --> 01:14:31,415
but it's three-dimensional.

1576
01:14:31,415 --> 01:14:32,740
It's different.

1577
01:14:32,740 --> 01:14:35,760
Different questions.

1578
01:14:35,760 --> 01:14:38,050
Hinged dissections span
so many dimensions,

1579
01:14:38,050 --> 01:14:40,580
I don't know where to put it.

1580
01:14:40,580 --> 01:14:43,285
So four, hinged dissections.

1581
01:14:47,970 --> 01:14:49,950
This didn't used to be
a topic in this class,

1582
01:14:49,950 --> 01:14:51,491
and it's not a topic
in our textbook,

1583
01:14:51,491 --> 01:14:54,450
because the result was proved
like last year or the year

1584
01:14:54,450 --> 01:14:56,260
before out of this class.

1585
01:14:56,260 --> 01:14:58,370
Three years ago.

1586
01:14:58,370 --> 01:15:14,750
And the theorem is you take
any finite set of polygons,

1587
01:15:14,750 --> 01:15:28,630
of the same area,
they can be folded

1588
01:15:28,630 --> 01:15:43,670
from one chain of polygons
without collision.

1589
01:15:46,900 --> 01:15:51,470
I'm going to say this
first, and but let me

1590
01:15:51,470 --> 01:15:54,260
show you a simple example.

1591
01:15:54,260 --> 01:15:56,216
Very simple.

1592
01:15:56,216 --> 01:15:57,715
This is going to
be a little boring.

1593
01:15:57,715 --> 01:16:00,610
Let me show you more
interesting example.

1594
01:16:00,610 --> 01:16:02,970
Let's take an
equilateral triangle.

1595
01:16:02,970 --> 01:16:05,790
I'm going to cut it
into two triangles,

1596
01:16:05,790 --> 01:16:09,210
and then I'm going to
hinge them together here.

1597
01:16:09,210 --> 01:16:13,550
So this is now a hinged
chain of polygons.

1598
01:16:13,550 --> 01:16:17,820
If I open it up a little
bit, it looks like this.

1599
01:16:17,820 --> 01:16:19,635
It looks a lot like
those vertex unfoldings

1600
01:16:19,635 --> 01:16:21,890
I was showing before.

1601
01:16:21,890 --> 01:16:27,570
And you can fold this
thing into one other shape.

1602
01:16:27,570 --> 01:16:30,615
It's going to look like this.

1603
01:16:33,532 --> 01:16:35,945
Not a very good drawing,
but you get the idea.

1604
01:16:35,945 --> 01:16:37,820
Something like that.

1605
01:16:37,820 --> 01:16:39,790
So this is what we call
a hinged dissection

1606
01:16:39,790 --> 01:16:43,200
from this equilateral triangle
into this other triangle.

1607
01:16:43,200 --> 01:16:45,940
And it's been an open
question for about 100 years.

1608
01:16:45,940 --> 01:16:48,430
Can you always do this for
polygons of the same area?

1609
01:16:48,430 --> 01:16:50,180
Turns out yes, not
only for two polygons--

1610
01:16:50,180 --> 01:16:53,240
you can take seven
polygons of equal area

1611
01:16:53,240 --> 01:16:56,650
and there's one of
these chains of blocks

1612
01:16:56,650 --> 01:16:59,080
that can fold into all of them.

1613
01:16:59,080 --> 01:17:04,380
And I think I have
a little teaser

1614
01:17:04,380 --> 01:17:06,980
picture of how the
construction goes.

1615
01:17:06,980 --> 01:17:08,310
It's really crazy.

1616
01:17:08,310 --> 01:17:09,860
You have this chain of blocks.

1617
01:17:09,860 --> 01:17:12,200
It's changed in a particular
way and you say, well

1618
01:17:12,200 --> 01:17:14,750
I really want that green piece--
so think of it like this.

1619
01:17:14,750 --> 01:17:20,180
You just have three triangles or
three pieces that you can fold.

1620
01:17:20,180 --> 01:17:22,600
You say, well I really like
that green piece to be up top.

1621
01:17:22,600 --> 01:17:25,094
Turns out you can make
all these crazy cuts.

1622
01:17:25,094 --> 01:17:27,260
And if you fold it this
way, you get the green piece

1623
01:17:27,260 --> 01:17:27,843
on the bottom.

1624
01:17:27,843 --> 01:17:30,360
If you fold it this way, you
get the green piece on the top.

1625
01:17:30,360 --> 01:17:33,060
It's one hinged dissection that
can move the pieces around,

1626
01:17:33,060 --> 01:17:36,130
and basically pretend as if
the hinges weren't there,

1627
01:17:36,130 --> 01:17:37,820
and make anything you want.

1628
01:17:37,820 --> 01:17:39,610
So it's a bit complicated,
but it's really

1629
01:17:39,610 --> 01:17:43,120
cool theory, something that
we worked on for like 10 years

1630
01:17:43,120 --> 01:17:46,030
before finally solving.

1631
01:17:46,030 --> 01:17:47,334
And it's practical.

1632
01:17:47,334 --> 01:17:49,500
I mean this particular
construction isn't practical,

1633
01:17:49,500 --> 01:17:52,020
but this is another way
to build transformers.

1634
01:17:52,020 --> 01:17:53,030
Also to build sculpture.

1635
01:17:53,030 --> 01:17:56,351
This is a sculpture by Laurie
Palmer that we collaborated on.

1636
01:17:56,351 --> 01:17:57,600
It's an interactive sculpture.

1637
01:17:57,600 --> 01:18:01,300
You pick up gloves, and
this is one chain of blocks.

1638
01:18:01,300 --> 01:18:03,890
And the theorem is, this
particular chain of blocks--

1639
01:18:03,890 --> 01:18:07,070
about 1,000 blocks here--
can make any shape made out

1640
01:18:07,070 --> 01:18:12,666
of about 250 cubes.

1641
01:18:12,666 --> 01:18:16,180
It's slightly skewed, but so
there's one hinged dissection

1642
01:18:16,180 --> 01:18:19,500
that can fold into exponentially
many different shapes-- again,

1643
01:18:19,500 --> 01:18:21,770
anything made out
of little cubes.

1644
01:18:21,770 --> 01:18:23,710
And this is the
sculpture version.

1645
01:18:23,710 --> 01:18:26,730
We also have various
prototypes of a robot that's

1646
01:18:26,730 --> 01:18:30,190
a long chain of little cubes
or cube-like shapes, that

1647
01:18:30,190 --> 01:18:32,506
can fold itself into
any 3D shape you want.

1648
01:18:32,506 --> 01:18:34,380
It's, again, a different
kind of transformer,

1649
01:18:34,380 --> 01:18:36,450
more inspired by
proteins, the way nature

1650
01:18:36,450 --> 01:18:41,420
does it by long chains,
compared to the sheet-folding,

1651
01:18:41,420 --> 01:18:43,180
origami-inspired approach.

1652
01:18:43,180 --> 01:18:46,450
But it's all pretty
exciting and fun.

1653
01:18:46,450 --> 01:18:48,530
Any questions?

1654
01:18:48,530 --> 01:18:49,910
Jean?

1655
01:18:49,910 --> 01:18:52,160
AUDIENCE: This is a 3D version
you were talking about.

1656
01:18:52,160 --> 01:18:55,610
PROFESSOR: Yeah, I
pulled a fast one.

1657
01:18:55,610 --> 01:18:57,040
That picture's 3D.

1658
01:18:57,040 --> 01:18:58,310
This is about 2D.

1659
01:18:58,310 --> 01:19:00,280
For 3D, you need the same area.

1660
01:19:00,280 --> 01:19:02,240
You need one other
condition just

1661
01:19:02,240 --> 01:19:05,750
for them to have a dissection.

1662
01:19:05,750 --> 01:19:08,890
It's impossible to go from a
regular tetrahedron to a cube.

1663
01:19:08,890 --> 01:19:11,530
This was one of Hilbert's
problems from 1900.

1664
01:19:11,530 --> 01:19:14,870
And volumetrically
it's impossible to go

1665
01:19:14,870 --> 01:19:18,090
from regular
tetrahedron to a cube,

1666
01:19:18,090 --> 01:19:20,460
because the angles are wrong.

1667
01:19:20,460 --> 01:19:22,190
It's an algebraic thing.

1668
01:19:22,190 --> 01:19:24,800
And if you make things out
of little cubes, it's fine.

1669
01:19:24,800 --> 01:19:25,500
You get that.

1670
01:19:25,500 --> 01:19:26,479
There's no problem.

1671
01:19:26,479 --> 01:19:28,270
But in general, there's
some tricky issues.

1672
01:19:28,270 --> 01:19:31,190
It's easier to state for
2D, which is why I did,

1673
01:19:31,190 --> 01:19:33,530
but it works in 3D, too, with
a little extra condition.

1674
01:19:33,530 --> 01:19:36,560
And we'll talk about
that in the future.

1675
01:19:36,560 --> 01:19:38,930
Other questions?

1676
01:19:38,930 --> 01:19:41,710
All right, that's the
end of lecture one.