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PROFESSOR: Hey, everybody.

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Welcome to 6.849, Geometric
Folding Algorithms.

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I am Erik Demaine.

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You can call me Erik.

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And we have as TA Jayson
Lynch, who's right there.

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And this class is a bit
unusual, at least for me,

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because I'm trying for the first
time a new experiment, which

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is inverted lecturing.

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And I wrote this on the
poster for the class,

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and everyone started asking
me, what's inverted lectures?

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Well, it's not a new idea, but
I've never tried it before.

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The concept is to make these
in-class times, where we're

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all here together,
more interactive

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by taking the lecture component
of the class, which is covering

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all the material into videos
that you watch online.

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So this class is basically
going to alternate

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between real in-person things,
as you are here-- of course,

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we are also being
video recorded,

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so slightly a
contradiction in terms--

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but that's for the
people in the interwebs

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to be able to watch this.

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So we'll alternate between the
real in-class part and video

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lectures from the
last time I taught

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this class, which was fall 2010.

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So this is actually
what the course web

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page looks like
as of an hour ago.

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So we've got these-- L01,
L02, L03-- those are lectures.

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That's content, lots
of material packed

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in there, which is how I'm
used to teaching a class-- put

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as much material as
I can for a lecture.

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Those are already available.

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You can start
watching those videos.

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Lecture one is optional.

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I'm going to basically cover a
shorter version of that today.

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It says right there, optional.

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Then the Cs, which
don't exist yet,

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because we're
doing it right now.

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The Cs are the in-class
sessions, the contact hours.

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Some of those may be
recorded, like this one,

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we're going to record.

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But some will not be, and
you need to attend both.

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So you've got to
watch the videos,

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and you've got attend
all the classes.

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That's the set up.

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So this is a bit unusual, so
I'll spend a few more minutes

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on how this class is formatted.

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So-- requirements.

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We've got-- watch
video lectures.

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I'll talk a little bit
about timing of that.

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And then we've got
attend classes.

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And then to the actual
material, a regular part

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of the class that
you'll be graded on

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consists of two parts:
problem sets and a project.

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The project also
has a presentation.

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This is a project focus class,
so the bulk of your grade

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is based on the project
and presentation.

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Project can take on
many different forms.

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This is a theoretical
computer science class,

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so there's the typical
kinds of projects, like,

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to read a bunch of papers and
summarize them, survey them.

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Though we ask that you
keep them disjoint,

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more or less-- avoid
stuff that is well

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covered in this book,
Geometric Folding Algorithms,

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or in the class itself.

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But otherwise you
can survey material.

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In addition to the
standard survey paper,

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or instead of a
standard survey paper,

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you can write a bunch
of Wikipedia articles

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about folding stuff.

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So where survey stuff
should be new material

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that you haven't
seen, Wikipedia stuff

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could be material that
you've seen in this class

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but is not well
covered in Wikipedia.

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And that will help take over
the world, our usual goal.

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If you see a cool
algorithm, in this class

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you could implement
it if you're a coder

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and see how it
actually works, make it

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demoable, which will be fun.

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Or you can work on
an open problem.

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So if there's
interest, we'll run

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an optional open problem-solving
session, where we'll all

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get together and try
to solve open problems.

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I might also do them
in these class times,

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because now we have a lot of
freedom to be more attractive,

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to actually fold paper
or to have fun in class.

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And instead of me trying
to pack as much material

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as I can-- because
that's already

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been done in the videos-- you
could pose an open problem.

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Especially if you come
from another field,

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you may have some interesting
folding-related problems.

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Those could be really cool.

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Or you could build something,
a physical sculpture,

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physical structure of
some kind, furniture,

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architecture, whatever.

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Or you could do it
in a virtual world

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and just design
something interesting.

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Though there you might
want to do more than one,

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make it harder for yourself.

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All of these are
possible projects.

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There's a lot of different
options for different types

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of people, different
backgrounds, whatever.

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Then we'll also have problem
sets, which are weekly,

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roughly, and they
shouldn't be too long.

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And they will also
have an option

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of not doing all the problems.

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So every problem
set will probably

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have the rule that
you can drop, we

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will drop the
lowest grade on one

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of the problems that's
on the problem set.

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So that means if you just
don't want to do a problem,

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you can skip it
and your grade will

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be determined by the others.

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If you want you could
try to do all of them,

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and we'll just drop
the lowest grade.

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So if there's some problems
are easier for you,

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harder for others,
you can mix and match.

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Cool.

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There's a lot more details
about this stuff on the website.

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If you click on
Project, you'll get

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lots of details about these
different styles of project

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implementation-- survey,
open problems, whatever.

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Because what I said
was pretty brief.

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So look at that
for more details.

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Project, you don't have
to worry about right away,

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but one of the luxuries
of having video lectures

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is you could skip
ahead a little bit

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to see what looks
interesting to you

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and work on a project
related to that.

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I wanted to briefly show you a
little bit about this website

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and how it works, so
you get a flavor for it.

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So let's say the
very next thing you

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should do, right after you
leave here, sometime between now

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and noon on Monday, you
should watch lecture two.

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So you can just click on
lecture two here on the left.

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And this is the way it's set up.

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You could zoom to fit
everything on your screen.

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On the upper left
you've, got your video.

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And this should start playing.

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[VIDEO PLAYBACK]

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-All right, welcome
back to 6.849.

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[END VIDEO PLAYBACK]

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PROFESSOR: Hey it's me.

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I'm even wearing
the right t-shirt.

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I had a coincidence.

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The fun thing about these
videos, as a few nice features.

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One is as you jump
around in the video,

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you see the slide down here
in the lower left updates

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to whatever I'm covering.

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And also if I jump farther,
the page handwritten

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notes that I'm covering
on the right changes.

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So as I scrabble around, I think
is the term-- too many buttons.

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Who designed this website?

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All right, I forgot this.

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Reload.

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So to jump around.

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I'll turn down the volume.

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You can also speed up the video.

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This can be pretty entertaining
if you go really fast.

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But you can go 1.1, 1.2,
that's pretty comfortable.

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1.5.

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Yesterday I listened to a video
to at 2x. it's interesting.

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[VIDEO PLAYBACK]

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-In general, when
I make that fold,

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it might come out of here.

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I'd have to wrap around
that [? cone ?], sometimes.

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[END VIDEO PLAYBACK]

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PROFESSOR: Thankfully, I do
not sound like a Chipmunk.

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But if you haven't seen lecture
before, it's a little harder.

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But I'll be watching
the lectures with you,

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experiencing the same
total number of hours

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you have to experience,
So we're in this together.

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Let me know if you have any
comments on the website.

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It's all been done by
hand, so if there's

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anything you want changed,
it's easy to change.

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You can also do fun things.

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Like when it's paused,
you can jump around

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in pages on the right and
say, oh, yeah, I didn't really

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understand that page.

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And you click on
this time, and it

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will start playing
from that time.

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At full speed.

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We could slow it
down, too, which

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can be pretty entertaining.

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Anyway, you get the idea.

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Let me know how it goes.

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I want to make it
as easy as possible.

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In fact, if you don't
like my software,

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you can just skip it
all, download this video,

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like 720p version
or the 360p version.

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Play it in VLC with whatever
fancy speed up, speed downs,

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you want to use, or put on your
iPad or other tablet, whatever.

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This page should
work on an iPad,

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but I haven't tested it lately.

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If it doesn't, let me know.

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Cool.

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So that's what it looks like.

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And then at the
top of this page,

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you see here a link
to completion form.

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When you finish watching the
video, you click on this form.

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It's pretty minimal.

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You enter basically
your name, your username

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or your email address, you
say yes, I watched the video,

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and then that's all that's
required, you click Submit.

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But we highly
encourage you to say

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what you think at this point.

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If you have any questions about
the lecture, that wasn't clear,

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or anything you
didn't understand.

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Or I a briefly mention something
that sounded cool to you,

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and you want to
know more about it.

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This is your chance to
influence what I cover in class,

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and this is why
you have to submit

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this form by noon
the previous day.

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So watching video lectures
and filling out that form

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is due by noon on
Mondays and Wednesdays,

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the day before class.

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That will give me 22 hours
to prepare class, which

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should be enough,
we'll find out.

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And so I can adapt
the in-class time

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to be whatever people
care about most.

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So it's kind of like a poll.

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If lots of people say I didn't
understand x, I will cover x.

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If lots of people are curious
about y, I will cover y.

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So whatever you
want to know about,

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typically related things, but
if you have unrelated things you

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want to bring up as well,
feel free to put it in there.

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And we'll try to schedule it in
when it's time or when it fits.

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Yeah, that's the plan.

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I've never done this before,
so if you don't like the form,

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tell me.

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Anything you don't like or think
should change, just send email.

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Or put it in the form,
but email's good, too.

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Question?

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AUDIENCE: I noticed
you have a "no" option.

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Is that for, like, oh,
shoot, it's 11:45 on Monday,

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and I haven't watched this.

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Should I still submit
the form and say, "no?"

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PROFESSOR: Interesting.

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You're so honest.

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Why would you fill
out "no" on this form?

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I guess so.

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You can fill out the
form multiple times.

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So you could you could say
no and give your excuse here,

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if you want it.

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I'll watch it tomorrow,
maybe, and then

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later on fill it in with yes.

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So in particular, if
you also have questions,

247
00:11:33,480 --> 00:11:35,902
like you watched the video
early on and then later

248
00:11:35,902 --> 00:11:38,360
on you have a question about
it, if it's still before class

249
00:11:38,360 --> 00:11:40,272
time, feel free to
fill this out again.

250
00:11:40,272 --> 00:11:41,980
It just makes another
entry in our table.

251
00:11:41,980 --> 00:11:43,480
This is a Google form.

252
00:11:43,480 --> 00:11:46,210
Another question?

253
00:11:46,210 --> 00:11:47,900
I guess that's
the reason for no.

254
00:11:47,900 --> 00:11:51,810
Maybe also we couldn't
put a single option.

255
00:11:51,810 --> 00:11:55,480
No, it's to force you
to claim, honestly.

256
00:11:55,480 --> 00:11:57,560
This all honor
system, right, so this

257
00:11:57,560 --> 00:12:00,310
is like forcing you to make
a decision to be dishonest,

258
00:12:00,310 --> 00:12:01,100
I guess.

259
00:12:01,100 --> 00:12:04,320
Or to be honest, rather.

260
00:12:04,320 --> 00:12:06,792
Cool.

261
00:12:06,792 --> 00:12:08,750
To make this work, you've
got to ask questions,

262
00:12:08,750 --> 00:12:09,916
you've got a request topics.

263
00:12:09,916 --> 00:12:14,650
Otherwise I'll just fill your
class times with more material.

264
00:12:14,650 --> 00:12:16,270
That would be my
natural temptation,

265
00:12:16,270 --> 00:12:19,390
so you've got to hold me back
and ask lots of questions.

266
00:12:19,390 --> 00:12:22,190
One particular request
I wanted to highlight

267
00:12:22,190 --> 00:12:26,410
is to send me cool
things to cover.

268
00:12:26,410 --> 00:12:29,900
In particular, you
all surf the web.

269
00:12:29,900 --> 00:12:34,550
And if you see cool
folding things, or any kind

270
00:12:34,550 --> 00:12:45,060
of folding-related thing,
on the web, send it to me.

271
00:12:49,960 --> 00:12:50,980
By email.

272
00:12:50,980 --> 00:12:55,581
And the plan is to do one
a day, or maybe two a day,

273
00:12:55,581 --> 00:12:56,330
for in class time.

274
00:12:56,330 --> 00:12:57,871
There's lots of fun
things out there,

275
00:12:57,871 --> 00:12:59,360
and it's hard to see it all.

276
00:12:59,360 --> 00:13:01,580
So when you discover
things, tell

277
00:13:01,580 --> 00:13:04,612
me and I will schedule it
in throughout the semester.

278
00:13:04,612 --> 00:13:06,820
You can think of this as a
standing homework problem.

279
00:13:06,820 --> 00:13:09,355
You should do it sometime
before the end of semester.

280
00:13:13,210 --> 00:13:14,040
Another experiment.

281
00:13:14,040 --> 00:13:15,940
I think to be fun.

282
00:13:15,940 --> 00:13:20,200
That is the end of the
class organizationally.

283
00:13:20,200 --> 00:13:23,261
Unless there are any questions
about requirements or style

284
00:13:23,261 --> 00:13:23,760
or format.

285
00:13:26,620 --> 00:13:31,880
Then in the rest of this class,
I'm going to cover, as I said,

286
00:13:31,880 --> 00:13:34,740
a short version of
lecture one, which is just

287
00:13:34,740 --> 00:13:36,080
an overview of the whole class.

288
00:13:36,080 --> 00:13:38,640
So I want to give you a flavor
of what the course is about,

289
00:13:38,640 --> 00:13:41,370
what kind of theorems we
prove, what kind of algorithms

290
00:13:41,370 --> 00:13:43,946
we develop, so you can
decide whether you are

291
00:13:43,946 --> 00:13:45,820
interested in this class,
whether to take it.

292
00:13:45,820 --> 00:13:50,066
There is also a
survey that, where

293
00:13:50,066 --> 00:13:51,440
you put your name,
email address,

294
00:13:51,440 --> 00:13:52,815
will add you to
the mailing list.

295
00:13:52,815 --> 00:13:55,009
And I announce every
lecture, so if you're

296
00:13:55,009 --> 00:13:56,550
listening to the
class, you just want

297
00:13:56,550 --> 00:13:59,190
to listen to things
you're interested in,

298
00:13:59,190 --> 00:14:01,550
come to class for
interesting things,

299
00:14:01,550 --> 00:14:06,900
just listen to that email and
see when cool things come up.

300
00:14:06,900 --> 00:14:09,310
And also there's a little
survey on that piece of paper.

301
00:14:09,310 --> 00:14:12,410
Does anyone not have
the piece of paper?

302
00:14:12,410 --> 00:14:15,100
Couple people, so we
will get them to you.

303
00:14:15,100 --> 00:14:17,900
A little survey just
to get your background.

304
00:14:17,900 --> 00:14:20,350
There's no required
background for this class,

305
00:14:20,350 --> 00:14:22,500
because lots of people
come from different areas.

306
00:14:22,500 --> 00:14:25,320
Hopefully, you know at least
one of the background areas,

307
00:14:25,320 --> 00:14:27,900
but anything can
kind of be filled in.

308
00:14:27,900 --> 00:14:30,590
And one of the points
of this questionnaire

309
00:14:30,590 --> 00:14:34,840
is if there's something
you don't understand, like,

310
00:14:34,840 --> 00:14:38,096
oh, gosh I should
know XYZ algorithm,

311
00:14:38,096 --> 00:14:40,220
and you assumed it in
lecture, but I don't know it,

312
00:14:40,220 --> 00:14:41,210
could you cover it?

313
00:14:41,210 --> 00:14:44,290
And if enough people
request that, I will.

314
00:14:44,290 --> 00:14:45,980
So that's the format.

315
00:14:45,980 --> 00:14:51,180
Out of curiosity, how many
people here are a course six?

316
00:14:51,180 --> 00:14:53,780
And how many people
are course 18?

317
00:14:53,780 --> 00:14:57,920
And how many people are other?

318
00:14:57,920 --> 00:14:59,730
OK.

319
00:14:59,730 --> 00:15:02,250
So reasonably balanced.

320
00:15:02,250 --> 00:15:05,500
A lot of 6, though.

321
00:15:05,500 --> 00:15:09,620
So we are talking about
geometric folding algorithms.

322
00:15:09,620 --> 00:15:13,180
So there's geometry,
there's folding,

323
00:15:13,180 --> 00:15:14,760
and there's algorithms.

324
00:15:14,760 --> 00:15:21,340
Algorithms are how we compute
things at least theoretically,

325
00:15:21,340 --> 00:15:26,590
and in general what
this field is about

326
00:15:26,590 --> 00:15:30,070
is both mathematics
and computer science.

327
00:15:30,070 --> 00:15:33,590
Say, the mathematics
and algorithms

328
00:15:33,590 --> 00:15:35,560
that underlie how things fold.

329
00:15:48,650 --> 00:15:51,885
And the technical term for
things is geometric objects.

330
00:15:56,690 --> 00:15:58,190
And we're interested
not only in how

331
00:15:58,190 --> 00:16:00,290
they fold, but also
how they unfold.

332
00:16:00,290 --> 00:16:02,660
In general, you have
some geometric structure

333
00:16:02,660 --> 00:16:04,540
they can reconfigure,
like, my arm here

334
00:16:04,540 --> 00:16:06,169
can reconfigure in
interesting ways.

335
00:16:06,169 --> 00:16:08,710
We want to know-- what are all
the ways they can reconfigure,

336
00:16:08,710 --> 00:16:10,780
what all the way it could fold?

337
00:16:10,780 --> 00:16:14,015
This piece of paper is
another geometric object.

338
00:16:14,015 --> 00:16:16,140
We want to know what all
the ways that it can fold.

339
00:16:21,280 --> 00:16:23,640
Before I talk about more
formally what this means,

340
00:16:23,640 --> 00:16:26,400
I want to show you that
folding comes up everywhere,

341
00:16:26,400 --> 00:16:26,900
pretty much.

342
00:16:26,900 --> 00:16:28,420
Almost any discipline
you can name,

343
00:16:28,420 --> 00:16:31,900
there's some, if it involves
physical things at least,

344
00:16:31,900 --> 00:16:33,160
has geometric objects.

345
00:16:33,160 --> 00:16:35,752
Geometric objects
tend to comply.

346
00:16:35,752 --> 00:16:37,960
Even if you're working with
completely rigid objects,

347
00:16:37,960 --> 00:16:39,990
you don't want them to fold.

348
00:16:39,990 --> 00:16:43,640
That's a folding problem.

349
00:16:43,640 --> 00:16:47,960
So I have here a list of a bunch
of different application areas,

350
00:16:47,960 --> 00:16:50,640
and I'm going to show you some
pictures and videos related

351
00:16:50,640 --> 00:16:51,810
to them.

352
00:16:51,810 --> 00:16:53,540
So the first one is robotics.

353
00:16:53,540 --> 00:16:56,670
So if you have things
like a robotic arm,

354
00:16:56,670 --> 00:16:58,140
it starts in one configuration.

355
00:16:58,140 --> 00:17:00,480
You want to continuously
move it to reach

356
00:17:00,480 --> 00:17:02,970
some other configuration,
so it can pick up somebody,

357
00:17:02,970 --> 00:17:05,079
drop it off over here.

358
00:17:05,079 --> 00:17:06,530
I want to know,
how should I plan

359
00:17:06,530 --> 00:17:09,349
the motion of my robotic arm?

360
00:17:09,349 --> 00:17:11,319
A different kind of
robotic application

361
00:17:11,319 --> 00:17:14,569
is to make sheets of
material that are themselves

362
00:17:14,569 --> 00:17:17,920
robots that told
themselves into origami.

363
00:17:17,920 --> 00:17:22,339
This is a self folding sheet
developed by collaboration

364
00:17:22,339 --> 00:17:25,560
with MIT and Harvard.

365
00:17:25,560 --> 00:17:28,790
And you just send a little
bit of electrical current.

366
00:17:28,790 --> 00:17:31,420
These little metal
pieces heat up,

367
00:17:31,420 --> 00:17:34,150
which causes them to pull
the creases shut, and, boom,

368
00:17:34,150 --> 00:17:38,680
you get your origami boat--
with no origamist required.

369
00:17:38,680 --> 00:17:42,230
The same sheet can fold
into many different shapes,

370
00:17:42,230 --> 00:17:44,356
and the underlying
algorithm here,

371
00:17:44,356 --> 00:17:45,730
or the underlying
mathematics, is

372
00:17:45,730 --> 00:17:48,370
that this pattern of
creases, square grid

373
00:17:48,370 --> 00:17:50,330
with altering
diagonals, can fold

374
00:17:50,330 --> 00:17:55,070
into essentially any shape
that is made out of cubes.

375
00:17:55,070 --> 00:17:57,670
Here we're making a paper
plane out of the same sheet,

376
00:17:57,670 --> 00:17:59,460
sending it different signals.

377
00:17:59,460 --> 00:18:04,790
It does not fly but
probably floats.

378
00:18:04,790 --> 00:18:06,220
So that's the idea.

379
00:18:06,220 --> 00:18:09,960
And you can imagine a
sheet like this could just

380
00:18:09,960 --> 00:18:13,690
be arbitrarily reprogrammed
to fold into this gadget,

381
00:18:13,690 --> 00:18:15,690
so I don't have to bring
around so many gadgets.

382
00:18:15,690 --> 00:18:17,790
Maybe my laptop
could fold smaller,

383
00:18:17,790 --> 00:18:20,430
and later could unfold in
something to my desktop.

384
00:18:20,430 --> 00:18:22,940
And I don't know, my
phone could reconfigure

385
00:18:22,940 --> 00:18:24,330
into something else.

386
00:18:24,330 --> 00:18:25,202
That's the vision.

387
00:18:25,202 --> 00:18:27,160
This is what we call
programmable matter, where

388
00:18:27,160 --> 00:18:28,312
you can change shape.

389
00:18:28,312 --> 00:18:30,020
In the same way that
we program software,

390
00:18:30,020 --> 00:18:33,680
we want to be able
to program shape.

391
00:18:33,680 --> 00:18:36,990
Obviously, we're just
getting there now.

392
00:18:36,990 --> 00:18:38,160
So that's robotics.

393
00:18:38,160 --> 00:18:40,380
Next up is graphics.

394
00:18:40,380 --> 00:18:43,590
One example is you want
to animate a character

395
00:18:43,590 --> 00:18:44,860
from one position to another.

396
00:18:44,860 --> 00:18:50,240
That's a key frame animation
that's a folding problem.

397
00:18:50,240 --> 00:18:53,680
I have here an example of
the two dimensional analog,

398
00:18:53,680 --> 00:18:57,610
where you have one polygon
and another polygon,

399
00:18:57,610 --> 00:18:59,170
and you want to
morph continuously

400
00:18:59,170 --> 00:19:00,430
from one to the other.

401
00:19:00,430 --> 00:19:03,250
And these animations
are all found

402
00:19:03,250 --> 00:19:06,310
by algorithms that are
motivated by folding stuff

403
00:19:06,310 --> 00:19:08,770
that we will cover
in this class.

404
00:19:08,770 --> 00:19:11,380
And that's not so easy, but
it's lots of fun things.

405
00:19:11,380 --> 00:19:14,190
All of these motions
avoid collision,

406
00:19:14,190 --> 00:19:17,310
and they approximately preserve
the edge lengths, also,

407
00:19:17,310 --> 00:19:18,030
if it's possible.

408
00:19:18,030 --> 00:19:19,696
If they match on the
left and the right,

409
00:19:19,696 --> 00:19:21,860
they will preserve
those edge links.

410
00:19:24,640 --> 00:19:26,215
So that's graphics, morphing.

411
00:19:29,010 --> 00:19:33,200
Mechanics motivated a lot of
mechanical linkage problems.

412
00:19:33,200 --> 00:19:35,790
In particular is this
great book from 1877

413
00:19:35,790 --> 00:19:37,466
called How to Draw
a Straight Line.

414
00:19:37,466 --> 00:19:39,590
You may think you know how
to draw a straight line,

415
00:19:39,590 --> 00:19:44,290
but the point of this
book is to figure out--

416
00:19:44,290 --> 00:19:49,430
by turning a circular crank,
can I draw a straight line?

417
00:19:49,430 --> 00:19:52,610
And this is motivated by steam
engines, where you have a steam

418
00:19:52,610 --> 00:19:54,760
piston is moving up and
down in a straight line,

419
00:19:54,760 --> 00:19:58,920
and you want to turn
a wheel on your train.

420
00:19:58,920 --> 00:20:00,110
So how do you do that?

421
00:20:00,110 --> 00:20:02,310
Well, these are two
old ways to do it,

422
00:20:02,310 --> 00:20:08,420
and here's what they look like
this in computational land.

423
00:20:08,420 --> 00:20:09,930
These are the original drawings.

424
00:20:09,930 --> 00:20:15,340
So we have on the top a design
by Watt, the unit, in 1784,

425
00:20:15,340 --> 00:20:16,840
and it's approximately straight.

426
00:20:16,840 --> 00:20:19,230
It's not perfect, but this
highlighted point moves

427
00:20:19,230 --> 00:20:20,690
along a circle,
and the green point

428
00:20:20,690 --> 00:20:21,912
moves along the figure eight.

429
00:20:21,912 --> 00:20:24,370
And the figure eight is fairly
straight here in the middle.

430
00:20:24,370 --> 00:20:26,500
So if you just use that
part, it works pretty well.

431
00:20:26,500 --> 00:20:28,820
You lose a little
bit from the wiggle,

432
00:20:28,820 --> 00:20:33,510
but it was used in a lot
of locomotive engines.

433
00:20:33,510 --> 00:20:35,850
Down here, you have the
first correct solution

434
00:20:35,850 --> 00:20:39,670
by [? Percy ?] in 1864,
almost 100 years later,

435
00:20:39,670 --> 00:20:42,520
and if you move the highlighted
a point along the circle,

436
00:20:42,520 --> 00:20:44,370
this point moves
along the red line.

437
00:20:46,606 --> 00:20:48,980
And there's a whole bunch of
mathematics related to this,

438
00:20:48,980 --> 00:20:53,460
generalizing this result, and
we will cover it in this class.

439
00:20:53,460 --> 00:20:56,050
Next up, we have manufacturing.

440
00:20:56,050 --> 00:21:01,360
This is a fun example of
bending a piece of wire.

441
00:21:01,360 --> 00:21:05,065
This is a machine called DIWire,
do-it-yourself wire bending.

442
00:21:05,065 --> 00:21:06,690
So you can build one
of these machines.

443
00:21:06,690 --> 00:21:09,230
Here, it's making
a planar shape.

444
00:21:09,230 --> 00:21:12,200
You can also make
three-dimensional shapes.

445
00:21:12,200 --> 00:21:13,750
Lots of things are possible.

446
00:21:13,750 --> 00:21:14,880
This is a recent machine.

447
00:21:14,880 --> 00:21:16,463
There's actually a
lot of wire bending

448
00:21:16,463 --> 00:21:17,740
machines that are out there.

449
00:21:17,740 --> 00:21:19,900
This is a nice and simple one.

450
00:21:19,900 --> 00:21:21,490
But can you make everything?

451
00:21:21,490 --> 00:21:24,570
No, because this piece of
wire can't collide with itself

452
00:21:24,570 --> 00:21:26,290
and can't collide
with the machine.

453
00:21:26,290 --> 00:21:27,470
That's a constraint.

454
00:21:27,470 --> 00:21:31,400
And you want to figure out,
what shapes can you make?

455
00:21:31,400 --> 00:21:35,320
Can I make my pair of glasses,
or am I limited somehow?

456
00:21:35,320 --> 00:21:38,900
That's a folding problem,
which we will talk about.

457
00:21:38,900 --> 00:21:41,530
Next up, medical.

458
00:21:41,530 --> 00:21:43,660
So one example of a
medical application

459
00:21:43,660 --> 00:21:45,670
is to build a stent.

460
00:21:45,670 --> 00:21:49,360
This is called an origami
stent, developed at Oxford

461
00:21:49,360 --> 00:21:50,600
about two years ago.

462
00:21:50,600 --> 00:21:53,300
And the idea is you want to do
non-intrusive heart surgery,

463
00:21:53,300 --> 00:21:55,490
so you want to take
this big thing,

464
00:21:55,490 --> 00:21:59,150
fold it down really small,
like this, and stick it

465
00:21:59,150 --> 00:22:01,370
through some small blood
vessels in your body

466
00:22:01,370 --> 00:22:03,245
until you get up near
your heart where you've

467
00:22:03,245 --> 00:22:06,320
got nice big vessels
you want to de-clog.

468
00:22:06,320 --> 00:22:09,596
And so you want to expand it
back out to its larger size.

469
00:22:09,596 --> 00:22:11,470
So it's essentially a
transportation problem.

470
00:22:11,470 --> 00:22:13,000
You want to make
something small,

471
00:22:13,000 --> 00:22:17,510
until it gets where you need
it, and then you make it big.

472
00:22:17,510 --> 00:22:22,462
That's one example of
a medical application.

473
00:22:22,462 --> 00:22:24,420
Related to that kind of
transportation problem,

474
00:22:24,420 --> 00:22:27,400
you also have
motivation of aerospace.

475
00:22:27,400 --> 00:22:30,740
So you want to send a
large object into space,

476
00:22:30,740 --> 00:22:33,530
and your space shuttle
isn't big enough to do it.

477
00:22:33,530 --> 00:22:36,180
You'd like to fold it
down to become smaller.

478
00:22:36,180 --> 00:22:38,980
This is an example by Robert
Lang, who is pictured here,

479
00:22:38,980 --> 00:22:40,770
leading origami designer.

480
00:22:40,770 --> 00:22:45,000
And this is a prototype.

481
00:22:45,000 --> 00:22:47,050
It's only five meters in size.

482
00:22:47,050 --> 00:22:50,470
The goal is to make 100-meter
telescope lens, which

483
00:22:50,470 --> 00:22:53,040
definitely does not fit
in your space station.

484
00:22:53,040 --> 00:22:55,130
So you do lots of
foldings like this,

485
00:22:55,130 --> 00:22:57,030
until it fits in
your space shuttle.

486
00:22:57,030 --> 00:22:57,550
Send it out.

487
00:22:57,550 --> 00:22:59,466
When you're in space,
you've got lots of room,

488
00:22:59,466 --> 00:23:00,490
and you can unfold out.

489
00:23:00,490 --> 00:23:04,900
And you get your giant lens,
much larger than Hubble's lens,

490
00:23:04,900 --> 00:23:05,460
for example.

491
00:23:07,940 --> 00:23:10,190
In general, this area is
called deployable structures,

492
00:23:10,190 --> 00:23:13,450
when you want to make something
small for transportation.

493
00:23:13,450 --> 00:23:15,430
Then we have biology.

494
00:23:15,430 --> 00:23:18,250
Protein folding
is a big problem.

495
00:23:18,250 --> 00:23:19,580
A lot of people work on it.

496
00:23:19,580 --> 00:23:23,620
And a protein folds kind of like
a robotic arm, geometrically.

497
00:23:23,620 --> 00:23:26,560
It's a little different , and
it's not fully understood how

498
00:23:26,560 --> 00:23:27,190
it works.

499
00:23:27,190 --> 00:23:29,580
But there's a lot of cool
folding problems, geometry

500
00:23:29,580 --> 00:23:34,670
problems, that we've studied,
motivated by how proteins fold

501
00:23:34,670 --> 00:23:35,970
and figuring that out.

502
00:23:35,970 --> 00:23:38,190
Applications, things
like drug design,

503
00:23:38,190 --> 00:23:41,840
you want to kill a virus
without killing the host.

504
00:23:41,840 --> 00:23:44,300
Design a proteins that
folds into the right shape,

505
00:23:44,300 --> 00:23:47,490
so it kills one thing
and not the other.

506
00:23:47,490 --> 00:23:50,890
That's the sort of general
goal, and have some cool things

507
00:23:50,890 --> 00:23:55,760
which we will cover in
this class related to that.

508
00:23:55,760 --> 00:23:58,840
Next is sculpture,
and origami design

509
00:23:58,840 --> 00:24:01,764
is an obvious motivation
for why you might be here.

510
00:24:01,764 --> 00:24:03,430
This is why we initially
got interested.

511
00:24:03,430 --> 00:24:05,420
Origami has reached
incredible heights.

512
00:24:05,420 --> 00:24:09,950
Both of these are folded
from one square, no cuts.

513
00:24:09,950 --> 00:24:12,840
So just one square paper, you
can make a three headed dog.

514
00:24:12,840 --> 00:24:16,160
You can make, I guess I
shouldn't call him a man,

515
00:24:16,160 --> 00:24:19,510
but you can make a
Nazgul on his horse.

516
00:24:19,510 --> 00:24:20,749
Or is it one entity?

517
00:24:20,749 --> 00:24:21,290
I don't know.

518
00:24:21,290 --> 00:24:24,760
But it's one square
paper, that much I know.

519
00:24:24,760 --> 00:24:26,780
This is designed
by Jason Ku who's

520
00:24:26,780 --> 00:24:29,700
the President of the origami
club at MIT, OrigaMIT, which

521
00:24:29,700 --> 00:24:31,030
you should all check out.

522
00:24:31,030 --> 00:24:33,442
They meet on Sundays at 3:00.

523
00:24:36,580 --> 00:24:38,500
I know there's a bunch
of origami people here.

524
00:24:38,500 --> 00:24:40,672
So how are these possible?

525
00:24:40,672 --> 00:24:42,630
These are possible through
mathematics and kind

526
00:24:42,630 --> 00:24:45,486
of algorithms that we
will cover in this class.

527
00:24:45,486 --> 00:24:46,860
In fact there, is
a guest lecture

528
00:24:46,860 --> 00:24:49,256
by Jason Ku, which you'll
be watching, I think,

529
00:24:49,256 --> 00:24:52,010
in lecture six.

530
00:24:52,010 --> 00:24:53,930
And he's not actually
in town, but still he

531
00:24:53,930 --> 00:24:57,140
can give the video lecture,
because he already gave it.

532
00:24:57,140 --> 00:25:00,180
It's like time travel.

533
00:25:00,180 --> 00:25:04,000
I have just a few examples
of some of our sculpture.

534
00:25:04,000 --> 00:25:07,500
This is with Martin Demaine,
who's your cameraman here.

535
00:25:07,500 --> 00:25:10,600
And these are based on
curve creases, which

536
00:25:10,600 --> 00:25:12,656
we may talk about to some
extent in this class.

537
00:25:12,656 --> 00:25:14,780
And we didn't talk about
it too much two years ago,

538
00:25:14,780 --> 00:25:16,050
but a little bit.

539
00:25:16,050 --> 00:25:18,520
Curve creases are not very
well understood mathematically.

540
00:25:18,520 --> 00:25:21,010
Almost everything we will cover
is based on straight crease

541
00:25:21,010 --> 00:25:23,100
design, like most origami.

542
00:25:23,100 --> 00:25:24,910
Curve creases are
pretty amazing, though.

543
00:25:24,910 --> 00:25:27,660
They do some really cool
shapes, and we're still

544
00:25:27,660 --> 00:25:30,285
trying to figure them out-- how
to design them algorithmically.

545
00:25:33,670 --> 00:25:35,400
These pieces are
on display in DC

546
00:25:35,400 --> 00:25:37,485
right now if you want
to check them out,

547
00:25:37,485 --> 00:25:39,090
for the next several months.

548
00:25:42,510 --> 00:25:47,450
These pieces of paper are
circles with a circular hole,

549
00:25:47,450 --> 00:25:48,810
and there's more than one.

550
00:25:48,810 --> 00:25:50,610
This has two pieces.

551
00:25:50,610 --> 00:25:51,925
This has three.

552
00:25:51,925 --> 00:25:55,240
This has, I think, five.

553
00:25:55,240 --> 00:25:55,740
OK.

554
00:25:55,740 --> 00:25:58,060
So that was a brief
sculpture origami design.

555
00:25:58,060 --> 00:26:00,940
Lots of different other possible
sculptures, kinetic sculpture,

556
00:26:00,940 --> 00:26:03,380
you might want to try building.

557
00:26:03,380 --> 00:26:06,180
Also architecture,
reconfigurable architecture.

558
00:26:06,180 --> 00:26:08,210
I think this is an
underexplored area.

559
00:26:08,210 --> 00:26:10,590
How many people here
are from architecture?

560
00:26:10,590 --> 00:26:12,780
A few.

561
00:26:12,780 --> 00:26:15,060
You might want to make some
reconfigurable buildings.

562
00:26:15,060 --> 00:26:19,040
Hoberman is one example of
someone embracing this a lot.

563
00:26:19,040 --> 00:26:21,140
He started in
folding toy design,

564
00:26:21,140 --> 00:26:23,700
but now he does mostly
folding architecture.

565
00:26:23,700 --> 00:26:27,490
This is one example, from his
company Hoberman Associates,

566
00:26:27,490 --> 00:26:31,870
from the winter Olympics
in Salt Lake City 2002.

567
00:26:31,870 --> 00:26:32,860
It's very dramatic.

568
00:26:32,860 --> 00:26:37,070
You have this folding structure
they can almost completely

569
00:26:37,070 --> 00:26:39,810
close up, except for
a little circle here,

570
00:26:39,810 --> 00:26:42,630
and when it opens up,
you've got this huge stage.

571
00:26:42,630 --> 00:26:45,280
You can see the
scale, of people.

572
00:26:45,280 --> 00:26:49,290
He co-designed this
giant U2 stage.

573
00:26:49,290 --> 00:26:52,760
If you've seen U2 any time in
the last few years, like I did.

574
00:26:52,760 --> 00:26:54,790
This incredible
folding structure

575
00:26:54,790 --> 00:26:57,730
is based on similar principles.

576
00:26:57,730 --> 00:27:00,240
And Chuck Hoberman is
actually teaching a class

577
00:27:00,240 --> 00:27:04,830
at the Graduate School of Design
at Harvard on Monday afternoon,

578
00:27:04,830 --> 00:27:07,560
and I think anyone's welcome
if you want to check it out.

579
00:27:07,560 --> 00:27:09,990
You should go see it.

580
00:27:09,990 --> 00:27:11,650
Talk about his techniques.

581
00:27:11,650 --> 00:27:13,540
Maybe we'll get them
to do a guest lecture.

582
00:27:13,540 --> 00:27:14,750
We'll see.

583
00:27:14,750 --> 00:27:17,310
So that was a brief
overview of a bunch

584
00:27:17,310 --> 00:27:18,790
of different applications.

585
00:27:18,790 --> 00:27:19,810
There's tons more.

586
00:27:19,810 --> 00:27:22,530
In fact, if you have more,
please send them to me.

587
00:27:22,530 --> 00:27:26,060
I know a bunch more,
and we will get to them,

588
00:27:26,060 --> 00:27:28,460
but this is a little
survey of different fields

589
00:27:28,460 --> 00:27:30,050
that folding touches on.

590
00:27:30,050 --> 00:27:33,020
Now I'm gonna switch over to
more mathematical stuff, which

591
00:27:33,020 --> 00:27:34,500
is the bulk of the class.

592
00:27:34,500 --> 00:27:37,380
Unless there are questions?

593
00:27:37,380 --> 00:27:38,980
Cool.

594
00:27:38,980 --> 00:27:44,660
So we move to what
kind of things

595
00:27:44,660 --> 00:27:46,470
we're interested in, in folding.

596
00:27:46,470 --> 00:27:51,500
There are essentially
three types.

597
00:27:51,500 --> 00:27:58,980
Linkages, which we
usually think of as one

598
00:27:58,980 --> 00:27:59,960
dimensional structures.

599
00:27:59,960 --> 00:28:02,850
So we have one dimensional,
usually straight segments,

600
00:28:02,850 --> 00:28:05,950
connected together at
hinges, and here I've

601
00:28:05,950 --> 00:28:07,130
drawn it in two dimensions.

602
00:28:07,130 --> 00:28:09,070
It could also live
in three dimensions.

603
00:28:09,070 --> 00:28:12,450
Like-- my arm is made up
of one dimensional bones,

604
00:28:12,450 --> 00:28:14,750
and then there's
sockets for them

605
00:28:14,750 --> 00:28:18,750
to join, essentially
universal joints.

606
00:28:18,750 --> 00:28:20,140
How can those things fold?

607
00:28:20,140 --> 00:28:21,630
What shapes can they fold into?

608
00:28:21,630 --> 00:28:23,770
General constraints
here are that the edges

609
00:28:23,770 --> 00:28:27,950
should stay the same length--
I can't stretch my arm longer--

610
00:28:27,950 --> 00:28:30,260
and they have to stay
connected at the joints.

611
00:28:30,260 --> 00:28:33,050
I can't detach my elbow
and later reattach it,

612
00:28:33,050 --> 00:28:35,340
at least not ideally.

613
00:28:35,340 --> 00:28:39,760
The mathematical version
is you're not allowed to.

614
00:28:39,760 --> 00:28:43,110
One dimension up, we have
sheet folding, which we usually

615
00:28:43,110 --> 00:28:46,660
refer to as paper
folding, but it

616
00:28:46,660 --> 00:28:50,270
could be sheet metal, anything.

617
00:28:50,270 --> 00:28:53,290
You have your sheet
of material, and you

618
00:28:53,290 --> 00:28:57,770
want to fold into things like
origami, like space stations,

619
00:28:57,770 --> 00:29:01,000
like telescope lenses, whatever.

620
00:29:01,000 --> 00:29:02,090
General rules here.

621
00:29:02,090 --> 00:29:05,140
Like this piece of paper,
you cannot stretch the paper.

622
00:29:05,140 --> 00:29:07,370
It can't get any longer.

623
00:29:07,370 --> 00:29:12,280
It can't collide with itself,
and you can't tear the paper.

624
00:29:12,280 --> 00:29:15,200
You're not allowed to
cut it, because that

625
00:29:15,200 --> 00:29:17,360
tends to make things too easy.

626
00:29:17,360 --> 00:29:19,740
So the sort of
mathematically pure version,

627
00:29:19,740 --> 00:29:22,591
and also the modern
origami pure version,

628
00:29:22,591 --> 00:29:24,840
is that you're not allowed
to cut your piece of paper.

629
00:29:24,840 --> 00:29:30,930
So just folding, no
stretching no crossing.

630
00:29:30,930 --> 00:29:33,640
If you want to go
another dimension up,

631
00:29:33,640 --> 00:29:36,716
we typically call this
polyhedron folding.

632
00:29:40,890 --> 00:29:46,190
So a polyhedron is made up of a
bunch of polygons, like a cube.

633
00:29:46,190 --> 00:29:51,760
And typically we're interested
in unfolding a polyhedron.

634
00:29:51,760 --> 00:29:55,410
So think of this as just a
surface that's hollow inside.

635
00:29:55,410 --> 00:29:57,470
If you wanted to
build that surface,

636
00:29:57,470 --> 00:30:03,120
you'd like to find a shape,
like this cross, that

637
00:30:03,120 --> 00:30:06,170
folds into that cube.

638
00:30:06,170 --> 00:30:07,830
Because if you have
sheet material,

639
00:30:07,830 --> 00:30:10,640
you want to start from
some original shape that's

640
00:30:10,640 --> 00:30:13,961
flat that can fold
into your shape.

641
00:30:13,961 --> 00:30:15,710
You can also consider
the reverse problem,

642
00:30:15,710 --> 00:30:16,730
which is folding.

643
00:30:16,730 --> 00:30:19,510
Suppose I give you this cross,
what shapes can it fold into?

644
00:30:19,510 --> 00:30:21,924
And we've looked
at both, and I'll

645
00:30:21,924 --> 00:30:23,090
talk about them in a second.

646
00:30:25,710 --> 00:30:28,125
So in general, for each
of these categories,

647
00:30:28,125 --> 00:30:30,000
and you'll see these
categories over and over

648
00:30:30,000 --> 00:30:31,540
because they're on the book.

649
00:30:31,540 --> 00:30:35,040
Here it says linkages,
origami, and polyhedra.

650
00:30:35,040 --> 00:30:36,660
And there's three
parts to the book.

651
00:30:36,660 --> 00:30:39,099
There's linkages,
origami, and polyhedra.

652
00:30:39,099 --> 00:30:40,390
Part one, part two, part three.

653
00:30:40,390 --> 00:30:41,750
It's just like this.

654
00:30:41,750 --> 00:30:44,280
Our class will not follow
this order, though.

655
00:30:44,280 --> 00:30:48,332
After today, I think we'll
talk about paper for a while,

656
00:30:48,332 --> 00:30:49,873
then go to linkages
for a while, then

657
00:30:49,873 --> 00:30:52,420
go to polyhedra for a while,
and then we'll repeat,

658
00:30:52,420 --> 00:30:55,177
go to paper for a
while, and so on.

659
00:30:55,177 --> 00:30:57,510
Different parts will be of
interest to different people,

660
00:30:57,510 --> 00:30:59,310
so we'll do lots of
different coverage.

661
00:30:59,310 --> 00:31:03,430
But I think paper's the most
exciting, so we'll start there.

662
00:31:03,430 --> 00:31:06,080
In general, for
each of these types

663
00:31:06,080 --> 00:31:07,870
of things you want
to fold, there

664
00:31:07,870 --> 00:31:12,990
are two kinds of problems
we're generally interested in.

665
00:31:12,990 --> 00:31:15,310
And based around the
idea that, well, there's

666
00:31:15,310 --> 00:31:18,040
some folding structure
that you're interested in,

667
00:31:18,040 --> 00:31:19,770
and then there's
the way it can fold.

668
00:31:23,480 --> 00:31:27,030
And you can start
from either one.

669
00:31:27,030 --> 00:31:28,940
If you start from a
folding structure,

670
00:31:28,940 --> 00:31:31,300
and you want to know
what it can fold into,

671
00:31:31,300 --> 00:31:34,490
this we typically call
a foldability problem,

672
00:31:34,490 --> 00:31:37,240
or you could call it
an analysis problem.

673
00:31:37,240 --> 00:31:46,290
I have something
like this, and I

674
00:31:46,290 --> 00:31:47,800
want to know what
it can fold into.

675
00:31:47,800 --> 00:31:48,990
That's an example.

676
00:31:48,990 --> 00:31:52,420
What can the cross fold into?

677
00:31:52,420 --> 00:31:55,179
Or I have some linkage,
and I have a robotic arm.

678
00:31:55,179 --> 00:31:55,970
It's already fixed.

679
00:31:55,970 --> 00:31:57,428
I've already designed
it, built it.

680
00:31:57,428 --> 00:31:59,490
What shapes can it fold into?

681
00:31:59,490 --> 00:32:03,820
The reverse problem is a design
problem or synthesis problem.

682
00:32:03,820 --> 00:32:07,840
So there I start with what
foldings I would like to have.

683
00:32:07,840 --> 00:32:10,340
So for example, I want
to fold the butterfly.

684
00:32:10,340 --> 00:32:13,420
So I design a butterfly,
and then I'd say,

685
00:32:13,420 --> 00:32:16,815
well, I have a rectangle paper,
can I fold it into a butterfly?

686
00:32:16,815 --> 00:32:17,940
So that's a design problem.

687
00:32:17,940 --> 00:32:20,150
Here you want to design
a folding structure that

688
00:32:20,150 --> 00:32:22,960
achieves your goals, like shape.

689
00:32:22,960 --> 00:32:26,250
So that's generically
what we like to do.

690
00:32:26,250 --> 00:32:30,920
And then there's three types
of results we typically get.

691
00:32:30,920 --> 00:32:34,110
I'm giving a super
high level first,

692
00:32:34,110 --> 00:32:36,305
and then we'll get to
some actual examples.

693
00:32:39,870 --> 00:32:42,710
So these are typical results.

694
00:32:45,360 --> 00:32:52,320
So three types are universality,
decision, and hardness,

695
00:32:52,320 --> 00:32:57,260
and these relate to
what is possible.

696
00:32:57,260 --> 00:32:59,754
In particular, is
everything possible?

697
00:32:59,754 --> 00:33:02,045
And this makes sense both
from a foldability standpoint

698
00:33:02,045 --> 00:33:03,660
and from a design standpoint.

699
00:33:03,660 --> 00:33:05,820
If I give you a robotic
arm, I'd like to know,

700
00:33:05,820 --> 00:33:08,790
can it folding to all the
possible configurations,

701
00:33:08,790 --> 00:33:10,470
or are there some
that it can't reach?

702
00:33:10,470 --> 00:33:12,610
That, if they can
reach all of them,

703
00:33:12,610 --> 00:33:14,430
we have a universality result.

704
00:33:14,430 --> 00:33:18,410
Typically when you can prove
that you can make anything,

705
00:33:18,410 --> 00:33:20,250
you actually get an
algorithm to do it.

706
00:33:20,250 --> 00:33:22,470
So you could say, oh,
well, how do I fold into x?

707
00:33:22,470 --> 00:33:23,640
Well, here's how you do it.

708
00:33:23,640 --> 00:33:25,515
There's an algorithm to
compute that for you.

709
00:33:25,515 --> 00:33:27,550
Also, in design,
I'd like to know,

710
00:33:27,550 --> 00:33:31,330
OK, I can make a butterfly,
and I can make a Nazgul.

711
00:33:31,330 --> 00:33:32,640
Can I make everything?

712
00:33:32,640 --> 00:33:35,240
If yes, you get a
universality result.

713
00:33:35,240 --> 00:33:37,810
That's a design universality.

714
00:33:37,810 --> 00:33:41,164
If universality is not true,
if you can't make everything,

715
00:33:41,164 --> 00:33:42,580
the next best thing
you could hope

716
00:33:42,580 --> 00:33:45,660
for is a decision
algorithm, an algorithm

717
00:33:45,660 --> 00:33:50,680
that tells you quickly, is this
possible, is this impossible.

718
00:33:50,680 --> 00:33:52,150
So this is like a
characterization

719
00:33:52,150 --> 00:33:55,510
of what's possible, and what's
impossible using algorithms.

720
00:33:55,510 --> 00:33:57,140
Sometimes, though,
that's not possible.

721
00:33:57,140 --> 00:33:59,390
There is no good algorithm
to tell whether something's

722
00:33:59,390 --> 00:34:01,490
foldable, and then
we aim for hardness

723
00:34:01,490 --> 00:34:04,240
result to prove that
there is no good algorithm

724
00:34:04,240 --> 00:34:07,710
to solve your problem.

725
00:34:07,710 --> 00:34:10,564
So that gets in
complexity theory.

726
00:34:10,564 --> 00:34:11,980
Not everyone's
going to know that.

727
00:34:11,980 --> 00:34:14,440
We will be reviewing it
in the video lectures,

728
00:34:14,440 --> 00:34:16,920
and in here as needed.

729
00:34:16,920 --> 00:34:19,260
So those are the sort
of typical outcomes

730
00:34:19,260 --> 00:34:22,403
for any of these
kinds of problems.

731
00:34:22,403 --> 00:34:23,944
Let's see some
examples, unless there

732
00:34:23,944 --> 00:34:25,980
are generic questions
at this point,

733
00:34:25,980 --> 00:34:29,001
only generic questions
about generic material.

734
00:34:29,001 --> 00:34:30,500
Next we'll get to
specific material,

735
00:34:30,500 --> 00:34:31,958
and you can ask
specific questions.

736
00:34:47,820 --> 00:34:49,210
So first up is linkages.

737
00:34:52,889 --> 00:34:55,679
And, of course,
there's lots of results

738
00:34:55,679 --> 00:34:56,679
in each of these fields.

739
00:34:56,679 --> 00:34:59,340
I'm just going to show
you a couple in each.

740
00:34:59,340 --> 00:35:01,840
And the first question
you are typically

741
00:35:01,840 --> 00:35:05,260
interested in about a
linkage is, is it rigid?

742
00:35:05,260 --> 00:35:06,890
Does it move it all?

743
00:35:06,890 --> 00:35:08,740
Let me give you some
examples of this.

744
00:35:15,600 --> 00:35:18,290
Do a little pop quiz.

745
00:35:18,290 --> 00:35:19,750
It's OK to get it wrong.

746
00:35:19,750 --> 00:35:23,545
Just makes for some
fun interaction.

747
00:35:28,750 --> 00:35:32,080
So, yes or no, is this rigid?

748
00:35:32,080 --> 00:35:34,729
Or, rigid or flexible,
I should say.

749
00:35:34,729 --> 00:35:35,395
AUDIENCE: Rigid.

750
00:35:35,395 --> 00:35:36,103
PROFESSOR: Rigid.

751
00:35:36,103 --> 00:35:37,140
Everyone agrees.

752
00:35:37,140 --> 00:35:39,000
Good.

753
00:35:39,000 --> 00:35:40,110
Correct.

754
00:35:40,110 --> 00:35:41,326
Rigid or flexible?

755
00:35:41,326 --> 00:35:42,247
AUDIENCE: [INAUDIBLE]

756
00:35:42,247 --> 00:35:43,080
PROFESSOR: Flexible?

757
00:35:43,080 --> 00:35:43,990
AUDIENCE: [INAUDIBLE]

758
00:35:43,990 --> 00:35:45,489
PROFESSOR: Depends
on the dimension.

759
00:35:45,489 --> 00:35:45,990
Very good.

760
00:35:45,990 --> 00:35:48,000
It's rigid or flexible.

761
00:35:48,000 --> 00:35:48,600
It depends.

762
00:35:48,600 --> 00:35:53,430
It's rigid in 2D, because
these are triangles.

763
00:35:53,430 --> 00:35:56,900
But it's flexible
in 3D, because you

764
00:35:56,900 --> 00:36:00,465
can rotate one triangle
around this hinge.

765
00:36:03,070 --> 00:36:04,580
And this is rigid or flexible?

766
00:36:04,580 --> 00:36:05,717
AUDIENCE: [INAUDIBLE]

767
00:36:05,717 --> 00:36:06,550
PROFESSOR: Flexible.

768
00:36:06,550 --> 00:36:07,910
Everyone agrees and is correct.

769
00:36:12,390 --> 00:36:15,300
In general, given a
structure like this linkage,

770
00:36:15,300 --> 00:36:17,330
you want to know, is it
rigid or is it flexible?

771
00:36:17,330 --> 00:36:18,480
This has lots of applications.

772
00:36:18,480 --> 00:36:19,460
Like, you want to
build a bridge,

773
00:36:19,460 --> 00:36:20,543
you don't want it to move.

774
00:36:20,543 --> 00:36:24,110
You're building a building,
you don't want it to move.

775
00:36:24,110 --> 00:36:26,720
So here's the mathematical
state of affairs.

776
00:36:26,720 --> 00:36:30,240
Distinguishing this boundary,
which is a rigid in 2D

777
00:36:30,240 --> 00:36:33,017
versus flexible in 2D, we
understand super well as

778
00:36:33,017 --> 00:36:34,100
great algorithms to do it.

779
00:36:34,100 --> 00:36:35,420
We will cover them.

780
00:36:35,420 --> 00:36:39,140
This boundary between rigid
in 3D and flexible in 3D,

781
00:36:39,140 --> 00:36:40,674
we don't really understand.

782
00:36:40,674 --> 00:36:42,590
And there aren't great
algorithms to solve it.

783
00:36:42,590 --> 00:36:44,423
It's a little disconcerting
given the number

784
00:36:44,423 --> 00:36:45,800
buildings and
bridges we live in,

785
00:36:45,800 --> 00:36:49,140
but that's the
state of the world.

786
00:36:49,140 --> 00:36:53,570
And we will talk about all that.

787
00:36:53,570 --> 00:36:55,670
So that is rigidity.

788
00:36:55,670 --> 00:36:59,560
Next up we have universality.

789
00:36:59,560 --> 00:37:01,720
So this is the robotic
arm question I mentioned.

790
00:37:01,720 --> 00:37:02,720
I have a robotic arm.

791
00:37:02,720 --> 00:37:09,560
Does it fold into
everything, into

792
00:37:09,560 --> 00:37:13,110
every possible configuration?

793
00:37:13,110 --> 00:37:16,960
And the answer, again, depends
what dimension you live in.

794
00:37:16,960 --> 00:37:20,380
In two dimensions,
there are cool ways

795
00:37:20,380 --> 00:37:22,030
to reconfigure your robotic arm.

796
00:37:22,030 --> 00:37:24,715
So this is actually a polygon,
a closed arm, if you will,

797
00:37:24,715 --> 00:37:28,085
a closed chain, and this
is one way to unfold it.

798
00:37:28,085 --> 00:37:29,460
And then, in
principle, you could

799
00:37:29,460 --> 00:37:32,510
refold that into any
other shape you want.

800
00:37:32,510 --> 00:37:34,750
And we will cover-- this
is one algorithm to do it.

801
00:37:34,750 --> 00:37:36,850
There's actually three
algorithms to do this.

802
00:37:36,850 --> 00:37:38,870
This is the most recent one.

803
00:37:38,870 --> 00:37:41,570
This was originally
my Ph.D. thesis,

804
00:37:41,570 --> 00:37:43,880
to figure out whether
this was possible.

805
00:37:43,880 --> 00:37:45,820
It's called the
carpenter's rule problem.

806
00:37:45,820 --> 00:37:49,060
This algorithm preserves the
fivefold rotational symmetry

807
00:37:49,060 --> 00:37:51,640
of the polygon,
which is pretty cool.

808
00:37:51,640 --> 00:37:54,130
And in general, you
take any 2D polygon,

809
00:37:54,130 --> 00:37:56,100
you can unfold it while
preserving all the edge

810
00:37:56,100 --> 00:37:57,641
lengths, keeping
all the connections,

811
00:37:57,641 --> 00:38:00,290
and avoiding collision.

812
00:38:00,290 --> 00:38:04,020
I'll show you some
other fun examples.

813
00:38:04,020 --> 00:38:08,420
This is a kind of
spider, 500 vertices.

814
00:38:08,420 --> 00:38:09,680
We're kind of zooming out.

815
00:38:09,680 --> 00:38:12,190
As we expand here, you can see
the lengths get really tiny,

816
00:38:12,190 --> 00:38:15,790
but, in reality, all the
lengths are staying the same

817
00:38:15,790 --> 00:38:18,730
throughout time.

818
00:38:18,730 --> 00:38:21,420
And we fold.

819
00:38:21,420 --> 00:38:23,040
So this is the state
of affairs in 2D.

820
00:38:23,040 --> 00:38:24,630
We have good algorithms.

821
00:38:24,630 --> 00:38:25,869
It's always possible.

822
00:38:25,869 --> 00:38:26,410
Universality.

823
00:38:30,910 --> 00:38:33,710
So what about 3D?

824
00:38:33,710 --> 00:38:38,430
3D is harder, in particular
because of examples like this.

825
00:38:38,430 --> 00:38:39,810
We call this knitting needles.

826
00:38:39,810 --> 00:38:41,490
You've got two blue
segments, which

827
00:38:41,490 --> 00:38:43,250
you can think of
as long needles,

828
00:38:43,250 --> 00:38:46,920
and then a short, purple
thread connecting them.

829
00:38:46,920 --> 00:38:49,570
And there's no way to undo this.

830
00:38:49,570 --> 00:38:52,300
In particular, if you started
with a straight, robotic arm,

831
00:38:52,300 --> 00:38:55,130
you could not hold it in
the shape, or vice versa.

832
00:38:55,130 --> 00:38:57,360
We call this a
locked configuration,

833
00:38:57,360 --> 00:38:59,360
and in general lock
configurations exist,

834
00:38:59,360 --> 00:39:00,750
obviously.

835
00:39:00,750 --> 00:39:03,626
And we don't know how to
distinguish good or bad.

836
00:39:03,626 --> 00:39:06,250
So here, we actually don't know
whether there's a good decision

837
00:39:06,250 --> 00:39:10,369
algorithm or hardness, but
it's definitely not universal.

838
00:39:10,369 --> 00:39:11,910
And 3D's very
interesting, because it

839
00:39:11,910 --> 00:39:14,450
relates to robotic arms
in real life and protein

840
00:39:14,450 --> 00:39:16,055
folding and all these things.

841
00:39:16,055 --> 00:39:17,180
I'll be talking about that.

842
00:39:17,180 --> 00:39:17,917
In 4D.

843
00:39:17,917 --> 00:39:20,000
If you're lucky enough to
live in four dimensions,

844
00:39:20,000 --> 00:39:22,000
you get universality,
everything's great.

845
00:39:22,000 --> 00:39:25,510
And we'll talk about that too.

846
00:39:25,510 --> 00:39:29,920
So next up, we
have paper folding.

847
00:39:29,920 --> 00:39:31,725
That was our brief
overview of linkages.

848
00:39:36,330 --> 00:39:38,560
So paper folding.

849
00:39:38,560 --> 00:39:43,920
I'm gonna mention one problem
in the foldability world

850
00:39:43,920 --> 00:39:46,790
and some problems
in the design world.

851
00:39:46,790 --> 00:39:50,572
This is probably my favorite
are of paper, origami design.

852
00:39:50,572 --> 00:39:52,030
But foldability's
interesting, too.

853
00:39:52,030 --> 00:39:55,840
The obvious and
the first question

854
00:39:55,840 --> 00:39:58,870
you might wonder about
in origami foldability

855
00:39:58,870 --> 00:40:02,780
is, if what crease
patterns fold flat?

856
00:40:02,780 --> 00:40:06,270
So if you take a classic
flapping bird, you unfold it,

857
00:40:06,270 --> 00:40:07,570
these are the creases you get.

858
00:40:07,570 --> 00:40:10,028
But in general, if I gave you
some pattern of creases drawn

859
00:40:10,028 --> 00:40:12,240
on your piece of paper,
does it fold into anything?

860
00:40:12,240 --> 00:40:13,948
And here, to make it
interesting, we say,

861
00:40:13,948 --> 00:40:15,340
does it fold into anything flat?

862
00:40:15,340 --> 00:40:16,840
Flat origami.

863
00:40:16,840 --> 00:40:18,600
You have to fold
along all the creases.

864
00:40:18,600 --> 00:40:23,650
Sadly, deciding this is a
problem that we call NP-hard.

865
00:40:23,650 --> 00:40:25,960
So there's basically no good
algorithm to solve this,

866
00:40:25,960 --> 00:40:33,050
and we will prove that in a
few classes, a few lectures.

867
00:40:33,050 --> 00:40:35,960
So that was the
foldability problem.

868
00:40:35,960 --> 00:40:37,470
Now onto design.

869
00:40:37,470 --> 00:40:40,390
If I give you a square paper,
what shapes can I make?

870
00:40:40,390 --> 00:40:42,520
That's the obvious
origami design problem.

871
00:40:42,520 --> 00:40:45,040
And it turns out you
can make any polygon,

872
00:40:45,040 --> 00:40:46,951
so you can make any
flat shape you want.

873
00:40:46,951 --> 00:40:49,200
If you have a piece of paper
that's white on one side,

874
00:40:49,200 --> 00:40:51,780
and patterned or colored
black on the other,

875
00:40:51,780 --> 00:40:53,740
you can make any two-color
pattern you want.

876
00:40:53,740 --> 00:40:56,920
If it's polygonal, it's
made of straight sides.

877
00:40:56,920 --> 00:40:58,870
If you want to make
a 3D thing, you

878
00:40:58,870 --> 00:41:00,630
could make any 3D
polyhedron, also

879
00:41:00,630 --> 00:41:02,480
with two color patterns,
whatever you want.

880
00:41:02,480 --> 00:41:04,135
We proved this way back in 1999.

881
00:41:04,135 --> 00:41:05,760
I think it was actually
the first paper

882
00:41:05,760 --> 00:41:08,300
to use the term
"computational origami."

883
00:41:08,300 --> 00:41:10,220
And it's not hard to prove.

884
00:41:10,220 --> 00:41:12,470
We'll prove it in a couple
of lectures, I think.

885
00:41:15,420 --> 00:41:17,647
We don't have a
practical way to do this.

886
00:41:17,647 --> 00:41:20,230
There is an algorithm here, but
it doesn't give you a good way

887
00:41:20,230 --> 00:41:21,940
to fold anything.

888
00:41:21,940 --> 00:41:25,290
And so the quest continues for
a good way to fold everything.

889
00:41:25,290 --> 00:41:27,775
And one such approach
is called Origamizer.

890
00:41:30,870 --> 00:41:33,170
You give it an
arbitrary 3D surface,

891
00:41:33,170 --> 00:41:34,710
it designs a crease pattern.

892
00:41:34,710 --> 00:41:37,980
This, you fold from a square
piece of paper-- solids are

893
00:41:37,980 --> 00:41:41,495
mountains, dashed are valleys--
and you get this bunny.

894
00:41:41,495 --> 00:41:43,620
This is a classic model in
computer graphics called

895
00:41:43,620 --> 00:41:47,350
the Stanford bunny, and this
takes about 10 hours to fold.

896
00:41:47,350 --> 00:41:49,472
If you're Tomohiro
Tachi, who designed it.

897
00:41:49,472 --> 00:41:50,930
And there's free
software available

898
00:41:50,930 --> 00:41:52,660
called Origamizer by Tomohiro.

899
00:41:52,660 --> 00:41:54,920
And he was just here
a couple weeks ago,

900
00:41:54,920 --> 00:41:57,980
and we've been proving that
this algorithm always works.

901
00:41:57,980 --> 00:42:00,450
And it also seems very
practical, most of the material

902
00:42:00,450 --> 00:42:02,930
here gets used in the surface.

903
00:42:02,930 --> 00:42:05,030
You don't need a very
large square of paper

904
00:42:05,030 --> 00:42:07,960
to make your bunny.

905
00:42:07,960 --> 00:42:11,017
I have one little
example here, sort

906
00:42:11,017 --> 00:42:12,600
of classic in the
origami world, which

907
00:42:12,600 --> 00:42:14,340
is to make a checkerboard.

908
00:42:14,340 --> 00:42:16,710
And this is a four-by-four
checkerboard, obviously not

909
00:42:16,710 --> 00:42:17,860
the full eight-by-eight.

910
00:42:17,860 --> 00:42:19,440
But you can fold
an eight-by-eight.

911
00:42:19,440 --> 00:42:21,760
We'll talk about
different ways to do it.

912
00:42:21,760 --> 00:42:23,810
This one is folded
from one square paper,

913
00:42:23,810 --> 00:42:25,750
white on one side,
green on the other.

914
00:42:25,750 --> 00:42:27,791
And, actually, it has a
pretty good scale factor,

915
00:42:27,791 --> 00:42:30,556
just collapses by
factor of two here.

916
00:42:30,556 --> 00:42:34,360
See if I can get it
back into correct state.

917
00:42:34,360 --> 00:42:35,310
So there you go.

918
00:42:35,310 --> 00:42:37,870
You can make your own
origami checkerboards

919
00:42:37,870 --> 00:42:40,360
if you ran out of your
regular wooden ones.

920
00:42:43,580 --> 00:42:46,370
Next example is what
we call maze folding.

921
00:42:46,370 --> 00:42:49,030
This is the poster.

922
00:42:49,030 --> 00:42:52,860
If you saw it, this is a
folding of this crease pattern,

923
00:42:52,860 --> 00:42:57,060
and the design was-- In general,
you take any orthogonal graph,

924
00:42:57,060 --> 00:43:00,220
meaning horizontal and
vertical lines on a grid,

925
00:43:00,220 --> 00:43:04,490
and you want to extrude
that graph out of the plane,

926
00:43:04,490 --> 00:43:05,930
then there's an algorithm.

927
00:43:05,930 --> 00:43:06,630
It's online.

928
00:43:06,630 --> 00:43:07,560
You can play with it.

929
00:43:07,560 --> 00:43:10,480
I think erikdemaine.org/maze
will get you there.

930
00:43:10,480 --> 00:43:11,629
You just draw that pattern.

931
00:43:11,629 --> 00:43:13,170
It will give you
this crease pattern.

932
00:43:13,170 --> 00:43:17,720
You print it out, and--
eight hours, maybe-- you

933
00:43:17,720 --> 00:43:20,070
can fold it into realities.

934
00:43:20,070 --> 00:43:23,160
So this is by [? Jenny ?] and
[? Eli, ?] who I think are

935
00:43:23,160 --> 00:43:24,807
students in this class.

936
00:43:24,807 --> 00:43:26,140
They got started a little early.

937
00:43:26,140 --> 00:43:30,690
And I didn't bring it
here, but at some point

938
00:43:30,690 --> 00:43:34,310
I will bring to the real 3D one.

939
00:43:34,310 --> 00:43:36,620
So that's maze folding.

940
00:43:36,620 --> 00:43:40,180
And the next one I have
is called folding cut.

941
00:43:40,180 --> 00:43:42,720
This is actually the first
problem we worked on, so kind

942
00:43:42,720 --> 00:43:44,830
of a personal favorite.

943
00:43:44,830 --> 00:43:51,400
You take a rectangle of paper,
and then you fold it flat,

944
00:43:51,400 --> 00:43:54,600
and you take your scissors and
make one complete straight cut.

945
00:43:54,600 --> 00:43:57,910
This is a magic trick
performed by Harry Houdini

946
00:43:57,910 --> 00:44:01,984
and other magicians, and you
get, in this case, two pieces.

947
00:44:01,984 --> 00:44:03,650
And the question is,
what shapes can you

948
00:44:03,650 --> 00:44:05,980
make by folding and
one straight cut?

949
00:44:05,980 --> 00:44:09,890
Here I made a little one.

950
00:44:09,890 --> 00:44:14,540
OK, you're not impressed,
so I'll do a harder one.

951
00:44:14,540 --> 00:44:17,310
This one is actually
quite challenging to cut.

952
00:44:17,310 --> 00:44:22,408
I was folding it as you were
arriving, as you probably saw.

953
00:44:22,408 --> 00:44:24,110
That's about right.

954
00:44:27,530 --> 00:44:30,680
It's a little hard
to open up, but this

955
00:44:30,680 --> 00:44:33,410
should be the MIT logo.

956
00:44:33,410 --> 00:44:34,575
AUDIENCE: [INAUDIBLE]

957
00:44:34,575 --> 00:44:35,450
PROFESSOR: All right.

958
00:44:35,450 --> 00:44:35,949
Good.

959
00:44:35,949 --> 00:44:38,550
AUDIENCE: [APPLAUSE]

960
00:44:38,550 --> 00:44:39,805
PROFESSOR: So that's the idea.

961
00:44:39,805 --> 00:44:41,180
You can impress
all your friends.

962
00:44:41,180 --> 00:44:43,439
Print these out, and fold them.

963
00:44:43,439 --> 00:44:45,730
It's a good challenge to fold
along the crease pattern,

964
00:44:45,730 --> 00:44:47,771
and as you might guess
there's an algorithm here.

965
00:44:47,771 --> 00:44:50,300
Given any polygon or
collection of polygons,

966
00:44:50,300 --> 00:44:52,700
you can fold a piece
of paper to line up

967
00:44:52,700 --> 00:44:56,990
all the edges in that polygon,
cut along it, you're done.

968
00:44:56,990 --> 00:44:59,650
OK, that is paper

969
00:44:59,650 --> 00:45:03,355
Next I want to briefly
mention polyhedra.

970
00:45:13,690 --> 00:45:15,600
Actually, I have a
polyhedra right over here.

971
00:45:15,600 --> 00:45:18,320
Let's just go there.

972
00:45:18,320 --> 00:45:21,350
So, again, there's an unfolding
problem, and a folding problem.

973
00:45:21,350 --> 00:45:23,960
And I'll start by
showing you a little bit

974
00:45:23,960 --> 00:45:27,180
about the folding problem.

975
00:45:27,180 --> 00:45:28,680
Oh, this shows a
little bit of both.

976
00:45:28,680 --> 00:45:31,066
This a video we
made back in '99.

977
00:45:31,066 --> 00:45:31,940
I was a grad student.

978
00:45:31,940 --> 00:45:34,760
I had lots of hours to
make videos, animations.

979
00:45:34,760 --> 00:45:40,010
So this is one example of
an unfolding of a cube.

980
00:45:40,010 --> 00:45:42,080
But the unfolding we
want to consider here

981
00:45:42,080 --> 00:45:45,360
is this cross unfolding
that I keep talking about.

982
00:45:45,360 --> 00:45:47,720
It's kind of a classic
so fun to analyze.

983
00:45:47,720 --> 00:45:49,220
So if you take that
cross shape, you

984
00:45:49,220 --> 00:45:52,120
want to know what other shapes,
what other convex polyhedra,

985
00:45:52,120 --> 00:45:53,452
can it fold into?

986
00:45:53,452 --> 00:45:55,660
And here the rules are little
different from origami.

987
00:45:55,660 --> 00:45:58,960
You have to get exactly the
surface you want no, overlap.

988
00:45:58,960 --> 00:46:02,650
Here we happen to get a doubly
covered quadrilateral, which

989
00:46:02,650 --> 00:46:05,830
is kind of a convex surface.

990
00:46:05,830 --> 00:46:07,512
A little bit cheating.

991
00:46:07,512 --> 00:46:09,220
But if we change the
creases in this way,

992
00:46:09,220 --> 00:46:13,580
we get a five-sided polyhedron
with a plane of symmetry

993
00:46:13,580 --> 00:46:14,455
down this axis.

994
00:46:17,350 --> 00:46:26,250
Or we can change the creases in
this way and get a tetrahedron.

995
00:46:26,250 --> 00:46:28,560
And it's exact coverage,
so this little tab

996
00:46:28,560 --> 00:46:31,224
fits into exactly that pocket.

997
00:46:31,224 --> 00:46:32,140
Exactly what you want.

998
00:46:32,140 --> 00:46:33,806
As you might guess,
there's an algorithm

999
00:46:33,806 --> 00:46:35,760
that will enumerate all
the different convex

1000
00:46:35,760 --> 00:46:38,550
polyhedra you can make from
a given polygon of paper.

1001
00:46:38,550 --> 00:46:41,610
And it's based on a
theorem in Russian

1002
00:46:41,610 --> 00:46:43,680
by Alexandrov, which is
in the background here.

1003
00:46:43,680 --> 00:46:46,050
If you're curious what the
Cyrillic background is,

1004
00:46:46,050 --> 00:46:49,470
this theorem is the
key to the algorithm.

1005
00:46:49,470 --> 00:46:56,659
So that's what's possible
by folding a given polygon,

1006
00:46:56,659 --> 00:46:58,200
and then you can
ask about unfolding.

1007
00:46:58,200 --> 00:47:00,111
Let me give you
a brief overview.

1008
00:47:00,111 --> 00:47:00,610
Unfolding.

1009
00:47:30,790 --> 00:47:33,210
So when you're
unfolding, you might

1010
00:47:33,210 --> 00:47:35,020
be interested in
only cutting along

1011
00:47:35,020 --> 00:47:36,950
the edges of the
surface, which is what

1012
00:47:36,950 --> 00:47:40,240
happened in this
cross unfolding.

1013
00:47:40,240 --> 00:47:43,160
Cut along these edges.

1014
00:47:43,160 --> 00:47:46,110
Or you might allow cutting
anywhere on the surface.

1015
00:47:46,110 --> 00:47:49,524
And the answers depend
on those two versions.

1016
00:47:49,524 --> 00:47:50,940
You could try to
convex polyhedra,

1017
00:47:50,940 --> 00:47:53,200
or you could try to do
arbitrary polyhedra.

1018
00:47:53,200 --> 00:47:55,640
So here's the story.

1019
00:47:55,640 --> 00:47:57,490
This one, we don't know.

1020
00:47:57,490 --> 00:47:59,770
This one, we don't know.

1021
00:47:59,770 --> 00:48:01,290
This one, the answer is yes.

1022
00:48:01,290 --> 00:48:02,150
We'll prove it.

1023
00:48:02,150 --> 00:48:03,670
This one, the answer is no.

1024
00:48:03,670 --> 00:48:04,790
We'll prove it.

1025
00:48:04,790 --> 00:48:08,100
In fact, I can show
you some examples.

1026
00:48:08,100 --> 00:48:11,720
This is an edge unfolding
of a convex polyhedron.

1027
00:48:11,720 --> 00:48:15,540
The oldest set that we know
this from this 1525 book

1028
00:48:15,540 --> 00:48:17,120
by Albrecht Duerer.

1029
00:48:17,120 --> 00:48:18,989
And this is a snub cube.

1030
00:48:18,989 --> 00:48:20,530
He didn't have
Wikipedia at the time.

1031
00:48:20,530 --> 00:48:23,510
You couldn't just look
at the image and draw it.

1032
00:48:23,510 --> 00:48:25,040
He had to build
a physical model.

1033
00:48:25,040 --> 00:48:27,670
And so he cut out pieces of
paper to build physical models,

1034
00:48:27,670 --> 00:48:28,590
so he could paint it.

1035
00:48:28,590 --> 00:48:32,810
And various of his prints
have convex polyhedra in them.

1036
00:48:32,810 --> 00:48:35,240
And this is an example
of this open problem,

1037
00:48:35,240 --> 00:48:37,870
which we don't
know the answer to.

1038
00:48:37,870 --> 00:48:40,000
Can you edge unfold
convex polyhedra?

1039
00:48:40,000 --> 00:48:44,220
If you want to take non-convex
polyhedra you cannot always do

1040
00:48:44,220 --> 00:48:44,720
it.

1041
00:48:44,720 --> 00:48:47,990
This is an example of a spiky
tetrahedron that cannot be edge

1042
00:48:47,990 --> 00:48:48,500
unfolded.

1043
00:48:48,500 --> 00:48:49,870
We'll prove that.

1044
00:48:49,870 --> 00:48:51,460
But it can be
generally unfolded.

1045
00:48:51,460 --> 00:48:55,460
If you can cut anywhere, you
can find a one-piece unfolding.

1046
00:48:55,460 --> 00:48:57,690
And so big open
problem is, can you

1047
00:48:57,690 --> 00:49:01,680
do any polyhedron
with general cuts?

1048
00:49:01,680 --> 00:49:03,170
I think the answer is yes.

1049
00:49:03,170 --> 00:49:06,380
Very cool problem.

1050
00:49:06,380 --> 00:49:07,950
And that's polyhedra.

1051
00:49:07,950 --> 00:49:10,590
And the last thing I wanted to
show you is hinged dissection.

1052
00:49:10,590 --> 00:49:12,580
This is one more thing.

1053
00:49:12,580 --> 00:49:16,650
After one, two, and three, we
have four, hinged dissection.

1054
00:49:16,650 --> 00:49:21,280
This is a chain of blocks, which
are kind of two dimensional,

1055
00:49:21,280 --> 00:49:22,770
but it's not really
paper folding.

1056
00:49:22,770 --> 00:49:26,460
So it's like a thick
linkage between one and two,

1057
00:49:26,460 --> 00:49:27,250
if you will.

1058
00:49:27,250 --> 00:49:30,140
And this chain folds into
an equilateral triangle

1059
00:49:30,140 --> 00:49:34,780
or a square, and it's
over 100 years old.

1060
00:49:34,780 --> 00:49:36,060
Can you do anything?

1061
00:49:36,060 --> 00:49:37,917
Turns out there's a
universality result.

1062
00:49:37,917 --> 00:49:40,000
This is just a picture of
part of the proof, which

1063
00:49:40,000 --> 00:49:42,570
we will cover in
a later lecture.

1064
00:49:42,570 --> 00:49:46,860
But you can fold, you can
take any set of polygons

1065
00:49:46,860 --> 00:49:50,410
of the same area, there's
one chain of blocks which

1066
00:49:50,410 --> 00:49:53,150
will fold into each
of those polygons.

1067
00:49:53,150 --> 00:49:55,370
So you can get a
universality result.

1068
00:49:55,370 --> 00:49:58,640
You can do an equilateral
triangle to a regular hexagon

1069
00:49:58,640 --> 00:50:00,710
to a swan to whatever.

1070
00:50:00,710 --> 00:50:03,970
As long as they're all the
same area, you can do it.

1071
00:50:03,970 --> 00:50:06,370
And this also generalizes to 3D.

1072
00:50:06,370 --> 00:50:08,470
We use that to make a
little sculpture here,

1073
00:50:08,470 --> 00:50:09,600
little sculpture.

1074
00:50:09,600 --> 00:50:12,950
About 1,000 blocks, you pick up
with your gloves, these blocks,

1075
00:50:12,950 --> 00:50:13,590
rearrange them.

1076
00:50:13,590 --> 00:50:16,170
And the theorem is, you can
make any shape you want out

1077
00:50:16,170 --> 00:50:21,190
of the blocks by this
kind of intersection.

1078
00:50:21,190 --> 00:50:23,670
And that was a brief overview
of a few of the results

1079
00:50:23,670 --> 00:50:25,420
that we'll be talking
about in this class.

1080
00:50:25,420 --> 00:50:26,480
There's a lot more.

1081
00:50:26,480 --> 00:50:28,880
And you can check
out the 2010 web page

1082
00:50:28,880 --> 00:50:31,370
if you really want to know
everything that's covered,

1083
00:50:31,370 --> 00:50:34,170
but we'll be posting
lectures up there.

1084
00:50:34,170 --> 00:50:37,000
And let me know if you
have any questions.