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PROFESSOR: All right,
let's get started.

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If you haven't already, there's
two handouts on the left,

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00:00:08,290 --> 00:00:11,520
and you should take
two pieces of paper.

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So we'll doing some
actual origami folding.

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We'll be folding 6.849
today, just like this.

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It'll only take us
eight hours or so.

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This is the Jenny
and Eli folding

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you've seen on the poster.

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Pretty awesome.

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No, we'll be folding
letters more like this.

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So thanks everyone for
giving so many cool questions

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and comments, and the
feedback is really helpful.

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I didn't know what I was
going to cover this class.

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I had too many ideas,
and your questions really

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helped narrow it down
into exciting thing.

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So the structures
going to be, I'm

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going to go through
questions, so it's

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a funny kind of interactivity
where there's a whole day

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in between where you
ask the questions

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and where I answer them.

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But feel free to ask more
questions, follow-up things.

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But there's a lot already
here, so it should be fun.

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These are not questions
but funnier comments.

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If you like double
rainbow jokes,

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this lecture happened
a couple months

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after the double rainbow fiasco.

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There's a lot more, as I recall.

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It was a running theme
throughout the whole semester,

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so look out for that.

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I know this is entertaining.

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I'm getting used to listening
to myself at double speeds.

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This class definitely
is nice in the way

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that with a fairly
simple technique

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you can prove a very
powerful theorem.

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Sorry, I just remembered I
need to push a different button

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here.

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That's hopefully a theme
throughout the class,

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but definitely
especially nice here.

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It is cool, though,
strip folding,

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it was an open problem for
a couple years, at least,

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of no one thinking, how
do we fold any shape?

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That seems really tough.

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The point of the
strip folding approach

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is that once you
have the right idea,

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to start with a really
long rectangle, somehow

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it becomes easy.

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There's still a lot of details
in getting that to work,

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but it's kind of neat
how that works out.

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Now we get to actual proposals.

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So folding practice, I was
planning on doing this,

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but-- this is an explicit
comment to that effect.

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So we're going to fold some
letters of the alphabet.

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You have in your
packet instructions

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for making the individual digits
six, four, eight, and nine,

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which are all pretty easy.

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You also have a diagram for this
crazy design called Typeset.

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This same folding
can make any letter

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of the alphabet and any digit.

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So just to show
you what they will

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look like these are my foldings
of six, eight, four, and nine,

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according to the
first set of diagrams.

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And then this is my folding
of, I think the number six,

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out of Jason Ku's design.

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So if you wanted
to reconfigure it,

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to-- I guess eight is kind
of hard-- let me do four.

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OK, so for four we've
gotta fold this guy under.

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Fold this here, this tab,
I think goes back here.

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So the advanced origami
folders in this class

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can definitely do
Jason Ku's design,

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but it takes a little while.

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There, I've got a four.

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Got it?

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A little hard to
hold in position,

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but, at least in theory, it
will make any letter all out

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of one folding.

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You just have to move
all the tabs around.

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So it's kind of neat, but it
takes at least half an hour

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or so to fold that,
unless you're really fast.

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So I would recommend--
pick one of these.

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Work in groups if you like.

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If you want you can
form a group of four,

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and make six, eight,
four, and nine.

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Follow the diagrams.

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This is an exercise of following
diagrams, one of the other.

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We'll only have time to
maybe make one digit each,

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but have fun with it.

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And if you have questions,
raise your hand.

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I can tell you, the first
step in six, eight, and nine,

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is to make an
eight-by-eight grid.

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There's a lot of ways to
make an eight-by-eight grid,

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but an easy one is shown here.

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So you take your sheet.

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You fold the bottom
edge to the top edge--

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I think they want to do it
white side up-- so they're all

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valleys.

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We align those edges.

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You'll get a nice bisector.

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Then you repeat, folding the
bottom edge to the middle.

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And once you do that, to
save a little bit of time,

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you could then fold that new
bottom edge to the middle

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again.

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That will do
eighths on one side.

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It's a little bit
inaccurate, because you're

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folding through two layers--
but it's a little faster.

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Time is of the essence.

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Then you do the same
thing on the bottom,

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and you'll get eighths
in one dimension.

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But then you have to fold it in
eighths in the other dimension.

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Once you have eighths,
it's like three steps.

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It's really easy.

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For six, eight, and nine--
Four uses a different approach,

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it's a little more free hand.

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If you want to be more
creative, try the four.

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You've got to eyeball what looks
and feels good for number four.

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So out of curiosity,
how many people

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have an eight-by-eight
grid at this point?

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Who wanted to?

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Cool.

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So that's the top half
of these diagrams,

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the eight by eight grid.

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Then it's mostly folding
up over individual edges

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and some corner
faults, but they're

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all simple folds in this world.

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So you're folding through, I
think, always all the layers.

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Oh, no.

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This is only folding
through one layer.

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But these would always fall into
the some layers, simple folds

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category.

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So for example, make a six.

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Fold this bottom edge up.

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Fold the left
three squares over.

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Fold this corner up.

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Fold this corner down.

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And fold this down, this over.

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There's a really big six.

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And these numbers are all pretty
much proportioned correctly.

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The four, you have
to be-- it helps

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to have a reference model
of one of the other digits

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to make it the right height, but
it'll end up roughly correct,

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anyway.

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The eight's a little bit
narrow in this design,

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slightly narrower
than six and the nine.

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But otherwise they're
nice compatible digits.

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There's a whole alphabet
on the website that's

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linked from this slide,
so you go check it out.

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Origami club.

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So it's kind of fun to think
about font design and alphabet

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design.

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There's actually a lot of
origami alphabets out there.

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This is one of the simplest.

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It has digits.

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Anyone have questions?

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How many people folded
a letter, a digit?

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OK.

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You can keep folding, but
maybe I will continue on.

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It's going to be a
lot of fun crackling.

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Anyone folded this one?

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Anyone working on it?

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A bunch of people, cool.

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Let me know when you finish.

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It's kind of fun.

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It's not that hard.

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AUDIENCE: Yes, it is.

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PROFESSOR: I wanted to point
at the OrigaMIT website.

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OrigaMIT is the origami
club at MIT, and at the top

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you see a different alphabet.

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This is a four-fold
alphabet designed

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by Jeannine Mosley
who's an MIT alum

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and came to this
class two years ago.

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And so that's the
reference design in 2002.

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I'm not sure if it
has digits, though.

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At least, the diagrams
we found for the letters,

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do not also have digits.

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So I think an
interesting challenge

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is to design a
four-fold digit set

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to complement her letter set.

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If you're interested, that could
be a cool project to work on.

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Folding design,
minimal fold alphabets.

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You could try a three-fold
alphabet, two-fold alphabet.

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Also on this website are
the meeting schedules.

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It always Sundays, sometimes
2, sometimes 3 o'clock.

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You should check it out.

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And there's a
convention coming up,

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our very own origami
convention on October 27th.

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Jason Ku wanted me to remind
you about that coming up.

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Even if you've never
done origami before,

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other than today, you
should check it out.

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It'll be fun, lots of different
sessions, from simple models

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to complicated models.

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This is a cool design by
Brain Chan, another MIT alum.

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It's one square paper folded
into the Mens et Manus logo,

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but, here, instead
of the oil lamp,

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you've got a little
origami crane.

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Cool stuff.

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So we proceed on
to other questions.

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This is a pretty simple one.

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At the top of the notes, it says
folding, AKA silhouette folding

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and gift wrapping, it has
a couple of references.

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00:11:22,770 --> 00:11:25,680
So where do those terms
come from, is the question.

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00:11:25,680 --> 00:11:29,670
And one answer is it's the title
of the paper, "Folding Flat

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00:11:29,670 --> 00:11:32,120
Silhouettes and Wrapping
Polyhedral Packages,"

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00:11:32,120 --> 00:11:33,940
but that's not the real answer.

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00:11:33,940 --> 00:11:37,680
So this is the two of
us and Joe Mitchell.

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00:11:37,680 --> 00:11:40,110
It's also, I think, the
introduction of the term

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00:11:40,110 --> 00:11:44,390
computational origami,
but those terms

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come from earlier references.

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00:11:45,970 --> 00:11:50,010
So in that paper, there's a
sentence, classic open question

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00:11:50,010 --> 00:11:51,900
origami mathematics,
and we don't really

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00:11:51,900 --> 00:11:53,380
know where it came about.

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00:11:53,380 --> 00:11:56,590
But it was first formally
posed by Bern and Hayes

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00:11:56,590 --> 00:11:58,940
in this SODA '96
paper, which we'll

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00:11:58,940 --> 00:12:03,240
be talking about in the
next lecture, lecture three.

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00:12:03,240 --> 00:12:05,930
And this is a quote
from their paper,

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00:12:05,930 --> 00:12:08,530
"Is every simple polygon, when
skilled sufficiently small,

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00:12:08,530 --> 00:12:10,790
the silhouette of
a flat origami?"

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00:12:10,790 --> 00:12:12,920
The point of saying
the word silhouette

217
00:12:12,920 --> 00:12:18,687
is that when you fold something,
like this number six--

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00:12:18,687 --> 00:12:20,520
there's a whole bunch
of layers, and there's

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00:12:20,520 --> 00:12:22,020
a lot of complexity
to this folding.

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00:12:22,020 --> 00:12:25,190
By saying silhouette, we just
mean, collapse all layers,

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00:12:25,190 --> 00:12:27,180
ignore the coloring, and
just take the outline.

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00:12:27,180 --> 00:12:30,910
So the silhouette of this
thing is a rectangle.

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00:12:30,910 --> 00:12:33,210
And, in general, that's
sort of the transformation

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00:12:33,210 --> 00:12:35,418
to throw away the complexity
of the folding, and say,

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00:12:35,418 --> 00:12:36,830
I just care about the shape.

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00:12:36,830 --> 00:12:39,411
Can I get the desired shape?

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00:12:39,411 --> 00:12:41,660
There's some other interesting
questions here, though,

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00:12:41,660 --> 00:12:43,118
which haven't been
fully addressed.

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00:12:43,118 --> 00:12:44,875
How many creases are
necessary to fold?

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00:12:44,875 --> 00:12:47,060
We'll actually get to
that in a later question.

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00:12:47,060 --> 00:12:50,070
How thick must the origami be?

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00:12:50,070 --> 00:12:52,712
The strip method
shows this if you

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00:12:52,712 --> 00:12:54,920
start from a rectangle of
paper, the number of layers

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00:12:54,920 --> 00:12:56,050
can be very small.

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00:12:56,050 --> 00:12:59,760
I think two or three is enough
for all the gadgets that we

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00:12:59,760 --> 00:13:02,197
use, probably three .

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00:13:02,197 --> 00:13:03,780
If you start from a
square, though, we

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00:13:03,780 --> 00:13:06,340
don't know the answer
to that question.

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00:13:06,340 --> 00:13:10,010
And in practice, folding
through many layers is tough.

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00:13:10,010 --> 00:13:12,970
So that is a silhouette problem.

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00:13:12,970 --> 00:13:15,307
The gift wrapping
problem, the motivation

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00:13:15,307 --> 00:13:16,640
is you have a weird-shaped gift.

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00:13:16,640 --> 00:13:19,540
You want to wrap it
with a piece of paper.

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00:13:19,540 --> 00:13:21,900
And this is posed
to us in a talk

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00:13:21,900 --> 00:13:25,230
by Jin Akiyama at a Canadian
geometry conference.

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00:13:25,230 --> 00:13:28,970
Jin Akiyama is a
really cool guy.

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00:13:28,970 --> 00:13:34,632
He has had for many, many years
a mathematics TV show in Japan.

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00:13:34,632 --> 00:13:37,090
And he's known throughout Japan,
because everyone in school

249
00:13:37,090 --> 00:13:38,880
watches his videos.

250
00:13:38,880 --> 00:13:41,876
And it covers really
interesting mathematics.

251
00:13:41,876 --> 00:13:43,250
Some of the results
in this class

252
00:13:43,250 --> 00:13:45,520
are actually in
his videos as well.

253
00:13:45,520 --> 00:13:48,630
It's mostly in Japanese, so it's
a little hard for most of us

254
00:13:48,630 --> 00:13:49,130
watch.

255
00:13:49,130 --> 00:13:50,700
But there's some
subtitled versions,

256
00:13:50,700 --> 00:13:51,709
and they're really fun.

257
00:13:51,709 --> 00:13:54,000
Maybe we can have a movie
night, and watch one of them,

258
00:13:54,000 --> 00:13:56,840
if I get permission.

259
00:13:56,840 --> 00:13:58,690
So there isn't a great
reference for that.

260
00:13:58,690 --> 00:14:00,290
I mean, he's written some
papers about different kinds

261
00:14:00,290 --> 00:14:02,331
of wrapping problems, but
mostly it was this talk

262
00:14:02,331 --> 00:14:05,530
that he gave in 1997, which is
when I was just starting out

263
00:14:05,530 --> 00:14:07,270
in computational geometry.

264
00:14:07,270 --> 00:14:09,700
That's where the
terms come from.

265
00:14:09,700 --> 00:14:11,250
A lot of words.

266
00:14:11,250 --> 00:14:12,860
Have you ever actually
folded a model

267
00:14:12,860 --> 00:14:18,240
using this method of zigzagging
and folding with the strip?

268
00:14:18,240 --> 00:14:20,620
Any real or sensible or
pretty origami models,

269
00:14:20,620 --> 00:14:22,760
or is it purely for the
sake of universality?

270
00:14:22,760 --> 00:14:26,290
My knee jerk reaction was,
no way is this practical.

271
00:14:26,290 --> 00:14:28,150
This is just for universality.

272
00:14:28,150 --> 00:14:30,940
And the point of this theorem
has always been, in my mind,

273
00:14:30,940 --> 00:14:32,780
to prove that
everything is possible,

274
00:14:32,780 --> 00:14:35,020
but then the challenge
is to find good foldings,

275
00:14:35,020 --> 00:14:36,280
for some notion of good.

276
00:14:36,280 --> 00:14:41,960
But, actually, there are a bunch
of examples of strip folding,

277
00:14:41,960 --> 00:14:44,550
not a lot of different folds,
not terribly many gadgets,

278
00:14:44,550 --> 00:14:48,460
but there's some cool things,
especially with strip weaving.

279
00:14:48,460 --> 00:14:51,710
These are just a few examples
of woven colored strips.

280
00:14:51,710 --> 00:14:54,480
You can make fun things
like Space Invaders.

281
00:14:54,480 --> 00:14:57,470
You could weave
together baskets,

282
00:14:57,470 --> 00:14:59,570
and wrap your
packages, and so on.

283
00:15:02,150 --> 00:15:04,279
This is a little
bit more origamic.

284
00:15:04,279 --> 00:15:05,820
So there were no
real folds in there,

285
00:15:05,820 --> 00:15:07,630
except at the edges of the cube.

286
00:15:07,630 --> 00:15:09,130
This one is a modular origami.

287
00:15:09,130 --> 00:15:10,750
It involves a bunch
of different folds

288
00:15:10,750 --> 00:15:13,240
to get all the pieces
to lock together.

289
00:15:13,240 --> 00:15:16,460
Modular origami means you have
a bunch of identical pieces.

290
00:15:16,460 --> 00:15:18,330
They kind of weave
together through folding,

291
00:15:18,330 --> 00:15:20,610
and then you can make
a nice little crown.

292
00:15:20,610 --> 00:15:22,040
This is a very classic model.

293
00:15:22,040 --> 00:15:23,520
You've probably
seen at some point.

294
00:15:23,520 --> 00:15:27,220
Here that the paper's
slit, and then it's woven.

295
00:15:27,220 --> 00:15:29,220
But there's some
folds down here.

296
00:15:29,220 --> 00:15:32,120
Not a lot of folding, but
strips are pretty neat.

297
00:15:32,120 --> 00:15:35,740
You can definitely use them in
all sorts of different designs.

298
00:15:35,740 --> 00:15:41,060
Here's some more
sculptural designing models

299
00:15:41,060 --> 00:15:44,320
from taking strips of paper.

300
00:15:44,320 --> 00:15:45,780
This has no glue
in it, so I think

301
00:15:45,780 --> 00:15:51,020
there's more strips at the
end locking this together.

302
00:15:51,020 --> 00:15:54,949
And this guy, Zachary
Futterer, took

303
00:15:54,949 --> 00:15:56,740
a bunch of these kinds
of units and started

304
00:15:56,740 --> 00:15:59,610
weaving them together to make
really complicated shapes.

305
00:15:59,610 --> 00:16:04,240
So you can definitely do cool
things with strip folding.

306
00:16:04,240 --> 00:16:07,520
And another common
one around these days

307
00:16:07,520 --> 00:16:12,000
is taking gum or candy
wrappers, and folding them

308
00:16:12,000 --> 00:16:14,300
down into little strips
and weaving them together

309
00:16:14,300 --> 00:16:16,695
to make handbags
and other things.

310
00:16:16,695 --> 00:16:18,320
This has become kind
of a fashion trend

311
00:16:18,320 --> 00:16:20,760
over the last few years.

312
00:16:20,760 --> 00:16:24,080
So those are things
you can do a strips.

313
00:16:24,080 --> 00:16:27,440
We have used it in one paper
where the goal is actually

314
00:16:27,440 --> 00:16:31,580
to be efficient and use
a small piece of paper,

315
00:16:31,580 --> 00:16:34,564
and not just prove some
universality result.

316
00:16:34,564 --> 00:16:36,730
This is in our paper, folding
a better checkerboard,

317
00:16:36,730 --> 00:16:40,260
which we'll talk about
in two lectures, if I

318
00:16:40,260 --> 00:16:42,940
recall correctly,
in more detail.

319
00:16:42,940 --> 00:16:44,490
But this is sort of a baseline.

320
00:16:44,490 --> 00:16:47,270
This is not the better method
that's developed in this paper,

321
00:16:47,270 --> 00:16:49,590
but it's the starting point.

322
00:16:49,590 --> 00:16:52,400
You take a square, so this
actually starts with a square.

323
00:16:52,400 --> 00:16:54,020
You do this pleating.

324
00:16:54,020 --> 00:16:56,370
And this is with bi-color paper.

325
00:16:56,370 --> 00:16:58,180
It's dark on one side,
light on the other.

326
00:16:58,180 --> 00:17:01,270
You get this strip of
squares in color pattern.

327
00:17:01,270 --> 00:17:03,740
And then you take that strip--
a huge number of layers

328
00:17:03,740 --> 00:17:05,569
in the middle, so it's
not super practical,

329
00:17:05,569 --> 00:17:07,460
but it's actually pretty
efficient in terms

330
00:17:07,460 --> 00:17:09,280
of how big a square
you start with.

331
00:17:09,280 --> 00:17:12,720
To make an n-by-n, this
is obviously not to scale.

332
00:17:12,720 --> 00:17:16,140
You need more squares here
in order to make this thing.

333
00:17:16,140 --> 00:17:18,312
And then you just snake
your path back and forth.

334
00:17:18,312 --> 00:17:20,020
You could use turn
gadgets, or here we're

335
00:17:20,020 --> 00:17:21,910
just using 45 degree folds.

336
00:17:21,910 --> 00:17:25,030
And this is pretty close
to what was believed

337
00:17:25,030 --> 00:17:26,910
to be the best way to
fold a checkerboard,

338
00:17:26,910 --> 00:17:29,609
and then this paper shows how
to do a factor of two better.

339
00:17:29,609 --> 00:17:30,900
So we'll talk about that later.

340
00:17:30,900 --> 00:17:33,240
But there are some
uses for strip folding.

341
00:17:33,240 --> 00:17:34,710
This is a little
bit theoretical,

342
00:17:34,710 --> 00:17:37,620
but it's actually pretty
competitive against the best

343
00:17:37,620 --> 00:17:40,720
n-by-n checkerboard foldings
in the origami world,

344
00:17:40,720 --> 00:17:43,690
like the one I
showed last class.

345
00:17:43,690 --> 00:17:48,080
So that's practicality
of strip folding.

346
00:17:48,080 --> 00:17:50,377
Next question's more
about strip folding.

347
00:17:50,377 --> 00:17:52,710
There are a couple things
that are in the lecture notes,

348
00:17:52,710 --> 00:17:55,180
the handwritten lecture
notes, but were not even

349
00:17:55,180 --> 00:18:00,040
mentioned in the audio
part of the lecture.

350
00:18:00,040 --> 00:18:03,840
So a few people asked,
what are these things?

351
00:18:03,840 --> 00:18:06,570
Said pseudopolynomial
upper bound.

352
00:18:06,570 --> 00:18:09,870
Pseudopolynomial is a fun term.

353
00:18:09,870 --> 00:18:12,200
Let me tell you a
little bit about it.

354
00:18:12,200 --> 00:18:14,310
It's from the algorithms
world, but even a lot

355
00:18:14,310 --> 00:18:16,220
of algorithms people
don't know it.

356
00:18:16,220 --> 00:18:21,210
So let me tell you.

357
00:18:21,210 --> 00:18:24,760
So maybe first I should
tell you about polynomial.

358
00:18:24,760 --> 00:18:27,510
In general, what
these terms are about

359
00:18:27,510 --> 00:18:30,870
is measuring how
fast an algorithm is.

360
00:18:30,870 --> 00:18:37,140
So the idea is you plot,
conceptually, n, this

361
00:18:37,140 --> 00:18:38,360
is the problem size.

362
00:18:44,440 --> 00:18:48,304
If you wanted to fold
an arbitrary polyhedron,

363
00:18:48,304 --> 00:18:49,970
the one way to think
of the problem size

364
00:18:49,970 --> 00:18:51,960
is the number vertices,
edges, and faces.

365
00:18:51,960 --> 00:18:55,000
Just the total number of
things you're given as input,

366
00:18:55,000 --> 00:18:56,600
and then your
output is whatever.

367
00:18:56,600 --> 00:18:59,300
But n is supposed to be
the input problem size.

368
00:18:59,300 --> 00:19:02,590
And then on the y-axis,
you want to plot

369
00:19:02,590 --> 00:19:05,860
the running time
of your algorithm.

370
00:19:05,860 --> 00:19:08,750
So this is how long it
takes to compute the way

371
00:19:08,750 --> 00:19:12,420
to fold your square paper
into your desired shape,

372
00:19:12,420 --> 00:19:14,720
and generally this
is going to increase.

373
00:19:14,720 --> 00:19:18,270
And the question is, does it
increase in a reasonable way

374
00:19:18,270 --> 00:19:22,460
or in a crazy way that
goes exponentially high?

375
00:19:22,460 --> 00:19:25,530
So you want to know, how does
the running time grow with n?

376
00:19:25,530 --> 00:19:27,930
Polynomial is a
sense of good growth,

377
00:19:27,930 --> 00:19:29,590
and it just means
you grow, like,

378
00:19:29,590 --> 00:19:32,530
n to the c, where
c is some constant.

379
00:19:36,750 --> 00:19:42,470
So ideally you'd have n--
or maybe you have n squared,

380
00:19:42,470 --> 00:19:44,490
or n cubed, or n to the
fourth-- all these are

381
00:19:44,490 --> 00:19:47,250
considered good running times.

382
00:19:47,250 --> 00:19:49,440
Not quite as good as
polynomial is pseudopolynomial.

383
00:19:57,940 --> 00:20:02,220
And I would conjecture, for this
problem of folding an arbitrary

384
00:20:02,220 --> 00:20:07,290
given polyhedron, you cannot
achieve a polynomial number

385
00:20:07,290 --> 00:20:08,320
of folds, let's say.

386
00:20:08,320 --> 00:20:09,650
So there are two things
we could measure here,

387
00:20:09,650 --> 00:20:11,730
the running time of the
algorithm-- we could measure,

388
00:20:11,730 --> 00:20:13,438
actually, three things--
we could measure

389
00:20:13,438 --> 00:20:16,700
the number of folds you make,
the number operations you

390
00:20:16,700 --> 00:20:20,430
do on the paper, and a third
thing would be scale factor.

391
00:20:20,430 --> 00:20:22,930
How big a square do I have
to start with in order

392
00:20:22,930 --> 00:20:27,110
to make a desired polygon?

393
00:20:27,110 --> 00:20:35,430
And pseudopolynomial
means n times r to the c.

394
00:20:35,430 --> 00:20:36,900
What's r?

395
00:20:36,900 --> 00:20:46,310
R is some geometric parameter,
geometric ratio, in the input.

396
00:20:46,310 --> 00:20:49,990
And in particular for this
problem what makes sense for r

397
00:20:49,990 --> 00:20:58,770
is basically the longest length
divided by the shortest length.

398
00:20:58,770 --> 00:21:01,190
This is typically
what r refers to.

399
00:21:01,190 --> 00:21:03,130
This will come up in
later lectures as well.

400
00:21:06,220 --> 00:21:08,350
So for example, you
take your entire shape,

401
00:21:08,350 --> 00:21:11,860
you measure the diameter of the
shape, the two farthest points.

402
00:21:11,860 --> 00:21:13,100
That's your longest length.

403
00:21:13,100 --> 00:21:15,420
Shortest length
would be-- maybe you

404
00:21:15,420 --> 00:21:18,260
have a triangle,
something like this,

405
00:21:18,260 --> 00:21:20,690
in the target polyhedron
you want to make.

406
00:21:20,690 --> 00:21:23,069
This would be your
shortest distance.

407
00:21:23,069 --> 00:21:24,860
This is actually called
the minimum feature

408
00:21:24,860 --> 00:21:28,300
size in computational geometry
or the minimum altitude

409
00:21:28,300 --> 00:21:30,720
of any of your triangles.

410
00:21:30,720 --> 00:21:32,950
OK, so that's just some number.

411
00:21:32,950 --> 00:21:37,950
And you can have a triangle,
which is super, super narrow.

412
00:21:37,950 --> 00:21:40,761
And so it's this ratio r
could be arbitrarily large

413
00:21:40,761 --> 00:21:43,010
even though you only have
three vertices, three edges,

414
00:21:43,010 --> 00:21:44,290
one face.

415
00:21:44,290 --> 00:21:46,756
So n and r not
necessarily comparable,

416
00:21:46,756 --> 00:21:48,630
so that's why in
pseudopolynomial we put them

417
00:21:48,630 --> 00:21:52,300
both together, and then we raise
them to some constant power.

418
00:21:52,300 --> 00:21:54,720
That's a pseudopolynomial
running time.

419
00:21:54,720 --> 00:21:58,510
So the question that's
being posed here

420
00:21:58,510 --> 00:22:01,980
is, can you get a
pseudopolynomial upper bound,

421
00:22:01,980 --> 00:22:04,109
and can you get a
pseudopolynomial lower bound?

422
00:22:04,109 --> 00:22:05,650
And it doesn't say
for what, but it's

423
00:22:05,650 --> 00:22:08,300
for all three
problems-- running time,

424
00:22:08,300 --> 00:22:11,070
number of folds, scale factor.

425
00:22:11,070 --> 00:22:13,410
And not all of these are open.

426
00:22:13,410 --> 00:22:16,520
So in the original
paper, there's

427
00:22:16,520 --> 00:22:19,830
this theorem that says lots of
things-- you can fold anything.

428
00:22:19,830 --> 00:22:22,340
Then it says here, the folding
requires a number of folds,

429
00:22:22,340 --> 00:22:27,480
polynomial of n
and the ratio, r.

430
00:22:27,480 --> 00:22:30,940
So it already claims it there
is a pseudopolynomial bound

431
00:22:30,940 --> 00:22:32,280
on the number of folds.

432
00:22:32,280 --> 00:22:35,370
It doesn't say what that
pseudopolynomial bound is.

433
00:22:35,370 --> 00:22:37,250
Is it n times r?

434
00:22:37,250 --> 00:22:38,950
Is it n plus r?

435
00:22:38,950 --> 00:22:40,690
Is it n times r squared?

436
00:22:40,690 --> 00:22:45,110
I would guess one of the first
two-- n times r or n plus r.

437
00:22:45,110 --> 00:22:47,280
So that's the upper
bound question.

438
00:22:47,280 --> 00:22:49,420
Maybe we can work on this
in a problem session.

439
00:22:49,420 --> 00:22:51,940
A lower bound
question is, do you

440
00:22:51,940 --> 00:22:54,790
prove that you need some
dependence, both on n and r?

441
00:22:54,790 --> 00:22:56,920
Which I would guess
is pretty easy.

442
00:22:56,920 --> 00:22:58,910
If you want to take a
square and fold it down

443
00:22:58,910 --> 00:23:00,510
to a really, really
skinny triangle,

444
00:23:00,510 --> 00:23:04,540
I think you need at
least r folds, roughly.

445
00:23:04,540 --> 00:23:06,900
And similarly you should
need at least n folds,

446
00:23:06,900 --> 00:23:10,040
so there should be a
lower bound like n plus r.

447
00:23:10,040 --> 00:23:12,210
But none of these have been
written down explicitly,

448
00:23:12,210 --> 00:23:14,690
so that's what those
open questions are.

449
00:23:14,690 --> 00:23:16,130
Then there was
another slide, six,

450
00:23:16,130 --> 00:23:18,690
which was completely uncovered.

451
00:23:18,690 --> 00:23:20,080
Stop me if there are questions.

452
00:23:20,080 --> 00:23:23,890
I should maybe take a
brief moment to breathe.

453
00:23:23,890 --> 00:23:29,640
So the next part of the lecture
notes ask about seam placement.

454
00:23:29,640 --> 00:23:32,560
So seam placement has the
following kind of issue.

455
00:23:32,560 --> 00:23:36,650
When you fold, like this
number-- is this a six?

456
00:23:36,650 --> 00:23:39,040
This is a nine.

457
00:23:39,040 --> 00:23:40,510
Fold this number nine.

458
00:23:40,510 --> 00:23:43,400
In addition to seeing the color
pattern, if you look closely

459
00:23:43,400 --> 00:23:45,090
there's also these
kinds of seams.

460
00:23:45,090 --> 00:23:48,460
This white square is
not just a white square.

461
00:23:48,460 --> 00:23:53,370
You can see on the top
layer this crease line.

462
00:23:53,370 --> 00:23:55,880
And here there's a seam,
here there's a seam.

463
00:23:55,880 --> 00:23:58,400
These are like visible lines.

464
00:23:58,400 --> 00:24:01,600
Of course, you have to have
seams at the color transitions,

465
00:24:01,600 --> 00:24:03,920
but there's other seams as well.

466
00:24:03,920 --> 00:24:06,282
Maybe you want to minimize
the seams you want to get,

467
00:24:06,282 --> 00:24:08,240
you want to place the
scenes in a cool pattern.

468
00:24:11,010 --> 00:24:12,510
When you fold
checkerboards, there's

469
00:24:12,510 --> 00:24:14,690
a such thing as a seamless
checkerboard, where

470
00:24:14,690 --> 00:24:16,880
every square is a
whole square paper.

471
00:24:16,880 --> 00:24:20,640
There's no visible crease
lines on the top layer.

472
00:24:20,640 --> 00:24:23,890
So this is an extension of
the universality result,

473
00:24:23,890 --> 00:24:26,830
to also get sort of
universal seem placement.

474
00:24:26,830 --> 00:24:30,230
And what the
original paper proves

475
00:24:30,230 --> 00:24:33,360
is that you can place
the seams however

476
00:24:33,360 --> 00:24:36,020
you want, provided the
seam regions, the regions

477
00:24:36,020 --> 00:24:39,930
between the seams, are
convex polygons-- which

478
00:24:39,930 --> 00:24:41,160
is almost always the case.

479
00:24:41,160 --> 00:24:44,250
You look at a
typical model-- here,

480
00:24:44,250 --> 00:24:48,140
the seam regions are all
rectangles and triangles.

481
00:24:48,140 --> 00:24:51,680
So you could achieve
exactly this seam pattern

482
00:24:51,680 --> 00:24:52,620
if you wanted it.

483
00:24:52,620 --> 00:24:54,620
You could also say, oh,
here's a nice rectangle.

484
00:24:54,620 --> 00:24:56,570
I'll make that a seam region.

485
00:24:56,570 --> 00:24:59,330
Here's a nice rectangle,
I'll make that same region.

486
00:24:59,330 --> 00:25:01,640
But you could not make
the entire number nine

487
00:25:01,640 --> 00:25:05,940
here a seam region,
because it's non convex.

488
00:25:05,940 --> 00:25:07,980
At least, you can't do
it with this technique.

489
00:25:07,980 --> 00:25:10,260
We don't know,
necessarily, whether this

490
00:25:10,260 --> 00:25:12,330
is possible by
some other folding.

491
00:25:12,330 --> 00:25:16,780
I would guess no, but
it is possible to make

492
00:25:16,780 --> 00:25:19,760
some non-convex seam regions.

493
00:25:19,760 --> 00:25:26,340
For example I could take this
page and fold the corner over,

494
00:25:26,340 --> 00:25:30,120
and now I've got a
non-convex seam region here.

495
00:25:30,120 --> 00:25:32,780
So some non-convexing
regions are possible.

496
00:25:32,780 --> 00:25:38,360
Open question is-- if
I give you a polygon--

497
00:25:38,360 --> 00:25:40,050
we know every
polygon's possible.

498
00:25:40,050 --> 00:25:43,004
Now I give you a polygon and I
subdivide it into seam regions.

499
00:25:43,004 --> 00:25:44,170
Which of those are possible?

500
00:25:44,170 --> 00:25:46,750
Not everything is
possible, I'm pretty sure,

501
00:25:46,750 --> 00:25:49,590
though I'm not sure we
have a proof of that.

502
00:25:49,590 --> 00:25:52,780
Some things like this little
heart shape are possible.

503
00:25:52,780 --> 00:25:53,960
Characterize.

504
00:25:53,960 --> 00:25:59,340
This another cool possible
problem for problem session.

505
00:25:59,340 --> 00:26:00,250
Questions about that?

506
00:26:05,090 --> 00:26:07,920
I have a little bit about the
proof of how this is done.

507
00:26:07,920 --> 00:26:11,040
If you wanted to just
do convex regions.

508
00:26:11,040 --> 00:26:15,530
So the general
approach here is you

509
00:26:15,530 --> 00:26:18,490
want to visit all of the
regions in some order.

510
00:26:18,490 --> 00:26:20,726
This is called the tour.

511
00:26:20,726 --> 00:26:22,100
It's pretty easy
to just-- I mean

512
00:26:22,100 --> 00:26:24,558
you're allowed to visit regions
more than once, so you just

513
00:26:24,558 --> 00:26:28,310
keep going, keep trying to visit
some unvisited seam region.

514
00:26:28,310 --> 00:26:31,360
When you visit a seam region,
it's a convex polygon.

515
00:26:31,360 --> 00:26:34,290
So what we're going to do is
make our strip fairly wide,

516
00:26:34,290 --> 00:26:37,150
actually, wide enough
to completely cover

517
00:26:37,150 --> 00:26:39,400
that seam region.

518
00:26:39,400 --> 00:26:41,250
And then at this
moment, we basically

519
00:26:41,250 --> 00:26:44,420
need to turn to do the next one.

520
00:26:44,420 --> 00:26:48,530
We know how to change the
direction of the strip using

521
00:26:48,530 --> 00:26:50,470
a turn gadget.

522
00:26:50,470 --> 00:26:52,850
Then we have to change
the width of the strip.

523
00:26:52,850 --> 00:26:53,990
Maybe it needs to be wider.

524
00:26:53,990 --> 00:26:56,080
Maybe it needs to be thinner.

525
00:26:56,080 --> 00:27:01,630
And then we need to shift the
strip one way or the other.

526
00:27:01,630 --> 00:27:05,940
So if we just end here, we
turn, we might be misaligned.

527
00:27:05,940 --> 00:27:09,380
We need to shift it over,
expand it, then do the next one,

528
00:27:09,380 --> 00:27:13,860
then turn, then shift it
over and set the right width.

529
00:27:13,860 --> 00:27:16,910
Keep going like that.

530
00:27:16,910 --> 00:27:17,520
OK.

531
00:27:17,520 --> 00:27:19,230
That's pretty messy
and complicated,

532
00:27:19,230 --> 00:27:21,550
but you can do with
these two gadgets.

533
00:27:21,550 --> 00:27:23,880
Strip width gadget--
you take a strip,

534
00:27:23,880 --> 00:27:28,750
and you could make it
anywhere between 1/2 and 100%

535
00:27:28,750 --> 00:27:30,637
of its original width.

536
00:27:30,637 --> 00:27:33,220
So the idea is you start with a
really wide strip, wide enough

537
00:27:33,220 --> 00:27:35,030
to cover all the polygons.

538
00:27:35,030 --> 00:27:37,840
Then you do this gadget and
keep shrinking it by half

539
00:27:37,840 --> 00:27:40,041
until it's roughly
the right size.

540
00:27:40,041 --> 00:27:41,540
And then when you're
almost correct,

541
00:27:41,540 --> 00:27:43,420
you shrink it by a
little bit more--

542
00:27:43,420 --> 00:27:46,040
here shrinking to a third.

543
00:27:46,040 --> 00:27:48,560
And then you get
your shrunken strip,

544
00:27:48,560 --> 00:27:51,680
and it happens right at
the line that you specify.

545
00:27:51,680 --> 00:27:54,770
So you can basically on
a dime shrink your strip,

546
00:27:54,770 --> 00:27:59,550
and then by doing the
reverse can grow it back.

547
00:27:59,550 --> 00:28:03,260
This is maybe not with
simple folds, though.

548
00:28:03,260 --> 00:28:05,290
And then the other
gadget is a shift gadget,

549
00:28:05,290 --> 00:28:08,770
where you're at this position
and you want to shift up.

550
00:28:08,770 --> 00:28:13,110
So that's pretty easy, you
just do two turn gadgets.

551
00:28:13,110 --> 00:28:18,680
So that's at a high level how
making a desired seam pattern

552
00:28:18,680 --> 00:28:20,460
works.

553
00:28:20,460 --> 00:28:23,580
Go on to the next question.

554
00:28:23,580 --> 00:28:27,210
A lot of people asked
about this-- and this

555
00:28:27,210 --> 00:28:29,340
is an open problem that
I mentioned orally.

556
00:28:29,340 --> 00:28:33,040
It's not written in the notes--
which is, can you actually

557
00:28:33,040 --> 00:28:37,380
do the things we said we
can do with simple folds?

558
00:28:37,380 --> 00:28:40,280
So can you get a
universal folding

559
00:28:40,280 --> 00:28:46,160
of a polygon, two-color pattern
polyhedron, using simple folds?

560
00:28:46,160 --> 00:28:50,600
And I thought it'd be fun to
actually work on this, here,

561
00:28:50,600 --> 00:28:53,484
live, because I think
this is an easy problem.

562
00:28:53,484 --> 00:28:55,150
And there's a bunch
of possible answers,

563
00:28:55,150 --> 00:28:58,850
and there are even two suggested
ideas from the comments field.

564
00:28:58,850 --> 00:29:03,410
So let me just remind you of
the issue, what's happening,

565
00:29:03,410 --> 00:29:06,980
and then I need your input,
what's going to work here.

566
00:29:06,980 --> 00:29:11,890
So general picture for the strip
method was we do one triangle,

567
00:29:11,890 --> 00:29:16,950
we end here, then we do a
bunch of folds like this,

568
00:29:16,950 --> 00:29:20,870
and then we end here,
maybe, and then we zigzag.

569
00:29:20,870 --> 00:29:24,520
And the trouble is we've already
made this triangle over here.

570
00:29:24,520 --> 00:29:27,857
When we make this triangle,
we have this excess stuff,

571
00:29:27,857 --> 00:29:29,440
which I haven't drawn
very accurately.

572
00:29:29,440 --> 00:29:33,300
If you recall, it looks
something like that.

573
00:29:33,300 --> 00:29:36,000
Or maybe even more like that
if we use right angle turn

574
00:29:36,000 --> 00:29:37,730
gadgets.

575
00:29:37,730 --> 00:29:39,960
And then we want to
fold it underneath

576
00:29:39,960 --> 00:29:41,790
and we're doing
that, the way I said,

577
00:29:41,790 --> 00:29:44,700
with height gadget
mountain folds.

578
00:29:44,700 --> 00:29:47,210
But the model's
simple folds, which

579
00:29:47,210 --> 00:29:48,850
I should make more explicit.

580
00:29:54,250 --> 00:29:57,720
You're not allowed to
collide during the motion.

581
00:29:57,720 --> 00:29:59,350
The idea with a simple
fold is that you

582
00:29:59,350 --> 00:30:08,420
should be folding
along one line segment,

583
00:30:08,420 --> 00:30:12,100
and you should fold
by-- at least the model

584
00:30:12,100 --> 00:30:16,070
that we defined back in the
day, I'll talk more about where

585
00:30:16,070 --> 00:30:20,600
this notion comes from-- you
fold by plus or minus 180

586
00:30:20,600 --> 00:30:24,110
degrees, which means
after you do the fold,

587
00:30:24,110 --> 00:30:26,050
you'll be flat again.

588
00:30:26,050 --> 00:30:29,745
And no collision
during the motion.

589
00:30:36,767 --> 00:30:39,100
If we folded this triangle,
and then we folded this one,

590
00:30:39,100 --> 00:30:41,900
and this stuff is on the
top-- we can't mountain fold.

591
00:30:41,900 --> 00:30:44,920
That's not considered
a simple fold.

592
00:30:44,920 --> 00:30:48,260
Now one proposal
is, could we just

593
00:30:48,260 --> 00:30:53,530
make the next triangle
underneath the previous one?

594
00:30:53,530 --> 00:30:55,890
A different proposal
is valley fold.

595
00:30:55,890 --> 00:30:57,520
These are actually
different proposals,

596
00:30:57,520 --> 00:30:59,920
because, especially
for two color patterns,

597
00:30:59,920 --> 00:31:02,150
it'll make a difference.

598
00:31:02,150 --> 00:31:05,050
If we valley fold
here there's going

599
00:31:05,050 --> 00:31:07,392
to be some junk
on the front side,

600
00:31:07,392 --> 00:31:09,600
especially if you want to
get a desired seam pattern.

601
00:31:09,600 --> 00:31:11,582
But maybe we'll leave
seam patterns for later.

602
00:31:11,582 --> 00:31:13,040
If you want to get
a color pattern,

603
00:31:13,040 --> 00:31:14,870
you might reveal
some wrong color

604
00:31:14,870 --> 00:31:17,600
when you do that valley fold.

605
00:31:17,600 --> 00:31:19,870
So I haven't really thought
about this idea yet.

606
00:31:19,870 --> 00:31:21,960
I think it might be good.

607
00:31:26,580 --> 00:31:30,940
The idea there would
be-- so you've already

608
00:31:30,940 --> 00:31:34,040
made this triangle.

609
00:31:34,040 --> 00:31:35,350
You mountain fold everything.

610
00:31:35,350 --> 00:31:38,930
Now when you go and
you do these zigzags,

611
00:31:38,930 --> 00:31:42,180
you want to be underneath
everything that you've done.

612
00:31:42,180 --> 00:31:44,190
AUDIENCE: Can you fold
the one you've already

613
00:31:44,190 --> 00:31:47,650
made, like as a
unit, over the side?

614
00:31:47,650 --> 00:31:50,167
And then make it and then
just put it back on top?

615
00:31:50,167 --> 00:31:50,750
PROFESSOR: OK.

616
00:31:50,750 --> 00:31:54,900
Different idea is you basically
fold this out of the way,

617
00:31:54,900 --> 00:31:58,130
do this thing, height gadget,
and then fold it back.

618
00:31:58,130 --> 00:31:59,810
Maybe.

619
00:31:59,810 --> 00:32:01,110
I've wondered about that, too.

620
00:32:01,110 --> 00:32:03,193
Is it's going to get a
little challenging, though.

621
00:32:03,193 --> 00:32:05,680
In general, there's a
huge set of triangles,

622
00:32:05,680 --> 00:32:08,680
so unless you can like go
far away, make your triangle,

623
00:32:08,680 --> 00:32:12,630
and then plop it down--
maybe it's possible.

624
00:32:12,630 --> 00:32:16,780
I guess we can pursue that
idea, but maybe first we

625
00:32:16,780 --> 00:32:19,280
should exhaust the easier ideas.

626
00:32:19,280 --> 00:32:21,800
I mean, that is definitely
plausible that that's possible.

627
00:32:21,800 --> 00:32:25,080
Doing that with simple folds
and not leaving any garbage

628
00:32:25,080 --> 00:32:26,580
is going to be a
little challenging,

629
00:32:26,580 --> 00:32:29,450
but it might be doable.

630
00:32:29,450 --> 00:32:31,610
This to me is the simplest
idea, so we should first

631
00:32:31,610 --> 00:32:32,590
see if it works.

632
00:32:32,590 --> 00:32:34,850
Does anyone see
problems with this plan?

633
00:32:37,470 --> 00:32:39,480
I have some strips.

634
00:32:39,480 --> 00:32:41,320
We could think
about what it means

635
00:32:41,320 --> 00:32:47,360
to be doing a turn
gadget underneath here.

636
00:32:47,360 --> 00:32:50,480
So I don't know
quite about, well,

637
00:32:50,480 --> 00:32:53,090
let's suppose we
are already here.

638
00:32:53,090 --> 00:32:56,252
And now maybe I do
some as the turn gadget

639
00:32:56,252 --> 00:32:57,210
goes to mountain folds.

640
00:32:59,800 --> 00:33:06,310
Then at 90 degrees, and then
a valley fold like that.

641
00:33:06,310 --> 00:33:08,240
And then I do a mountain fold.

642
00:33:08,240 --> 00:33:11,180
I think they might be OK,
because turn gadgets start

643
00:33:11,180 --> 00:33:13,860
with a mountain fold.

644
00:33:13,860 --> 00:33:15,790
So if you're
underneath everything,

645
00:33:15,790 --> 00:33:17,990
that's going to avoid collision.

646
00:33:17,990 --> 00:33:20,880
And then the valley
fold brings it back.

647
00:33:20,880 --> 00:33:24,100
Again, we're using just
the space below everything

648
00:33:24,100 --> 00:33:26,095
that we see.

649
00:33:26,095 --> 00:33:28,220
So then we make that strip,
we keep turning around,

650
00:33:28,220 --> 00:33:32,590
and then later on we're going
to mountain fold this behind,

651
00:33:32,590 --> 00:33:34,075
somehow, to meet this edge.

652
00:33:37,800 --> 00:33:39,650
Seems OK.

653
00:33:39,650 --> 00:33:41,080
Yeah?

654
00:33:41,080 --> 00:33:42,119
Question?

655
00:33:42,119 --> 00:33:42,952
AUDIENCE: I'm sorry.

656
00:33:42,952 --> 00:33:44,270
I'm confused about something.

657
00:33:44,270 --> 00:33:47,210
When you're doing
the turn gadget,

658
00:33:47,210 --> 00:33:51,130
while the paper is above the
first triangle you go over,

659
00:33:51,130 --> 00:33:54,070
you're doing a bunch
of turns where you're

660
00:33:54,070 --> 00:33:57,420
taking a piece of paper up
another one, folding just that

661
00:33:57,420 --> 00:33:58,850
part Is that a s--

662
00:33:58,850 --> 00:34:01,310
PROFESSOR: Going from
one triangle next.

663
00:34:01,310 --> 00:34:01,810
Yeah.

664
00:34:01,810 --> 00:34:04,230
So we could think
about that, too.

665
00:34:04,230 --> 00:34:05,560
It's very, you're right.

666
00:34:05,560 --> 00:34:08,164
I mean, I'm just looking
at the turning around part

667
00:34:08,164 --> 00:34:10,080
for making a single
triangle, but there's also

668
00:34:10,080 --> 00:34:12,446
the turn gadget going
from here to here.

669
00:34:12,446 --> 00:34:14,070
It's actually slightly
more complicated

670
00:34:14,070 --> 00:34:16,111
than we've covered in
class, but not really much.

671
00:34:16,111 --> 00:34:18,350
It's just a slightly
generalized turn gadget.

672
00:34:18,350 --> 00:34:21,170
So you're coming here, you
basically want to turn around.

673
00:34:24,967 --> 00:34:26,550
Let's just think
about a way to do it.

674
00:34:26,550 --> 00:34:32,354
You could imagine first
doing a turn like this.

675
00:34:37,370 --> 00:34:39,540
It's not exactly a
pure turn gadget.

676
00:34:39,540 --> 00:34:46,239
And then turning around
to get to next place--

677
00:34:46,239 --> 00:34:50,070
this is really hard
to do on a blackboard.

678
00:34:50,070 --> 00:34:52,734
Strips just tend not
to stay together well.

679
00:34:52,734 --> 00:34:54,984
OK, now we're going parallel
to the correct direction,

680
00:34:54,984 --> 00:34:57,740
and then we turn back and forth.

681
00:34:57,740 --> 00:34:59,750
But each of those is
just using turn gadgets.

682
00:34:59,750 --> 00:35:01,750
As long as a turn
gadget works fine,

683
00:35:01,750 --> 00:35:04,655
a turn gadget is going to
be a mountain fold, which

684
00:35:04,655 --> 00:35:06,720
is going to go
behind everything,

685
00:35:06,720 --> 00:35:11,740
and then a valley fold to bring
it to a desired direction.

686
00:35:11,740 --> 00:35:14,650
Those are all using
everything behind the board.

687
00:35:14,650 --> 00:35:17,410
So it seems like all
those operations are OK.

688
00:35:17,410 --> 00:35:19,240
Now we should also
check the color reversal

689
00:35:19,240 --> 00:35:21,510
gadget, which--
it's funny thing.

690
00:35:21,510 --> 00:35:26,010
I remember everything I
did before 2000 or so,

691
00:35:26,010 --> 00:35:29,240
so I still have memorized the
color reversal gadget, at least

692
00:35:29,240 --> 00:35:29,820
I think so.

693
00:35:29,820 --> 00:35:33,770
I should probably color
this piece of paper,

694
00:35:33,770 --> 00:35:35,550
so you can see
the colors change.

695
00:35:35,550 --> 00:35:37,950
I don't remember anything
I've done since 2000,

696
00:35:37,950 --> 00:35:41,530
but anything up to 2000 I'm OK.

697
00:35:41,530 --> 00:35:45,449
This is 1998, I think.

698
00:35:45,449 --> 00:35:47,490
I remember folding lots
of-- this is ticker tape,

699
00:35:47,490 --> 00:35:49,360
they use it for or
not ticker tape,

700
00:35:49,360 --> 00:35:52,190
but they use this for
like adding machines.

701
00:35:52,190 --> 00:35:54,150
So I think it's a 90
degree mountain fold,

702
00:35:54,150 --> 00:35:57,680
then you fold up
like that, and then

703
00:35:57,680 --> 00:36:00,639
you fold back down with a
mountain fold like that.

704
00:36:00,639 --> 00:36:01,680
You get a color reversal.

705
00:36:01,680 --> 00:36:06,990
And all of those folds we're
working behind my plane

706
00:36:06,990 --> 00:36:09,517
here, so should avoid
collision with everything.

707
00:36:09,517 --> 00:36:11,350
I think you could do
color reversal and turn

708
00:36:11,350 --> 00:36:13,370
gadgets behind.

709
00:36:13,370 --> 00:36:15,070
And so the suggestion works.

710
00:36:15,070 --> 00:36:17,600
Who made this suggestion?

711
00:36:17,600 --> 00:36:18,240
Good idea.

712
00:36:21,087 --> 00:36:23,420
Unless there are any objections,
I think that will work.

713
00:36:23,420 --> 00:36:28,510
I had a different plan, which
was to use this second idea

714
00:36:28,510 --> 00:36:32,050
and set up the turn gadgets
so there is no-- when you fold

715
00:36:32,050 --> 00:36:35,550
this with the valley fold,
there is no ugly colors.

716
00:36:35,550 --> 00:36:37,280
So you could maybe
modify the turn gadget

717
00:36:37,280 --> 00:36:40,190
to be completely solidly
colored on both sides,

718
00:36:40,190 --> 00:36:42,045
but I think this is much easier.

719
00:36:42,045 --> 00:36:44,170
It's probably why I didn't
write it down the notes,

720
00:36:44,170 --> 00:36:45,270
but I'm not sure.

721
00:36:45,270 --> 00:36:45,986
Yeah?

722
00:36:45,986 --> 00:36:48,610
AUDIENCE: I think you could also
just do a thing where you just

723
00:36:48,610 --> 00:36:50,901
take it far away, because
you have a really long strip.

724
00:36:50,901 --> 00:36:54,806
So you can just take
that strip, and go

725
00:36:54,806 --> 00:36:58,141
to where there are no things,
fold it, and then take it out

726
00:36:58,141 --> 00:36:58,890
and put it on top.

727
00:36:58,890 --> 00:36:59,150
PROFESSOR: OK.

728
00:36:59,150 --> 00:37:00,358
How do you do that last part?

729
00:37:03,446 --> 00:37:07,817
AUDIENCE: Accordion fold
it with the right lengths.

730
00:37:07,817 --> 00:37:09,900
PROFESSOR: So you have to
do it with simple folds.

731
00:37:09,900 --> 00:37:12,110
That's the main,
that's the challenge.

732
00:37:12,110 --> 00:37:14,642
So the idea is
you're way out here,

733
00:37:14,642 --> 00:37:16,100
you have a triangle
out here, which

734
00:37:16,100 --> 00:37:18,729
you want to bring over here.

735
00:37:18,729 --> 00:37:20,020
Maybe I should do it like this.

736
00:37:22,640 --> 00:37:24,930
You could do
something like this,

737
00:37:24,930 --> 00:37:27,050
so now the triangle's over here.

738
00:37:27,050 --> 00:37:31,090
Then maybe you want to
go almost all the way

739
00:37:31,090 --> 00:37:35,481
here and then fold it back, and
then fold it forth, and back,

740
00:37:35,481 --> 00:37:37,730
until you get your triangle
exactly where you want it.

741
00:37:37,730 --> 00:37:38,810
AUDIENCE: Figure
out from the base--

742
00:37:38,810 --> 00:37:39,480
PROFESSOR: It seems plausible.

743
00:37:39,480 --> 00:37:41,140
AUDIENCE: --where
you need to fold it.

744
00:37:41,140 --> 00:37:43,970
PROFESSOR: My only concern
would be when you do this thing,

745
00:37:43,970 --> 00:37:46,660
there might be a little corner.

746
00:37:46,660 --> 00:37:48,660
Depends how you fold this
thing, and then you've

747
00:37:48,660 --> 00:37:50,100
got to hide that corner.

748
00:37:50,100 --> 00:37:52,910
And if there's triangles
all around here,

749
00:37:52,910 --> 00:37:54,045
there may not be room.

750
00:37:54,045 --> 00:37:56,170
I mean, maybe if there's
triangles all around here,

751
00:37:56,170 --> 00:37:59,130
it's OK to have that corner,
but maybe the triangles

752
00:37:59,130 --> 00:38:00,670
are different colors.

753
00:38:00,670 --> 00:38:03,562
So I do believe that
should be possible,

754
00:38:03,562 --> 00:38:05,520
but I think it is a little
bit more complicated

755
00:38:05,520 --> 00:38:08,180
because you have to hide
one last piece after you

756
00:38:08,180 --> 00:38:10,630
get in position.

757
00:38:10,630 --> 00:38:12,505
Anyway, I think there
are at least three ways

758
00:38:12,505 --> 00:38:13,551
to solve this problem.

759
00:38:13,551 --> 00:38:14,050
Yeah?

760
00:38:14,050 --> 00:38:15,550
AUDIENCE: Are you
allowed to unfold?

761
00:38:15,550 --> 00:38:17,091
PROFESSOR: Are you
allowed to unfold?

762
00:38:17,091 --> 00:38:18,350
That's a good question.

763
00:38:18,350 --> 00:38:22,210
I don't remember whether
the original model says

764
00:38:22,210 --> 00:38:23,890
whether you're
allowed to unfold.

765
00:38:23,890 --> 00:38:27,170
So if there are two versions,
simple folds and unfolds,

766
00:38:27,170 --> 00:38:29,530
or just simple folds.

767
00:38:29,530 --> 00:38:33,332
I don't think we actually
said unfolding is allowed.

768
00:38:33,332 --> 00:38:35,540
Though we're definitely
thinking about at some point,

769
00:38:35,540 --> 00:38:38,632
it's probably not in
the model as defined.

770
00:38:38,632 --> 00:38:39,215
Any questions?

771
00:38:42,380 --> 00:38:44,350
What's next?

772
00:38:44,350 --> 00:38:46,357
This is the paper that
introduced simple folds.

773
00:38:46,357 --> 00:38:47,940
It's called "When
Can You Fold a Map?"

774
00:38:47,940 --> 00:38:51,920
because it originally was
motivated by map folding.

775
00:38:51,920 --> 00:38:57,870
And it had a bunch of reasons
for introducing simple folds,

776
00:38:57,870 --> 00:39:02,070
among them is this quote, which
if you watched L1 was in there.

777
00:39:02,070 --> 00:39:05,500
I think the easiest way to
refold a roadmap is differently

778
00:39:05,500 --> 00:39:07,930
and her goal was
to make it easier

779
00:39:07,930 --> 00:39:11,100
to refold your
roadmap correctly.

780
00:39:11,100 --> 00:39:13,770
So here's one quote from
that paper as motivation.

781
00:39:13,770 --> 00:39:15,570
So it's origami motivation.

782
00:39:15,570 --> 00:39:17,920
But we're also wondering
about applications,

783
00:39:17,920 --> 00:39:20,720
like sheet metal bending,
cardboard folding, things

784
00:39:20,720 --> 00:39:23,660
like that where you want
to manufacture things

785
00:39:23,660 --> 00:39:25,190
using a machine.

786
00:39:25,190 --> 00:39:28,610
And while origamists can
do complicated folds,

787
00:39:28,610 --> 00:39:31,225
non-simple folds,
to make art work,

788
00:39:31,225 --> 00:39:33,200
in practical
manufacturing, you want

789
00:39:33,200 --> 00:39:35,140
to have the simplest
possible machine.

790
00:39:35,140 --> 00:39:38,100
So if you can get away with just
simple folds, as defined here,

791
00:39:38,100 --> 00:39:39,350
that would be great.

792
00:39:39,350 --> 00:39:42,780
Now you don't really need, some
of these are maybe artificial.

793
00:39:42,780 --> 00:39:46,717
You probably don't need
the 180 degree condition,

794
00:39:46,717 --> 00:39:49,050
because most of the things
you want to fold aren't flat.

795
00:39:49,050 --> 00:39:52,567
We introduced that just to keep
things simple mathematically.

796
00:39:52,567 --> 00:39:54,900
But you'd like to fold along
just one segment at a time,

797
00:39:54,900 --> 00:39:55,840
ideally.

798
00:39:55,840 --> 00:39:57,670
You definitely don't
want collision.

799
00:39:57,670 --> 00:39:59,410
You don't want
material to hit things.

800
00:39:59,410 --> 00:40:01,110
Whereas in origami,
you can do tucks,

801
00:40:01,110 --> 00:40:06,560
you can do things that
are not simple folds.

802
00:40:06,560 --> 00:40:08,550
That's a lot harder
with a machine

803
00:40:08,550 --> 00:40:10,900
that doesn't have any feedback.

804
00:40:10,900 --> 00:40:13,740
So here's a very simple machine.

805
00:40:13,740 --> 00:40:19,010
This is a brake folder.

806
00:40:19,010 --> 00:40:20,890
We actually have a
brake folder in CC,

807
00:40:20,890 --> 00:40:23,490
although this one
is Electrabrake,

808
00:40:23,490 --> 00:40:25,670
so this has an electric assist.

809
00:40:25,670 --> 00:40:30,380
So the idea is you slide your
sheet in, and you hold here.

810
00:40:30,380 --> 00:40:32,570
You pull up, and,
in this case, he's

811
00:40:32,570 --> 00:40:34,170
bending to a 90 degree angle.

812
00:40:34,170 --> 00:40:38,370
You can adjust it to
different angles and so on.

813
00:40:38,370 --> 00:40:40,180
There are lots of
automated machines,

814
00:40:40,180 --> 00:40:42,940
it's a little hard to get
photos and videos of them,

815
00:40:42,940 --> 00:40:45,310
but they're based
on this principle.

816
00:40:45,310 --> 00:40:48,340
Maybe you push in
a v, and you end up

817
00:40:48,340 --> 00:40:49,640
with the crease in one spot.

818
00:40:49,640 --> 00:40:51,940
And you'd like to just make
a sort of conveyor belt

819
00:40:51,940 --> 00:40:54,220
with lots of different
pushes and pulls

820
00:40:54,220 --> 00:40:56,940
and do a bunch of
simple folds, basically,

821
00:40:56,940 --> 00:40:59,137
except for this 180
degree constraint.

822
00:40:59,137 --> 00:41:00,720
And so we're just
curious about what's

823
00:41:00,720 --> 00:41:02,530
possible by simple
folds, and that led us

824
00:41:02,530 --> 00:41:04,420
into the map
folding stuff, where

825
00:41:04,420 --> 00:41:06,370
it's fairly easy
to characterize.

826
00:41:06,370 --> 00:41:10,790
Other things where it's harder,
we'll see in lecture three.

827
00:41:10,790 --> 00:41:13,430
I thought I'd show you
some examples of things

828
00:41:13,430 --> 00:41:15,670
people make with pretty
much simple folds,

829
00:41:15,670 --> 00:41:17,870
other than this 180 condition.

830
00:41:17,870 --> 00:41:21,250
Out of things like--
this is folding wood.

831
00:41:21,250 --> 00:41:23,030
You take a sheet of material.

832
00:41:23,030 --> 00:41:24,410
You start bending
these parts up,

833
00:41:24,410 --> 00:41:26,410
and you can make a little
chair, a little table.

834
00:41:26,410 --> 00:41:28,159
And you could fold it
back when you're not

835
00:41:28,159 --> 00:41:29,600
using your living room.

836
00:41:29,600 --> 00:41:31,760
You can hide everything.

837
00:41:31,760 --> 00:41:33,902
So you could imagine also
having multiple sheets,

838
00:41:33,902 --> 00:41:35,610
and sometimes your
room is a living room,

839
00:41:35,610 --> 00:41:38,140
other times it's-- whatever
furniture you need,

840
00:41:38,140 --> 00:41:40,420
you just unfold the
appropriate thing.

841
00:41:40,420 --> 00:41:42,602
That's the vision.

842
00:41:42,602 --> 00:41:44,060
Here's a cute little
folding chair.

843
00:41:44,060 --> 00:41:46,480
There's a huge number
of folding chairs,

844
00:41:46,480 --> 00:41:50,410
but this one is pretty
much simple folds.

845
00:41:50,410 --> 00:41:52,010
The one thing I'm
not sure whether it

846
00:41:52,010 --> 00:41:55,020
falls under simple
folds is this fold.

847
00:41:55,020 --> 00:41:58,980
You do fold along one line, but
it's in two different pieces.

848
00:41:58,980 --> 00:42:00,730
I'm not sure we'd call
that a simple fold.

849
00:42:00,730 --> 00:42:03,350
There's, of course, lots of
slits in the material here.

850
00:42:03,350 --> 00:42:05,584
But of course it has
all the same advantages

851
00:42:05,584 --> 00:42:06,250
of simple folds.

852
00:42:06,250 --> 00:42:09,910
This is easy to execute
one step at time.

853
00:42:09,910 --> 00:42:11,410
Here's some more
complicated design.

854
00:42:11,410 --> 00:42:12,993
Some of these are
computer renderings.

855
00:42:12,993 --> 00:42:14,100
Some of these are real.

856
00:42:14,100 --> 00:42:17,330
Again, taking furniture
out of flat walls.

857
00:42:20,010 --> 00:42:21,330
And here's some table designs.

858
00:42:21,330 --> 00:42:24,119
These are sheet metal.

859
00:42:24,119 --> 00:42:24,785
I like this one.

860
00:42:24,785 --> 00:42:25,850
It's very simple.

861
00:42:25,850 --> 00:42:28,820
Take the square of sheet
metal, put in some slips do,

862
00:42:28,820 --> 00:42:32,510
some very simple folds--
boom, you've got a table.

863
00:42:32,510 --> 00:42:36,470
This one's also pretty simple.

864
00:42:36,470 --> 00:42:39,060
Again, here we're
folding along one line,

865
00:42:39,060 --> 00:42:40,540
but it's in two
different pieces.

866
00:42:40,540 --> 00:42:41,610
So is that a simple fold?

867
00:42:41,610 --> 00:42:43,568
It's definitely harder
to build such a machine,

868
00:42:43,568 --> 00:42:44,600
but it's doable.

869
00:42:44,600 --> 00:42:47,340
Here we have something that's
definitely not a simple fold,

870
00:42:47,340 --> 00:42:49,810
but it's also fairly
easy to execute.

871
00:42:49,810 --> 00:42:54,420
Using a roller, you can
kind of curve one segment.

872
00:42:54,420 --> 00:42:56,055
I mean, when you
go to reality, you

873
00:42:56,055 --> 00:42:57,860
can change the model all
sorts of different ways

874
00:42:57,860 --> 00:42:59,318
and still have
something practical.

875
00:42:59,318 --> 00:43:03,740
And no one rule set is gospel.

876
00:43:03,740 --> 00:43:07,200
But mathematically we have to
hone in on at least one model

877
00:43:07,200 --> 00:43:10,010
at a time, and then we can
see how changing the rules

878
00:43:10,010 --> 00:43:12,760
changes what you can make.

879
00:43:12,760 --> 00:43:14,300
OK, next question.

880
00:43:14,300 --> 00:43:17,020
This is actually about the
definition of simple folds,

881
00:43:17,020 --> 00:43:20,359
so it was probably
answered already.

882
00:43:20,359 --> 00:43:22,900
Is it allowed to bend the rest
of paper to get it out the way

883
00:43:22,900 --> 00:43:23,770
and avoid collision?

884
00:43:23,770 --> 00:43:24,970
The answer is no.

885
00:43:24,970 --> 00:43:27,570
In simple folds,
at least, you're

886
00:43:27,570 --> 00:43:29,899
only allowed to move
that one segment.

887
00:43:29,899 --> 00:43:31,440
We have actually
lately been thinking

888
00:43:31,440 --> 00:43:32,856
about a different
model, where you

889
00:43:32,856 --> 00:43:37,120
do allow this, but simple folds,
you can't move other parts.

890
00:43:37,120 --> 00:43:40,730
You can just move the single
hinge that you're folding.

891
00:43:40,730 --> 00:43:44,824
And the end product has to
be flat, yes, in our model.

892
00:43:44,824 --> 00:43:46,490
Though, it would be
interesting to think

893
00:43:46,490 --> 00:43:48,781
without this condition,
because you're doing 180 degree

894
00:43:48,781 --> 00:43:50,840
operations before
you do the next one,

895
00:43:50,840 --> 00:43:52,364
you'll be flat at all times.

896
00:43:52,364 --> 00:43:53,780
1D or 2D, according
to whether you

897
00:43:53,780 --> 00:43:58,280
started with a 1D piece of
paper or 2D piece of paper.

898
00:43:58,280 --> 00:43:58,780
OK.

899
00:44:01,780 --> 00:44:04,410
The second half of the
lecture was basically

900
00:44:04,410 --> 00:44:08,530
about proving, characterizing
flat foldability

901
00:44:08,530 --> 00:44:11,110
of 1D segments.

902
00:44:11,110 --> 00:44:14,160
And it showed in particular
that simple folds are universal,

903
00:44:14,160 --> 00:44:21,530
that if you have some
mountain valley pattern,

904
00:44:21,530 --> 00:44:24,700
and it's foldable at all,
if it's flat foldable,

905
00:44:24,700 --> 00:44:28,220
it will be flat foldable
via simple folds.

906
00:44:28,220 --> 00:44:30,875
And in particular using
crimps and n folds.

907
00:44:36,410 --> 00:44:37,860
And it was a bit
of a messy prove,

908
00:44:37,860 --> 00:44:41,320
partly because I've made a
mistake in lecture, as you saw.

909
00:44:41,320 --> 00:44:43,010
I kind of corrected
for it on the fly,

910
00:44:43,010 --> 00:44:44,570
but it's maybe not
the best written.

911
00:44:44,570 --> 00:44:48,660
So I wanted to go through
a couple quick examples

912
00:44:48,660 --> 00:44:53,220
to make clear all
the issues there.

913
00:44:53,220 --> 00:44:55,470
So here are the ones I prepared.

914
00:44:55,470 --> 00:44:58,530
We can certainly do more
if it's still not clear.

915
00:45:02,660 --> 00:45:05,660
So here's a simple
mountain-valley pattern,

916
00:45:05,660 --> 00:45:09,300
and it's got some long
segments and, let's just

917
00:45:09,300 --> 00:45:11,110
say, equidistant segments here.

918
00:45:11,110 --> 00:45:13,320
Three valleys, then a mountain.

919
00:45:13,320 --> 00:45:18,510
So first question
is, is this mingling?

920
00:45:18,510 --> 00:45:21,130
And then the ultimate question
is, is it flat foldable?

921
00:45:28,140 --> 00:45:29,220
So is it mingling?

922
00:45:29,220 --> 00:45:32,680
Well, maybe you
could answer for me.

923
00:45:32,680 --> 00:45:33,860
Just yes or no.

924
00:45:33,860 --> 00:45:34,950
50% chance.

925
00:45:37,910 --> 00:45:41,620
So maybe the definition of
mingling is not super clear.

926
00:45:41,620 --> 00:45:42,990
Let me review it.

927
00:45:42,990 --> 00:45:45,760
So you look at each,
I mean, generally

928
00:45:45,760 --> 00:45:47,410
of a sequence of
mountains and valleys,

929
00:45:47,410 --> 00:45:49,301
you look at a chunk
of all valleys,

930
00:45:49,301 --> 00:45:51,050
then you look at a
chunk of all mountains,

931
00:45:51,050 --> 00:45:52,091
and chunk of all valleys.

932
00:45:52,091 --> 00:45:55,090
Here there's only two chunks--
three valleys and one mountain.

933
00:45:55,090 --> 00:45:57,760
And the definition is a little
awkward for a single crease,

934
00:45:57,760 --> 00:46:00,160
but let's start
with the valleys.

935
00:46:00,160 --> 00:46:04,860
The point is to check-- for
the first segment between two

936
00:46:04,860 --> 00:46:10,050
valleys versus the segment just
before it, which is bigger?

937
00:46:10,050 --> 00:46:14,030
And this is the bad case this
is the non-mingling situation,

938
00:46:14,030 --> 00:46:16,550
because this thing
is bigger than this.

939
00:46:16,550 --> 00:46:21,140
Strictly bigger, the notation
we use in the lecture

940
00:46:21,140 --> 00:46:23,640
is an open square bracket.

941
00:46:23,640 --> 00:46:26,410
So square bracket meant
that this is bigger.

942
00:46:26,410 --> 00:46:29,700
Round bracket would mean this
is less than or equal to this.

943
00:46:29,700 --> 00:46:31,050
That's just the definition.

944
00:46:31,050 --> 00:46:33,010
Over on this side
of the valleys,

945
00:46:33,010 --> 00:46:35,890
this length is equal
to this length.

946
00:46:35,890 --> 00:46:38,240
So the last distance
between two values

947
00:46:38,240 --> 00:46:40,140
is equal to the
one right after it.

948
00:46:40,140 --> 00:46:45,820
And that's a good case, so we
write a closed, round bracket.

949
00:46:45,820 --> 00:46:47,486
Then we have a
sequence of mountains,

950
00:46:47,486 --> 00:46:48,860
and here it's a
little confusing,

951
00:46:48,860 --> 00:46:51,490
but it's the same idea.

952
00:46:51,490 --> 00:46:53,159
So this is the very
first mountain.

953
00:46:53,159 --> 00:46:55,450
You look at the length right
after it versus the length

954
00:46:55,450 --> 00:46:58,070
right before it,
and this is smaller.

955
00:46:58,070 --> 00:46:59,240
And that's a good case.

956
00:46:59,240 --> 00:47:03,280
We write an open, round bracket
for this mountain group.

957
00:47:03,280 --> 00:47:04,430
And then same thing.

958
00:47:04,430 --> 00:47:06,390
Now we're comparing
the same two distances,

959
00:47:06,390 --> 00:47:08,700
but it's now bad, because
this one is strictly longer

960
00:47:08,700 --> 00:47:09,320
than this one.

961
00:47:09,320 --> 00:47:11,620
So we write a closed,
square bracket.

962
00:47:11,620 --> 00:47:13,890
So that's the notation
in this example.

963
00:47:13,890 --> 00:47:17,030
Any questions about that?

964
00:47:17,030 --> 00:47:19,140
So you just have to
check-- in general,

965
00:47:19,140 --> 00:47:23,590
you have a whole group of
valleys-- these are all

966
00:47:23,590 --> 00:47:26,510
valleys, or all
mountains-- and you

967
00:47:26,510 --> 00:47:29,840
want to compare this
one versus this one.

968
00:47:29,840 --> 00:47:32,760
And square or closed,
according to which is bigger.

969
00:47:32,760 --> 00:47:36,010
And you want to look at the last
one versus the after last one.

970
00:47:38,530 --> 00:47:40,070
So that's the notation.

971
00:47:40,070 --> 00:47:43,410
And the point of the proof
was to argue that, either

972
00:47:43,410 --> 00:47:47,410
if you're going to be
flat foldable at all,

973
00:47:47,410 --> 00:47:49,820
actually if you're
mingling-- mingling meant

974
00:47:49,820 --> 00:47:52,570
that for each of these
intervals, at least one

975
00:47:52,570 --> 00:47:54,780
of the sides was round.

976
00:47:54,780 --> 00:47:56,180
That was considered good.

977
00:47:56,180 --> 00:48:00,470
So this crease
pattern is mingling,

978
00:48:00,470 --> 00:48:02,360
because there's two regions.

979
00:48:02,360 --> 00:48:03,590
This one has a round bracket.

980
00:48:03,590 --> 00:48:05,320
This one has a round bracket.

981
00:48:05,320 --> 00:48:08,740
And what we argue is
it if you're mingling,

982
00:48:08,740 --> 00:48:11,140
which was necessary, if
you're flat foldable,

983
00:48:11,140 --> 00:48:12,780
you have to be mingling.

984
00:48:12,780 --> 00:48:14,740
It's a necessary but
not sufficient condition

985
00:48:14,740 --> 00:48:16,760
for flat foldability.

986
00:48:16,760 --> 00:48:19,300
If you're mingling,
either you have a pattern

987
00:48:19,300 --> 00:48:22,140
like this, close round
bracket, open round bracket.

988
00:48:22,140 --> 00:48:26,590
That's good because this
is a crimp that you can do.

989
00:48:26,590 --> 00:48:27,470
You see it up here.

990
00:48:27,470 --> 00:48:29,070
This is a crimp.

991
00:48:29,070 --> 00:48:31,610
You valley fold, mountain fold,
and you don't hide anything

992
00:48:31,610 --> 00:48:33,360
when you make that operation.

993
00:48:33,360 --> 00:48:35,820
Or there's an end fold, which
corresponded to an open round

994
00:48:35,820 --> 00:48:37,236
bracket at the
beginning or closed

995
00:48:37,236 --> 00:48:38,510
round bracket at the end.

996
00:48:38,510 --> 00:48:39,610
So here there's a crimp.

997
00:48:39,610 --> 00:48:42,260
Let's do the crimp.

998
00:48:42,260 --> 00:48:47,090
So when we do the crimp, let's
keep this part of paper fixed.

999
00:48:47,090 --> 00:48:52,070
So this we go over
to here, hopefully,

1000
00:48:52,070 --> 00:48:56,470
then we valley fold,
then we mountain fold,

1001
00:48:56,470 --> 00:48:59,490
and we keep going from there.

1002
00:48:59,490 --> 00:49:02,660
That segment is that segment.

1003
00:49:02,660 --> 00:49:07,930
So we still have this valley.

1004
00:49:07,930 --> 00:49:09,950
This was a valley
we just folded.

1005
00:49:09,950 --> 00:49:12,030
This is the mountain
we just folded.

1006
00:49:12,030 --> 00:49:14,472
Now conceptually, we just
sort of fuse this back

1007
00:49:14,472 --> 00:49:16,430
into the paper, because
those creases are done.

1008
00:49:16,430 --> 00:49:17,846
We don't need to
think about them.

1009
00:49:17,846 --> 00:49:22,160
The point is, in that region
there were no extra creases.

1010
00:49:22,160 --> 00:49:23,910
These round parentheses
will guarantee

1011
00:49:23,910 --> 00:49:27,530
there's nothing here, no
creases here, here, or here.

1012
00:49:27,530 --> 00:49:31,050
Could be creases farther away
but by these inequalities,

1013
00:49:31,050 --> 00:49:34,140
that this length is less
than or equal to this one,

1014
00:49:34,140 --> 00:49:38,220
and this length is less
than or equal to this one.

1015
00:49:38,220 --> 00:49:41,050
Sorry, greater than or equal to.

1016
00:49:41,050 --> 00:49:42,430
Then you know this is OK.

1017
00:49:47,430 --> 00:49:50,120
So there's two valleys left.

1018
00:49:50,120 --> 00:49:52,799
So now we have two valleys,
we have a long segment,

1019
00:49:52,799 --> 00:49:53,590
and a long segment.

1020
00:49:53,590 --> 00:49:55,970
And this is something
that can't be made.

1021
00:49:55,970 --> 00:49:58,470
Because there's no folded
state of this thing,

1022
00:49:58,470 --> 00:50:02,340
never mind simple folds, because
it's going to cross like that.

1023
00:50:02,340 --> 00:50:03,570
So this is not flat foldable.

1024
00:50:06,340 --> 00:50:09,680
It's also not mingling,
because if you

1025
00:50:09,680 --> 00:50:11,842
look at these two valleys--

1026
00:50:11,842 --> 00:50:13,300
You look at the
distance over here.

1027
00:50:13,300 --> 00:50:15,160
It's bigger than this
one, so that's bad,

1028
00:50:15,160 --> 00:50:17,110
so you have an open
square bracket.

1029
00:50:17,110 --> 00:50:21,121
And this one is also bigger than
this, imagine these as fused,

1030
00:50:21,121 --> 00:50:22,620
and so it's a closed
square bracket.

1031
00:50:22,620 --> 00:50:25,140
And so this group of
values is not mingling.

1032
00:50:25,140 --> 00:50:26,930
So it's not mingling.

1033
00:50:26,930 --> 00:50:29,160
So ultimately this
pattern we started

1034
00:50:29,160 --> 00:50:32,300
with is not flat foldable,
because one of the things we

1035
00:50:32,300 --> 00:50:36,279
proved is doing a crimp never
changes flat foldability.

1036
00:50:36,279 --> 00:50:37,570
It's always a safe thing to do.

1037
00:50:40,302 --> 00:50:41,760
You might wonder,
oh, maybe there's

1038
00:50:41,760 --> 00:50:44,340
some other fold I could to
do that eventually works,

1039
00:50:44,340 --> 00:50:47,380
but we proved crimps
are always safe to do.

1040
00:50:47,380 --> 00:50:49,280
So we did it, and we got stock.

1041
00:50:49,280 --> 00:50:51,220
That means this was
not flat foldable,

1042
00:50:51,220 --> 00:50:52,680
even though it was mingling.

1043
00:50:52,680 --> 00:50:55,050
And so the mingling
forever property just

1044
00:50:55,050 --> 00:50:57,490
means, if it's mingling
and then you do a crimp,

1045
00:50:57,490 --> 00:50:59,185
and it's still mingling, and
if you keep doing crimps,

1046
00:50:59,185 --> 00:51:00,601
and it stays
mingling all the way,

1047
00:51:00,601 --> 00:51:01,947
then you were flat foldable.

1048
00:51:01,947 --> 00:51:03,780
It's not a very satisfying
characterization,

1049
00:51:03,780 --> 00:51:07,660
but it is a thing.

1050
00:51:07,660 --> 00:51:15,030
Maybe I'll do one more
example where it works,

1051
00:51:15,030 --> 00:51:18,110
so we're super clear.

1052
00:51:18,110 --> 00:51:22,605
Whoa, we're out of time, so
I won't do another example.

1053
00:51:22,605 --> 00:51:23,480
All right, ambitious.

1054
00:51:29,930 --> 00:51:31,800
I gotta work on my timing.

1055
00:51:31,800 --> 00:51:32,300
There are a

1056
00:51:32,300 --> 00:51:33,700
Couple other fun questions here.

1057
00:51:33,700 --> 00:51:35,824
I would encourage you to
read the notes about them.

1058
00:51:35,824 --> 00:51:38,030
In particular, there's
an algorithmic question.

1059
00:51:38,030 --> 00:51:40,050
How do you actually
compute this efficiently?

1060
00:51:40,050 --> 00:51:43,030
You could do it very
efficiently in linear time.

1061
00:51:43,030 --> 00:51:47,320
So n, where c is one, just n
time-- instead of n squared

1062
00:51:47,320 --> 00:51:50,384
or something-- using
a pretty simple idea.

1063
00:51:50,384 --> 00:51:52,050
Basically just look
for the first crimp,

1064
00:51:52,050 --> 00:51:54,360
do it, and then see if
there are crimps nearby,

1065
00:51:54,360 --> 00:51:55,927
and keep going
forward, and you can

1066
00:51:55,927 --> 00:51:57,260
prove that it takes linear time.

1067
00:52:00,220 --> 00:52:02,660
There is this fun question
I enjoy thinking about.

1068
00:52:02,660 --> 00:52:05,280
Can you make any mountain-valley
pattern flat foldable

1069
00:52:05,280 --> 00:52:06,260
by adding creases?

1070
00:52:06,260 --> 00:52:07,197
The answer is yes.

1071
00:52:07,197 --> 00:52:08,530
You can think of it as a puzzle.

1072
00:52:08,530 --> 00:52:10,180
There is one proposed
way to do it here.

1073
00:52:10,180 --> 00:52:11,513
I have another one in the notes.

1074
00:52:11,513 --> 00:52:13,592
You can think about it.

1075
00:52:13,592 --> 00:52:15,050
And the last
question, is what does

1076
00:52:15,050 --> 00:52:17,300
it possibly mean to fold
something in four dimensions?

1077
00:52:17,300 --> 00:52:18,430
How do you imagine it?

1078
00:52:18,430 --> 00:52:21,360
Hard to imagine, but
you can think about it.

1079
00:52:21,360 --> 00:52:23,390
You have a d-dimensional
piece of paper,

1080
00:52:23,390 --> 00:52:25,820
you fold it through
d plus 1 dimensions.

1081
00:52:25,820 --> 00:52:29,910
If you want it flat folded, it
ends up back in d dimensions,

1082
00:52:29,910 --> 00:52:32,890
and your creases are
d minus 1 dimensional.

1083
00:52:32,890 --> 00:52:34,660
And the rest you just
have to visualize.

1084
00:52:34,660 --> 00:52:36,565
I have one example
folding a solid cube

1085
00:52:36,565 --> 00:52:38,070
in half in the notes.

1086
00:52:38,070 --> 00:52:39,432
That's certainly possible.

1087
00:52:39,432 --> 00:52:41,140
That's not very well
studied, and there's

1088
00:52:41,140 --> 00:52:42,556
lots of interesting
open questions

1089
00:52:42,556 --> 00:52:44,380
about higher
dimensional folding.

1090
00:52:44,380 --> 00:52:47,860
Any questions before we go?

1091
00:52:47,860 --> 00:52:48,950
All right.

1092
00:52:48,950 --> 00:52:51,700
Watch lecture three, and
please send your feedback.

1093
00:52:51,700 --> 00:52:53,390
It was really helpful.