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PROFESSOR: All right.

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Today I think is the
last lecture at least

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for the while about
origami, and I'm

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going to talk about
where I got started

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thinking about
organic mathematics.

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The folding cup
problem, and this

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is sort of motivated
by a magic trick.

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The idea is you take
a piece of paper.

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You fold it flat.

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You make one complete
straight cut.

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You cut along a line.

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And you unfold the
pieces, and the question

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is what shapes can you
get by that process.

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So this is like a magic trick.

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I showed you making a
swan which I have here

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just for-- to jog your memory.

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You have a rectangle paper, and
you can see the swan on there

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and you can see a
bunch of creases.

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You fold along all the
creases, not the swan lines,

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and you end up with all
the edges of the swan lying

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right along that line.

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You cut along the line,
and you get your swan,

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as we did before.

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And you also get the anti-swan.

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The other piece-- I didn't
show that last time.

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But it's really--
it's not like we're

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making we're not allowed
to make any extra creases.

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We really want the swan.

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So we cut along exactly
the edges of the swan

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by lining them up onto a line.

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So really, you could think
of this as a magic trick

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in cutting, but you
can also think of it

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as an origami
problem, which is I

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want to line up all
these edges by folding.

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How do I do it?

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And that way it connects to a
lot origami design problems.

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This problem has an old history.

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It goes back to 1721.

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This is the oldest
reference we know.

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This is a Japanese puzzle
book, Wakoku Chiyekurabe

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by Kan Chu Sen.

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And I think this is
kind of like-- it's

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called "Mathematical
Contests" is the translation.

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And it's sort of
like the old version

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of like these big
problem solving sessions

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that kids do these days
to get better at math,

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and one of the pages
poses this problem.

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I have this Japanese
emblem shape.

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Can I make it from
a piece of paper

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by folding in one straight
cut, and the answer is yes.

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And this is a solution
if you cheat and look

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in the back of the book.

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So I'll let you read
that for minute.

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[LAUGHS]

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You're making folds
along lines that end up

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lining up other
parts of the shape

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so that in the end everything
lies along the line,

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then you cut along the line.

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We learned about
this a bit later

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than the-- we learned about
this problem from Martin Gardner

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originally, and he knew
about the magic world.

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So Houdini, before he
was an escape artist,

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he was a general magician, and
in 1922, he wrote or probably

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had ghost written this book,
"Houdini's Paper Magic".

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And one of the pages is about
folding, and it says here,

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you can take a square
paper, fold it flat,

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make one straight cut, and get
a regular five-pointed star.

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And that was pretty cool,
and then other magicians

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picked it up, in particular
this guy Gerald Lowe, who

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wrote this book "Paper Capers".

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It's more like a
very small book.

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Magic book.

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And he could make all
sorts of different things

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and he would incorporate
them into magic tricks.

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And he was primarily
using simple folds.

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He would just fold along
one line at a time,

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and make one
straight cut, and go

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to make all these cool patterns.

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Like here, I have one
of his examples redone.

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I start from a rectangle,
I fold, I fold, I fold,

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I fold-- these are
all simple folds.

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I take my scissors, and I make
one complete straight cut.

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And usually when I perform
this trick, I say look!

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I made an isosceles triangle.

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Wow!

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I made five isosceles triangles.

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Amazing!

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And then I made
everything, except the five

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isosceles triangles.

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So you saw that coming.

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And you could make an
arrangement of five of these

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if you want.

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All that sort of
very symmetric stuff

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is easy to do by simple folds.

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I'll talk more about
simple folds later,

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but we were really curious
about the general challenge.

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So this is the-- you can
download this from my web page

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if you want to make one.

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It's pretty-- it's a fun trick.

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Good.

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So I have some more
interesting examples.

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See here I have a
rectangle folded,

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and I make one
complete straight cut.

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And this one has actually
a line of symmetry,

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so I fold in half
at the beginning.

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So I get an angel fish.

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Ooh!

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All right, you're not impressed.

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Keep going.

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Here's another one.

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One straight cut.

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We could go all day here.

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I mean, the point
of today's lecture

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is to see how this is done.

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In general.

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Here we have a butterfly.

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All right.

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You guys are tough
to impress here.

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Here we go.

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This one is deemed thematic.

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It's almost October, so
it's sort of appropriate.

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We'll one straight cut.

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Haven't done this
one for quite awhile.

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Get tons of pieces, and if I'm
lucky-- open it up the right

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way around.

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Jack-o-lantern.

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Wow, it's amazing!

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How is it possible?

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Obviously you can make
many shapes all at once.

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That's the general idea here.

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I'll admit, I cheated
here because I wanted kids

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to be able to fold this.

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The outer octagon
was cut initially.

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Yeah, so it's not
from a rectangle.

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Just to make it easier
to fold, but you

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could do it all at once.

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And now, the big demo.

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In fact, it's so
big I think I might

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want to use the exacto knife.

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Uh, we'll try with scissors.

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Got a lot of layers.

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Oh, yeah.

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This is why I usually
bring my own scissors.

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Just checking the other side.

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They're better scissors.

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That's what's special
about my scissors.

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All right, time for
the exacto knife.

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Make sure I'm only
cutting on the boundary

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between red and white.

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Yeah, my lecture notes.

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Yeah, who needs those?

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It's flat.

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A little more cutting.

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I believe all the pieces in
this case are rectangles.

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Exciting.

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All right, straight cut, right?

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I don't remember which
way this one opens.

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This should be the MIT logo.

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This one I encourage
you to try at home.

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It's pretty crazy.

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It's definitely a hard folding.

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I'm getting used to it now.

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All right, so you could
make anything is the point.

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Some are a little more difficult
because they have more layers

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and so on.

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If I made it out of thinner
paper like some of the math

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magicians do, it is super easy.

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Onion skin paper or something.

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I think I have some pictures
of the piece patterns

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if you want to see
what these look like.

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And these are all
available for download.

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You want to impress
your friends, go for it.

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They all use slightly
different color codings,

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but it's mountain or valley.

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And so we want to see
how to make these.

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So let me state
the theorem first.

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We have our good friend,
the universality result,

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which is any set of line
segments on your piece of paper

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can be-- I'll phrase it as can
be aligned by flat folding.

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A line means when you
make the flat folding,

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all the segments come
onto a common line.

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Nothing else comes to that
line, and therefore you

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cut a long line, you get
exactly those line segments.

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And there are two methods
for solving this problem.

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The first one is what I call
straight skeleton method,

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and second one I'll call
the disk packing method.

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By slightly different authors.

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This one was me, my dad, and
my advisor, Anna [INAUDIBLE].

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This one was Marshall Burn,
me, David Epstein, Barry Hayes.

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This one's slightly
after this one.

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This is sort of my first
computational origami paper.

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And they're quite different.

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I mean at a high level,
this one's practical.

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This one is theoretically
good, but impractical.

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This one's actually
theoretically

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bad in a few situations
which we will get to,

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but it works very
well almost always.

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In a formal sense
of almost always.

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This one always works, but
it's a challenge to fold.

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All of the examples
I showed you are

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made with the straight
skeleton method.

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So that's the idea.

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That's where we're going
to talk about both of them.

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And first, I have a warm
up-- three warm ups.

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Suppose you had a
square paper and you

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wanted to make a single cut.

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What folding do I do?

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Nothing!

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Yeah, that was easy.

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Let's say I have two
lines I want to make.

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What should I do?

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Fold between them.

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Fold the lines onto each other,
which is angular bisector.

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If I extend the lines, and
I bisect the angle there,

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then I will fold one
line to the other.

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I brought one just to
be totally obvious.

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You have two lines
fold along the angular

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bisector of their extensions.

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It lines up the lines,
and nothing else.

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OK, a little more exciting.

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What if I had a triangle?

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How would you line up the edges
the triangle and nothing else?

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Rabbit ear.

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Yes.

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Rabbit ear is you fold
along the three angular

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bisectors of the triangle.

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This is something we talked
about in the tree method.

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This is one of the
sort of gadgets

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we use in one of-- where
these were active paths.

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And we fold along those
angular bisectors.

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Angular bisectors
intuitively are

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very good because locally
they line up edges.

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So if I fold along
all three of them,

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and you may know that they
always meet at a point,

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then I kind of line
up all those edges.

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If I just fold along
those three edges,

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it won't be flat foldable.

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It's like this floppy thing.

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And so I've added in
these perpendicular--

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the purple lines--
to give me my hinges

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so I can manipulate these arms.

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You don't have to
use all of them.

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Then you flatten it,
and now along this line

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are all the black guys.

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All the black lines.

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We've got a bunch
of flat foldings.

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All right, so a general idea,
and in the straight skeleton

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method, what we are going
to use are angular bisectors

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and perpendicular folds.

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Perpendicular folds are also
good because locally folds

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are a reflection
if you fold flat.

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So this line will fold
on top of this line.

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So perpendiculars are good.

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Angular bisectors are good.

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00:13:14,540 --> 00:13:17,690
If you fold along all of them,
you should line everything up.

256
00:13:17,690 --> 00:13:22,730
That's the intuition,
and the question

257
00:13:22,730 --> 00:13:25,400
is how do I fold a long
enough angular bisectors

258
00:13:25,400 --> 00:13:29,090
and perpendiculars so that
it actually folds flat.

259
00:13:29,090 --> 00:13:31,605
And that is the straight
skeleton, or one way

260
00:13:31,605 --> 00:13:33,105
to do that is the
straight skeleton.

261
00:13:44,020 --> 00:13:47,910
And this actually was
invented a few years earlier.

262
00:13:47,910 --> 00:13:52,390
95, 96 by a couple of Austrians.

263
00:13:52,390 --> 00:13:54,790
That was in [INAUDIBLE]
and actually four people.

264
00:13:59,950 --> 00:14:01,530
So let me write
down a definition,

265
00:14:01,530 --> 00:14:07,040
and then we will-- I'll
show you what it means.

266
00:14:07,040 --> 00:14:09,810
We've seen a thing
like this very briefly

267
00:14:09,810 --> 00:14:11,990
in the universal
molecule, but this

268
00:14:11,990 --> 00:14:13,945
is going to be more
general in some sense.

269
00:14:31,660 --> 00:14:35,450
And actually, why don't
I put up an image of one

270
00:14:35,450 --> 00:14:38,430
while we're defining it.

271
00:15:19,764 --> 00:15:22,999
All right, so it is trajectory
of the vertices of the desired

272
00:15:22,999 --> 00:15:24,290
cut pattern-- that's our input.

273
00:15:24,290 --> 00:15:28,230
The graph of edges
we want to cut out.

274
00:15:28,230 --> 00:15:30,300
As we simultaneously
shrink each region.

275
00:15:30,300 --> 00:15:33,490
That's every face
outlined by those cuts.

276
00:15:33,490 --> 00:15:36,340
Keeping edges parallel and a
uniform perpendicular distance

277
00:15:36,340 --> 00:15:38,760
from the original edges.

278
00:15:38,760 --> 00:15:45,900
So, a bit of a mouthful, but
let me draw some pictures.

279
00:15:45,900 --> 00:15:48,610
Well, let's maybe
start with a triangle.

280
00:15:48,610 --> 00:15:51,330
So the idea with
a triangle is you

281
00:15:51,330 --> 00:15:54,210
shrink-- this is really the
wrong order to do things.

282
00:15:56,637 --> 00:15:58,970
I want to shrink-- there's
two regions for the triangle.

283
00:15:58,970 --> 00:16:00,500
There's the inside
and the outside.

284
00:16:00,500 --> 00:16:04,760
If I shrink the inside, I
get these parallel lines.

285
00:16:04,760 --> 00:16:07,420
All-- I want all of these
distances to be equal.

286
00:16:07,420 --> 00:16:09,900
Those are the
perpendicular distances.

287
00:16:09,900 --> 00:16:12,910
I keep shrinking,
and at some point

288
00:16:12,910 --> 00:16:15,280
I can't shrink anymore
because I get a single point.

289
00:16:15,280 --> 00:16:16,950
The in-center of the triangle.

290
00:16:16,950 --> 00:16:19,490
If I watch where did the
vertices go during that time,

291
00:16:19,490 --> 00:16:21,450
it's along angular bisectors.

292
00:16:21,450 --> 00:16:24,606
Hey, our good friends,
angular bisectors.

293
00:16:24,606 --> 00:16:26,230
Now I do the same
thing on the outside.

294
00:16:26,230 --> 00:16:30,420
Shrinking the outside region
is like expanding the triangle.

295
00:16:30,420 --> 00:16:35,120
As I expand the triangle, the
outside gets smaller area.

296
00:16:35,120 --> 00:16:37,390
So actually, it's
just the same thing.

297
00:16:37,390 --> 00:16:40,170
I get concentric
triangles, and I just

298
00:16:40,170 --> 00:16:42,430
keep going along the
angular bisectors.

299
00:16:42,430 --> 00:16:43,280
So that's it.

300
00:16:43,280 --> 00:16:45,460
It's not going to give us
the perpendicular fold.

301
00:16:45,460 --> 00:16:47,470
It's just the angular
bisector parts.

302
00:16:47,470 --> 00:16:49,185
Still see a triangle
in there somewhere.

303
00:16:51,730 --> 00:16:53,570
OK, that's a really
simple example,

304
00:16:53,570 --> 00:16:55,280
and the only event
that happened is

305
00:16:55,280 --> 00:16:57,640
that the polygon disappeared.

306
00:16:57,640 --> 00:17:03,950
When that happens, you stop
with that particular polygon.

307
00:17:03,950 --> 00:17:08,191
In general, there are three
things that can happen.

308
00:17:08,191 --> 00:17:09,065
We call these events.

309
00:17:12,150 --> 00:17:12,900
Three interesting.

310
00:17:12,900 --> 00:17:14,589
Things.

311
00:17:14,589 --> 00:17:17,145
One is that an edge disappears.

312
00:17:21,550 --> 00:17:24,880
So for example, locally, if
I have a picture like this

313
00:17:24,880 --> 00:17:32,500
and I shrink, and I shrink,
and I shrink, at some point--

314
00:17:32,500 --> 00:17:36,730
what's happening is these
angular bisectors are meeting,

315
00:17:36,730 --> 00:17:40,410
and now I lost-- this edge
shrinks to zero length.

316
00:17:40,410 --> 00:17:41,810
So just forget about the edge.

317
00:17:41,810 --> 00:17:42,935
Pretend it was never there.

318
00:17:42,935 --> 00:17:45,790
Just keeps shrinking now
what is these two edges.

319
00:17:45,790 --> 00:17:50,900
And I shrink, and I
shrink, and I shrink.

320
00:17:50,900 --> 00:17:53,550
What happens is, in some
sense, these vertices

321
00:17:53,550 --> 00:17:56,640
merge-- these edges
merge, and now I

322
00:17:56,640 --> 00:18:00,400
have one guy going
straight up there.

323
00:18:00,400 --> 00:18:01,710
And what is that edge doing?

324
00:18:01,710 --> 00:18:04,280
It's not an angular
bisector of this or this,

325
00:18:04,280 --> 00:18:07,825
but it's an angular bisector
of the extension of these two

326
00:18:07,825 --> 00:18:08,325
lines.

327
00:18:11,620 --> 00:18:15,950
Because if you look at this--
one of these two edges--

328
00:18:15,950 --> 00:18:17,620
they are parallel
to the two original.

329
00:18:17,620 --> 00:18:19,930
So if you bisect those
parallel offsets,

330
00:18:19,930 --> 00:18:22,810
it's the same thing as
bisecting the original edges

331
00:18:22,810 --> 00:18:24,490
and extension.

332
00:18:24,490 --> 00:18:27,650
So this looks good because
these two folds will line up

333
00:18:27,650 --> 00:18:30,785
those two edges, or this fold
will line up these two edge.

334
00:18:30,785 --> 00:18:32,410
This fold will line
up those two edges.

335
00:18:32,410 --> 00:18:34,414
This fold will line
up these two edges.

336
00:18:34,414 --> 00:18:36,080
And we're doing kind
of extra alignment,

337
00:18:36,080 --> 00:18:38,070
but everything looks kosher.

338
00:18:40,830 --> 00:18:43,580
Good.

339
00:18:43,580 --> 00:18:46,240
So when an edge disappears,
you just forget about it.

340
00:18:50,650 --> 00:18:53,480
All right, forget-- it's
probably "forgedaboutit"--

341
00:18:53,480 --> 00:18:56,092
something like that.

342
00:18:56,092 --> 00:18:57,900
All right.

343
00:18:57,900 --> 00:19:01,825
Then we have-- a
region can disappear.

344
00:19:05,850 --> 00:19:08,970
That's what happened
with the triangle.

345
00:19:08,970 --> 00:19:11,480
And then again, you
forget about it.

346
00:19:15,307 --> 00:19:16,390
There may be many regions.

347
00:19:16,390 --> 00:19:18,265
You have to keep shrinking
the other regions,

348
00:19:18,265 --> 00:19:21,310
but when one
disappears, you're done.

349
00:19:21,310 --> 00:19:25,335
Third thing that can happen
is that a region splits.

350
00:19:29,950 --> 00:19:32,590
So let's look at an
interesting polygon.

351
00:19:36,250 --> 00:19:42,580
This one-- the straight edges.

352
00:19:42,580 --> 00:19:48,130
And when I shrink this
guy-- see, what's happening

353
00:19:48,130 --> 00:19:56,440
is this edge is approaching
that vertex, and at some point

354
00:19:56,440 --> 00:19:57,065
they will meet.

355
00:20:00,670 --> 00:20:04,120
And what we're left with are
two triangles in this case.

356
00:20:04,120 --> 00:20:06,060
And generally split
into two parts,

357
00:20:06,060 --> 00:20:08,380
you just keep
shrinking the parts.

358
00:20:08,380 --> 00:20:11,710
So it's not really like
you're stopping at any point.

359
00:20:11,710 --> 00:20:13,464
Just the same thing
over and over.

360
00:20:13,464 --> 00:20:15,380
But if you're implementing
this on a computer,

361
00:20:15,380 --> 00:20:17,046
you really have to
realize that happens.

362
00:20:17,046 --> 00:20:18,990
Otherwise you'd shrink
them beyond each other

363
00:20:18,990 --> 00:20:22,002
and it would be self
intersecting and ugly.

364
00:20:22,002 --> 00:20:23,460
But you do the
obvious thing, which

365
00:20:23,460 --> 00:20:25,990
is you cut where they split.

366
00:20:25,990 --> 00:20:28,410
And what will happen in this
case with straight skeleton

367
00:20:28,410 --> 00:20:30,910
is you keep going along
an angular bisector

368
00:20:30,910 --> 00:20:32,880
until that little
triangle stops.

369
00:20:40,430 --> 00:20:43,434
One more edge here.

370
00:20:43,434 --> 00:20:45,600
So that's what the straight
skeleton will look like.

371
00:20:45,600 --> 00:20:48,560
This edge is an angular bisector
of this one and this one.

372
00:20:48,560 --> 00:20:51,180
This edge is an angular bisector
of this one and this one.

373
00:20:51,180 --> 00:20:54,090
In general, if you
look at an edge,

374
00:20:54,090 --> 00:20:57,390
and you see what
original edge can I

375
00:20:57,390 --> 00:21:00,600
reach without crossing
another skeleton edge,

376
00:21:00,600 --> 00:21:02,020
those are the two
that you bisect.

377
00:21:02,020 --> 00:21:03,210
So it's really easy to see.

378
00:21:03,210 --> 00:21:04,220
Look at this guy.

379
00:21:04,220 --> 00:21:06,427
The only two I can reach
are this one and this one.

380
00:21:06,427 --> 00:21:08,260
I look at this one, the
only two I can reach

381
00:21:08,260 --> 00:21:09,343
are that one and that one.

382
00:21:09,343 --> 00:21:11,480
So it's an angular
bisector of those two.

383
00:21:11,480 --> 00:21:14,085
In fact, in general, if you
finish the outside here too,

384
00:21:14,085 --> 00:21:16,750
it's the same deal.

385
00:21:16,750 --> 00:21:19,740
And then-- yeah,
all right, enough.

386
00:21:19,740 --> 00:21:21,740
Here's a bigger example.

387
00:21:21,740 --> 00:21:25,180
Little turtle drawn
on a triangular grid,

388
00:21:25,180 --> 00:21:26,950
and you can see there's
angular bisectors.

389
00:21:26,950 --> 00:21:28,500
This is a straight skeleton.

390
00:21:28,500 --> 00:21:30,650
This guy, for example,
bisects this horizontal edge

391
00:21:30,650 --> 00:21:33,000
and this horizontal edge has a
little bit of a boundary case

392
00:21:33,000 --> 00:21:35,541
we have to think about, but this
is the right interpretation.

393
00:21:37,740 --> 00:21:41,770
It's like an angle of 180,
so you bisect it to 90.

394
00:21:41,770 --> 00:21:43,124
Other fun features.

395
00:21:43,124 --> 00:21:44,540
Here we get a
little bit of action

396
00:21:44,540 --> 00:21:46,170
on the outside of the polygon.

397
00:21:46,170 --> 00:21:48,110
So far, we haven't seen that.

398
00:21:48,110 --> 00:21:50,890
So, like these guys
meet, and there

399
00:21:50,890 --> 00:21:56,680
is some bigger-- there's a
bigger turtle here somewhere.

400
00:21:56,680 --> 00:21:58,630
It's hard to draw.

401
00:21:58,630 --> 00:22:01,780
Anyway, what's
happening is this edge

402
00:22:01,780 --> 00:22:06,080
is shrinking to zero while this
one is offsetting down this way

403
00:22:06,080 --> 00:22:07,680
and this is offsetting
down this way.

404
00:22:07,680 --> 00:22:09,820
So the new turtle ends
up being like that.

405
00:22:12,170 --> 00:22:12,670
And so on.

406
00:22:16,780 --> 00:22:18,860
You're shrinking every face.

407
00:22:18,860 --> 00:22:21,360
So in general, you have a
whole bunch of polygons.

408
00:22:21,360 --> 00:22:24,840
Or in general we're
allowing crazy things

409
00:22:24,840 --> 00:22:29,090
like this as this pattern
of cuts I want to make.

410
00:22:29,090 --> 00:22:31,970
Maybe you want to cut your
square into five pieces.

411
00:22:31,970 --> 00:22:35,320
I'm going to shrink each of
them separately, or in parallel.

412
00:22:35,320 --> 00:22:37,514
It doesn't actually matter.

413
00:22:37,514 --> 00:22:38,930
So in this case
there's just going

414
00:22:38,930 --> 00:22:41,410
to be five angular bisectors.

415
00:22:41,410 --> 00:22:44,090
In general, there are several
regions you shrink all of them.

416
00:22:44,090 --> 00:22:45,890
A lot of the time we
think about polygons,

417
00:22:45,890 --> 00:22:47,860
and then there's two
regions to shrink.

418
00:22:47,860 --> 00:22:50,422
And it looks like you're
expanding the turtle to go out,

419
00:22:50,422 --> 00:22:52,380
but really you're shrinking
the outside region.

420
00:22:52,380 --> 00:22:54,324
It just happens to be--
there's one infinite

421
00:22:54,324 --> 00:22:55,490
region that one looks weird.

422
00:22:59,250 --> 00:23:02,130
A couple other special
cases, because I

423
00:23:02,130 --> 00:23:05,380
want to do any graph
and not just polygons.

424
00:23:05,380 --> 00:23:08,990
You could have something
that just terminates.

425
00:23:08,990 --> 00:23:12,550
So a degree one vertex only
has one [? edge ?] into it.

426
00:23:12,550 --> 00:23:13,940
In this case, it's
not quite well

427
00:23:13,940 --> 00:23:16,430
defined what to do
because you offset this

428
00:23:16,430 --> 00:23:19,340
and you offset this,
but what happens here?

429
00:23:19,340 --> 00:23:20,970
And there it's
sort of arbitrary.

430
00:23:20,970 --> 00:23:23,136
You can do whatever you
want, but the simplest thing

431
00:23:23,136 --> 00:23:25,920
you can do is to make
a-- imagine that there's

432
00:23:25,920 --> 00:23:28,740
a little vertical segment here
that happens to be length zero,

433
00:23:28,740 --> 00:23:34,680
and it expands into the
edge of a rectangle.

434
00:23:34,680 --> 00:23:40,910
So you end up with these
245 degree angular bisectors

435
00:23:40,910 --> 00:23:44,760
between this vertical edge
and the horizontal one.

436
00:23:44,760 --> 00:23:47,000
But you have some
flexibility there.

437
00:23:47,000 --> 00:23:48,780
You can design it how you want.

438
00:23:48,780 --> 00:23:50,940
The other case you can
have is a degree 0 vertex.

439
00:23:50,940 --> 00:23:53,812
There are no edges here.

440
00:23:53,812 --> 00:23:55,770
This is a little funny
in the way I defined it.

441
00:23:55,770 --> 00:23:58,030
I just said I wanted
to align line segments.

442
00:23:58,030 --> 00:24:01,050
You could also align points
if you really feel like it,

443
00:24:01,050 --> 00:24:03,140
and that would be
represented by a dot that

444
00:24:03,140 --> 00:24:04,630
has no cuts next to it.

445
00:24:04,630 --> 00:24:07,560
If you want to cut
out just this point--

446
00:24:07,560 --> 00:24:08,880
I need to make it something.

447
00:24:08,880 --> 00:24:11,260
You could think of it as a
tiny triangle for consistency

448
00:24:11,260 --> 00:24:12,230
with this picture.

449
00:24:12,230 --> 00:24:13,980
We think of it as
a little square.

450
00:24:13,980 --> 00:24:16,990
And so, when you
expand it, or when

451
00:24:16,990 --> 00:24:20,870
you shrink the outside region,
you get four 45 degrees folds.

452
00:24:20,870 --> 00:24:25,590
This is actually how
[? Eichholtz ?] et al defined

453
00:24:25,590 --> 00:24:30,400
it back in '96, and
it's a fine definition.

454
00:24:30,400 --> 00:24:33,780
But you have flexibility
here in your design process.

455
00:24:33,780 --> 00:24:35,494
They'll all work.

456
00:24:35,494 --> 00:24:37,910
And this would let you take a
whole bunch of points, align

457
00:24:37,910 --> 00:24:40,871
them onto a common line, and
nothing else is on that line.

458
00:24:40,871 --> 00:24:42,870
Because these folds are
going to push everything

459
00:24:42,870 --> 00:24:47,910
that's surrounding the
point away from the line.

460
00:24:47,910 --> 00:24:49,270
All right.

461
00:24:49,270 --> 00:24:50,820
Some fun facts.

462
00:24:50,820 --> 00:24:52,410
Straight skeleton
is nice and small.

463
00:24:52,410 --> 00:24:56,740
If you have n original
points and line segments

464
00:24:56,740 --> 00:24:59,770
in your desired cut pattern,
the straight skeleton

465
00:24:59,770 --> 00:25:03,700
has a linear n number of
line segments-- linear number

466
00:25:03,700 --> 00:25:04,570
of creases.

467
00:25:04,570 --> 00:25:05,200
So order n.

468
00:25:11,980 --> 00:25:15,100
Other fun facts-- there is
a one-to-one correspondence

469
00:25:15,100 --> 00:25:20,100
between the edges you
want to cut along,

470
00:25:20,100 --> 00:25:23,380
like let me pick
one over here maybe.

471
00:25:23,380 --> 00:25:25,880
Like this-- this is an
edge I want to cut along,

472
00:25:25,880 --> 00:25:28,180
and regions of the
straight skeleton.

473
00:25:28,180 --> 00:25:29,200
So here's a region.

474
00:25:29,200 --> 00:25:31,300
A face of the straight skeleton.

475
00:25:31,300 --> 00:25:31,930
This guy.

476
00:25:31,930 --> 00:25:34,870
There's exactly one cut
edge inside of that.

477
00:25:34,870 --> 00:25:36,060
That's always the case.

478
00:25:36,060 --> 00:25:37,477
You look everywhere here.

479
00:25:37,477 --> 00:25:39,310
Every region of the
straight skeleton-- it's

480
00:25:39,310 --> 00:25:41,480
more obvious if I color
them different colors.

481
00:25:41,480 --> 00:25:43,720
There's one cut edge inside.

482
00:25:43,720 --> 00:25:48,590
And all of those guys that
surround that cut edge

483
00:25:48,590 --> 00:25:52,052
bisect that edge
and another one.

484
00:25:52,052 --> 00:25:54,010
And the other one is the
one on the other side.

485
00:25:54,010 --> 00:25:56,343
In general, you take one of
the straight skeleton edges.

486
00:25:56,343 --> 00:25:57,209
There are two sides.

487
00:25:57,209 --> 00:25:59,000
There's two faces of
the straight skeleton.

488
00:25:59,000 --> 00:26:00,820
This one's crazy and non convex.

489
00:26:00,820 --> 00:26:03,980
This one's just a little
infinite triangle down here,

490
00:26:03,980 --> 00:26:06,442
and that edge bisects
those two cut edges.

491
00:26:06,442 --> 00:26:08,400
So it's very easy to walk
around the structure.

492
00:26:08,400 --> 00:26:10,410
See what it bisects.

493
00:26:10,410 --> 00:26:11,950
lots of things get bisected.

494
00:26:11,950 --> 00:26:16,360
But, it's not flat
foldable, so we're not done.

495
00:26:16,360 --> 00:26:20,660
And that's where we
need the perpendiculars.

496
00:26:20,660 --> 00:26:21,160
So--

497
00:26:39,580 --> 00:26:41,550
I'll write down the
definition, and maybe

498
00:26:41,550 --> 00:26:43,990
show the picture
we're going for.

499
00:27:43,996 --> 00:27:45,370
There's a lot of
structures here.

500
00:27:45,370 --> 00:27:46,980
There is what I
call the cut graph--

501
00:27:46,980 --> 00:27:48,347
the things we're trying to make.

502
00:27:48,347 --> 00:27:49,805
Then there's the
straight skeleton.

503
00:27:49,805 --> 00:27:52,289
You should think of it as
a graph drawn on a paper.

504
00:27:52,289 --> 00:27:53,830
It has vertices of
straight skeleton,

505
00:27:53,830 --> 00:27:55,612
which is called
skeleton vertices.

506
00:27:55,612 --> 00:27:57,070
Regions of straight
skeleton, which

507
00:27:57,070 --> 00:27:58,324
is called skeleton regions.

508
00:27:58,324 --> 00:27:59,990
Edges of the cut
graph, which are called

509
00:27:59,990 --> 00:28:01,399
cut edges, and so on.

510
00:28:01,399 --> 00:28:02,940
We're going to add
a new graph, which

511
00:28:02,940 --> 00:28:04,940
is the perpendicular graph.

512
00:28:04,940 --> 00:28:07,650
Which you can think
of as hinges from tree

513
00:28:07,650 --> 00:28:10,380
method of origami design.

514
00:28:10,380 --> 00:28:13,460
So what is this--
what does it mean?

515
00:28:13,460 --> 00:28:15,540
We started a straight
skeleton vertex.

516
00:28:15,540 --> 00:28:18,550
Usually there are three
skeleton edges coming together.

517
00:28:18,550 --> 00:28:21,060
Vertex-- sometimes they're
more like this guy.

518
00:28:21,060 --> 00:28:22,100
Four.

519
00:28:22,100 --> 00:28:25,530
And if there's three
edges coming together,

520
00:28:25,530 --> 00:28:27,090
there are three
skeleton regions.

521
00:28:27,090 --> 00:28:29,500
For each one-- each
of those regions--

522
00:28:29,500 --> 00:28:31,720
has one cut edge
in it, so we try

523
00:28:31,720 --> 00:28:36,360
to walk perpendicular
and toward that cut edge.

524
00:28:36,360 --> 00:28:37,690
So here I walk perpendicular.

525
00:28:37,690 --> 00:28:39,130
I meet at right angles here.

526
00:28:39,130 --> 00:28:41,400
I just go off to infinity.

527
00:28:41,400 --> 00:28:46,330
Here I walk perpendicular to
this cut edge, and that's cool,

528
00:28:46,330 --> 00:28:49,990
but then I leave
the skeleton region.

529
00:28:49,990 --> 00:28:52,230
At that point I enter
a new skeleton region,

530
00:28:52,230 --> 00:28:55,110
which is this one,
this non-convex thing.

531
00:28:55,110 --> 00:28:58,080
It contains one et edge, and
when I-- where was I here?

532
00:28:58,080 --> 00:28:59,000
I entered.

533
00:28:59,000 --> 00:29:01,810
Now I want to move
perpendicular to that cut edge,

534
00:29:01,810 --> 00:29:04,699
so that when I cross it, I
cross it perpendicularly.

535
00:29:04,699 --> 00:29:05,740
Now I enter a new region.

536
00:29:05,740 --> 00:29:07,940
It contains this cut edge.

537
00:29:07,940 --> 00:29:12,410
I move perpendicular to it,
and I don't actually cross it,

538
00:29:12,410 --> 00:29:13,635
but I enter a new region.

539
00:29:13,635 --> 00:29:15,260
Now I'm in the region
of this cut edge,

540
00:29:15,260 --> 00:29:16,880
so I move perpendicular to that.

541
00:29:16,880 --> 00:29:19,100
And wow, I hit another
skeleton vertex.

542
00:29:19,100 --> 00:29:20,550
I stop.

543
00:29:20,550 --> 00:29:22,880
OK this example is because
it's on a triangular grid,

544
00:29:22,880 --> 00:29:24,670
there's lots of
degeneracies like that.

545
00:29:24,670 --> 00:29:26,720
Usually you'd eventually
go off to infinity,

546
00:29:26,720 --> 00:29:29,030
or come around to meet yourself.

547
00:29:29,030 --> 00:29:31,440
Here I happen to hit
a different vertex.

548
00:29:31,440 --> 00:29:34,250
You do that all the vertices.

549
00:29:34,250 --> 00:29:35,540
All the skeleton vertices.

550
00:29:35,540 --> 00:29:39,750
Now there's some weird
ones like this one.

551
00:29:39,750 --> 00:29:43,420
Notice there are no purple
lines coming out from here,

552
00:29:43,420 --> 00:29:44,990
and that's because
every region you

553
00:29:44,990 --> 00:29:47,910
try to enter you
immediately leave.

554
00:29:47,910 --> 00:29:50,130
So if I tried-- there's
four regions here.

555
00:29:50,130 --> 00:29:51,220
Try each one of them.

556
00:29:51,220 --> 00:29:53,800
Like this region
has this cut edge.

557
00:29:53,800 --> 00:29:55,400
If I try to go
perpendicular to it,

558
00:29:55,400 --> 00:29:58,260
I'd enter a different region,
so I can actually go at all.

559
00:29:58,260 --> 00:30:00,970
Like I move and then
I instantly stop.

560
00:30:00,970 --> 00:30:05,130
So you could think of there
being like a zero radius

561
00:30:05,130 --> 00:30:05,874
thing there.

562
00:30:05,874 --> 00:30:07,290
That's sort of the
degenerate case

563
00:30:07,290 --> 00:30:10,400
of a river being a disc
Is it being a circle.

564
00:30:10,400 --> 00:30:11,840
Same thing going on.

565
00:30:11,840 --> 00:30:13,320
All of-- in general
when you have

566
00:30:13,320 --> 00:30:15,554
reflex vertices and their
regular bisectors meeting,

567
00:30:15,554 --> 00:30:17,220
you're going to lose
some perpendiculars

568
00:30:17,220 --> 00:30:20,000
because you can't enter them.

569
00:30:20,000 --> 00:30:22,884
Here's another one where I
just have one perpendicular

570
00:30:22,884 --> 00:30:23,550
edge coming out.

571
00:30:23,550 --> 00:30:26,375
This one I can
reach, but if I tried

572
00:30:26,375 --> 00:30:28,000
to be perpendicular
to either of these,

573
00:30:28,000 --> 00:30:31,320
I enter the wrong region.

574
00:30:31,320 --> 00:30:33,840
That's the perpendicular
folds, and that's pretty much

575
00:30:33,840 --> 00:30:34,660
the crease pattern.

576
00:30:34,660 --> 00:30:36,285
There are technically
a few other folds

577
00:30:36,285 --> 00:30:38,160
you have to deal
with, but that is--

578
00:30:38,160 --> 00:30:40,550
if you want to make
something right now,

579
00:30:40,550 --> 00:30:44,750
just apply those two algorithms
and you get your shape.

580
00:30:44,750 --> 00:30:46,260
Just fold along
that crease pattern.

581
00:30:46,260 --> 00:30:52,410
It will be flat
foldable almost always.

582
00:30:52,410 --> 00:30:54,430
Why is it flat foldable?

583
00:30:54,430 --> 00:30:57,810
So one thing we can check is
local flat foldability at least

584
00:30:57,810 --> 00:31:02,860
satisfies Kawasaki's condition
because at a typical vertex

585
00:31:02,860 --> 00:31:07,180
you're going to have three
skeleton edges coming together.

586
00:31:07,180 --> 00:31:08,720
And so there are
three faces here.

587
00:31:08,720 --> 00:31:14,610
Each of them has a cut edge
somewhere-- probably draw

588
00:31:14,610 --> 00:31:16,990
this reasonably well-- and
should have the property

589
00:31:16,990 --> 00:31:22,250
that when I extend these--
didn't draw that so well.

590
00:31:22,250 --> 00:31:26,600
And I extend these-- these
are angular bisectors.

591
00:31:26,600 --> 00:31:31,260
We know the skeleton edges will
angularly bisect two cut edges.

592
00:31:31,260 --> 00:31:34,110
The two cut edges that
are defined by these guys.

593
00:31:34,110 --> 00:31:39,390
So I should get an angular
bisector here, and those meet.

594
00:31:39,390 --> 00:31:42,412
An angular bisector here.

595
00:31:42,412 --> 00:31:44,120
And then I also have
perpendicular folds.

596
00:31:44,120 --> 00:31:48,590
So they may not actually meet
this guy, but if they did,

597
00:31:48,590 --> 00:31:51,311
they certainly
meet the extension.

598
00:31:51,311 --> 00:31:53,790
Hey, that's our good
friend, the rabbit ear.

599
00:31:53,790 --> 00:31:55,630
Just regular triangle fold.

600
00:31:55,630 --> 00:32:03,510
And in particular, you can see
that these angles are equal.

601
00:32:03,510 --> 00:32:06,880
I call this three prime.

602
00:32:06,880 --> 00:32:10,971
It's like 180 minus
that, I think.

603
00:32:10,971 --> 00:32:12,930
If I'm not mistaken.

604
00:32:12,930 --> 00:32:15,332
And this is one prime.

605
00:32:15,332 --> 00:32:18,000
Not the best notation.

606
00:32:18,000 --> 00:32:21,480
And these are two
prime and two prime.

607
00:32:21,480 --> 00:32:24,044
And so I've got these
nice angle pairings.

608
00:32:24,044 --> 00:32:26,460
That means if I add the odd
angles, I get the even angles.

609
00:32:26,460 --> 00:32:27,120
Same thing.

610
00:32:27,120 --> 00:32:30,280
So I definitely have
Kawasaki's theorem everywhere.

611
00:32:30,280 --> 00:32:31,780
You could check--
it works even when

612
00:32:31,780 --> 00:32:35,280
you have these degenerate
situations where

613
00:32:35,280 --> 00:32:38,207
more than three skeleton
edges come together.

614
00:32:38,207 --> 00:32:39,040
For the same reason.

615
00:32:39,040 --> 00:32:40,052
You still get pairing.

616
00:32:40,052 --> 00:32:41,260
Just more than three of them.

617
00:32:43,770 --> 00:32:45,740
All right.

618
00:32:45,740 --> 00:32:48,000
But some exciting
things can happen.

619
00:32:48,000 --> 00:32:50,660
So I'm going to look
at proving foldability,

620
00:32:50,660 --> 00:32:52,710
but one exciting
thing that can happen

621
00:32:52,710 --> 00:32:56,570
is you get a lot of
perpendicular folds at a very

622
00:32:56,570 --> 00:32:59,840
few original cut lines.

623
00:32:59,840 --> 00:33:02,070
So here I'm trying to make
this weird pinwheel shape.

624
00:33:02,070 --> 00:33:04,910
I want to cut out the bold
lines of the cut line.

625
00:33:04,910 --> 00:33:06,860
So I want to cut
out this square,

626
00:33:06,860 --> 00:33:09,934
and then these four squares
arranged in a pinwheel pattern

627
00:33:09,934 --> 00:33:10,600
around that one.

628
00:33:10,600 --> 00:33:11,610
Why you'd want to do that?

629
00:33:11,610 --> 00:33:13,568
I don't know, but we're
mathematicians so we're

630
00:33:13,568 --> 00:33:15,710
going to consider all the cases.

631
00:33:15,710 --> 00:33:18,980
So the straight skeleton
is the thin black lines,

632
00:33:18,980 --> 00:33:20,000
and that's linear size.

633
00:33:20,000 --> 00:33:23,370
That's nice, but
the perpendiculars--

634
00:33:23,370 --> 00:33:25,150
if this piece of
paper's infinite,

635
00:33:25,150 --> 00:33:27,371
the number of
perpendiculars is infinite.

636
00:33:27,371 --> 00:33:28,870
If you have a finite
piece of paper,

637
00:33:28,870 --> 00:33:31,470
which is what you
usually buy in the store,

638
00:33:31,470 --> 00:33:33,590
then it's a finite
number of creases.

639
00:33:33,590 --> 00:33:36,090
So in any finite region, this
is a finite number of creases,

640
00:33:36,090 --> 00:33:38,250
but it's a lot of them.

641
00:33:38,250 --> 00:33:39,280
So that's one sad thing.

642
00:33:39,280 --> 00:33:40,990
You can't bound the
number of creases

643
00:33:40,990 --> 00:33:43,650
as a function of the
number of cut lines.

644
00:33:43,650 --> 00:33:45,280
But I think that's
actually necessary.

645
00:33:45,280 --> 00:33:47,460
I don't think it's possible
to solve this problem

646
00:33:47,460 --> 00:33:48,960
while bounding the
number of creases

647
00:33:48,960 --> 00:33:50,640
in terms of the
number of cut lines.

648
00:33:50,640 --> 00:33:53,650
That's one of the open
problems getting down

649
00:33:53,650 --> 00:33:56,910
on one of these pictures.

650
00:33:56,910 --> 00:34:00,495
One of these slide--
lecture notes.

651
00:34:00,495 --> 00:34:02,952
There's something else
even more annoying, though.

652
00:34:02,952 --> 00:34:06,300
It happens even in a
finite piece of paper,

653
00:34:06,300 --> 00:34:09,679
and it's even more obscure
why you'd want to make this.

654
00:34:09,679 --> 00:34:13,600
But the bold blue
lines are the cuts,

655
00:34:13,600 --> 00:34:16,820
and then the thin black lines
are the straight skeleton.

656
00:34:16,820 --> 00:34:18,320
You could tell this
spans many years

657
00:34:18,320 --> 00:34:21,190
because I keep changing
notational style,

658
00:34:21,190 --> 00:34:24,080
and this is from the textbook.

659
00:34:24,080 --> 00:34:26,730
And then you get these
perpendicular folds.

660
00:34:26,730 --> 00:34:28,980
So I haven't drawn all
of them but these dash

661
00:34:28,980 --> 00:34:31,409
lines, the light blue.

662
00:34:31,409 --> 00:34:36,350
And this example is set up so
that-- let me get this right.

663
00:34:36,350 --> 00:34:41,870
This width is an
irrational multiple

664
00:34:41,870 --> 00:34:44,590
of this width or this width.

665
00:34:44,590 --> 00:34:46,270
One of those.

666
00:34:46,270 --> 00:34:50,630
Things are irrational, so
they're not very nice numbers.

667
00:34:50,630 --> 00:34:53,130
And what I need to do
to finish this picture

668
00:34:53,130 --> 00:34:55,305
is these guys go-- they
enter a new skeleton region.

669
00:34:55,305 --> 00:34:57,180
They're actually going
to keep going straight

670
00:34:57,180 --> 00:34:59,138
because there's two cut
lines that are parallel

671
00:34:59,138 --> 00:34:59,810
to each other.

672
00:34:59,810 --> 00:35:02,226
It's going to go up there, and
it's going to cycle around.

673
00:35:02,226 --> 00:35:04,310
Let me do one round.

674
00:35:04,310 --> 00:35:05,720
Who did I move?

675
00:35:05,720 --> 00:35:07,210
This guy.

676
00:35:07,210 --> 00:35:09,020
So this guy I just
extended down.

677
00:35:09,020 --> 00:35:12,070
He's going to turn around,
make 180 degree turn,

678
00:35:12,070 --> 00:35:13,860
and you can check each of these.

679
00:35:13,860 --> 00:35:16,740
Your setups do 180 degree
turn around this axis,

680
00:35:16,740 --> 00:35:20,270
around this axis on the bottom,
and on the top around this axis

681
00:35:20,270 --> 00:35:21,380
and around this axis.

682
00:35:21,380 --> 00:35:23,550
Depending wherever you
enter, it's like a racetrack.

683
00:35:23,550 --> 00:35:25,190
Keep going around and around.

684
00:35:25,190 --> 00:35:28,460
And if you follow that one
guy a little bit farther,

685
00:35:28,460 --> 00:35:29,680
it looks like that.

686
00:35:29,680 --> 00:35:32,460
And a little bit farther,
and it looks like that.

687
00:35:32,460 --> 00:35:34,720
It never finishes.

688
00:35:34,720 --> 00:35:38,180
So in fact you completely
filled this region with creases.

689
00:35:38,180 --> 00:35:40,289
It's like a dense
region of creases.

690
00:35:40,289 --> 00:35:41,705
Now this would be
a bitch to fold.

691
00:35:44,910 --> 00:35:46,610
I don't recommend you try it.

692
00:35:46,610 --> 00:35:49,900
And this is really a situation
where this algorithm fails.

693
00:35:49,900 --> 00:35:53,520
The good news is if I move
any of these vertices--

694
00:35:53,520 --> 00:35:57,250
the cut vertices the tiniest
amount-- this will disappear.

695
00:35:57,250 --> 00:36:00,520
I really had to be very careful
and get lots of degeneracy

696
00:36:00,520 --> 00:36:01,610
for this to happen.

697
00:36:04,272 --> 00:36:05,980
We don't actually know
how to prove that.

698
00:36:05,980 --> 00:36:10,830
It's conjecture that if you take
any cuts-- any graph of cuts

699
00:36:10,830 --> 00:36:13,530
you want to make, and
you perturb the vertices

700
00:36:13,530 --> 00:36:17,320
in a tiny epsilon disc then
the resulting thing will not

701
00:36:17,320 --> 00:36:19,070
have this density behavior.

702
00:36:19,070 --> 00:36:22,690
I'm totally sure it's true,
but we don't have the proof.

703
00:36:22,690 --> 00:36:23,960
So that's life.

704
00:36:23,960 --> 00:36:28,420
This is why I said skeleton
method works almost always.

705
00:36:28,420 --> 00:36:31,170
There are these
annoying situations

706
00:36:31,170 --> 00:36:34,090
where it doesn't really
give your crease pattern.

707
00:36:34,090 --> 00:36:36,130
So if you feel like
unless you somehow

708
00:36:36,130 --> 00:36:39,172
think this is legitimate to
make infinitely many creases.

709
00:36:39,172 --> 00:36:39,880
I don't think so.

710
00:36:44,120 --> 00:36:48,780
All right, let me tell
you a few more things

711
00:36:48,780 --> 00:36:51,270
to make this practical for you.

712
00:36:51,270 --> 00:36:52,710
You want to-- what
you really want

713
00:36:52,710 --> 00:36:54,110
is a mountain valley assignment.

714
00:36:54,110 --> 00:36:56,860
Before I showed you lots of
perpendicular and skeleton

715
00:36:56,860 --> 00:37:00,820
edges, and basically
the way it works

716
00:37:00,820 --> 00:37:06,380
is if you look at any
skeleton edge like this one,

717
00:37:06,380 --> 00:37:09,350
it's bisecting in this
case a convex angle.

718
00:37:09,350 --> 00:37:11,050
So I make it a mountain.

719
00:37:11,050 --> 00:37:13,760
Here, red is mountain,
blue is valley.

720
00:37:13,760 --> 00:37:15,680
Dot dashes mountain,
dash is valley.

721
00:37:15,680 --> 00:37:17,030
That is the standard.

722
00:37:17,030 --> 00:37:19,590
Whenever I'm bisecting
a reflex angle,

723
00:37:19,590 --> 00:37:24,350
then I make the
skeleton edge a valley.

724
00:37:24,350 --> 00:37:27,310
And that's basically true.

725
00:37:27,310 --> 00:37:29,780
Convex angles, mountains,
reflex angles, valleys.

726
00:37:29,780 --> 00:37:31,410
That's for the straight
skeleton edges.

727
00:37:31,410 --> 00:37:31,910
Yeah.

728
00:37:37,740 --> 00:37:42,060
Like this fold, or this one?

729
00:37:42,060 --> 00:37:43,179
Oh, this guy.

730
00:37:43,179 --> 00:37:44,970
All right, this is a
bit of a special case.

731
00:37:44,970 --> 00:37:48,350
Here I'm really bisecting
an angle of zero.

732
00:37:48,350 --> 00:37:50,350
To extend these guys out,
they need an infinite,

733
00:37:50,350 --> 00:37:52,516
and they form a zero angle
because they're parallel.

734
00:37:52,516 --> 00:37:56,980
And I call zero a convex angle,
but I just defined it that way,

735
00:37:56,980 --> 00:37:58,850
and so this is a mountain.

736
00:37:58,850 --> 00:38:05,170
Whereas this guy bisecting
is barely convex.

737
00:38:05,170 --> 00:38:08,140
Is that really a mountain?

738
00:38:08,140 --> 00:38:09,790
No, that's a typo.

739
00:38:09,790 --> 00:38:10,460
Good.

740
00:38:10,460 --> 00:38:12,390
This one should be a valley.

741
00:38:12,390 --> 00:38:14,940
Pretty sure.

742
00:38:14,940 --> 00:38:20,610
Yeah, should be a valley-- wow,
this is a weird crease pattern.

743
00:38:20,610 --> 00:38:22,250
That's not a straight
skeleton there.

744
00:38:22,250 --> 00:38:25,350
Never mind this picture-- I
knew there was always a bug.

745
00:38:25,350 --> 00:38:29,133
I think there's a typo
in the book as we say.

746
00:38:35,299 --> 00:38:37,340
How did I make the initial
pattern of the turtle?

747
00:38:37,340 --> 00:38:40,140
I just drew something
that looked like a turtle.

748
00:38:40,140 --> 00:38:41,994
Anything.

749
00:38:41,994 --> 00:38:43,410
I happened to draw
this on a grid,

750
00:38:43,410 --> 00:38:44,701
but there's no reason I had to.

751
00:39:00,840 --> 00:39:01,920
I designed them.

752
00:39:01,920 --> 00:39:06,650
So in this example, I designed
the ratios to be really nasty.

753
00:39:06,650 --> 00:39:09,080
Like root 2 over 10
ratio or something.

754
00:39:12,614 --> 00:39:15,030
The whole thing will-- if I
perturb these vertices at all,

755
00:39:15,030 --> 00:39:16,410
the whole thing will fall apart.

756
00:39:16,410 --> 00:39:18,050
I won't get these
180 degree turns.

757
00:39:18,050 --> 00:39:20,330
Things will end up
going off to infinity.

758
00:39:20,330 --> 00:39:23,490
The hard part here is
actually-- rational ratios

759
00:39:23,490 --> 00:39:24,710
are quite common.

760
00:39:24,710 --> 00:39:26,940
What's uncommon
in this picture is

761
00:39:26,940 --> 00:39:29,980
that this thing is closed
up, and you never escape.

762
00:39:29,980 --> 00:39:32,380
Almost always, there'll
be a little gap

763
00:39:32,380 --> 00:39:34,120
and you'll eventually
reach the gap

764
00:39:34,120 --> 00:39:35,470
and then go off to infinity.

765
00:39:35,470 --> 00:39:37,660
So that's what happens
in a typical case.

766
00:39:42,569 --> 00:39:45,110
So if you drew a picture on the
grid this would never happen.

767
00:39:45,110 --> 00:39:46,570
That you can prove.

768
00:39:46,570 --> 00:39:48,670
Yeah.

769
00:39:48,670 --> 00:39:50,364
Square grid.

770
00:39:50,364 --> 00:39:51,780
Probably also the
triangular grid.

771
00:39:51,780 --> 00:39:53,571
You need to be a little
careful because you

772
00:39:53,571 --> 00:39:55,850
want all these constructions
to stay on the grid,

773
00:39:55,850 --> 00:39:58,310
but I think something
like that is true.

774
00:39:58,310 --> 00:40:04,779
OK, let's move on to how we
construct a folded state.

775
00:40:04,779 --> 00:40:07,070
When this algorithm works,
when it gives a valid crease

776
00:40:07,070 --> 00:40:10,560
pattern, you know it's
locally flat foldable

777
00:40:10,560 --> 00:40:12,200
because it satisfies
Kawasaki, but how

778
00:40:12,200 --> 00:40:14,230
do we actually know that
it globally folds flat?

779
00:40:14,230 --> 00:40:15,771
To do that, you have
to describe what

780
00:40:15,771 --> 00:40:17,680
it looks like after it's folded.

781
00:40:17,680 --> 00:40:23,140
And the idea here is to look at
what we call corridors but are

782
00:40:23,140 --> 00:40:27,030
essentially discrete versions
of rivers from tree theory.

783
00:40:27,030 --> 00:40:30,100
So you have these
constant width strips

784
00:40:30,100 --> 00:40:33,740
that turn at skeleton
edges, and they

785
00:40:33,740 --> 00:40:36,030
could go off to
infinity on both sides.

786
00:40:36,030 --> 00:40:38,850
In general they could loop
around and meet themselves

787
00:40:38,850 --> 00:40:44,410
again, but in this case they
actually all go to infinity.

788
00:40:44,410 --> 00:40:46,780
And if you look at
one of those strips--

789
00:40:46,780 --> 00:40:51,260
you could actually just cut
this out of your textbook.

790
00:40:51,260 --> 00:40:55,290
Just slice it up and look
at how that's folding.

791
00:40:55,290 --> 00:40:59,112
Well, it meets a skeleton
edge, and then maybe it

792
00:40:59,112 --> 00:40:59,820
meets a cut edge.

793
00:40:59,820 --> 00:41:01,120
Usually you don't fold those.

794
00:41:01,120 --> 00:41:02,619
Then it meets another
skeleton edge.

795
00:41:02,619 --> 00:41:04,500
It just meets edges
one at a time.

796
00:41:04,500 --> 00:41:06,920
It's never
complicated because we

797
00:41:06,920 --> 00:41:08,820
divided along all these
perpendicular folds.

798
00:41:08,820 --> 00:41:12,710
You really only meet
one edge at a time.

799
00:41:12,710 --> 00:41:14,960
Which is good.

800
00:41:14,960 --> 00:41:19,290
In fact, if you look at
one of these skeleton edges

801
00:41:19,290 --> 00:41:22,160
it's creased along,
normally you think

802
00:41:22,160 --> 00:41:26,560
of that as an angular bisector
of these two cut edges,

803
00:41:26,560 --> 00:41:29,020
but you can also think of
it as an angular bisector

804
00:41:29,020 --> 00:41:31,510
of these two
perpendicular folds.

805
00:41:31,510 --> 00:41:34,650
Because if you bisect
these two things,

806
00:41:34,650 --> 00:41:37,750
you also bisect two things
that are perpendicular to them.

807
00:41:37,750 --> 00:41:40,830
It's like two wrongs
make a right somehow.

808
00:41:40,830 --> 00:41:44,130
So this guy bisects
those two creases.

809
00:41:44,130 --> 00:41:46,000
So if you fold along
here-- actually

810
00:41:46,000 --> 00:41:47,915
you align these creases.

811
00:41:47,915 --> 00:41:50,040
It's a duality-- you are
aligning the perpendicular

812
00:41:50,040 --> 00:41:50,840
folds.

813
00:41:50,840 --> 00:41:53,580
That fold along here, you line
up this fold with that one,

814
00:41:53,580 --> 00:41:54,750
this fold with that one.

815
00:41:54,750 --> 00:41:57,290
I fold here, I lined up
this fold with that one.

816
00:41:57,290 --> 00:42:00,797
There's like a zero
length fold that's there.

817
00:42:00,797 --> 00:42:02,130
You fold along all these things.

818
00:42:02,130 --> 00:42:05,210
You line up this with
itself, and so on.

819
00:42:05,210 --> 00:42:06,540
So you follow along this thing.

820
00:42:06,540 --> 00:42:13,260
This corridor folds down
to basically a rectangle.

821
00:42:13,260 --> 00:42:15,480
It's got some rough edges
in the top and the bottom,

822
00:42:15,480 --> 00:42:20,960
but it lies in the strip in 3D,
and I have a picture of that.

823
00:42:20,960 --> 00:42:26,670
So I took the one over
here-- this blue corridor--

824
00:42:26,670 --> 00:42:29,790
and if you fold it up,
it looks like this.

825
00:42:29,790 --> 00:42:32,090
OK, now in particular, you
can check at this point.

826
00:42:32,090 --> 00:42:33,480
It's pretty easy
to check because

827
00:42:33,480 --> 00:42:37,340
of all this bisectorness--
Bisector goodness--

828
00:42:37,340 --> 00:42:40,760
that you bring into
alignment all the cut edges.

829
00:42:40,760 --> 00:42:44,100
So for example, this guy,
because it bisects that cut

830
00:42:44,100 --> 00:42:46,640
edge and that cut edge, it
brings them into alignment.

831
00:42:46,640 --> 00:42:48,598
And you can see that
somewhere in this picture.

832
00:42:48,598 --> 00:42:51,200
I think it's these two guys.

833
00:42:51,200 --> 00:42:53,190
It can be a little
more complicated.

834
00:42:53,190 --> 00:42:56,340
Like over here I
have a cut edge,

835
00:42:56,340 --> 00:42:58,920
then I have a bisector, and
then a bisector, and then

836
00:42:58,920 --> 00:43:01,230
a bisector, and then
another cut edge.

837
00:43:01,230 --> 00:43:04,920
But if you think about it right,
it's-- I don't happen to meet

838
00:43:04,920 --> 00:43:08,042
these cut edges, but I'm
effectively bringing this

839
00:43:08,042 --> 00:43:10,500
into alignment with this, and
then this into alignment with

840
00:43:10,500 --> 00:43:12,374
this, and then this into
alignment with that.

841
00:43:12,374 --> 00:43:15,270
So in the end, I
line this with that,

842
00:43:15,270 --> 00:43:20,910
and that's what's happening
up here on the left.

843
00:43:20,910 --> 00:43:22,830
Where I don't quite
come all the way down,

844
00:43:22,830 --> 00:43:24,610
but I still end up
lining everything up.

845
00:43:24,610 --> 00:43:26,890
So this is the solution to
the fold-and-cut problem

846
00:43:26,890 --> 00:43:29,662
as long as it actually faults.

847
00:43:29,662 --> 00:43:31,370
And to show that
actually folds, you just

848
00:43:31,370 --> 00:43:35,556
need to show that these
corridors-- I forget,

849
00:43:35,556 --> 00:43:36,930
I think we call
these accordions.

850
00:43:36,930 --> 00:43:38,000
It's been a long time.

851
00:43:38,000 --> 00:43:42,230
Since '98 I wrote this paper

852
00:43:42,230 --> 00:43:44,060
You take these
accordions and you just

853
00:43:44,060 --> 00:43:46,250
want to see how
they fit together.

854
00:43:46,250 --> 00:43:48,170
And low and behold,
they fit together

855
00:43:48,170 --> 00:43:51,620
in a tree in this picture.

856
00:43:51,620 --> 00:43:54,750
It gets more complicated
in another picture, which

857
00:43:54,750 --> 00:43:57,310
I will show you in a moment.

858
00:43:57,310 --> 00:44:00,980
But in this situation where
every quarter goes off

859
00:44:00,980 --> 00:44:05,610
to infinity on both sides,
you get a nice tree structure.

860
00:44:05,610 --> 00:44:08,870
And as long as you could
fold this tree flat,

861
00:44:08,870 --> 00:44:12,350
then you can fold
this thing flat.

862
00:44:12,350 --> 00:44:15,250
Because each of these
edges of the tree

863
00:44:15,250 --> 00:44:16,960
is this very simple
accordion structure

864
00:44:16,960 --> 00:44:20,502
which trivially falls flat.

865
00:44:20,502 --> 00:44:22,085
The other thing you
have to check here

866
00:44:22,085 --> 00:44:23,835
is you actually alternate
mountain valley.

867
00:44:23,835 --> 00:44:25,460
That's a little more
subtle, especially

868
00:44:25,460 --> 00:44:27,790
when I don't draw the
mountain valley assignment.

869
00:44:27,790 --> 00:44:30,380
But it turns out you always
alternate between mountain

870
00:44:30,380 --> 00:44:32,570
and valley in this
picture, which is great.

871
00:44:32,570 --> 00:44:34,850
That's the thing we
know always folds flats.

872
00:44:34,850 --> 00:44:36,720
It's like a 1D folding.

873
00:44:36,720 --> 00:44:38,250
So these are super easy to fold.

874
00:44:38,250 --> 00:44:40,690
You can fold each of them if
you cut along all the dash

875
00:44:40,690 --> 00:44:42,940
lines You can fold
each separately.

876
00:44:42,940 --> 00:44:44,490
Then you need to
join them together

877
00:44:44,490 --> 00:44:47,760
where the edges here-- just like
in tree theory-- the edges here

878
00:44:47,760 --> 00:44:50,180
correspond to these rivers.

879
00:44:50,180 --> 00:44:52,820
And now you need to
somehow attach them here.

880
00:44:52,820 --> 00:44:57,360
Check that where you attach
them, there's no crossings.

881
00:44:57,360 --> 00:45:00,140
I'm not going to describe,
but it's pretty easy.

882
00:45:00,140 --> 00:45:02,210
Plain area essentially
of that diagram,

883
00:45:02,210 --> 00:45:04,660
and then you need to
fold this tree flat.

884
00:45:04,660 --> 00:45:07,160
Folding a tree flat
is actually kind

885
00:45:07,160 --> 00:45:08,800
of a segue into
next lecture, which

886
00:45:08,800 --> 00:45:11,720
will be about folding linkages.

887
00:45:11,720 --> 00:45:13,300
In this case, it's really easy.

888
00:45:13,300 --> 00:45:16,360
You just pick up some root,
like the letter "A" over there,

889
00:45:16,360 --> 00:45:18,020
and you hang the tree.

890
00:45:18,020 --> 00:45:19,455
Pull up.

891
00:45:19,455 --> 00:45:21,830
Technically this is like a
[? depth ?] [? first ?] search

892
00:45:21,830 --> 00:45:22,920
of the tree.

893
00:45:22,920 --> 00:45:25,380
So you just walk
down-- aways walk down

894
00:45:25,380 --> 00:45:27,260
until you're finished,
then you go back up.

895
00:45:27,260 --> 00:45:28,720
Walk down some more branches.

896
00:45:28,720 --> 00:45:32,900
You end up drawing everything
downward away from A.

897
00:45:32,900 --> 00:45:34,200
And it will be a flat folding.

898
00:45:34,200 --> 00:45:36,190
There won't be any
crossings here.

899
00:45:36,190 --> 00:45:37,962
And then this is a
1D representation

900
00:45:37,962 --> 00:45:39,420
of what's really
going on, which is

901
00:45:39,420 --> 00:45:44,340
that above each of these
edges is really an accordion,

902
00:45:44,340 --> 00:45:46,150
and you need to adjoin
them together there.

903
00:45:46,150 --> 00:45:48,060
So we'll basically do
this modular folding

904
00:45:48,060 --> 00:45:49,770
where you fold each
accordion separately,

905
00:45:49,770 --> 00:45:51,460
put them together
according to this.

906
00:45:51,460 --> 00:45:52,580
Boom, you get your flat folding.

907
00:45:52,580 --> 00:45:54,954
From this picture, you could
read out the mountain valley

908
00:45:54,954 --> 00:45:57,021
assignments before the
perpendicular folds.

909
00:45:57,021 --> 00:45:58,520
This looks like a
valley, this looks

910
00:45:58,520 --> 00:46:01,000
like a valley-- this looks
like a perpendicular fold

911
00:46:01,000 --> 00:46:03,810
I didn't use because
there's no crease there.

912
00:46:03,810 --> 00:46:05,910
That's flat.

913
00:46:05,910 --> 00:46:10,880
Whereas a mountain-- mountain's
probably at the top at A.

914
00:46:10,880 --> 00:46:16,137
What really happens-- if
you want to know-- really

915
00:46:16,137 --> 00:46:18,720
what we're deciding is whether
this starts mountain or valley,

916
00:46:18,720 --> 00:46:21,390
and then it will actually
alternate back and forth

917
00:46:21,390 --> 00:46:23,772
as you move along
the perpendicular.

918
00:46:23,772 --> 00:46:27,760
So that's basically how you
construct a folded state.

919
00:46:27,760 --> 00:46:31,460
In this situation of
so-called linear corridors.

920
00:46:31,460 --> 00:46:35,400
Now there are a bunch of
things I haven't mentioned,

921
00:46:35,400 --> 00:46:37,750
but I think-- I don't
want to talk about them

922
00:46:37,750 --> 00:46:39,990
too much so I get out
to the other topics.

923
00:47:07,590 --> 00:47:12,220
So, what I just talked
about is something called

924
00:47:12,220 --> 00:47:14,410
a linear corridor
case, which is really

925
00:47:14,410 --> 00:47:19,440
where it's most beautiful--
this construction

926
00:47:19,440 --> 00:47:23,445
of a folded state-- and
linear corridor intuitively

927
00:47:23,445 --> 00:47:24,450
is something like this.

928
00:47:24,450 --> 00:47:26,810
It goes off to
infinite on both sides.

929
00:47:26,810 --> 00:47:28,370
Has constant width all the way.

930
00:47:28,370 --> 00:47:29,660
Of course, it's really
a discrete thing.

931
00:47:29,660 --> 00:47:30,451
Not a smooth thing.

932
00:47:34,060 --> 00:47:40,690
Let me say
conjecture-- if you're

933
00:47:40,690 --> 00:47:46,290
cut graph has that
maximum degree

934
00:47:46,290 --> 00:47:51,810
2-- it means at most two
edges at every vertex.

935
00:47:51,810 --> 00:47:53,227
This is a very common scenario.

936
00:47:53,227 --> 00:47:54,810
This is if you want
to make a polygon,

937
00:47:54,810 --> 00:47:56,189
every vertex has degree 2.

938
00:47:56,189 --> 00:47:57,730
If you want to make
several polygons,

939
00:47:57,730 --> 00:47:59,260
every vertex has degree 2.

940
00:47:59,260 --> 00:48:02,950
I'll even let you have vertices
of degree 1 or 0 for free,

941
00:48:02,950 --> 00:48:05,790
but mainly we're thinking
about degree 2 everywhere.

942
00:48:08,620 --> 00:48:20,315
Then, we almost always
have a linear corridor.

943
00:48:24,329 --> 00:48:26,120
So this is why the
situation's interesting,

944
00:48:26,120 --> 00:48:28,203
although unfortunately
this is still a conjecture.

945
00:48:28,203 --> 00:48:30,150
I'm sure this is
true, but proving it--

946
00:48:30,150 --> 00:48:33,500
I don't quite know
the right techniques.

947
00:48:33,500 --> 00:48:37,820
So in this typical situation,
you take any picture you want.

948
00:48:37,820 --> 00:48:41,250
You slightly perturb very
vertex randomly, say,

949
00:48:41,250 --> 00:48:44,510
then with probability
1-- 100% probability--

950
00:48:44,510 --> 00:48:46,395
you will get only
linear corridors.

951
00:48:49,510 --> 00:48:52,060
And that's the situation
where it turns into a tree.

952
00:48:52,060 --> 00:48:55,750
It's easy to fold life is good.

953
00:48:55,750 --> 00:48:59,235
The annoying case is
circular corridor case.

954
00:49:05,560 --> 00:49:07,060
This takes a lot
more work to prove,

955
00:49:07,060 --> 00:49:09,155
and I'm not going to
talk about it much.

956
00:49:09,155 --> 00:49:15,550
A circular corridor
looks like this.

957
00:49:15,550 --> 00:49:18,120
Here we have these
in with rivers also.

958
00:49:18,120 --> 00:49:19,590
So you just loop around.

959
00:49:19,590 --> 00:49:23,737
Still constant width everywhere,
but you meet yourself.

960
00:49:23,737 --> 00:49:24,945
You don't go out to infinity.

961
00:49:27,730 --> 00:49:29,010
It's harder.

962
00:49:29,010 --> 00:49:30,560
Why is it harder?

963
00:49:30,560 --> 00:49:37,180
Well in particular, if you
look at how one corridor folds,

964
00:49:37,180 --> 00:49:39,360
it's no longer-- it's
like the same situation we

965
00:49:39,360 --> 00:49:41,970
had in like lectures
two and three

966
00:49:41,970 --> 00:49:45,560
where we had on the one hand
a 1D folding was really easy.

967
00:49:45,560 --> 00:49:47,210
But then when you
made it circular--

968
00:49:47,210 --> 00:49:49,460
you're just folding a circle
instead of folding a line

969
00:49:49,460 --> 00:49:51,760
segment-- now you had
this wrap around issue.

970
00:49:51,760 --> 00:49:54,310
So, like, these guys
would have to line up.

971
00:49:54,310 --> 00:49:56,410
It turns out they will
line up because everything

972
00:49:56,410 --> 00:49:57,809
is bisecting and whatnot.

973
00:49:57,809 --> 00:49:59,350
These edges will
line up, but now you

974
00:49:59,350 --> 00:50:00,940
have to join them together.

975
00:50:00,940 --> 00:50:04,160
And if this part went all the
way down here and came back up,

976
00:50:04,160 --> 00:50:06,920
then you'd get an intersection.

977
00:50:06,920 --> 00:50:12,840
And it turns out, in general,
I get a choice of who's first

978
00:50:12,840 --> 00:50:13,484
and who's last.

979
00:50:13,484 --> 00:50:14,900
I have a circular
order of things,

980
00:50:14,900 --> 00:50:17,850
and I get to choose where
I break that circular order

981
00:50:17,850 --> 00:50:19,540
and make it a linear order.

982
00:50:19,540 --> 00:50:21,000
Where I do the wrap around.

983
00:50:21,000 --> 00:50:22,860
There's some
circular corridors--

984
00:50:22,860 --> 00:50:27,590
you can't even break
them and make it work.

985
00:50:27,590 --> 00:50:30,200
Kind of depressing.

986
00:50:30,200 --> 00:50:32,679
So this is definitely harder
and that sometimes it's

987
00:50:32,679 --> 00:50:33,220
not possible.

988
00:50:36,680 --> 00:50:40,450
But we can save a
little bit, which

989
00:50:40,450 --> 00:50:43,800
is-- I don't have
an example handy.

990
00:50:43,800 --> 00:50:44,940
I wish I did.

991
00:50:44,940 --> 00:50:46,530
I'll have to reconstruct it.

992
00:50:46,530 --> 00:50:47,530
This is so long ago.

993
00:50:50,810 --> 00:50:53,010
Here's a way to make
it definitely work.

994
00:50:53,010 --> 00:50:54,435
Fold all the cut edges.

995
00:50:57,300 --> 00:50:59,030
So far in the pictures
I've been drawing,

996
00:50:59,030 --> 00:51:00,910
I didn't fold
along the cut edges

997
00:51:00,910 --> 00:51:03,992
because I really wanted to
separate the green region here

998
00:51:03,992 --> 00:51:04,950
from the yellow region.

999
00:51:04,950 --> 00:51:08,007
If I folded this up-- this
is quite a complicated one

1000
00:51:08,007 --> 00:51:12,320
to fold-- you get the cut lines
somewhere like over there,

1001
00:51:12,320 --> 00:51:14,910
and then the green stuff will
be always above the cut line

1002
00:51:14,910 --> 00:51:17,150
and the yellow stuff will
be below the cut line.

1003
00:51:17,150 --> 00:51:18,770
In general, this is
called a side assignment.

1004
00:51:18,770 --> 00:51:20,160
You have a bunch of
regions you decide

1005
00:51:20,160 --> 00:51:21,868
which ones you want
to be above and which

1006
00:51:21,868 --> 00:51:23,190
ones you want to be below.

1007
00:51:23,190 --> 00:51:25,060
And usually, you have
polygons, and you

1008
00:51:25,060 --> 00:51:28,530
say the interiors are above
and the exteriors are below.

1009
00:51:28,530 --> 00:51:31,770
But in general, you could
ask for anything you want.

1010
00:51:31,770 --> 00:51:34,282
You could say maybe I want
both of these to be above.

1011
00:51:34,282 --> 00:51:35,740
If you make both
of them above, you

1012
00:51:35,740 --> 00:51:39,314
have to valley fold along
all of the cut lines.

1013
00:51:39,314 --> 00:51:41,480
So if you do that-- you say
I want all regions to be

1014
00:51:41,480 --> 00:51:43,990
above the cut line-- you
can still line them up.

1015
00:51:43,990 --> 00:51:47,010
You end up folding along all
the cut edges with valleys.

1016
00:51:47,010 --> 00:51:50,966
And then wrap around
is super easy.

1017
00:51:50,966 --> 00:51:53,090
We take this thing, and in
fact, everyone's folding

1018
00:51:53,090 --> 00:51:54,520
along the black lines.

1019
00:51:54,520 --> 00:51:57,550
So really everybody
comes down to the floor,

1020
00:51:57,550 --> 00:52:00,090
and then the wrap around is
just underneath the floor

1021
00:52:00,090 --> 00:52:02,610
and life is good.

1022
00:52:02,610 --> 00:52:04,876
So that's one way to
deal with this case,

1023
00:52:04,876 --> 00:52:06,250
and you have to
prove that works.

1024
00:52:06,250 --> 00:52:06,958
It's complicated.

1025
00:52:09,190 --> 00:52:13,270
I would rather go
onto other topics.

1026
00:52:13,270 --> 00:52:17,040
I do think this would
make a fun project.

1027
00:52:17,040 --> 00:52:20,200
It's not easy to make
these crease patterns.

1028
00:52:20,200 --> 00:52:22,050
Currently we draw
them always by hand,

1029
00:52:22,050 --> 00:52:23,570
meaning with the fancy
drawing program that

1030
00:52:23,570 --> 00:52:24,986
knows how to do
angular bisectors.

1031
00:52:27,470 --> 00:52:30,590
And it's an art to
move around the points

1032
00:52:30,590 --> 00:52:32,930
so that you get
lots of alignments.

1033
00:52:32,930 --> 00:52:35,866
Like when you get four straight
skeleton edges coming together,

1034
00:52:35,866 --> 00:52:37,240
that means you
get fewer creases.

1035
00:52:37,240 --> 00:52:39,850
That's a good thing whenever
you can make that happen.

1036
00:52:39,850 --> 00:52:42,630
So when you could give
me software to help

1037
00:52:42,630 --> 00:52:44,540
do that, I would love you.

1038
00:52:47,980 --> 00:52:51,199
So let's move on.

1039
00:52:51,199 --> 00:52:53,240
Any questions about the
straight skeleton method?

1040
00:52:53,240 --> 00:52:55,860
Now I'm going to switch
over to disk packing.

1041
00:52:58,758 --> 00:53:01,180
All right.

1042
00:53:01,180 --> 00:53:04,480
Same problem, but
now I'll give you

1043
00:53:04,480 --> 00:53:10,300
a method that always
works in theory.

1044
00:53:10,300 --> 00:53:14,440
Just difficult in practice.

1045
00:53:19,060 --> 00:53:23,710
I think this is good
chalk because it's yellow.

1046
00:53:23,710 --> 00:53:25,410
Generally, if it's
yellow on the outside

1047
00:53:25,410 --> 00:53:28,320
it is railroad chalk,
which is the good stuff.

1048
00:53:28,320 --> 00:53:32,040
But, what the problem
is-- we have bad erasers.

1049
00:53:32,040 --> 00:53:33,304
It's persistence of vision.

1050
00:53:33,304 --> 00:53:35,220
You just get to remember
what I used to write.

1051
00:53:51,001 --> 00:53:52,750
I don't want to write
down the algorithms.

1052
00:53:52,750 --> 00:53:55,040
It's complicated.

1053
00:53:55,040 --> 00:53:59,240
I want to give you a visual
overview of the main steps.

1054
00:53:59,240 --> 00:54:02,910
I guess there's nice
figures do that for us.

1055
00:54:02,910 --> 00:54:06,980
So we start with a
very complicated shape

1056
00:54:06,980 --> 00:54:09,740
we want to make, like
this quadrilateral.

1057
00:54:09,740 --> 00:54:13,510
And you can see disk
packing is involved.

1058
00:54:13,510 --> 00:54:16,660
The very first thing we do-- and
I'll tell you why in a moment--

1059
00:54:16,660 --> 00:54:19,292
is thicken the graph
you want to build.

1060
00:54:19,292 --> 00:54:20,750
So maybe it's a
polygon, maybe it's

1061
00:54:20,750 --> 00:54:22,875
a graph-- I'll think about
the polygon case for now

1062
00:54:22,875 --> 00:54:25,610
because it's easier, then I'll
come back to the general case.

1063
00:54:25,610 --> 00:54:28,949
I thicken it by tiny epsilon.

1064
00:54:28,949 --> 00:54:30,240
Just offset in both directions.

1065
00:54:30,240 --> 00:54:32,400
Just like as if you're starting
the straight skeleton method,

1066
00:54:32,400 --> 00:54:34,090
but then you stop after epsilon.

1067
00:54:34,090 --> 00:54:38,880
No events happen
in epsilon time.

1068
00:54:38,880 --> 00:54:45,250
OK, then I have this picture--
the purple stuff, pink stuff,

1069
00:54:45,250 --> 00:54:45,750
whatever.

1070
00:54:45,750 --> 00:54:46,960
Magenta.

1071
00:54:46,960 --> 00:54:48,160
50% magenta.

1072
00:54:48,160 --> 00:54:51,390
I happen to know I
drew this figure.

1073
00:54:51,390 --> 00:54:55,460
Now I'm going to take
some 50% cyan disks

1074
00:54:55,460 --> 00:54:57,370
and pack them to
fill this region.

1075
00:54:57,370 --> 00:54:59,930
Now what I want-- there's
three properties I want.

1076
00:54:59,930 --> 00:55:01,930
One is that every
vertex, but you

1077
00:55:01,930 --> 00:55:03,680
have to think about
each region separately

1078
00:55:03,680 --> 00:55:05,290
which is a little bit confusing.

1079
00:55:05,290 --> 00:55:06,120
Let's think about the outside.

1080
00:55:06,120 --> 00:55:07,100
It's a little easier.

1081
00:55:07,100 --> 00:55:08,680
Bigger.

1082
00:55:08,680 --> 00:55:12,180
Every vertex of this
graph on the outside

1083
00:55:12,180 --> 00:55:14,040
should be the center of a disk.

1084
00:55:14,040 --> 00:55:16,100
There's a disk center here.

1085
00:55:16,100 --> 00:55:17,520
There's a disk center here.

1086
00:55:17,520 --> 00:55:19,930
there's a disk center here,
and a disk center there.

1087
00:55:19,930 --> 00:55:23,680
On the inside it's also true,
they're just different disks.

1088
00:55:23,680 --> 00:55:26,370
Then also I want the
edges of the graph

1089
00:55:26,370 --> 00:55:29,855
to be covered by
a radii of disks.

1090
00:55:29,855 --> 00:55:31,950
And so here's a
radius of one disk.

1091
00:55:31,950 --> 00:55:33,800
Here's a diameter of a disc.

1092
00:55:33,800 --> 00:55:36,820
Here's a radius of a disk
that covers the edge.

1093
00:55:36,820 --> 00:55:39,390
So in other words, I
want to fill-- along

1094
00:55:39,390 --> 00:55:41,820
the edge I want to
have a bunch of centers

1095
00:55:41,820 --> 00:55:44,680
so that I completely
cover that edge with blue.

1096
00:55:44,680 --> 00:55:46,320
You'll see why later.

1097
00:55:46,320 --> 00:55:46,820
Question?

1098
00:55:51,960 --> 00:55:54,630
The disks have to
be non-overlapping.

1099
00:55:54,630 --> 00:55:57,709
These properties are actually
quite challenging to achieve.

1100
00:55:57,709 --> 00:55:59,500
Your question is why
do we use small disks.

1101
00:56:03,360 --> 00:56:05,680
Disk, because if I
had a big disk here,

1102
00:56:05,680 --> 00:56:07,050
it would intersect this disk.

1103
00:56:09,990 --> 00:56:11,790
Now I didn't have to
make that disk so big,

1104
00:56:11,790 --> 00:56:13,248
but if I made that
one smaller, I'd

1105
00:56:13,248 --> 00:56:16,349
have to have more disks here.

1106
00:56:16,349 --> 00:56:17,890
Or here there's also
two small disks.

1107
00:56:17,890 --> 00:56:23,420
That one I probably could have
gotten away with a bigger disk.

1108
00:56:23,420 --> 00:56:26,404
Oh, no, but then on the
inside you have a problem.

1109
00:56:26,404 --> 00:56:28,070
So these guys actually
have to match up.

1110
00:56:28,070 --> 00:56:29,250
That's another constraint.

1111
00:56:29,250 --> 00:56:30,730
And the inside and the
outside have to match up.

1112
00:56:30,730 --> 00:56:32,310
Here there's a slight
change in radii

1113
00:56:32,310 --> 00:56:33,560
to compensate for the epsilon.

1114
00:56:33,560 --> 00:56:35,810
Along the edges they're
exactly the same.

1115
00:56:35,810 --> 00:56:37,850
So if I made this
one a big disk,

1116
00:56:37,850 --> 00:56:39,030
it would overlap this one.

1117
00:56:39,030 --> 00:56:41,340
So I could make that one
smaller, but then-- other

1118
00:56:41,340 --> 00:56:42,779
problems.

1119
00:56:42,779 --> 00:56:44,320
So you have to
simultaneously balance

1120
00:56:44,320 --> 00:56:48,270
all these constraints,
which is a bit exciting.

1121
00:56:48,270 --> 00:56:48,860
What else?

1122
00:56:48,860 --> 00:56:50,300
The discs don't overlap.

1123
00:56:50,300 --> 00:56:53,240
And the last property is that
the gaps between the disks

1124
00:56:53,240 --> 00:56:56,970
have always three or four sides.

1125
00:56:56,970 --> 00:56:58,010
Why?

1126
00:56:58,010 --> 00:56:59,970
Because I want it to.

1127
00:56:59,970 --> 00:57:01,770
It will make life easier.

1128
00:57:01,770 --> 00:57:03,730
You could try to
deal with more sides,

1129
00:57:03,730 --> 00:57:08,370
but three and four is nice.

1130
00:57:08,370 --> 00:57:10,530
Yeah, I'll get to that.

1131
00:57:10,530 --> 00:57:12,240
Why do we care about
number of sides?

1132
00:57:12,240 --> 00:57:14,820
Because I'm going
to draw a graph.

1133
00:57:14,820 --> 00:57:17,030
I'm going to subdivide
my original graph here

1134
00:57:17,030 --> 00:57:21,210
with these red lines to
say whenever disks kiss,

1135
00:57:21,210 --> 00:57:23,815
I will draw-- connect the
centers of those disks.

1136
00:57:26,660 --> 00:57:31,537
And because these gaps always
have three or four sides--

1137
00:57:31,537 --> 00:57:32,620
it's not the best example.

1138
00:57:32,620 --> 00:57:34,230
Here's like three sides.

1139
00:57:34,230 --> 00:57:36,990
The red lines I draw
will outline a triangle.

1140
00:57:36,990 --> 00:57:39,800
Whenever I have three sides,
whenever I have four sides,

1141
00:57:39,800 --> 00:57:41,520
I'll have a quadrilateral.

1142
00:57:41,520 --> 00:57:44,040
So I've subdivided my
regions into triangles

1143
00:57:44,040 --> 00:57:45,820
and quadrilaterals.

1144
00:57:45,820 --> 00:57:48,300
OK, you have to believe
that this is possible.

1145
00:57:48,300 --> 00:57:50,700
I can sketch an
algorithm for you,

1146
00:57:50,700 --> 00:57:54,945
which is you draw a tiny
disk at each of the corners,

1147
00:57:54,945 --> 00:57:57,070
and then you draw lots of
tiny this along the edges

1148
00:57:57,070 --> 00:57:58,850
to fill the edges.

1149
00:57:58,850 --> 00:58:00,400
And that will
satisfy everything.

1150
00:58:00,400 --> 00:58:01,670
I mean the disks will
be non-overlapping

1151
00:58:01,670 --> 00:58:02,795
because they're super tiny.

1152
00:58:02,795 --> 00:58:05,390
They won't get near any other
disks from some other side.

1153
00:58:05,390 --> 00:58:07,650
And what other good things?

1154
00:58:07,650 --> 00:58:09,580
Oh, but the regions
will be ginormous.

1155
00:58:09,580 --> 00:58:11,080
They won't have
three or four sides.

1156
00:58:11,080 --> 00:58:14,000
They'll have 100 sides, a
million sides-- who knows.

1157
00:58:14,000 --> 00:58:18,460
Well, whenever you have
some crazy region outlined

1158
00:58:18,460 --> 00:58:20,830
by disks-- might not be convex.

1159
00:58:20,830 --> 00:58:25,276
Whatever-- just draw the
biggest disk you can in there.

1160
00:58:25,276 --> 00:58:27,890
I'll get it to turn
into a disk eventually.

1161
00:58:27,890 --> 00:58:30,520
That does not intersect
anybody, but if it's

1162
00:58:30,520 --> 00:58:32,200
the biggest possible,
it will actually

1163
00:58:32,200 --> 00:58:34,510
touch at least three sides.

1164
00:58:34,510 --> 00:58:36,880
If you degenerate,
it might touch four.

1165
00:58:36,880 --> 00:58:39,610
But in general, it will
touch three sides, which

1166
00:58:39,610 --> 00:58:42,152
will subdivide that
region into three pieces,

1167
00:58:42,152 --> 00:58:44,360
and you can show that those
pieces are all little bit

1168
00:58:44,360 --> 00:58:47,480
smaller than what you had before
in terms of number of sides.

1169
00:58:47,480 --> 00:58:49,510
Except when you start
with a quadrilateral.

1170
00:58:49,510 --> 00:58:51,020
When there's four
sides, you'll get

1171
00:58:51,020 --> 00:58:52,200
quadrilaterals and triangle.

1172
00:58:52,200 --> 00:58:53,930
So you can't go
below three and four.

1173
00:58:53,930 --> 00:58:55,638
It'd be great if we
could always get down

1174
00:58:55,638 --> 00:58:57,531
to three sides in
every region, but we

1175
00:58:57,531 --> 00:58:58,780
can get down to three or four.

1176
00:58:58,780 --> 00:59:00,760
Anything bigger than three
or four you can show.

1177
00:59:00,760 --> 00:59:02,218
This will make it
strictly smaller.

1178
00:59:02,218 --> 00:59:03,420
So that is an algorithm.

1179
00:59:03,420 --> 00:59:06,160
It's not super efficient, but
it will find a disk packing

1180
00:59:06,160 --> 00:59:07,510
with all these properties.

1181
00:59:07,510 --> 00:59:08,914
Then we do the subdivision.

1182
00:59:08,914 --> 00:59:10,580
Now what do you think
we're going to do?

1183
00:59:10,580 --> 00:59:13,220
What do we do with
the triangles?

1184
00:59:13,220 --> 00:59:14,940
Rabbit ear.

1185
00:59:14,940 --> 00:59:17,760
That's the key phrase for today.

1186
00:59:17,760 --> 00:59:20,690
So remember our good friend,
the rabbit ear, and then

1187
00:59:20,690 --> 00:59:22,880
there was the universal
molecule-- Lang's

1188
00:59:22,880 --> 00:59:24,760
universal molecule
for the quadrilateral.

1189
00:59:24,760 --> 00:59:27,330
We're going to use that
for the quadrilaterals.

1190
00:59:27,330 --> 00:59:30,390
And it turns out there's
some nice properties here,

1191
00:59:30,390 --> 00:59:34,800
which is the perpendicular
folds of the rabbit ear

1192
00:59:34,800 --> 00:59:37,530
will always hit right at the
kissing point between the two

1193
00:59:37,530 --> 00:59:38,630
disks.

1194
00:59:38,630 --> 00:59:40,925
And same thing in here.

1195
00:59:40,925 --> 00:59:42,050
We've got these four disks.

1196
00:59:42,050 --> 00:59:45,610
We've got this quadrilateral
region in between.

1197
00:59:45,610 --> 00:59:50,990
and the perpendicular folds that
come out of these two points.

1198
00:59:50,990 --> 00:59:52,840
You may not remember
exactly what's

1199
00:59:52,840 --> 00:59:55,440
happening here as we shrink.

1200
00:59:55,440 --> 00:59:59,920
And then in the tree method,
this became an active path.

1201
00:59:59,920 --> 01:00:01,840
There's no notion of
active paths here,

1202
01:00:01,840 --> 01:00:04,510
but we just make that so.

1203
01:00:04,510 --> 01:00:08,580
That these perpendicular folds
will end up hitting kissing

1204
01:00:08,580 --> 01:00:12,760
disks, and we'll end up actually
with the four arm starfish.

1205
01:00:12,760 --> 01:00:18,090
In terms of the tree you get and
the articulatable flaps here,

1206
01:00:18,090 --> 01:00:20,630
these guys will all
meet at a point.

1207
01:00:20,630 --> 01:00:23,470
That's just the way this
works with disk packing.

1208
01:00:23,470 --> 01:00:26,160
And you can think of
there being disks here

1209
01:00:26,160 --> 01:00:29,070
and you're actually applying
the tree method to that flap

1210
01:00:29,070 --> 01:00:33,430
pattern, and that's probably the
easiest way to think about it.

1211
01:00:33,430 --> 01:00:37,080
But what's good for us--
do I have a picture?

1212
01:00:37,080 --> 01:00:37,740
Not yet.

1213
01:00:37,740 --> 01:00:39,890
But the point is, I have
perpendiculars coming out

1214
01:00:39,890 --> 01:00:40,442
of here.

1215
01:00:40,442 --> 01:00:42,150
I have perpendiculars
coming out of here.

1216
01:00:42,150 --> 01:00:46,710
They will meet because these
disks kiss at the same point

1217
01:00:46,710 --> 01:00:48,720
up from both sides.

1218
01:00:48,720 --> 01:00:49,780
Perpendiculars meet.

1219
01:00:49,780 --> 01:00:50,350
That's good.

1220
01:00:50,350 --> 01:00:51,891
That means I don't
get perpendiculars

1221
01:00:51,891 --> 01:00:53,324
bouncing all over the place.

1222
01:00:53,324 --> 01:00:55,490
So all this work is to make
sure that perpendiculars

1223
01:00:55,490 --> 01:00:56,645
are well behaved.

1224
01:00:56,645 --> 01:01:00,540
It's a lot of work to
do it, but it does it.

1225
01:01:00,540 --> 01:01:04,700
Now when you fold this
thing, what we end up doing

1226
01:01:04,700 --> 01:01:08,770
is lining up-- remember there
was two copies of my polygon.

1227
01:01:08,770 --> 01:01:11,160
There's the inner copy
and the outer copy.

1228
01:01:11,160 --> 01:01:17,960
I end up lining up
all of these guys--

1229
01:01:17,960 --> 01:01:19,390
I'm got to go back
to the picture.

1230
01:01:19,390 --> 01:01:20,870
I'm sorry.

1231
01:01:20,870 --> 01:01:24,440
So we have this inner copy,
and what these molecules end up

1232
01:01:24,440 --> 01:01:26,540
doing is lining up all
the edges of this quad,

1233
01:01:26,540 --> 01:01:29,200
all the edges of this quad,
all the edges of this triangle.

1234
01:01:29,200 --> 01:01:30,860
All those edges
on the inside will

1235
01:01:30,860 --> 01:01:33,020
become lined up on one edge.

1236
01:01:33,020 --> 01:01:34,640
All the lines on
the outside become

1237
01:01:34,640 --> 01:01:36,980
lined up on another line.

1238
01:01:36,980 --> 01:01:39,840
Turns out it will be
parallel to that line.

1239
01:01:39,840 --> 01:01:42,277
But what we really wanted
to line up were these edges,

1240
01:01:42,277 --> 01:01:44,860
and you can see why we had to
do the outside of the beginning,

1241
01:01:44,860 --> 01:01:48,030
because otherwise we'd get
tons of extra junk on our line.

1242
01:01:48,030 --> 01:01:50,860
We only want these
edges on our line.

1243
01:01:50,860 --> 01:01:53,330
So we did the offset
so that all this stuff

1244
01:01:53,330 --> 01:01:54,950
will come to one line.

1245
01:01:54,950 --> 01:01:57,870
All this stuff on the outside
will come to another line.

1246
01:01:57,870 --> 01:02:01,410
And then we get this picture.

1247
01:02:01,410 --> 01:02:03,640
So this is one
line at the bottom.

1248
01:02:03,640 --> 01:02:04,640
Another line at the top.

1249
01:02:04,640 --> 01:02:09,270
We really wanted to line up
stuff-- the blue stuff there.

1250
01:02:09,270 --> 01:02:10,950
And there's still
some junk on our line.

1251
01:02:10,950 --> 01:02:14,510
These cyan triangles
represent things

1252
01:02:14,510 --> 01:02:17,654
that come from down
here, but we really

1253
01:02:17,654 --> 01:02:19,070
don't want them
to cross our line.

1254
01:02:19,070 --> 01:02:20,390
They just happened to.

1255
01:02:20,390 --> 01:02:22,650
So we have to sync
them repeatedly.

1256
01:02:22,650 --> 01:02:25,730
Do lots of folds to make them
underneath that epsilon line.

1257
01:02:25,730 --> 01:02:28,040
Then we can cut along
our line, and we're done.

1258
01:02:28,040 --> 01:02:28,540
Easy.

1259
01:02:33,620 --> 01:02:37,880
To prove that this works,
we'll view a little sketch.

1260
01:02:37,880 --> 01:02:40,670
This is kind of fun,
and it's one piece

1261
01:02:40,670 --> 01:02:43,020
of what we're in the process
of doing for tree maker.

1262
01:02:43,020 --> 01:02:45,103
This is sort of like a
special case of tree maker.

1263
01:02:45,103 --> 01:02:47,260
You just have very
simple molecules

1264
01:02:47,260 --> 01:02:49,510
and a relatively simple way
in which they're combined.

1265
01:02:55,380 --> 01:02:57,370
Here I've done a
simpler example.

1266
01:02:57,370 --> 01:03:01,890
I want to make a square, and I
end up decomposing in this case

1267
01:03:01,890 --> 01:03:04,669
into nine molecules-- nine
quadrilateral molecules.

1268
01:03:04,669 --> 01:03:06,460
A very simple disk
packing which I have not

1269
01:03:06,460 --> 01:03:08,170
shown the disk packing here.

1270
01:03:08,170 --> 01:03:11,580
The idea is I'm going to
make some cuts in my paper

1271
01:03:11,580 --> 01:03:13,645
to make my problem easier.

1272
01:03:13,645 --> 01:03:16,020
I'm going to have to pay for
that, because later I really

1273
01:03:16,020 --> 01:03:17,100
want those edges joined.

1274
01:03:17,100 --> 01:03:18,641
I'll have to glue
them back together.

1275
01:03:18,641 --> 01:03:24,290
But to make it easier think
about, I cut those four edges.

1276
01:03:24,290 --> 01:03:27,170
So that the way in which
my molecules are connected

1277
01:03:27,170 --> 01:03:29,751
to each other is a tree,
because I like trees.

1278
01:03:29,751 --> 01:03:31,000
They're easier to think about.

1279
01:03:31,000 --> 01:03:32,490
Easier to do induction over.

1280
01:03:32,490 --> 01:03:35,580
So that's the blue line
connecting the centers

1281
01:03:35,580 --> 01:03:36,460
in a tree.

1282
01:03:36,460 --> 01:03:41,440
The other remaining edges in
the grid have been cut away.

1283
01:03:41,440 --> 01:03:44,690
Now the idea is--
it's kind of like what

1284
01:03:44,690 --> 01:03:47,190
I was drawing for the
linear corridor case.

1285
01:03:47,190 --> 01:03:51,020
You have a tree, you pick
up the tree from some node,

1286
01:03:51,020 --> 01:03:52,710
and just hang it down.

1287
01:03:52,710 --> 01:03:56,870
And in this case, we hang
it from this molecule.

1288
01:03:56,870 --> 01:03:59,760
The red edges are
mountain, so three of these

1289
01:03:59,760 --> 01:04:00,940
are going to be valley.

1290
01:04:00,940 --> 01:04:02,060
One mountain.

1291
01:04:02,060 --> 01:04:04,886
The idea is this thing reaches
around the next guy, which

1292
01:04:04,886 --> 01:04:06,760
reaches is around the
next guy, which reaches

1293
01:04:06,760 --> 01:04:08,930
around the next guy,
and there's actually--

1294
01:04:08,930 --> 01:04:11,326
there's two valleys
here-- two little pockets.

1295
01:04:11,326 --> 01:04:12,950
Each of-- this guy
goes in that pocket,

1296
01:04:12,950 --> 01:04:16,190
this guy goes in that pocket,
and recursively, it just works.

1297
01:04:16,190 --> 01:04:20,095
I think I have a picture of
what's actually happening here.

1298
01:04:20,095 --> 01:04:22,740
Yeah, is it's hard
to really draw,

1299
01:04:22,740 --> 01:04:25,990
but each of these
forearm starfish

1300
01:04:25,990 --> 01:04:28,970
has one mountain
and three valleys,

1301
01:04:28,970 --> 01:04:33,070
and you just nestle inside
your parent in the tree.

1302
01:04:33,070 --> 01:04:35,820
And this is really easy to show
that there's no crossings here

1303
01:04:35,820 --> 01:04:38,310
because just joined
to your parent,

1304
01:04:38,310 --> 01:04:40,820
and it's a nice
nesting structure.

1305
01:04:40,820 --> 01:04:44,270
It's just in the same way
that trees can be folded flat.

1306
01:04:44,270 --> 01:04:48,910
You can fold all
these molecules flat

1307
01:04:48,910 --> 01:04:50,960
and join them
together in a tree.

1308
01:04:50,960 --> 01:04:52,590
But we didn't
really have a tree.

1309
01:04:52,590 --> 01:04:54,840
We had all those extra cuts
that we have to re-suture.

1310
01:04:57,380 --> 01:04:59,369
So we have this picture,
and now we really

1311
01:04:59,369 --> 01:05:01,910
have to join up these edges and
think about what the mountain

1312
01:05:01,910 --> 01:05:03,520
valley assignment is there.

1313
01:05:03,520 --> 01:05:05,820
And it works.

1314
01:05:08,666 --> 01:05:13,140
This green thing
is the boundary,

1315
01:05:13,140 --> 01:05:15,300
and then I have
connected the dots.

1316
01:05:15,300 --> 01:05:17,830
Each of these dots corresponds
to one green edge here.

1317
01:05:17,830 --> 01:05:22,270
I forget whether it's this
edge or that one, I think.

1318
01:05:22,270 --> 01:05:25,200
It's just a single edge.

1319
01:05:25,200 --> 01:05:34,930
And so, for example, these
two are-- these two joins,

1320
01:05:34,930 --> 01:05:38,600
and then the joins above
that nestle around it.

1321
01:05:38,600 --> 01:05:40,950
And then the other
branch at the top are

1322
01:05:40,950 --> 01:05:44,464
joins, and in the leftmost
cut are these two joins.

1323
01:05:44,464 --> 01:05:46,130
You have to make these
joins, and really

1324
01:05:46,130 --> 01:05:47,710
all you need to
check is that these

1325
01:05:47,710 --> 01:05:50,650
joins form a non-crossing
picture like they do here.

1326
01:05:50,650 --> 01:05:53,200
And that's almost obvious
because this is a planer

1327
01:05:53,200 --> 01:05:56,660
diagram and we're cutting
along a planer tree,

1328
01:05:56,660 --> 01:05:59,330
and so this is again a depth
first search kind of thing.

1329
01:05:59,330 --> 01:06:03,510
So there's one tree we
call the dual tree here

1330
01:06:03,510 --> 01:06:05,840
that works because it's
a tree, and then there's

1331
01:06:05,840 --> 01:06:08,430
the cuts you make which are
different trees-- primal tree

1332
01:06:08,430 --> 01:06:12,410
if you want-- and that also
works because of planarity.

1333
01:06:12,410 --> 01:06:13,490
And it all works.

1334
01:06:13,490 --> 01:06:17,685
That's the hand wavy version,
and you can read the textbook

1335
01:06:17,685 --> 01:06:20,330
if you want more details.

1336
01:06:20,330 --> 01:06:21,430
Oh, gosh.

1337
01:06:21,430 --> 01:06:24,720
If you want to solve more
general graphs, you can do it.

1338
01:06:24,720 --> 01:06:28,360
In general, you have to
offset all of those cut lines,

1339
01:06:28,360 --> 01:06:34,240
and you get all these things--
along the pink lines here you

1340
01:06:34,240 --> 01:06:36,440
line things up,
but you really want

1341
01:06:36,440 --> 01:06:40,450
to line up these blue
lines-- purple lines.

1342
01:06:40,450 --> 01:06:42,860
And so you have to do more
syncing to get it to work.

1343
01:06:42,860 --> 01:06:44,651
Now I have all the
things I want to line up

1344
01:06:44,651 --> 01:06:45,850
on this line and this line.

1345
01:06:45,850 --> 01:06:47,808
I fold in the middle,
and now they're lined up.

1346
01:06:52,070 --> 01:06:54,422
That fold in the middle--
yeah, that will work.

1347
01:06:54,422 --> 01:06:55,530
Good.

1348
01:06:55,530 --> 01:06:56,780
Might have to do more syncing.

1349
01:07:00,180 --> 01:07:02,120
Whew.

1350
01:07:02,120 --> 01:07:03,750
Questions about
disk-packing method?

1351
01:07:03,750 --> 01:07:05,957
This is a bit of
a whirlwind tour,

1352
01:07:05,957 --> 01:07:07,540
but I wanted to get
through it quickly

1353
01:07:07,540 --> 01:07:09,340
to tell you about a new result.

1354
01:07:09,340 --> 01:07:11,670
Just got accepted
to a conference

1355
01:07:11,670 --> 01:07:16,490
to appear in October.

1356
01:07:16,490 --> 01:07:17,050
Pretty soon.

1357
01:07:22,780 --> 01:07:25,280
And it's a project that started
in this class in the problem

1358
01:07:25,280 --> 01:07:30,090
session three years ago,
and we just solved it.

1359
01:07:30,090 --> 01:07:33,310
Took awhile.

1360
01:07:33,310 --> 01:07:35,170
Took another
co-author to chime in.

1361
01:07:39,730 --> 01:07:42,720
And it goes back to the
early history of fold and cut

1362
01:07:42,720 --> 01:07:44,640
which is simple folds.

1363
01:07:44,640 --> 01:07:46,400
All the magicians were
using simple folds.

1364
01:07:46,400 --> 01:07:48,670
What can you make
with simple folds?

1365
01:07:48,670 --> 01:07:51,640
Now you've been wowed and dowed
that you could make anything

1366
01:07:51,640 --> 01:07:55,030
with arbitrary folds, but simple
folds you cannot make anything.

1367
01:07:55,030 --> 01:08:00,310
Because the first told
you make, say, this one,

1368
01:08:00,310 --> 01:08:02,785
has to be a line of
symmetry of your diagram.

1369
01:08:05,844 --> 01:08:09,045
Got to stop making my life hard.

1370
01:08:09,045 --> 01:08:11,170
If you can fold something,
you can never unfold it.

1371
01:08:11,170 --> 01:08:14,040
That's the usual
simple fold model.

1372
01:08:14,040 --> 01:08:15,910
This has to line
up-- the cuts you

1373
01:08:15,910 --> 01:08:17,702
want to line up over
here, that are exactly

1374
01:08:17,702 --> 01:08:19,243
be the cuts you want
to line up here.

1375
01:08:19,243 --> 01:08:20,950
So you can only make
symmetric diagrams.

1376
01:08:20,950 --> 01:08:24,990
The first fold has to be a
line of reflectional symmetry.

1377
01:08:24,990 --> 01:08:27,890
But is that the only
property you need?

1378
01:08:27,890 --> 01:08:28,510
No.

1379
01:08:28,510 --> 01:08:32,020
Kind of have to have symmetry
for while until you're done.

1380
01:08:32,020 --> 01:08:33,620
How do you formalize that?

1381
01:08:33,620 --> 01:08:36,090
Well, we came up
with an algorithm

1382
01:08:36,090 --> 01:08:40,340
that in polynomial time,
an efficient algorithm

1383
01:08:40,340 --> 01:08:47,090
will tell you whether a
given polygon can be made.

1384
01:08:47,090 --> 01:08:51,640
Like this polygon looks good.

1385
01:08:51,640 --> 01:08:56,430
I think-- yeah, I
think this can be made.

1386
01:08:56,430 --> 01:09:00,740
So I think I would fold
along and do a bisector here,

1387
01:09:00,740 --> 01:09:02,390
and then this
basically disappears.

1388
01:09:02,390 --> 01:09:03,670
Folding over.

1389
01:09:03,670 --> 01:09:05,710
Then I would fold along
and get a bisector here,

1390
01:09:05,710 --> 01:09:08,140
and then this
disappears into that,

1391
01:09:08,140 --> 01:09:15,050
and then maybe I can fold here.

1392
01:09:15,050 --> 01:09:16,580
Does that work?

1393
01:09:16,580 --> 01:09:17,122
Barely.

1394
01:09:17,122 --> 01:09:19,580
I mean I've got to make sure
that this blank paper does not

1395
01:09:19,580 --> 01:09:22,770
come onto that.

1396
01:09:22,770 --> 01:09:24,930
But if that's a
problem, I can probably

1397
01:09:24,930 --> 01:09:28,950
make-- I could make a
fold here, for example.

1398
01:09:28,950 --> 01:09:30,080
Shrinks that up.

1399
01:09:30,080 --> 01:09:32,130
There's lots of
things you can do.

1400
01:09:32,130 --> 01:09:34,529
This is a borderline case
whether it's yes or no.

1401
01:09:34,529 --> 01:09:36,720
I will give you an
algorithm that does it.

1402
01:09:39,410 --> 01:09:41,909
For polygons with margin.

1403
01:09:44,620 --> 01:09:47,000
Bit of a technical condition.

1404
01:09:47,000 --> 01:09:49,080
Something that is
pretty typical.

1405
01:09:49,080 --> 01:09:52,200
What I mean is the thing
you're trying to cut out

1406
01:09:52,200 --> 01:09:54,490
does not meet the
boundary of the paper.

1407
01:09:54,490 --> 01:09:55,680
There's no margin here.

1408
01:09:55,680 --> 01:09:57,240
It'd be hard to print out.

1409
01:09:57,240 --> 01:09:59,160
So I really want
something that has margin.

1410
01:09:59,160 --> 01:10:00,787
That's a typical
case we care about.

1411
01:10:00,787 --> 01:10:02,620
We actually need this
for the proof to work.

1412
01:10:02,620 --> 01:10:04,370
We also need that
it's a single polygon.

1413
01:10:09,160 --> 01:10:11,760
It does not work
with general graphs.

1414
01:10:11,760 --> 01:10:12,589
This algorithm.

1415
01:10:12,589 --> 01:10:14,380
Because more complicated
things can happen.

1416
01:10:14,380 --> 01:10:15,400
It might be [INAUDIBLE].

1417
01:10:15,400 --> 01:10:19,290
for all we know,
the general case.

1418
01:10:19,290 --> 01:10:25,450
So here's the algorithm that
I'll give you in number form.

1419
01:10:25,450 --> 01:10:28,730
First thing you do is
guess the first fold.

1420
01:10:28,730 --> 01:10:31,880
This is a powerful
idea that even

1421
01:10:31,880 --> 01:10:34,460
most algorithmisists
don't necessarily know.

1422
01:10:34,460 --> 01:10:37,394
The idea is what could
the first fold be?

1423
01:10:37,394 --> 01:10:39,060
Has to be a line of
reflection symmetry.

1424
01:10:39,060 --> 01:10:41,270
Turns out there's a linear
number [INAUDIBLE] most.

1425
01:10:41,270 --> 01:10:43,310
You can find them
in linear time.

1426
01:10:43,310 --> 01:10:45,120
All these good algorithms
for finding them.

1427
01:10:45,120 --> 01:10:47,390
But which fold do I make first?

1428
01:10:47,390 --> 01:10:48,970
The answer is I don't care.

1429
01:10:48,970 --> 01:10:51,190
Let's just try them
all one at a time.

1430
01:10:51,190 --> 01:10:52,510
This is what I call guessing.

1431
01:10:52,510 --> 01:10:54,990
Just imagine from now on
that we made the right guess,

1432
01:10:54,990 --> 01:10:57,250
but if you end up failing
later on this algorithm,

1433
01:10:57,250 --> 01:10:59,020
just go back here
try the next one.

1434
01:10:59,020 --> 01:11:00,937
There's only N of them to try.

1435
01:11:00,937 --> 01:11:02,770
So you're going to
multiply the running time

1436
01:11:02,770 --> 01:11:06,809
of the rest of the algorithm
by N, and if this is N to N

1437
01:11:06,809 --> 01:11:08,850
squared-- which I think
the rest of the algorithm

1438
01:11:08,850 --> 01:11:11,010
is N squared-- the whole
algorithm will be N cubed

1439
01:11:11,010 --> 01:11:12,635
because you just run
this over and over

1440
01:11:12,635 --> 01:11:13,894
for each possible first fold.

1441
01:11:13,894 --> 01:11:16,060
We don't have a great theory
to find the first fold.

1442
01:11:16,060 --> 01:11:18,270
Just try them all.

1443
01:11:18,270 --> 01:11:20,160
That's step one.

1444
01:11:20,160 --> 01:11:25,710
Step two-- that's the only
guess we're going to make.

1445
01:11:25,710 --> 01:11:28,840
Fold down to convex hull.

1446
01:11:28,840 --> 01:11:32,390
This is a central idea.

1447
01:11:32,390 --> 01:11:36,490
So we have this polygon
we want to make.

1448
01:11:36,490 --> 01:11:38,049
There's all this
extra blank paper.

1449
01:11:38,049 --> 01:11:39,340
I don't like extra blank paper.

1450
01:11:39,340 --> 01:11:40,440
Just get rid of it.

1451
01:11:40,440 --> 01:11:43,230
Make lots of folds to fold
the blank paper onto itself

1452
01:11:43,230 --> 01:11:46,720
until it gets so tiny it just
goes slightly around the convex

1453
01:11:46,720 --> 01:11:47,820
hull.

1454
01:11:47,820 --> 01:11:49,940
Convex hull is the
smallest convex polygon

1455
01:11:49,940 --> 01:11:51,714
that contains your shape.

1456
01:11:51,714 --> 01:11:52,630
So it'll be like that.

1457
01:11:52,630 --> 01:11:55,630
It'll reduce the
paper down to that.

1458
01:11:55,630 --> 01:11:57,880
And I do this a lot.

1459
01:11:57,880 --> 01:11:58,822
I might as well.

1460
01:11:58,822 --> 01:12:01,280
It makes the problem easier
because I have less blank space

1461
01:12:01,280 --> 01:12:01,740
to worry about.

1462
01:12:01,740 --> 01:12:03,739
Blank space is a problem
because if I [? fold ?]

1463
01:12:03,739 --> 01:12:06,120
blank space onto
a cut, it's bad.

1464
01:12:06,120 --> 01:12:06,880
It's not allowed.

1465
01:12:09,480 --> 01:12:16,380
So the next thing we do is make
the best possible safe fold.

1466
01:12:19,820 --> 01:12:23,276
I need to define
that, but a safe fold

1467
01:12:23,276 --> 01:12:24,900
is just the folds
we're trying to make,

1468
01:12:24,900 --> 01:12:27,630
which is locally they
are lines of symmetry.

1469
01:12:27,630 --> 01:12:31,160
So like this one was a good
fold after I made this fold.

1470
01:12:31,160 --> 01:12:34,010
So this is-- all the
right stuff here is gone.

1471
01:12:34,010 --> 01:12:36,580
It's a good fold because
it folds this on to this,

1472
01:12:36,580 --> 01:12:38,960
and it folds blank space
onto this blank space.

1473
01:12:38,960 --> 01:12:42,640
So anything that comes into-- on
top of each other is identical.

1474
01:12:42,640 --> 01:12:45,780
That's a safe fold.

1475
01:12:45,780 --> 01:12:50,140
And repeat.

1476
01:12:50,140 --> 01:12:52,720
That's the algorithm.

1477
01:12:52,720 --> 01:12:53,920
How does that work?

1478
01:12:58,750 --> 01:13:00,512
Why does this work?

1479
01:13:00,512 --> 01:13:02,220
So it's the obvious
algorithm, basically.

1480
01:13:02,220 --> 01:13:05,080
It says make safe folds
until you're done.

1481
01:13:05,080 --> 01:13:08,120
If you finish, then you're done.

1482
01:13:08,120 --> 01:13:09,470
And then the answer's yes.

1483
01:13:09,470 --> 01:13:11,720
If you don't finish-- which
is a little hard to check

1484
01:13:11,720 --> 01:13:14,480
because you can always
make microscopic folds--

1485
01:13:14,480 --> 01:13:16,600
but you take the
limit and-- if it's

1486
01:13:16,600 --> 01:13:18,960
possible to do this
in polynomial time--

1487
01:13:18,960 --> 01:13:20,130
you find that out.

1488
01:13:20,130 --> 01:13:23,750
Turns out I can't finish
by making safe folds.

1489
01:13:23,750 --> 01:13:25,340
Then you can show
that actually there

1490
01:13:25,340 --> 01:13:29,420
was no way to make that
pattern by simple folds.

1491
01:13:29,420 --> 01:13:32,790
So let me give you an
idea of why that's true.

1492
01:13:36,210 --> 01:13:39,930
After I make a single
fold-- the first fold--

1493
01:13:39,930 --> 01:13:41,990
my picture looks like this.

1494
01:13:41,990 --> 01:13:44,275
It's no longer a polygon.

1495
01:13:44,275 --> 01:13:45,700
It's like half a polygon.

1496
01:13:45,700 --> 01:13:48,204
It ends at the
boundary of the paper.

1497
01:13:48,204 --> 01:13:50,120
It begins and ends with
the boundary of paper,

1498
01:13:50,120 --> 01:13:51,703
and you have some
chain in the middle.

1499
01:13:51,703 --> 01:13:52,830
We call this a passage.

1500
01:13:52,830 --> 01:13:54,870
It's like a path
you wander along.

1501
01:13:54,870 --> 01:13:57,430
And I want to somehow bring
all those edges into alignment.

1502
01:13:59,970 --> 01:14:01,800
Here I'm already
using it as a polygon.

1503
01:14:01,800 --> 01:14:04,008
If it weren't a polygon,
there might be more than one

1504
01:14:04,008 --> 01:14:05,280
of these.

1505
01:14:05,280 --> 01:14:06,550
Now I want to make a fold.

1506
01:14:06,550 --> 01:14:09,210
For example, this fold is
safe because it folds this

1507
01:14:09,210 --> 01:14:11,740
on to this, and it folds
blank space onto blank space

1508
01:14:11,740 --> 01:14:12,990
if I do it right.

1509
01:14:18,140 --> 01:14:19,390
And keep doing that.

1510
01:14:19,390 --> 01:14:22,220
Now when I make a fold
like this, what happens

1511
01:14:22,220 --> 01:14:25,810
is I can think of this region
that I folded-- the smaller

1512
01:14:25,810 --> 01:14:28,240
side-- as disappearing.

1513
01:14:28,240 --> 01:14:30,940
It just got absorbed
into the paper here.

1514
01:14:30,940 --> 01:14:33,850
So the graph that I was
trying to line up got smaller.

1515
01:14:33,850 --> 01:14:35,570
That's clearly a good thing.

1516
01:14:35,570 --> 01:14:37,020
Makes my problem easier.

1517
01:14:37,020 --> 01:14:40,210
The piece of paper I was
folding also got smaller.

1518
01:14:40,210 --> 01:14:42,300
That's a good thing.

1519
01:14:42,300 --> 01:14:45,610
But that's not always true.

1520
01:14:45,610 --> 01:14:48,745
Suppose you had a piece
of paper like this,

1521
01:14:48,745 --> 01:14:51,020
which could happen
after a bunch folding,

1522
01:14:51,020 --> 01:14:53,270
and then you fold
along a line like that

1523
01:14:53,270 --> 01:14:54,880
because, for
example, your passage

1524
01:14:54,880 --> 01:14:56,700
looks like that or something.

1525
01:14:56,700 --> 01:14:59,910
When I make the fold--
this has to go off.

1526
01:14:59,910 --> 01:15:04,880
When I make the fold I
get this crazy thing.

1527
01:15:04,880 --> 01:15:06,600
Not drawn to scale.

1528
01:15:06,600 --> 01:15:09,380
And this polygon does not
fit inside this polygon.

1529
01:15:09,380 --> 01:15:11,910
So my paper got
bigger in some places.

1530
01:15:11,910 --> 01:15:14,110
And that's a worry, because
now I have the stuff--

1531
01:15:14,110 --> 01:15:15,960
maybe it happens
to be blank space.

1532
01:15:15,960 --> 01:15:18,240
Maybe there's other
junk that got out here.

1533
01:15:18,240 --> 01:15:20,027
And now have to worry
about collisions

1534
01:15:20,027 --> 01:15:21,360
with this bigger piece of paper.

1535
01:15:21,360 --> 01:15:23,960
And this is always
our sticking point,

1536
01:15:23,960 --> 01:15:28,550
but there's some
magic you can do.

1537
01:15:28,550 --> 01:15:30,410
In fact, the picture
cannot look like this.

1538
01:15:33,110 --> 01:15:36,820
Because, look, you've got
some portion of your passage

1539
01:15:36,820 --> 01:15:38,074
to the left of the crease.

1540
01:15:38,074 --> 01:15:40,240
You've got some portion of
the passage to the right.

1541
01:15:40,240 --> 01:15:42,460
One of them has to be shorter.

1542
01:15:42,460 --> 01:15:44,980
Plus, this is a
line of symmetry.

1543
01:15:44,980 --> 01:15:49,680
So wherever I have a portion
of my passage over here,

1544
01:15:49,680 --> 01:15:51,660
I will have a portion
of my passage over here

1545
01:15:51,660 --> 01:15:52,880
until I run out of length.

1546
01:15:52,880 --> 01:15:54,420
One of them is shorter.

1547
01:15:54,420 --> 01:15:57,360
So the shorter one like this
one gets totally absorbed

1548
01:15:57,360 --> 01:15:58,670
by the larger one.

1549
01:15:58,670 --> 01:16:01,410
So the shorter side
always disappears.

1550
01:16:01,410 --> 01:16:04,620
So in this picture, I have
the long side of my passage,

1551
01:16:04,620 --> 01:16:07,210
and it's really a
subset of the original.

1552
01:16:07,210 --> 01:16:10,330
If I reduce this
to the convex hull,

1553
01:16:10,330 --> 01:16:14,460
like this, this
stuff disappears.

1554
01:16:14,460 --> 01:16:18,430
And in general, if you do
this fold down to convex hull,

1555
01:16:18,430 --> 01:16:21,740
this repeat goes
back to step two.

1556
01:16:21,740 --> 01:16:23,240
If you fold down
to the convex hull,

1557
01:16:23,240 --> 01:16:25,573
you can show that not only
does your passage-- the thing

1558
01:16:25,573 --> 01:16:28,150
you want to cut out-- get
smaller, but you piece of paper

1559
01:16:28,150 --> 01:16:30,060
also gets smaller.

1560
01:16:30,060 --> 01:16:30,610
Guaranteed.

1561
01:16:30,610 --> 01:16:33,710
And once you know the paper get
smaller and your things you're

1562
01:16:33,710 --> 01:16:37,740
trying to align get smaller, you
know that every move is safe.

1563
01:16:37,740 --> 01:16:40,830
So you never get stuck by
following this algorithm.

1564
01:16:40,830 --> 01:16:43,440
This works for
polygons with margin,

1565
01:16:43,440 --> 01:16:45,760
but not in any other situation
as far as we can tell.

1566
01:16:48,626 --> 01:16:49,125
Cool.

1567
01:16:52,426 --> 01:16:54,050
The last thing I
wanted to leave you on

1568
01:16:54,050 --> 01:17:00,440
is going out a little way
from regular 2D flat sheets

1569
01:17:00,440 --> 01:17:01,530
of paper.

1570
01:17:01,530 --> 01:17:04,920
You can generalize
and go up a dimension

1571
01:17:04,920 --> 01:17:09,000
to folding polyhedra surface.

1572
01:17:09,000 --> 01:17:10,630
Here, a surface of a polyhedron.

1573
01:17:10,630 --> 01:17:11,713
You've probably done this.

1574
01:17:11,713 --> 01:17:15,730
You take a cereal box and
you can collapse it flat.

1575
01:17:15,730 --> 01:17:17,180
Is that always possible?

1576
01:17:17,180 --> 01:17:21,210
It's called the flattening
problem, and the answer is yes.

1577
01:17:21,210 --> 01:17:23,390
And you can think of it
as a fold and cut problem,

1578
01:17:23,390 --> 01:17:24,889
because of the fold
and cut problem,

1579
01:17:24,889 --> 01:17:27,070
you have some polygon
like this diamond.

1580
01:17:27,070 --> 01:17:29,170
You make some
collection of folds

1581
01:17:29,170 --> 01:17:32,406
that brings the boundary
of the diamond to align.

1582
01:17:32,406 --> 01:17:34,030
So if you forget
about what's happening

1583
01:17:34,030 --> 01:17:35,220
on the inside of
the paper-- you just

1584
01:17:35,220 --> 01:17:37,250
look at the boundary
of the paper-- you're

1585
01:17:37,250 --> 01:17:41,410
folding that one-dimensional
boundary so that it collapses

1586
01:17:41,410 --> 01:17:43,590
down to a single line.

1587
01:17:43,590 --> 01:17:46,840
What I want to do is
this up a dimension.

1588
01:17:46,840 --> 01:17:51,510
I take a 3D cube of
paper-- solid cube.

1589
01:17:51,510 --> 01:17:55,770
I have embedded within
it some polygons

1590
01:17:55,770 --> 01:18:00,220
that I want to bring
to a common plane.

1591
01:18:00,220 --> 01:18:04,690
And I want to fold the solid
cube through four dimensions,

1592
01:18:04,690 --> 01:18:07,520
but flat so it ends up
back in three dimensions.

1593
01:18:07,520 --> 01:18:11,280
I get a different 3D solid,
but with multiple layers

1594
01:18:11,280 --> 01:18:13,030
right on top of each
other-- little bit

1595
01:18:13,030 --> 01:18:14,727
a fourth dimension
hanging out there.

1596
01:18:14,727 --> 01:18:16,310
But if I just look
at what's happening

1597
01:18:16,310 --> 01:18:17,792
to the boundary
of my polyhedron--

1598
01:18:17,792 --> 01:18:20,000
say I start with a dodecahedron
or something embedded

1599
01:18:20,000 --> 01:18:22,670
in there-- and I want
to fold this thing

1600
01:18:22,670 --> 01:18:24,430
so all the sides
of the dodecahedron

1601
01:18:24,430 --> 01:18:25,980
come to a common plane.

1602
01:18:25,980 --> 01:18:28,210
That is the 3D
folding cup problem.

1603
01:18:28,210 --> 01:18:29,980
It remains unsolved.

1604
01:18:29,980 --> 01:18:33,390
I suspect it's possible to solve
even with straight skeletons

1605
01:18:33,390 --> 01:18:35,620
and perpendiculars,
but it's really hard

1606
01:18:35,620 --> 01:18:37,100
to draw the pictures.

1607
01:18:37,100 --> 01:18:40,320
So we have not resolved
that one way or the other.

1608
01:18:40,320 --> 01:18:41,930
But the boundary
problem-- forget

1609
01:18:41,930 --> 01:18:43,870
about what's happening to
the interior and the exterior

1610
01:18:43,870 --> 01:18:44,720
of the dodecahedron.

1611
01:18:44,720 --> 01:18:46,886
If you just look at the
surface of the dodecahedron,

1612
01:18:46,886 --> 01:18:49,630
that you can fold
in 3D-- we think--

1613
01:18:49,630 --> 01:18:53,500
and you can show
and burn in haze

1614
01:18:53,500 --> 01:18:56,330
from the complexity proof
a couple lectures ago,

1615
01:18:56,330 --> 01:19:00,460
and also on this disk-packing
method with our co-authors.

1616
01:19:00,460 --> 01:19:03,350
They prove just two
years ago that if you

1617
01:19:03,350 --> 01:19:08,720
have any orientable manifold,
which is things like polyhedron

1618
01:19:08,720 --> 01:19:13,315
but no Mobius strips, no Klein
bundles, and other ugly things.

1619
01:19:13,315 --> 01:19:14,940
They have to be
manifold, so you're not

1620
01:19:14,940 --> 01:19:20,760
allowed to join three triangles
together along a single edge.

1621
01:19:20,760 --> 01:19:23,010
That would be forbidden.

1622
01:19:23,010 --> 01:19:24,970
So it's locally flat.

1623
01:19:24,970 --> 01:19:27,870
In such a case, you
can flatten the thing.

1624
01:19:27,870 --> 01:19:32,530
And the proof is very similar to
the disk-packing method of fold

1625
01:19:32,530 --> 01:19:34,570
and cut, and in the
textbook we talk

1626
01:19:34,570 --> 01:19:36,722
about how do you apply
that to do something

1627
01:19:36,722 --> 01:19:37,805
that's just like a sphere.

1628
01:19:37,805 --> 01:19:39,240
A regular polyhedron.

1629
01:19:39,240 --> 01:19:41,490
That's pretty easy to do
with the disk-packing method.

1630
01:19:41,490 --> 01:19:42,989
They generalize it
to the case where

1631
01:19:42,989 --> 01:19:49,045
you have polyhedral doughnuts
and all sorts of fun things.

1632
01:19:49,045 --> 01:19:50,670
But there's tons of
open problems here.

1633
01:19:50,670 --> 01:19:52,128
So we know how to
flatten surfaces,

1634
01:19:52,128 --> 01:19:55,190
and that's useful for things
like folding airbags flat.

1635
01:19:55,190 --> 01:19:58,660
But can you fold
the 3D solid flat?

1636
01:19:58,660 --> 01:20:02,530
You can think of--
we have here 1D edges

1637
01:20:02,530 --> 01:20:06,890
which we are collapsing
to a 1D line.

1638
01:20:06,890 --> 01:20:09,900
There is also zero
dimensional points here,

1639
01:20:09,900 --> 01:20:12,610
which we don't bring
to a single point.

1640
01:20:12,610 --> 01:20:15,090
It'd be nice if you could--
the generalized fold and cup

1641
01:20:15,090 --> 01:20:17,430
problem is you take a
D dimensional thing,

1642
01:20:17,430 --> 01:20:20,180
and you have all of
these-- there's 0, 1, 2,

1643
01:20:20,180 --> 01:20:22,720
3-- up to D dimensional
parts to it--

1644
01:20:22,720 --> 01:20:24,340
or D minus
one-dimensional parts.

1645
01:20:24,340 --> 01:20:26,600
You want to bring each of
them down into alignment

1646
01:20:26,600 --> 01:20:29,550
so that all the vertices
come to common point,

1647
01:20:29,550 --> 01:20:31,702
all the edges come
to a common line,

1648
01:20:31,702 --> 01:20:33,910
all the two-dimensional
faces come to a common plane,

1649
01:20:33,910 --> 01:20:36,820
and so on up the
dimension hierarchy.

1650
01:20:36,820 --> 01:20:38,330
That is the ultimate
open problem.

1651
01:20:38,330 --> 01:20:42,890
I think we end the book with
it, and it's totally unsolved.

1652
01:20:42,890 --> 01:20:43,490
Any questions?

1653
01:20:46,256 --> 01:20:47,630
That's folding
and cutting paper,

1654
01:20:47,630 --> 01:20:50,000
and next time we'll
start linkages.