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PROFESSOR: I'm excited
about today's lecture

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because there's so
many fun topics.

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This is like many fun
things all in one lecture.

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We're going to start with a
cool little problem, which

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is about unfolding
and re-folding.

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You could think
of them-- they're

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kind of like hinged
dissections although they're

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from between surfaces
of polyhedra.

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You can also think of
them as common unfoldings.

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So general idea is,
you have over here

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in the left team you have
unfoldings-- polygons,

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let's say.

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Possible things that
could fold into polyhedra.

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And we'll think convex again.

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And when we were
thinking about unfolding,

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we we're thinking about
going this direction.

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When we were thinking
about folding,

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we were thinking
about this direction.

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But what if we do
both repeatedly?

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So I start, say, with a cube.

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And then I unfold it
and then I re-fold it.

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That's sort of what the
metamorphosis of the cube

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was about.

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And then I unfold
that and I re-fold it,

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and I unfold and I re-fold
and I unfold and re-fold.

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This is a fun tour to take.

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Is this space connected?

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Can I get from any
polyhedral surface

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to any other-- say, convex--
by unfolding and re-folding?

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We have no idea.

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So while each of
these operations

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we know a fair amount about--
we know how to generally unfold

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convex polyhedra,
we know how to find

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all of the gluings of a polygon
into a polyhedron-- repeating

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those processes,
we could explore

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but actually no ones' every
tried that algorithmically.

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It'd be fun to see what
crazy things you get.

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But we have lots of examples
and partial progress

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towards this question.

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So here's the first example.

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This appears in the book.

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It's by Hirata from 2000.

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He was one of the guys who
implemented a gluing algorithm.

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And this is an unfolding--
a common unfolding--

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of a cube-- ah, not quite--
and a regular tetrahedron.

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So it's actually
a box not a cube.

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One of the big open
questions in this area

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is, can you find
a common unfolding

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of a perfect cube and a regular
tetrahedron that's open?

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Marty posed that
question I think

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in 1998, in the good old days of
the beginning of folding stuff.

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And this is almost an answer
but it's not quite a cube,

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as a result.

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You could see-- I think
it's pretty easy to see

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how to fold it into a box just
with extending all the lines.

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And the dotted lines
show how to fold

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into a regular tetrahedron.

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That's one example.

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You can also fold-- this thing
makes both a regular octahedron

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instead of a regular tetrahedron
and some kind of tetrahedra.

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It's not regular.

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We call it a tetramonohedron,
meaning each of the sides

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are the same.

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So it's just one type of side,
but there's four of them.

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And I guess the red lines
are folding into that,

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the green lines fold into
a regular tetrahedron.

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I feel like I'm
getting forgetful.

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I don't even remember where
this example comes from.

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But it's in the book, it's
not attributed to anyone

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so I assume Joe invented it.

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But recently, like
just this year,

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inspired a whole bunch of
examples by these guys.

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I just saw the talk
last week in China--

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that's where I was traveling,
it was a geometry conference.

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And Horiyama and Uehara
talked about a bunch

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of different common unfoldings.

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This is, again, of
a regular octahedron

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and a tetramonohedron.

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But a different example.

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But they kept going.

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Said all right, let's do
all the platonic solids.

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So this is a common unfolding
of a cube and a tetramonohedron.

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Cube-- actually two of them.

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You can see how it makes a cube.

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That's obvious.

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But the tetramonohedron
are the blue lines.

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Then I have the octahedron,
we have the icosahedron.

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I don't think they
have a dodecahedron yet

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because all of these
approaches are actually

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based on dissection
techniques where

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you take a tiling of-- here
are equilateral triangles

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for the icosahedron,
and you take

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a tiling of some
triangle that's going

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to make the tetramonohedron,
and you align them

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so that things work out nicely.

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And that's not
always well defined,

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but it often leads
to good dissections,

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often hinged dissections.

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It's the way, for
example, you show

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you can hinge
dissect a rectangle

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into any other rectangle.

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You take tilings of
different rectangles

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and overlay them appropriately.

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Here, same approach,
but with a dodecahedron,

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regular pentagons
don't tile so it's

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very hard to use that approach.

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So this is as far
as they've gotten.

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Here they've measured
the proportions

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of the tetramonohedron
to give you an idea.

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It's close to regular.

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But again, open question,
can any two platonic solids--

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do any two have a
common unfolding?

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They proved in this paper,
if you restrict one of them

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to be an edge unfolding
and the other one

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can be a general unfolding
then it's not possible.

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But if they're both general
unfoldings, who knows.

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Here you see it's edge
unfolding for the icosahedron,

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but general unfolding
for the other.

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Cool.

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But very recently,
this is, like,

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hasn't even been released yet.

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This is the premier
of the following,

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I'll call it conjecture.

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I think it's not yet
approved or explicitly-- it's

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not certain that it's correct.

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But the claim is this gray
shape, taken to the limit--

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so this is the first
iteration of the fractal,

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this is the second iteration
of the fractal-- in the limit,

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will fold both to a regular
tetrahedron and a cube.

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This would be an
amazing breakthrough.

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Still, the conjecture
is that it is not

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possible to go from
a regular tetrahedron

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to a cube with a regular--
like a normal polygon

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with a finite number of sides.

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But with an infinite number of
sides, it seems to be possible.

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Which is awesome.

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This is-- I haven't
met this guy yet.

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He's a professional puzzler
in Japan, Toshihiro Shirakawa.

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So stay tuned for maybe a proof
of that theorem, that claim.

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The next topic--
this started back

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in '99, inspired
by this question.

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We thought well,
what about boxes?

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And this is a common unfolding
of two different size boxes

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which is cool.

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And it's generated in
a fairly intuitive way.

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You start with an
unfolding of a box.

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This makes a one
by one by three box

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if you think about it
for a little while.

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Just wrap around-- this
is like one top side, 1

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by 3 rectangle and wrap around.

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And then you-- you
draw the gluing-- here

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I've labeled which edges
glue to which edges.

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And for every letter,
like the two Gs

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here, I could imagine adding a
triangle here and removing it

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from here.

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That will still fold to
the same thing, right?

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So I get to choose
for each letter which

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guy goes in, which guy goes out.

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Then I get this shape as
a result of that process.

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And if I do it right,
when I rotate 45 degrees,

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I get a different orthogonal
unfolding of some other box.

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So here, if I started
with a root 2 by root 2

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by three root 2 box, I
get a 1 by 2 by 4 box.

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So this is a common
unfolding of both.

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I guess here is like the
center core of the 1 by 2 by 4

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and you wrap around again.

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Or this is probably center core.

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So that's pretty cool.

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This is an idea by Timothy Chan.

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Of course, natural
question is, do I

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need to do this rotation trick
or could I get away without it?

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And you can get away without it.

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This is an idea
of Therese Biedl.

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This is my Waterloo days.

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We had an open problem
session, kind of like the one

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we have in this class.

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It was the very
first one that I ran.

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I ran it with Therese.

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And she came up with this idea.

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This is a-- it's not obvious
exactly how this works

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or where she came up with this.

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It was kind of magical.

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But this the same polygon,
different creases, you

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make two different size boxes.

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Now they're going to be integers
because they're orthogonal

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aligned on the grid.

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Pretty cool.

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And for a long time, this
was the only example of this

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that we had.

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There's still an open question
which I give you A by C by-- A

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by B by C box and a D by E
by F box, is this possible?

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When is this possible?

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But at least now--
oh, here's, sorry.

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There are two examples.

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Of course, the surface
areas have to match.

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So the sum of A plus B plus
C equals D plus E plus F.

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I think that's right.

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Or no.

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It's more complicated.

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It's like A times
B plus-- and so on.

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Pairwise products.

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Now we know that
there are actually

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infinitely many examples.

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This is another fairly
new result by Uehara,

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again in 2008.

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This is-- Uehara translated
our textbook to Japanese,

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by the way.

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Another connection.

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He does a lot of
fun folding stuff.

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Just saw him last week.

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So this is an
example of taking a 1

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by 5 by 2k box and a common
unfolding of that with a 1

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by 1 by 6k plus 2 box.

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Which has a very
regular pattern.

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And this one, little more
complicated to copy and paste

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but it makes 1 by 1 by 8k
plus 11 box and a 1 by 3

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by 4k plus 5 box.

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Which is pretty cool.

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I haven't studied
these carefully,

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but there's two different
ways to fold each.

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So now we know
there are infinitely

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many of these examples.

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It's still not known
when it's possible,

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but also in this
paper Uehara came up

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with a sort of random
generation algorithm

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where he would take a
box, randomly unfold

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it, try to fold it
into other boxes.

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And he has now over 250,000
examples of this process.

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And I think these are maybe
inspired by his random search.

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I'm not sure, but this
is theory obviously.

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One particularly fun
example, this one

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actually tiles the plane.

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So it'd be efficient
if you're cutting it

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out of sheet material.

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You can make a common unfolding
of 1 by 2 by 5 box and a 1

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by 1 by 8 box.

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I think this one he
made into a puzzle

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with a paper manufacturer.

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I'm getting ahead of myself.

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Big open question
here that he asked is,

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are there-- is there a common
unfolding of three boxes?

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That remains open.

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And he feels like he's
very close to finding one

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with his random search
but hasn't yet found one.

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It's like it almost
folds into the third box.

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But it's strange how
hard it is to find them.

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Maybe they don't exist.

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That would be surprising.

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All right.

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00:11:29,430 --> 00:11:35,570
Next example on this theme,
common unfoldings, is this toy.

251
00:11:35,570 --> 00:11:40,950
So this is a 1 by 1 by 4 box,
also known as the-- as a tetra

252
00:11:40,950 --> 00:11:43,980
cubes, four cubes
joined edge to edge.

253
00:11:43,980 --> 00:11:48,480
And-- oh, I should
have practiced.

254
00:11:48,480 --> 00:11:51,160
You can unfold it.

255
00:11:51,160 --> 00:11:53,340
This is a commercially
produced toy.

256
00:11:53,340 --> 00:11:55,930
It's called Cubigami.

257
00:11:55,930 --> 00:12:00,210
And it's one net, one unfolding.

258
00:12:00,210 --> 00:12:03,100
But it can fold into
all tetra cubes.

259
00:12:03,100 --> 00:12:08,560
So that was the tube and, let's
see, what else can I make here?

260
00:12:13,870 --> 00:12:14,875
I should have practiced.

261
00:12:20,070 --> 00:12:22,840
Interesting.

262
00:12:22,840 --> 00:12:23,840
It's fun.

263
00:12:23,840 --> 00:12:25,550
You can play with it later.

264
00:12:25,550 --> 00:12:28,200
Oh, that looks interesting.

265
00:12:28,200 --> 00:12:28,700
Almost.

266
00:12:31,280 --> 00:12:31,870
Not quite.

267
00:12:31,870 --> 00:12:34,900
It's almost-- it looks
like a Tetris piece, right?

268
00:12:34,900 --> 00:12:39,140
Except I didn't quite
cover it right at the end.

269
00:12:39,140 --> 00:12:41,220
It's a good puzzle.

270
00:12:41,220 --> 00:12:42,930
Obviously.

271
00:12:42,930 --> 00:12:46,830
Maybe if I do it like--
that doesn't work.

272
00:12:46,830 --> 00:12:47,540
Wow.

273
00:12:47,540 --> 00:12:48,040
All right.

274
00:12:48,040 --> 00:12:50,310
You're going to have to
play with it yourself.

275
00:12:50,310 --> 00:12:51,980
I lose.

276
00:12:51,980 --> 00:12:56,970
But you can make all of
the four cube joinings

277
00:12:56,970 --> 00:13:06,595
except-- except for this one.

278
00:13:09,344 --> 00:13:10,760
See, the problem
with this example

279
00:13:10,760 --> 00:13:13,430
is it doesn't have the same
surface areas the rest.

280
00:13:13,430 --> 00:13:15,710
And you have to
conserve surface area.

281
00:13:15,710 --> 00:13:17,570
But if you think about
all the other four

282
00:13:17,570 --> 00:13:24,240
cubes-- like I was trying to
make this guy or this guy--

283
00:13:24,240 --> 00:13:31,580
it's like Tetris
pieces-- but also things

284
00:13:31,580 --> 00:13:34,484
like-- I can draw
better than I can

285
00:13:34,484 --> 00:13:35,650
solve the puzzle apparently.

286
00:13:38,330 --> 00:13:39,000
Like this.

287
00:13:39,000 --> 00:13:41,240
This is the one non-planar one.

288
00:13:45,713 --> 00:13:47,710
All right?

289
00:13:47,710 --> 00:13:50,045
So you've got all the Tetris
pieces except this guy.

290
00:13:50,045 --> 00:13:51,320
It's the wrong surface area.

291
00:13:51,320 --> 00:13:53,660
So, in general, it's
all the tree shape ones.

292
00:13:53,660 --> 00:13:55,445
No cycles.

293
00:13:55,445 --> 00:13:56,320
Can make all of them.

294
00:13:56,320 --> 00:14:01,870
This is an example found by
Donald Knuth, famous computer

295
00:14:01,870 --> 00:14:03,120
scientist.

296
00:14:03,120 --> 00:14:07,310
He sort of accidentally
at a cocktail party heard

297
00:14:07,310 --> 00:14:10,770
about this problem from George
Miller who's a toy designer--

298
00:14:10,770 --> 00:14:13,800
or puzzle designer.

299
00:14:13,800 --> 00:14:16,020
And so then Knuth
went back and he

300
00:14:16,020 --> 00:14:19,010
wrote a program to numerate
all unfoldings of these things

301
00:14:19,010 --> 00:14:21,410
and found there are a whole
bunch of common unfoldings.

302
00:14:21,410 --> 00:14:23,350
But it's found by
exhaustive search.

303
00:14:23,350 --> 00:14:26,750
This was the one that sort of
had the smallest bounding box,

304
00:14:26,750 --> 00:14:28,740
I believe, in terms of area.

305
00:14:28,740 --> 00:14:30,809
And so it's a nice
compact thing.

306
00:14:30,809 --> 00:14:32,850
And then there are various
manufacturing's of it.

307
00:14:32,850 --> 00:14:35,640
This is the latest one that's
sort of relatively easy

308
00:14:35,640 --> 00:14:36,760
to build I guess.

309
00:14:36,760 --> 00:14:40,660
So, play with that later.

310
00:14:40,660 --> 00:14:42,090
That's four cubes.

311
00:14:42,090 --> 00:14:45,050
What about five cubes?

312
00:14:45,050 --> 00:14:48,920
Well, that was the subject
of a paper also presented

313
00:14:48,920 --> 00:14:51,340
last week in China.

314
00:14:51,340 --> 00:14:54,620
And it turns out, there
is no common unfolding

315
00:14:54,620 --> 00:14:57,770
of all 24 pentacubes.

316
00:14:57,770 --> 00:14:59,540
That's five cubes
joined together.

317
00:14:59,540 --> 00:15:05,280
But there is a common unfolding
of 23 out of 24 of them.

318
00:15:05,280 --> 00:15:06,220
So this is the paper.

319
00:15:06,220 --> 00:15:07,810
There was a bunch of authors.

320
00:15:07,810 --> 00:15:11,030
There were basically
three teams that

321
00:15:11,030 --> 00:15:14,180
worked on this, two of
which were writing code.

322
00:15:14,180 --> 00:15:15,819
I was on one of
the coding teams.

323
00:15:15,819 --> 00:15:17,360
And the third one
was trying to think

324
00:15:17,360 --> 00:15:20,370
about the problem
using mathematics.

325
00:15:20,370 --> 00:15:22,805
And for a long time, the
thinkers were leading.

326
00:15:22,805 --> 00:15:24,680
But then, of course,
eventually the computers

327
00:15:24,680 --> 00:15:29,100
win because after a couple days
of just solid computation on I

328
00:15:29,100 --> 00:15:32,000
don't know now how many cores
with each of the groups,

329
00:15:32,000 --> 00:15:34,170
we numerated all
these unfoldings.

330
00:15:34,170 --> 00:15:40,570
And so now there's, I think,
over one trillion unfoldings

331
00:15:40,570 --> 00:15:43,320
of the 1 by 1 by 5 tube.

332
00:15:43,320 --> 00:15:45,960
But one trillions not that bad.

333
00:15:45,960 --> 00:15:48,542
Now, we could not
do this for six.

334
00:15:48,542 --> 00:15:52,070
But for five it wasn't so bad.

335
00:15:52,070 --> 00:15:55,410
And this is-- there's a bunch
of unfoldings of 23 out of 24.

336
00:15:55,410 --> 00:15:57,150
But that's the max you can get.

337
00:15:57,150 --> 00:15:58,220
Little disappointing.

338
00:15:58,220 --> 00:16:00,115
AUDIENCE: Which pentacube lost?

339
00:16:00,115 --> 00:16:01,990
PROFESSOR: I don't know
which pentacube lost.

340
00:16:01,990 --> 00:16:03,090
I should find out.

341
00:16:03,090 --> 00:16:05,580
That's a good question.

342
00:16:05,580 --> 00:16:07,260
But I do have some
other answers.

343
00:16:07,260 --> 00:16:09,550
So in some sense,
the hard ones I think

344
00:16:09,550 --> 00:16:11,180
are the non-planar guys.

345
00:16:11,180 --> 00:16:12,950
Like this guy.

346
00:16:12,950 --> 00:16:15,180
So they're planar
if all the cubes

347
00:16:15,180 --> 00:16:18,900
lie in a thickened
plane, thickened by one.

348
00:16:18,900 --> 00:16:22,180
This guy will fold into
all the non-planar cubes

349
00:16:22,180 --> 00:16:25,670
and also fold into
most of the pentacubes.

350
00:16:25,670 --> 00:16:28,110
There are only 24 total
so it's only missing two.

351
00:16:28,110 --> 00:16:30,310
And this'll fold into
all the non-planar guys.

352
00:16:30,310 --> 00:16:32,330
And what's cool is
this one is unique.

353
00:16:32,330 --> 00:16:36,210
There's only one common
unfolding of those guys.

354
00:16:36,210 --> 00:16:38,022
AUDIENCE: How many
non-planar [INAUDIBLE]?

355
00:16:38,022 --> 00:16:39,730
PROFESSOR: How many
non-planar are there?

356
00:16:39,730 --> 00:16:41,700
I don't remember.

357
00:16:41,700 --> 00:16:43,940
Maybe 10.

358
00:16:43,940 --> 00:16:46,680
I think it's a minority.

359
00:16:46,680 --> 00:16:48,180
But there are still
a bunch of them.

360
00:16:48,180 --> 00:16:49,929
And I think this would
make a good puzzle.

361
00:16:49,929 --> 00:16:52,075
If you're going to
manufacture-- if you guys want

362
00:16:52,075 --> 00:16:54,670
to do a start up, here you go.

363
00:16:54,670 --> 00:16:57,820
Freely available.

364
00:16:57,820 --> 00:16:59,800
If you want to do
the planar pentacubes

365
00:16:59,800 --> 00:17:02,430
there are a lot of common
unfoldings of them.

366
00:17:02,430 --> 00:17:04,780
This is one of the
fairly compact ones.

367
00:17:04,780 --> 00:17:07,186
And this one, I have
an animation of.

368
00:17:07,186 --> 00:17:08,560
Or, actually, not
this particular

369
00:17:08,560 --> 00:17:11,130
unfolding, but this concept.

370
00:17:11,130 --> 00:17:12,359
And that is here.

371
00:17:14,946 --> 00:17:16,839
Yes, I trust myself.

372
00:17:16,839 --> 00:17:20,270
So here is unfolding
and re-folding

373
00:17:20,270 --> 00:17:21,859
of that particular shape.

374
00:17:21,859 --> 00:17:23,849
And then I hit
Space bar and it's

375
00:17:23,849 --> 00:17:28,260
the same unfolding but
a different polycube.

376
00:17:28,260 --> 00:17:32,470
And this is going to go
through all of the flat ones.

377
00:17:32,470 --> 00:17:34,930
And that animation is
done heuristically.

378
00:17:34,930 --> 00:17:38,520
I don't have any theory
that that's always possible.

379
00:17:38,520 --> 00:17:40,630
But always keeping
the same unfolding,

380
00:17:40,630 --> 00:17:42,986
just changing where
the creases are.

381
00:17:42,986 --> 00:17:43,610
That's so cool.

382
00:17:47,416 --> 00:17:48,290
It's funny, actually.

383
00:17:48,290 --> 00:17:51,850
The person who implemented this
was the guy who was thinking,

384
00:17:51,850 --> 00:17:54,340
not the coder-- coder teams.

385
00:17:54,340 --> 00:17:55,510
I guess he had free time.

386
00:17:58,732 --> 00:18:00,190
This is by [? Karim ?]
[INAUDIBLE].

387
00:18:03,380 --> 00:18:06,357
I guess you can count them
and then subtract from 24

388
00:18:06,357 --> 00:18:08,440
and you'll get how many
non-planar ones there are.

389
00:18:08,440 --> 00:18:12,150
But I've lost track
of where we are.

390
00:18:12,150 --> 00:18:13,580
Always same unfolding.

391
00:18:13,580 --> 00:18:15,494
This is a fairly nice,
simple unfolding.

392
00:18:15,494 --> 00:18:16,660
It's mostly one dimensional.

393
00:18:20,270 --> 00:18:20,840
All right.

394
00:18:20,840 --> 00:18:23,500
You get the idea, right?

395
00:18:23,500 --> 00:18:25,830
Pretty fun.

396
00:18:25,830 --> 00:18:27,840
All right.

397
00:18:27,840 --> 00:18:31,990
So, right.

398
00:18:31,990 --> 00:18:33,110
Do I have anymore?

399
00:18:33,110 --> 00:18:35,450
I think that was the
end of my slides.

400
00:18:35,450 --> 00:18:38,510
I do have one more example in
the notes which is, if you just

401
00:18:38,510 --> 00:18:43,200
want to make the flat
guys-- so just the two

402
00:18:43,200 --> 00:18:47,030
dimensional
examples-- so we know

403
00:18:47,030 --> 00:18:52,040
we could do that with five,
but you can also do with six.

404
00:18:52,040 --> 00:18:55,440
We wrote another
program that looks

405
00:18:55,440 --> 00:18:58,150
at a very special
kind of unfolding.

406
00:18:58,150 --> 00:19:01,690
And in that special class,
it's possible to fold

407
00:19:01,690 --> 00:19:05,060
all planar six
cubes-- hexacubes.

408
00:19:05,060 --> 00:19:11,700
But that class does not work
for septacubes-- seven cubes.

409
00:19:11,700 --> 00:19:13,730
Open question, is there
a common unfolding

410
00:19:13,730 --> 00:19:17,030
of all planar seven cubes or
all non-planar seven cubes?

411
00:19:17,030 --> 00:19:18,440
Or how many can you make?

412
00:19:18,440 --> 00:19:23,475
There it was beyond exhaustion,
and we have to actually think.

413
00:19:23,475 --> 00:19:25,350
Unless you come up with
a good class and then

414
00:19:25,350 --> 00:19:27,870
you can exhaust
it in that class.

415
00:19:27,870 --> 00:19:31,740
One nice open question here
is, if you go to large N,

416
00:19:31,740 --> 00:19:35,600
are there some two polycubes
that have no common unfolding?

417
00:19:35,600 --> 00:19:37,180
At the moment, it
could be every pair

418
00:19:37,180 --> 00:19:39,170
of polycubes has a
common unfolding.

419
00:19:39,170 --> 00:19:40,870
Certainly there
are-- for example,

420
00:19:40,870 --> 00:19:43,020
in this class there
are four polycubes that

421
00:19:43,020 --> 00:19:45,260
do not have a common unfolding.

422
00:19:45,260 --> 00:19:48,370
I think we're not
sure about three.

423
00:19:48,370 --> 00:19:50,475
For size five.

424
00:19:50,475 --> 00:19:52,350
Of course, we don't even
know whether there's

425
00:19:52,350 --> 00:19:55,634
any polycube that
has no unfolding.

426
00:19:55,634 --> 00:19:56,800
I think there should be one.

427
00:19:56,800 --> 00:19:58,870
But at the moment, it
could be every polycube

428
00:19:58,870 --> 00:20:00,040
has an unfolding.

429
00:20:00,040 --> 00:20:02,470
But to make it a little harder,
what about two polycubes

430
00:20:02,470 --> 00:20:04,382
at once you want a
common unfolding?

431
00:20:04,382 --> 00:20:05,590
All these questions are open.

432
00:20:08,410 --> 00:20:09,810
All right.

433
00:20:09,810 --> 00:20:11,510
This is common unfolding.

434
00:20:11,510 --> 00:20:15,840
We're going to stick
with unfolding stuff

435
00:20:15,840 --> 00:20:18,550
but change around the
problem a little bit.

436
00:20:18,550 --> 00:20:21,790
So let me ask you a question.

437
00:20:21,790 --> 00:20:25,540
Can have a vote for those who
haven't read the notes already

438
00:20:25,540 --> 00:20:28,700
and know the answer.

439
00:20:28,700 --> 00:20:33,510
So let's say,
orthogonal polyhedron.

440
00:20:33,510 --> 00:20:36,270
We talk a lot about
orthogonal polyhedra.

441
00:20:36,270 --> 00:20:41,790
See all this polycube
stuff is in that genre.

442
00:20:41,790 --> 00:20:44,480
And so way back when we
talked about grid unfolding

443
00:20:44,480 --> 00:20:46,820
of orthogonal polyhedra.

444
00:20:46,820 --> 00:20:48,502
It's not known whether
that's possible,

445
00:20:48,502 --> 00:20:49,960
but we have epsilon
unfolding where

446
00:20:49,960 --> 00:20:52,580
you cut exponentially
many times and so on.

447
00:20:52,580 --> 00:20:56,100
But how do you define
orthogonal polyhedra?

448
00:20:56,100 --> 00:20:58,520
There are two
natural definitions.

449
00:20:58,520 --> 00:21:01,820
One is that the
angles are right.

450
00:21:05,410 --> 00:21:08,615
Right angles, let's
say, in the faces.

451
00:21:11,550 --> 00:21:15,560
The other natural definition
is that you look at two faces

452
00:21:15,560 --> 00:21:18,320
and the angles between
those should be right.

453
00:21:23,845 --> 00:21:25,220
So I guess right
dihedral angles.

454
00:21:30,080 --> 00:21:31,985
Right face angles or
right dihedral angles.

455
00:21:31,985 --> 00:21:36,830
The question is, are
these the same thing?

456
00:21:36,830 --> 00:21:39,830
Who thinks yes?

457
00:21:39,830 --> 00:21:41,431
Who thinks no?

458
00:21:41,431 --> 00:21:41,930
Yeah.

459
00:21:41,930 --> 00:21:42,950
I kind of set it up.

460
00:21:42,950 --> 00:21:45,935
Everyone thinks no.

461
00:21:45,935 --> 00:21:46,810
They're not the same.

462
00:21:49,480 --> 00:21:53,720
But for genus zero
polyhedra, they are the same.

463
00:21:53,720 --> 00:21:55,990
That's maybe the
surprising thing.

464
00:21:55,990 --> 00:22:01,180
This is a genus
eight polyhedron.

465
00:22:01,180 --> 00:22:02,450
No, seven.

466
00:22:02,450 --> 00:22:05,950
It's always a little
tricky to measure genus.

467
00:22:05,950 --> 00:22:09,950
If it just had one triangular
face that would be-- well,

468
00:22:09,950 --> 00:22:12,410
anyway, that's genus seven.

469
00:22:12,410 --> 00:22:14,704
I'm not going to
try to convince you.

470
00:22:14,704 --> 00:22:16,120
This is an unfolding
of the thing.

471
00:22:16,120 --> 00:22:17,647
So you can see,
in the unfolding,

472
00:22:17,647 --> 00:22:19,480
it looks like it's an
orthogonal polyhedron.

473
00:22:19,480 --> 00:22:20,580
All the angles are right.

474
00:22:20,580 --> 00:22:21,550
It looks perfect.

475
00:22:21,550 --> 00:22:24,000
And yet, it folds into
this non-orthogonal thing.

476
00:22:24,000 --> 00:22:27,740
Kind of a thickened octahedron.

477
00:22:27,740 --> 00:22:32,960
It's kind of critical that it
has a genus more than zero.

478
00:22:32,960 --> 00:22:34,940
In this paper-- this
is the original paper

479
00:22:34,940 --> 00:22:42,210
that considered this problem--
there's a lower bound of one.

480
00:22:42,210 --> 00:22:45,110
The genus 0 is not enough.

481
00:22:45,110 --> 00:22:50,730
Then we came along-- I don't
have the author list here.

482
00:22:50,730 --> 00:22:56,370
It's with the Waterloo gang, it
was Therese Biedl and others.

483
00:22:56,370 --> 00:23:03,050
And this is a smaller genus,
a genus six polyhedron.

484
00:23:03,050 --> 00:23:03,760
This guy.

485
00:23:03,760 --> 00:23:06,540
So it's almost the same
example turned inside out.

486
00:23:06,540 --> 00:23:09,210
So this like to half
octahedra and then joined

487
00:23:09,210 --> 00:23:11,230
in this cubicle frame
and then you put caps

488
00:23:11,230 --> 00:23:14,990
on the cubicle frame to make
there not be holes out there.

489
00:23:14,990 --> 00:23:17,120
And then you have
slightly smaller genus

490
00:23:17,120 --> 00:23:20,610
because you can go
around the backside.

491
00:23:20,610 --> 00:23:22,830
So that's the best known,
is a genus six example.

492
00:23:22,830 --> 00:23:25,770
We proved that you need
genus at least three

493
00:23:25,770 --> 00:23:28,031
in order for such
an example to exist.

494
00:23:28,031 --> 00:23:30,030
I have a little sketch
of the proof in the notes

495
00:23:30,030 --> 00:23:31,650
if you want to read it.

496
00:23:31,650 --> 00:23:34,610
It's a fun little boot
strapping argument but I would--

497
00:23:34,610 --> 00:23:36,740
I'm going to go on
to other fun topics

498
00:23:36,740 --> 00:23:39,810
here rather than prove this.

499
00:23:39,810 --> 00:23:41,589
Is that a good plan?

500
00:23:41,589 --> 00:23:42,630
I think that's good plan.

501
00:23:45,830 --> 00:23:46,330
OK.

502
00:23:46,330 --> 00:23:48,060
So this is a fun
kind of-- unfoldings

503
00:23:48,060 --> 00:23:51,430
can be orthogonal while the
three dimensional versions are

504
00:23:51,430 --> 00:23:52,055
not orthogonal.

505
00:23:55,300 --> 00:23:56,550
All right.

506
00:23:56,550 --> 00:23:57,135
Next topic.

507
00:24:01,870 --> 00:24:05,260
I'm going to go back to
Alexandrov's theorem.

508
00:24:05,260 --> 00:24:08,620
Turns out there is a
smooth version of that.

509
00:24:08,620 --> 00:24:11,310
So now we're going
to segue a little bit

510
00:24:11,310 --> 00:24:14,215
into smooth foldings
and smooth unfoldings.

511
00:24:16,730 --> 00:24:20,140
So instead of thinking of
polyhedron objects-- polygons

512
00:24:20,140 --> 00:24:25,180
and convex polyhedra-- I want
to think about smooth bodies.

513
00:24:25,180 --> 00:24:33,760
So, for example, suppose
I take a smooth convex

514
00:24:33,760 --> 00:24:36,820
shape like an
ellipse or something

515
00:24:36,820 --> 00:24:39,950
and I picked two
perimeter antipodes

516
00:24:39,950 --> 00:24:49,191
and I do perimeter having
gluing where this is defines--

517
00:24:49,191 --> 00:24:51,190
as I walk along the
perimeter in both directions

518
00:24:51,190 --> 00:24:53,460
it-- I obviously
didn't do it perfectly,

519
00:24:53,460 --> 00:24:56,474
they should meet at
exactly the same time.

520
00:24:56,474 --> 00:24:57,390
This defines a gluing.

521
00:24:57,390 --> 00:24:59,100
For every point there
is a corresponding point

522
00:24:59,100 --> 00:24:59,900
it gets glued to.

523
00:24:59,900 --> 00:25:04,780
This is a way to make what is
locally like a convex surface

524
00:25:04,780 --> 00:25:08,200
because at any point, at
most 360 degrees of material

525
00:25:08,200 --> 00:25:10,260
is glued there because
this thing was convex.

526
00:25:10,260 --> 00:25:13,020
And the same way as
for convex polygons,

527
00:25:13,020 --> 00:25:15,474
does this make a convex
something in three dimensions?

528
00:25:15,474 --> 00:25:17,390
It's not going to make
a polyhedron because it

529
00:25:17,390 --> 00:25:20,180
has this curve,
but indeed it does

530
00:25:20,180 --> 00:25:22,700
make a convex smooth surface.

531
00:25:22,700 --> 00:25:40,660
So every convex metric
that's topologically a sphere

532
00:25:40,660 --> 00:25:46,225
is realized by a
unique convex surface.

533
00:25:54,210 --> 00:25:56,860
So Alexandrov's theorem was
identical to this statement

534
00:25:56,860 --> 00:25:59,430
except it also had
polyhedral up here,

535
00:25:59,430 --> 00:26:02,740
that there were finite number
of points of nonzero curvature.

536
00:26:02,740 --> 00:26:04,890
Now we have
infinitely many points

537
00:26:04,890 --> 00:26:07,550
of slightly positive
curvature because, again, this

538
00:26:07,550 --> 00:26:08,950
is supposed to sum to 4 pi.

539
00:26:08,950 --> 00:26:10,230
That's still true.

540
00:26:10,230 --> 00:26:13,067
But it's now an integral
instead of a sum.

541
00:26:13,067 --> 00:26:15,650
So each of these things sort of
has an infinitesimal curvature

542
00:26:15,650 --> 00:26:19,250
and if you integrate
them it adds up to 4 pi.

543
00:26:19,250 --> 00:26:22,190
You can check that.

544
00:26:22,190 --> 00:26:24,670
So this will make something.

545
00:26:24,670 --> 00:26:26,754
I encourage you to try
this out, again, with tape.

546
00:26:26,754 --> 00:26:28,545
You're going to
approximate it because tape

547
00:26:28,545 --> 00:26:29,830
is going to be polygonal.

548
00:26:29,830 --> 00:26:31,520
But it works.

549
00:26:31,520 --> 00:26:33,140
It will make something.

550
00:26:33,140 --> 00:26:36,620
I particularly-- I should
mention the idea of this proof

551
00:26:36,620 --> 00:26:39,110
is just to take limits
of regular Alexandrov.

552
00:26:39,110 --> 00:26:41,599
Take closer and closer
polyhedral approximations.

553
00:26:41,599 --> 00:26:43,640
In the polyhedral world
we can use that induction

554
00:26:43,640 --> 00:26:45,170
that we talked about.

555
00:26:45,170 --> 00:26:47,720
In the smooth case, you
can't, but the smooth case

556
00:26:47,720 --> 00:26:51,180
is the limit of the polyhedral
case so it works out.

557
00:26:51,180 --> 00:26:55,580
This is proved by a student
of Alexandrov Pogorelov

558
00:26:55,580 --> 00:26:59,860
in the '70s, whereas he
proved his theorem in the '40s

559
00:26:59,860 --> 00:27:01,558
or '50s.

560
00:27:01,558 --> 00:27:03,730
One really fun thing you
could do with this theorem

561
00:27:03,730 --> 00:27:07,550
is, instead of using
just one convex shape,

562
00:27:07,550 --> 00:27:12,389
you could use two convex shapes
and zip them to each other.

563
00:27:12,389 --> 00:27:13,555
And this is called a D-form.

564
00:27:13,555 --> 00:27:16,820
It was invented by an
artist, Tony Wills.

565
00:27:16,820 --> 00:27:18,572
So here I've taken
two-- in this case,

566
00:27:18,572 --> 00:27:20,030
two identical
convect shapes-- they

567
00:27:20,030 --> 00:27:23,240
don't have to be identical--
picked two points to glue them

568
00:27:23,240 --> 00:27:26,626
to each other and then you
just zip around the perimeter.

569
00:27:26,626 --> 00:27:29,250
And again, because you're always
gluing convex angles to convex

570
00:27:29,250 --> 00:27:31,060
angles-- the smooth
convex angles--

571
00:27:31,060 --> 00:27:34,840
but they're still less than
180, so you glue them together,

572
00:27:34,840 --> 00:27:36,760
it's less than 360.

573
00:27:36,760 --> 00:27:39,360
And in this case, it's-- we
could actually figure out

574
00:27:39,360 --> 00:27:42,230
and think about what 3D shape
you get so we could build that

575
00:27:42,230 --> 00:27:43,620
in Mathematica.

576
00:27:43,620 --> 00:27:47,060
But in general, you get
some weird convex shape.

577
00:27:47,060 --> 00:27:49,394
For a while there were
some examples of this

578
00:27:49,394 --> 00:27:51,310
but no one knew exactly
what they looked like.

579
00:27:51,310 --> 00:27:54,800
Then, I think it was a
project in this class

580
00:27:54,800 --> 00:27:57,180
or it came out of the problem
session in this class three

581
00:27:57,180 --> 00:27:59,900
years ago, Greg
Price and I proved

582
00:27:59,900 --> 00:28:02,930
that these things
are nice in the sense

583
00:28:02,930 --> 00:28:05,270
that the only place
you get creases

584
00:28:05,270 --> 00:28:08,820
is along the seam,
the pink stuff.

585
00:28:08,820 --> 00:28:11,310
Both for-- these
are called D-forms,

586
00:28:11,310 --> 00:28:13,960
these have another-- I
think these are pita-forms,

587
00:28:13,960 --> 00:28:17,390
if I recall, where you take one
convex shape instead of two.

588
00:28:17,390 --> 00:28:20,060
Here you only get--
you can get the creases

589
00:28:20,060 --> 00:28:21,910
along the seam and
one extra crease,

590
00:28:21,910 --> 00:28:23,752
if I recall correctly here.

591
00:28:23,752 --> 00:28:25,460
And here you don't
get any extra creases.

592
00:28:25,460 --> 00:28:27,810
It's always smooth
except at the seam.

593
00:28:27,810 --> 00:28:31,020
And also, if you just take
the seam-- the pink part--

594
00:28:31,020 --> 00:28:35,030
and take the convex hull of that
thing, you get the same thing.

595
00:28:35,030 --> 00:28:37,050
So it doesn't go outside.

596
00:28:37,050 --> 00:28:40,240
It's really tightly
wrapped around wherever

597
00:28:40,240 --> 00:28:41,810
the seam happens to go.

598
00:28:41,810 --> 00:28:44,930
Remember, this thing is
unique by Pogorelov's theorem.

599
00:28:44,930 --> 00:28:48,060
So it's kind of nice to
know that structurally these

600
00:28:48,060 --> 00:28:49,530
are pretty nice and smooth.

601
00:28:49,530 --> 00:28:52,880
If your original shapes
are smooth convex bodies,

602
00:28:52,880 --> 00:28:55,340
you will get a
nice, smooth shape

603
00:28:55,340 --> 00:28:59,680
at the end that only has
creases at the seams.

604
00:28:59,680 --> 00:29:01,160
Cool.

605
00:29:01,160 --> 00:29:03,005
Again, please try to it at home.

606
00:29:03,005 --> 00:29:04,509
It's lots of fun.

607
00:29:04,509 --> 00:29:05,425
AUDIENCE: [INAUDIBLE].

608
00:29:17,355 --> 00:29:18,230
PROFESSOR: Let's see.

609
00:29:18,230 --> 00:29:21,760
This point should glue to here.

610
00:29:21,760 --> 00:29:25,170
I mean, it's
perfectly symmetric.

611
00:29:25,170 --> 00:29:27,570
AUDIENCE: I'm looking
at the 3D form.

612
00:29:27,570 --> 00:29:28,690
PROFESSOR: The 3D form.

613
00:29:28,690 --> 00:29:33,200
So one of the two shapes
is here and the other one

614
00:29:33,200 --> 00:29:37,170
is here and goes around
the back to there.

615
00:29:37,170 --> 00:29:39,340
And it's, again, symmetric.

616
00:29:39,340 --> 00:29:42,630
This is a nice example because
we have those parallel lines.

617
00:29:42,630 --> 00:29:45,330
Usually you wouldn't have that.

618
00:29:45,330 --> 00:29:49,126
And they're harder to
draw, is the challenge.

619
00:29:49,126 --> 00:29:50,500
Here, obviously,
we're doing sort

620
00:29:50,500 --> 00:29:51,710
of a polygonal approximation.

621
00:29:51,710 --> 00:29:54,330
You see all the lines.

622
00:29:54,330 --> 00:29:55,980
All right.

623
00:29:55,980 --> 00:30:00,230
This is a smooth version of
folding-- a smooth version

624
00:30:00,230 --> 00:30:01,630
of Alexandrov's theorem.

625
00:30:01,630 --> 00:30:05,500
We can also think about
smooth versions of unfolding.

626
00:30:05,500 --> 00:30:10,520
So here there is
exactly one paper

627
00:30:10,520 --> 00:30:17,150
by Nadia Benbernou whose in
the front row and Heather--

628
00:30:17,150 --> 00:30:17,982
AUDIENCE: Patricia.

629
00:30:17,982 --> 00:30:21,880
PROFESSOR: Patricia
Khan and Joe O'Rourke.

630
00:30:21,880 --> 00:30:25,720
So this is a prismatoid.

631
00:30:25,720 --> 00:30:27,440
It's a little hard to
see the bottom face

632
00:30:27,440 --> 00:30:31,720
but there's a convex polygon on
the bottom, a parallel convex

633
00:30:31,720 --> 00:30:32,890
polygon on the top.

634
00:30:32,890 --> 00:30:35,820
You take the convex hull--
sorry, they're not polygons.

635
00:30:35,820 --> 00:30:38,060
They're smooth convex bodies.

636
00:30:38,060 --> 00:30:40,640
So you've got one in the floor
plane, one in a parallel plane

637
00:30:40,640 --> 00:30:41,440
up top.

638
00:30:41,440 --> 00:30:44,300
You take the convex hull, you
get this nice smooth thing

639
00:30:44,300 --> 00:30:45,660
around the outside.

640
00:30:45,660 --> 00:30:48,570
Now, we have to generalize
our notion of unfolding

641
00:30:48,570 --> 00:30:50,690
because normally with
unfolding this would take

642
00:30:50,690 --> 00:30:53,410
an infinite number of cuts
to-- we have to cut everywhere

643
00:30:53,410 --> 00:30:54,160
there's curvature.

644
00:30:54,160 --> 00:30:55,993
Which means we're going
to have to cut along

645
00:30:55,993 --> 00:30:57,250
all those black lines.

646
00:30:57,250 --> 00:31:02,490
Which means we slice the thing
into very infinitesimal pieces.

647
00:31:02,490 --> 00:31:06,530
But if you think
of it as a limit

648
00:31:06,530 --> 00:31:11,630
of regular polyhedral things--
here I have a little example.

649
00:31:11,630 --> 00:31:16,370
Let's think about the limit in
a simpler example to start with.

650
00:31:16,370 --> 00:31:19,000
So imagine you want
to take a pyramid--

651
00:31:19,000 --> 00:31:23,886
so the top poly-- the top
convex shape is just a point.

652
00:31:23,886 --> 00:31:25,645
I guess it looks like
a cone but down here

653
00:31:25,645 --> 00:31:32,180
it could be anything, could be
an ellipse or any convex shape.

654
00:31:32,180 --> 00:31:35,480
Well, we can think of this as
a polyhedral approximation.

655
00:31:35,480 --> 00:31:40,420
So, for example, we take
a hexagon on the bottom,

656
00:31:40,420 --> 00:31:42,590
we want to unfold that.

657
00:31:42,590 --> 00:31:45,160
And in particular, we
thought about these--

658
00:31:45,160 --> 00:31:49,520
these are the
pyramids-- and we--

659
00:31:49,520 --> 00:31:53,830
one of the unfoldings of
them is the volcano unfolding

660
00:31:53,830 --> 00:31:56,894
if you remember back to
the unfolding lecture.

661
00:31:56,894 --> 00:31:57,560
Looks like that.

662
00:31:57,560 --> 00:32:01,210
All these triangles just fold
out, get all those parts.

663
00:32:01,210 --> 00:32:03,660
That never overlaps,
it's sort of trivial.

664
00:32:03,660 --> 00:32:06,430
You take the limit of that.

665
00:32:06,430 --> 00:32:09,510
In the limit, the floor becomes
whatever the convex shape.

666
00:32:09,510 --> 00:32:14,240
Maybe-- let's think about the
simple case where it's a disk.

667
00:32:14,240 --> 00:32:19,209
And in the limit you're getting
lots of really tiny triangles.

668
00:32:19,209 --> 00:32:21,500
You take that all the way to
limit, what you're getting

669
00:32:21,500 --> 00:32:26,950
is actually a concentric
circle around there.

670
00:32:26,950 --> 00:32:30,100
So really, these
become segments.

671
00:32:30,100 --> 00:32:33,970
Now, what's funny about
this limit is, normally--

672
00:32:33,970 --> 00:32:38,460
before you get to the end, you
see there's these big gaps.

673
00:32:38,460 --> 00:32:40,174
A lot of the area's missing.

674
00:32:40,174 --> 00:32:42,090
These things are getting
infinitesimally small

675
00:32:42,090 --> 00:32:44,570
and yet they have to
spread over this big range.

676
00:32:44,570 --> 00:32:46,650
So most of this area
is actually absent.

677
00:32:46,650 --> 00:32:49,066
When you take the limit you
can't really see that anymore.

678
00:32:49,066 --> 00:32:50,250
It just becomes filled in.

679
00:32:50,250 --> 00:32:55,860
And so in this unfolding,
the area of the green stuff,

680
00:32:55,860 --> 00:32:59,120
if you just took that area
measured on the floor,

681
00:32:59,120 --> 00:33:02,600
it's going to be larger
than the area in the pink.

682
00:33:02,600 --> 00:33:04,720
So this is not an area
preserving unfolding

683
00:33:04,720 --> 00:33:06,420
which makes it a little weird.

684
00:33:06,420 --> 00:33:07,920
But it's still well defined.

685
00:33:07,920 --> 00:33:10,311
I mean, you wouldn't
actually-- well,

686
00:33:10,311 --> 00:33:12,560
for example-- we're going
to get to this in a moment--

687
00:33:12,560 --> 00:33:15,850
if you actually took material
like this out of, say,

688
00:33:15,850 --> 00:33:18,570
tin foil, something
that can crinkle,

689
00:33:18,570 --> 00:33:22,290
then you should be able to
crinkle it into this shape.

690
00:33:22,290 --> 00:33:25,020
I'll talk about
how to prove that.

691
00:33:25,020 --> 00:33:27,020
That's actually most of
the rest of the lecture,

692
00:33:27,020 --> 00:33:29,090
is about that kind
of notion of folding.

693
00:33:29,090 --> 00:33:31,520
It's more like origami,
though, than unfolding

694
00:33:31,520 --> 00:33:35,020
because you're sort of
over covering stuff.

695
00:33:35,020 --> 00:33:37,030
And here-- actually,
here the hard part

696
00:33:37,030 --> 00:33:40,110
is not so much to show
these ribs don't intersect,

697
00:33:40,110 --> 00:33:42,110
although they don't and
it's not totally obvious

698
00:33:42,110 --> 00:33:44,950
because here they're kind
of going towards each other.

699
00:33:44,950 --> 00:33:48,849
But they won't actually
intersect in the unfolding.

700
00:33:48,849 --> 00:33:51,390
One of the tricky parts is to
show that the top face actually

701
00:33:51,390 --> 00:33:53,150
can fit somewhere,
so you actually

702
00:33:53,150 --> 00:33:56,400
get an unfolding of this thing.

703
00:33:56,400 --> 00:34:01,220
And I forget, this is choosing
some kind of extreme rib

704
00:34:01,220 --> 00:34:02,665
and putting it out there.

705
00:34:02,665 --> 00:34:06,240
It's a nice line of
separation if I recall.

706
00:34:09,230 --> 00:34:14,000
This is funny because here we
have a limit of some process.

707
00:34:14,000 --> 00:34:20,840
Now, for pyramids we know
the volcano unfolding works.

708
00:34:20,840 --> 00:34:23,850
For prismatoids,
regular prismatoids

709
00:34:23,850 --> 00:34:26,429
are when you take
some convex polygon

710
00:34:26,429 --> 00:34:30,250
and some other parallel convex
polygon, take the convex hull.

711
00:34:35,760 --> 00:34:38,620
We don't know whether
these have edge unfoldings.

712
00:34:38,620 --> 00:34:40,969
Still an open problem.

713
00:34:40,969 --> 00:34:44,440
And yet, for the
smooth case, we do.

714
00:34:44,440 --> 00:34:45,949
Kind of magical.

715
00:34:45,949 --> 00:34:47,219
I know why that's true.

716
00:34:47,219 --> 00:34:50,948
It's somehow the edge
constraint is harder.

717
00:34:50,948 --> 00:34:52,989
If you're not just worried
about edge unfoldings,

718
00:34:52,989 --> 00:34:55,260
we do know how to
unfold those obviously.

719
00:34:55,260 --> 00:34:58,310
You can do star unfolding,
source unfolding.

720
00:34:58,310 --> 00:35:00,946
Here, edge unfolding--
well, we have

721
00:35:00,946 --> 00:35:02,820
to cut along all those
infinitely many edges.

722
00:35:02,820 --> 00:35:04,610
That makes it easier,
much smoother.

723
00:35:08,132 --> 00:35:10,090
But it's funny when the
smooth case is actually

724
00:35:10,090 --> 00:35:12,160
easier than the discrete case.

725
00:35:12,160 --> 00:35:12,660
All right.

726
00:35:15,250 --> 00:35:18,872
The next problem I want to
talk about is chocolate.

727
00:35:22,680 --> 00:35:26,100
I have here a bunch of
spherical chocolates.

728
00:35:26,100 --> 00:35:29,650
The most-- apparently the only
perfectly spherical chocolate,

729
00:35:29,650 --> 00:35:32,310
at least according
to the advertisement,

730
00:35:32,310 --> 00:35:34,640
is this thing called
the Mozart Kugel.

731
00:35:34,640 --> 00:35:38,230
Now, Mozart Kugel-- how many
people have eaten them before.

732
00:35:38,230 --> 00:35:39,671
Just a couple.

733
00:35:39,671 --> 00:35:40,170
Good stuff.

734
00:35:40,170 --> 00:35:43,530
It's, I guess, more
a European thing.

735
00:35:43,530 --> 00:35:44,790
It's from Austria.

736
00:35:44,790 --> 00:35:49,150
It's invented in early
1900s, I forget exactly when.

737
00:35:49,150 --> 00:35:51,677
It was a big thing at the
time because making chocolate

738
00:35:51,677 --> 00:35:53,010
perfectly spherical is not easy.

739
00:35:53,010 --> 00:35:55,230
It's like, how do
you hold it and not

740
00:35:55,230 --> 00:35:56,460
get little indentations?

741
00:35:56,460 --> 00:35:57,620
This thing is
perfectly spherical.

742
00:35:57,620 --> 00:35:58,411
Here I'll show you.

743
00:36:00,870 --> 00:36:06,040
But, really, the question I
wonder, because as an origamist

744
00:36:06,040 --> 00:36:08,270
you wonder, how do
you wrap this thing?

745
00:36:08,270 --> 00:36:10,065
Oh, I ripped it.

746
00:36:10,065 --> 00:36:11,595
I'll try to do this
more carefully.

747
00:36:15,820 --> 00:36:18,786
It's not how you usually
eat chocolate, is it?

748
00:36:18,786 --> 00:36:21,520
Open very carefully.

749
00:36:21,520 --> 00:36:22,020
It's OK.

750
00:36:22,020 --> 00:36:24,840
I have more in case I fail.

751
00:36:24,840 --> 00:36:25,475
More chocolate.

752
00:36:29,000 --> 00:36:31,801
Not enough for you, I'm afraid.

753
00:36:31,801 --> 00:36:32,800
So ignore the chocolate.

754
00:36:36,410 --> 00:36:40,360
It's not the Droid
you're looking for.

755
00:36:40,360 --> 00:36:41,330
It's not the chocolate.

756
00:36:41,330 --> 00:36:42,830
So it's a rectangle.

757
00:36:42,830 --> 00:36:44,455
I didn't do a perfect job.

758
00:36:44,455 --> 00:36:46,070
There's a few
little tears there.

759
00:36:46,070 --> 00:36:48,320
But they fold-- Mirabell.

760
00:36:48,320 --> 00:36:52,737
Mirabell makes the most
Mozart Kugel in the world.

761
00:36:52,737 --> 00:36:54,320
Mozart Kugel is very
cool because it's

762
00:36:54,320 --> 00:36:56,900
built in-- built up
in concentric spheres.

763
00:36:56,900 --> 00:37:01,400
You have the marzipan layer,
then the milk chocolate layer,

764
00:37:01,400 --> 00:37:03,880
then the nougat layer, then
the dark chocolate layer,

765
00:37:03,880 --> 00:37:05,210
if I recall.

766
00:37:05,210 --> 00:37:08,530
Each brand has a
different recipe,

767
00:37:08,530 --> 00:37:11,324
but they're always
spherical on the outside.

768
00:37:11,324 --> 00:37:12,990
It's kind of like
glassblowing actually.

769
00:37:12,990 --> 00:37:15,430
It's kind of fun.

770
00:37:15,430 --> 00:37:19,340
And each company has a
slightly different way-- well,

771
00:37:19,340 --> 00:37:21,190
I've studied only
two main companies.

772
00:37:21,190 --> 00:37:25,320
Mirabell makes the most of them
but they are not the original.

773
00:37:25,320 --> 00:37:29,694
They are called the
echta which is--

774
00:37:29,694 --> 00:37:30,490
AUDIENCE: The real.

775
00:37:30,490 --> 00:37:32,360
PROFESSOR: The real.

776
00:37:32,360 --> 00:37:34,160
The authentic.

777
00:37:34,160 --> 00:37:37,880
But-- now, these
are hard to come by.

778
00:37:37,880 --> 00:37:41,470
These are brought to me by
my student, [INAUDIBLE],

779
00:37:41,470 --> 00:37:42,540
some of you know.

780
00:37:42,540 --> 00:37:44,519
Architecture.

781
00:37:44,519 --> 00:37:46,060
This is the-- oh,
it says in English.

782
00:37:46,060 --> 00:37:46,930
It's kind of easy.

783
00:37:46,930 --> 00:37:50,000
The original Salzburger
Mozart Kugel.

784
00:37:50,000 --> 00:37:53,790
Mozart was-- lived
in Salzburg, I guess.

785
00:37:53,790 --> 00:37:55,510
And so this is named after him.

786
00:37:55,510 --> 00:37:56,460
Oh, 1884.

787
00:37:56,460 --> 00:37:57,580
Wow.

788
00:37:57,580 --> 00:37:59,310
So that's when they started.

789
00:37:59,310 --> 00:38:05,026
Now, these guys--
I think I've only

790
00:38:05,026 --> 00:38:06,400
had like one of
these in my life.

791
00:38:06,400 --> 00:38:07,420
I'm really excited.

792
00:38:07,420 --> 00:38:09,962
The Mirabell's are
OK, but this stuff?

793
00:38:09,962 --> 00:38:10,462
Wow.

794
00:38:14,884 --> 00:38:16,300
Oh, it's so much
easier to unfold.

795
00:38:16,300 --> 00:38:18,810
I like-- this is, I
think, probably handmade.

796
00:38:18,810 --> 00:38:20,450
Much more handmade.

797
00:38:20,450 --> 00:38:23,980
In fact, you can kind of tell
it's not perfectly spherical.

798
00:38:23,980 --> 00:38:25,730
It has a little nub at the top.

799
00:38:25,730 --> 00:38:28,626
So this is probably more how
they were originally made.

800
00:38:28,626 --> 00:38:30,594
AUDIENCE: [INAUDIBLE].

801
00:38:30,594 --> 00:38:32,260
PROFESSOR: What, are
you getting hungry?

802
00:38:34,910 --> 00:38:36,880
Now, Furst-- Furst
is the company

803
00:38:36,880 --> 00:38:41,911
who made the first Mozart
Kugel-- they use squares.

804
00:38:41,911 --> 00:38:43,410
I claim they're a
little bit better.

805
00:38:43,410 --> 00:38:48,840
But interesting question is,
which uses more material?

806
00:38:48,840 --> 00:38:50,160
Something to think about.

807
00:38:50,160 --> 00:38:56,430
Meanwhile-- I won't
torture you more by eating.

808
00:38:59,400 --> 00:39:01,000
All right.

809
00:39:01,000 --> 00:39:02,470
So this is a Mozart
Kugel problem.

810
00:39:05,250 --> 00:39:12,660
This paper started when-- I
think we are in New Orleans.

811
00:39:12,660 --> 00:39:15,200
Some part of the gang
was in New Orleans

812
00:39:15,200 --> 00:39:17,620
and there was this
deadline the next day

813
00:39:17,620 --> 00:39:20,536
and it was for this workshop
on computational geometry--

814
00:39:20,536 --> 00:39:22,160
the European workshop--
and it happened

815
00:39:22,160 --> 00:39:26,500
to be in Austria that
year-- that coming year.

816
00:39:26,500 --> 00:39:28,330
We're like, what
paper could we write?

817
00:39:28,330 --> 00:39:30,770
And we're sitting there
in the little cafe eating

818
00:39:30,770 --> 00:39:33,040
these beignets of
doughnuts and thinking,

819
00:39:33,040 --> 00:39:34,665
what could we write
a paper about?

820
00:39:34,665 --> 00:39:36,620
It's like, how about food?

821
00:39:36,620 --> 00:39:39,350
And what does Austria have?

822
00:39:39,350 --> 00:39:42,950
Well, [SPEAKING GERMAN]
Well, how about Mozart Kugel?

823
00:39:42,950 --> 00:39:46,440
So-- this is how
research happens.

824
00:39:46,440 --> 00:39:48,690
But it's really cool.

825
00:39:48,690 --> 00:39:51,410
It's really cool because
this is not your--

826
00:39:51,410 --> 00:39:55,400
this is not the origami you
grew up with in this class.

827
00:39:55,400 --> 00:39:58,430
Because you can't fold a
sphere out of a square paper.

828
00:39:58,430 --> 00:40:01,970
And yet, this was a square
folded into a sphere.

829
00:40:01,970 --> 00:40:06,650
Somehow that was done using
a finite amount of effort.

830
00:40:06,650 --> 00:40:09,510
Whereas if you use regular
paper to fold something

831
00:40:09,510 --> 00:40:11,650
that has positive
curvature everywhere,

832
00:40:11,650 --> 00:40:13,760
you need infinitely
many creases.

833
00:40:13,760 --> 00:40:15,500
And yet this is fairly
practical to do.

834
00:40:15,500 --> 00:40:16,980
This is hand wrapped.

835
00:40:16,980 --> 00:40:18,910
In theory, I could re-wrap this.

836
00:40:18,910 --> 00:40:20,570
I've never actually tried.

837
00:40:20,570 --> 00:40:21,760
But something like this.

838
00:40:21,760 --> 00:40:23,215
It works pretty well.

839
00:40:23,215 --> 00:40:26,860
I even covered the tip.

840
00:40:26,860 --> 00:40:28,320
How did I do that
in finite effort?

841
00:40:28,320 --> 00:40:30,840
Well, of course it's
not a perfect sphere.

842
00:40:30,840 --> 00:40:33,490
But we need to model
this mathematically.

843
00:40:33,490 --> 00:40:35,625
It's annoying to
take approximations.

844
00:40:35,625 --> 00:40:37,500
I would like a model of
folding that actually

845
00:40:37,500 --> 00:40:40,470
does make it possible
to fold a sphere.

846
00:40:40,470 --> 00:40:44,490
So this is the idea of
contractive folding.

847
00:40:50,990 --> 00:40:54,210
We haven't been super rigorous
about defining origami,

848
00:40:54,210 --> 00:40:57,970
but I always say you
take a piece of paper,

849
00:40:57,970 --> 00:41:00,890
you can't stretch the paper,
you can't tear the paper,

850
00:41:00,890 --> 00:41:03,400
and you can't self-intersect.

851
00:41:03,400 --> 00:41:06,980
We're going to relax the
you can't stretch the paper.

852
00:41:06,980 --> 00:41:08,770
You still can't
stretch the paper,

853
00:41:08,770 --> 00:41:11,060
but now the paper can contract.

854
00:41:11,060 --> 00:41:13,110
So it can't get longer
but it can get shorter.

855
00:41:13,110 --> 00:41:15,640
It's like cables from
tensegrity theory.

856
00:41:15,640 --> 00:41:17,470
So you take any pair of points.

857
00:41:17,470 --> 00:41:20,300
There's some distance as
measured on the piece of paper.

858
00:41:20,300 --> 00:41:24,540
When you fold this thing
into your awesome sphere,

859
00:41:24,540 --> 00:41:26,750
you take the two points,
you measure the distance

860
00:41:26,750 --> 00:41:29,890
along the surface of
the piece of paper.

861
00:41:29,890 --> 00:41:31,910
Now, I'll-- normally
with origami,

862
00:41:31,910 --> 00:41:34,850
these two distances
should be equal.

863
00:41:34,850 --> 00:41:36,440
That's isometry.

864
00:41:36,440 --> 00:41:39,440
That's regular origami.

865
00:41:39,440 --> 00:41:45,190
But if I say that this one--
so write equal that way.

866
00:41:45,190 --> 00:41:51,490
If this one can be smaller,
then this is contractive.

867
00:41:51,490 --> 00:41:55,920
And that is paper that can
shrink but not stretch.

868
00:41:55,920 --> 00:42:06,364
So pairwise distances
don't increase.

869
00:42:06,364 --> 00:42:07,530
So they could stay the same.

870
00:42:07,530 --> 00:42:09,710
We could still do
regular origami.

871
00:42:09,710 --> 00:42:12,070
But we can also contract.

872
00:42:12,070 --> 00:42:15,140
And then it is possible
to fold a sphere.

873
00:42:15,140 --> 00:42:17,690
Now, intuitively,
what's going on

874
00:42:17,690 --> 00:42:21,380
is, when you fold out of
tinfoil, you can crinkle.

875
00:42:21,380 --> 00:42:26,840
And that lets you not have
to worry about-- I mean,

876
00:42:26,840 --> 00:42:28,790
this is pretty good
approximation to a sphere

877
00:42:28,790 --> 00:42:31,040
especially when I have
a good base underneath

878
00:42:31,040 --> 00:42:35,090
to approximate against.

879
00:42:35,090 --> 00:42:38,250
I'm doing lots of little
tiny creases in order

880
00:42:38,250 --> 00:42:43,080
to approximate that contractive
mapping, contractive folding.

881
00:42:43,080 --> 00:42:47,350
So, conveniently,
there's a theorem

882
00:42:47,350 --> 00:42:49,590
that tells us this
is always possible.

883
00:42:49,590 --> 00:42:51,340
I'll tell you the theorem.

884
00:42:51,340 --> 00:42:55,350
This is another theorem
by Burago and Zalgallar

885
00:42:55,350 --> 00:42:59,560
who proved the other
theorem that if you

886
00:42:59,560 --> 00:43:04,380
have any polyhedral
metric you can turn it

887
00:43:04,380 --> 00:43:08,230
into some polyhedron,
not necessarily convex.

888
00:43:08,230 --> 00:43:11,610
So they solved that problem
plus they solved this problem.

889
00:43:11,610 --> 00:43:16,580
If you take a contractive-- they
only solved the smooth case.

890
00:43:16,580 --> 00:43:19,650
I imagine this works
for piecewise smooth.

891
00:43:19,650 --> 00:43:23,070
You take a contractive
C2 immersion--

892
00:43:23,070 --> 00:43:25,700
it's just some fancy
word for folding,

893
00:43:25,700 --> 00:43:29,060
it has some extra constraints
but let's not worry about it.

894
00:43:29,060 --> 00:43:31,490
So C2 means continuous up
to the second derivative.

895
00:43:31,490 --> 00:43:32,440
So nice and smooth.

896
00:43:32,440 --> 00:43:35,450
Spheres are valid, for example.

897
00:43:35,450 --> 00:43:42,714
Then any one of those
things has a C0--

898
00:43:42,714 --> 00:43:44,505
I'll just call it
polyhedral approximation.

899
00:43:48,570 --> 00:43:51,090
And this is approximation
in the Hausdorff sense

900
00:43:51,090 --> 00:43:54,300
meaning you want
to fold the sphere.

901
00:43:54,300 --> 00:43:57,200
If you just thicken that
sphere by some epsilon,

902
00:43:57,200 --> 00:44:00,570
then you can sort of stay within
that epsilon thickened sphere

903
00:44:00,570 --> 00:44:02,290
and do some kind of wiggling.

904
00:44:02,290 --> 00:44:04,120
You'll be polyhedral
and you'll only

905
00:44:04,120 --> 00:44:06,390
have finitely many creases.

906
00:44:06,390 --> 00:44:10,180
So there-- if you have a nice,
smooth contracted folded state,

907
00:44:10,180 --> 00:44:14,250
you could turn it into a regular
origami isometry folded state

908
00:44:14,250 --> 00:44:16,610
that's arbitrarily
close to the sphere.

909
00:44:16,610 --> 00:44:17,460
So for any epsilon.

910
00:44:21,090 --> 00:44:25,560
So that's a convenient
theorem from 1996.

911
00:44:25,560 --> 00:44:26,730
So that's the model.

912
00:44:26,730 --> 00:44:30,870
Now we can think about, how
do I optimally wrap a sphere?

913
00:44:30,870 --> 00:44:35,010
Now, there are
many possible goals

914
00:44:35,010 --> 00:44:38,450
you might want to optimize
in wrapping a sphere.

915
00:44:38,450 --> 00:44:40,630
Presumably, you want your
shape to tile the plane.

916
00:44:40,630 --> 00:44:42,505
So you can cut out a
whole bunch of these out

917
00:44:42,505 --> 00:44:44,780
of one sheet of tinfoil.

918
00:44:44,780 --> 00:44:47,000
Obviously Mirabell
likes rectangles,

919
00:44:47,000 --> 00:44:49,590
Furst likes squares.

920
00:44:49,590 --> 00:44:51,430
Are those the
right things to do?

921
00:44:51,430 --> 00:44:53,230
What about triangles?

922
00:44:53,230 --> 00:44:54,230
What about other shapes?

923
00:44:54,230 --> 00:44:56,104
These are the sorts of
questions we consider.

924
00:44:56,104 --> 00:44:58,220
We don't have a complete
answer to this question.

925
00:44:58,220 --> 00:45:01,642
But we have a lot of
fun-- fun things to do.

926
00:45:01,642 --> 00:45:03,100
What should I tell
you about first?

927
00:45:08,449 --> 00:45:09,240
I'm getting hungry.

928
00:45:11,070 --> 00:45:12,820
Maybe I'll have the
rest of that Mirabell.

929
00:45:12,820 --> 00:45:14,210
It's just sitting there.

930
00:45:14,210 --> 00:45:17,410
Half a Mozart Kugel,
can't have that.

931
00:45:17,410 --> 00:45:17,910
All right.

932
00:45:17,910 --> 00:45:19,165
AUDIENCE: [INAUDIBLE].

933
00:45:19,165 --> 00:45:19,790
PROFESSOR: Yes.

934
00:45:19,790 --> 00:45:21,105
Question?

935
00:45:21,105 --> 00:45:22,230
AUDIENCE: I was wondering--

936
00:45:22,230 --> 00:45:24,790
PROFESSOR: You can have any.

937
00:45:24,790 --> 00:45:26,622
AUDIENCE: Because of
the fact that you've

938
00:45:26,622 --> 00:45:29,330
got this contracting
that occurs,

939
00:45:29,330 --> 00:45:32,645
I was wondering-- it's obvious
that because of contracting

940
00:45:32,645 --> 00:45:39,920
you need more than 4 pi
r surface area of paper

941
00:45:39,920 --> 00:45:44,590
to wrap a sphere--
a one unit sphere.

942
00:45:44,590 --> 00:45:47,790
I'm wondering, do you
know exactly how much

943
00:45:47,790 --> 00:45:50,041
paper do you require due to
the fact that [INAUDIBLE]?

944
00:45:50,041 --> 00:45:51,081
PROFESSOR: Good question.

945
00:45:51,081 --> 00:45:52,360
How much paper do you require?

946
00:45:52,360 --> 00:45:53,950
Area is not being
preserved here.

947
00:45:53,950 --> 00:45:57,170
Just like the continuous
unfolding stuff,

948
00:45:57,170 --> 00:45:58,790
you don't preserve area.

949
00:45:58,790 --> 00:46:00,040
This is really the same thing.

950
00:46:00,040 --> 00:46:02,075
I think you can prove
this is a contractive map.

951
00:46:02,075 --> 00:46:03,700
So if you had this
as a piece of paper,

952
00:46:03,700 --> 00:46:08,510
you really could fold
that smooth pyramid.

953
00:46:11,570 --> 00:46:14,790
But you lose material when
you do a contractive mapping.

954
00:46:14,790 --> 00:46:17,880
And in the inverse, when you're
unfolding, you gain material.

955
00:46:17,880 --> 00:46:21,395
How much material
do I have to gain?

956
00:46:21,395 --> 00:46:22,020
Think about it.

957
00:46:22,020 --> 00:46:23,830
You should be able to
answer that question.

958
00:46:23,830 --> 00:46:28,190
I'm going to tell you about some
stuff while you think about it.

959
00:46:28,190 --> 00:46:31,140
That was the original
question we started with.

960
00:46:31,140 --> 00:46:35,330
What's the minimum area
wrapping of a unit sphere?

961
00:46:35,330 --> 00:46:38,847
The surface area of a
sphere is 4 pi, FYI.

962
00:46:38,847 --> 00:46:39,930
How much more do you need?

963
00:46:42,570 --> 00:46:43,230
All right.

964
00:46:43,230 --> 00:46:46,960
To describe these unfoldings,
I guess they are, or wrappings,

965
00:46:46,960 --> 00:46:49,434
whichever way you're
thinking about going,

966
00:46:49,434 --> 00:46:51,100
it's useful to have
some structure, kind

967
00:46:51,100 --> 00:46:52,590
of a backbone to the unfolding.

968
00:46:52,590 --> 00:46:55,400
So we came up with this
idea of stretched path.

969
00:46:55,400 --> 00:46:57,770
And you can show that any
kind of optimal wrapping

970
00:46:57,770 --> 00:47:00,260
will have at least
one stretch path.

971
00:47:00,260 --> 00:47:02,820
And our idea is
actually-- let me tell you

972
00:47:02,820 --> 00:47:04,670
what a stretch path is.

973
00:47:04,670 --> 00:47:10,830
This is a, let's say, a
path between two vertices

974
00:47:10,830 --> 00:47:12,990
whose length does not decrease.

975
00:47:12,990 --> 00:47:19,850
So it actually is
preserved in the folding.

976
00:47:29,190 --> 00:47:32,150
So I have my sphere,
I have two points,

977
00:47:32,150 --> 00:47:33,960
take the shortest
path between them.

978
00:47:33,960 --> 00:47:35,660
That shortest path
is exactly what

979
00:47:35,660 --> 00:47:38,327
it is in the unfolding,
exactly the same length.

980
00:47:38,327 --> 00:47:39,910
So that's what we
call a stretch path.

981
00:47:39,910 --> 00:47:41,610
There's no crinkling
along that path.

982
00:47:41,610 --> 00:47:44,150
There maybe crinkling on
either side of the path,

983
00:47:44,150 --> 00:47:46,550
but not on it.

984
00:47:46,550 --> 00:47:50,500
Our idea is-- OK, that gives
you some kind of structure.

985
00:47:50,500 --> 00:47:52,740
Our idea is to completely
cover the surface

986
00:47:52,740 --> 00:47:55,777
of the sphere with
stretched paths.

987
00:47:55,777 --> 00:47:57,360
There's a lot of
ways you can do that.

988
00:47:57,360 --> 00:48:00,270
Some of them will be valid,
some of them won't be.

989
00:48:00,270 --> 00:48:03,469
The simplest one is
the source unfolding.

990
00:48:03,469 --> 00:48:05,510
Or I guess you could call
it the source wrapping.

991
00:48:08,720 --> 00:48:12,090
Wrapping somehow means
contractive I guess.

992
00:48:12,090 --> 00:48:15,550
Which is, you take your
sphere, take some point x,

993
00:48:15,550 --> 00:48:17,992
you take all the
shortest paths around x--

994
00:48:17,992 --> 00:48:19,200
it's a little tricky to draw.

995
00:48:22,697 --> 00:48:24,780
In other words, the Voronoi
diagram at that point.

996
00:48:24,780 --> 00:48:27,200
Just like we did with
source unfolding.

997
00:48:27,200 --> 00:48:31,180
And you call all of these edges,
all those straight lines--

998
00:48:31,180 --> 00:48:36,020
geodesics, whatever-- they go to
this opposite south pole here.

999
00:48:36,020 --> 00:48:40,130
All of those lines you
call stretch paths.

1000
00:48:40,130 --> 00:48:42,210
That determines an unfolding.

1001
00:48:42,210 --> 00:48:44,720
And it happens to
be contractive.

1002
00:48:44,720 --> 00:48:48,070
The unfolding will be a
big disk because you'll

1003
00:48:48,070 --> 00:48:51,750
have the center-- call this x.

1004
00:48:51,750 --> 00:48:54,500
This is 3D, this is 2D,
in case it wasn't clear.

1005
00:48:54,500 --> 00:48:56,250
They both look the same.

1006
00:48:56,250 --> 00:48:57,850
But now all those
paths are straight.

1007
00:49:00,960 --> 00:49:03,260
And it's clear what the
lengths of these paths are.

1008
00:49:03,260 --> 00:49:08,927
Each one of them is
half the circumference.

1009
00:49:08,927 --> 00:49:09,510
Circumference?

1010
00:49:09,510 --> 00:49:10,551
Is that what you call it?

1011
00:49:10,551 --> 00:49:13,050
Half of an equator
of the sphere.

1012
00:49:13,050 --> 00:49:16,350
Which for a unit sphere is pi.

1013
00:49:16,350 --> 00:49:24,020
So this is radius pi, which
means the area of this thing

1014
00:49:24,020 --> 00:49:25,750
is pi r squared.

1015
00:49:25,750 --> 00:49:28,380
Sorry, the area is pi cubed.

1016
00:49:31,110 --> 00:49:32,610
Interesting.

1017
00:49:32,610 --> 00:49:38,910
But pi cubed, you can
compute, is more than 4pi.

1018
00:49:38,910 --> 00:49:43,580
Pi squared is
almost-- is over nine.

1019
00:49:43,580 --> 00:49:45,480
It's over nine times pi.

1020
00:49:45,480 --> 00:49:47,130
We want four times pi.

1021
00:49:47,130 --> 00:49:49,340
This is an example of
not preserving area.

1022
00:49:49,340 --> 00:49:51,680
What's happening is
there's lots of crinkling

1023
00:49:51,680 --> 00:49:52,604
between these lines.

1024
00:49:52,604 --> 00:49:54,520
Now, there are infinitely
many of these lines.

1025
00:49:54,520 --> 00:49:58,340
Every point is covered by
one of these stretched paths.

1026
00:49:58,340 --> 00:50:00,460
But again, when I
unfold them, if I

1027
00:50:00,460 --> 00:50:03,760
did a limiting approximation,
there'd be lots of holes here.

1028
00:50:03,760 --> 00:50:05,326
But in the limit
there are no holes

1029
00:50:05,326 --> 00:50:06,700
and that's where
I waste material

1030
00:50:06,700 --> 00:50:08,510
and that's where
I get contraction.

1031
00:50:08,510 --> 00:50:09,495
Make sense?

1032
00:50:09,495 --> 00:50:13,022
I actually brought a
sphere, a couple spheres.

1033
00:50:13,022 --> 00:50:15,230
I brought some chocolate
spheres, but slightly bigger

1034
00:50:15,230 --> 00:50:18,120
sphere so we can
think about this.

1035
00:50:18,120 --> 00:50:22,840
In this case, we're making
stretch paths all around

1036
00:50:22,840 --> 00:50:25,145
from the north pole
down to the south pole

1037
00:50:25,145 --> 00:50:27,320
and kind of cutting
at the south pole.

1038
00:50:27,320 --> 00:50:29,812
And that's it.

1039
00:50:29,812 --> 00:50:30,788
All right.

1040
00:50:34,700 --> 00:50:39,680
Has anyone solved the problem
of what the minimum area

1041
00:50:39,680 --> 00:50:42,480
wrapping of the sphere is?

1042
00:50:42,480 --> 00:50:42,980
Yeah?

1043
00:50:42,980 --> 00:50:43,720
You solved it?

1044
00:50:43,720 --> 00:50:44,570
Janine.

1045
00:50:44,570 --> 00:50:45,360
Or an idea.

1046
00:50:45,360 --> 00:50:47,320
AUDIENCE: I think
it's 4pi plus epsilon.

1047
00:50:47,320 --> 00:50:49,219
PROFESSOR: 4pi plus
epsilon is correct.

1048
00:50:49,219 --> 00:50:50,260
How do you want to do it?

1049
00:50:53,680 --> 00:50:54,380
Exactly.

1050
00:50:54,380 --> 00:50:56,810
AUDIENCE: An apple
core-- or peeling.

1051
00:50:56,810 --> 00:50:59,480
PROFESSOR: Apple peeling, yes.

1052
00:50:59,480 --> 00:51:01,860
Or I prefer to think
of it as lecturer--

1053
00:51:01,860 --> 00:51:04,670
I don't know which one--
the one about strip folding.

1054
00:51:04,670 --> 00:51:06,630
AUDIENCE: Oh, yeah.

1055
00:51:06,630 --> 00:51:07,470
PROFESSOR: Yes.

1056
00:51:07,470 --> 00:51:09,260
Exactly.

1057
00:51:09,260 --> 00:51:11,700
So we had this result
about strip folding

1058
00:51:11,700 --> 00:51:17,364
if you remember way back in
the folding anything lecture.

1059
00:51:17,364 --> 00:51:19,030
We talked about this
and then orgamizer.

1060
00:51:19,030 --> 00:51:20,450
This was the inefficient method.

1061
00:51:20,450 --> 00:51:24,580
But if you started with a
very long rectangle of paper,

1062
00:51:24,580 --> 00:51:28,490
we could show that the
area of that piece of paper

1063
00:51:28,490 --> 00:51:30,540
is arbitrary-- could
be made arbitrarily

1064
00:51:30,540 --> 00:51:32,805
close to the area
of your polyhedron.

1065
00:51:36,380 --> 00:51:38,036
So that area plus some epsilon.

1066
00:51:38,036 --> 00:51:39,410
And you can make
epsilon as small

1067
00:51:39,410 --> 00:51:43,650
as you wanted just by
making this polygon longer.

1068
00:51:43,650 --> 00:51:46,780
If you take a polyhedral
approximation of a sphere

1069
00:51:46,780 --> 00:51:50,180
and apply this theorem, you'll
get very close to the surface

1070
00:51:50,180 --> 00:51:52,770
area of that approximation which
is very close to the surface

1071
00:51:52,770 --> 00:51:54,960
area of the sphere,
which is 4pi.

1072
00:51:58,720 --> 00:52:01,170
And then, if you take
an outer approximation,

1073
00:52:01,170 --> 00:52:04,430
then you can just sort of
crinkle all those vertices down

1074
00:52:04,430 --> 00:52:05,920
against the sphere.

1075
00:52:05,920 --> 00:52:08,460
And you'll do-- it's
maybe not totally obvious,

1076
00:52:08,460 --> 00:52:09,887
but it's not that hard.

1077
00:52:09,887 --> 00:52:12,470
You get a contractive map that
actually wraps the sphere using

1078
00:52:12,470 --> 00:52:16,465
not much more surface
area than what you had.

1079
00:52:16,465 --> 00:52:18,090
The same as what you
had, which was not

1080
00:52:18,090 --> 00:52:19,280
much more than the sphere.

1081
00:52:19,280 --> 00:52:21,020
So you can get 4pi plus epsilon.

1082
00:52:21,020 --> 00:52:23,860
So it turns out, minimizing
the area of your wrapping

1083
00:52:23,860 --> 00:52:26,272
is not that interesting
a question by itself

1084
00:52:26,272 --> 00:52:28,230
because there's kind of
this cheating solution.

1085
00:52:28,230 --> 00:52:30,146
Obviously, you don't
want to manufacture that.

1086
00:52:30,146 --> 00:52:31,380
I don't think.

1087
00:52:31,380 --> 00:52:35,660
Maybe she built a robot, did
the inverse orange peeling,

1088
00:52:35,660 --> 00:52:38,229
it would be pretty cool.

1089
00:52:38,229 --> 00:52:40,520
But if you're trying to sell
this idea to confectioners

1090
00:52:40,520 --> 00:52:43,499
around the world, I
wouldn't recommend

1091
00:52:43,499 --> 00:52:44,540
attempting strip folding.

1092
00:52:44,540 --> 00:52:46,030
So we would like
something that's

1093
00:52:46,030 --> 00:52:51,410
reasonably nice but-- yeah.

1094
00:52:51,410 --> 00:52:52,250
OK.

1095
00:52:52,250 --> 00:52:54,695
AUDIENCE: So if
you're trying sell

1096
00:52:54,695 --> 00:53:00,563
wrapper companies [INAUDIBLE]
square or rectangular amounts

1097
00:53:00,563 --> 00:53:03,008
would these be 4pi squared?

1098
00:53:03,008 --> 00:53:05,453
Because you're basically
producing something

1099
00:53:05,453 --> 00:53:09,320
that maybe would expand
the circumference and--

1100
00:53:09,320 --> 00:53:11,320
PROFESSOR: Well, OK, an
interesting question is,

1101
00:53:11,320 --> 00:53:14,524
what is the surface
area of these wrappings?

1102
00:53:14,524 --> 00:53:16,440
These actually have
pretty special properties,

1103
00:53:16,440 --> 00:53:18,970
the rectangle and
the-- I'm getting

1104
00:53:18,970 --> 00:53:20,440
hungry-- and the square.

1105
00:53:23,940 --> 00:53:28,236
Have you thought about which
of these has more surface area?

1106
00:53:28,236 --> 00:53:30,030
The rectangle has more?

1107
00:53:30,030 --> 00:53:32,110
Who thinks the
rectangle has more?

1108
00:53:32,110 --> 00:53:34,474
Who thinks the square has more?

1109
00:53:34,474 --> 00:53:35,640
Who thinks they're the same?

1110
00:53:39,140 --> 00:53:41,640
Seed you with that idea.

1111
00:53:41,640 --> 00:53:43,220
Well, they're actually the same.

1112
00:53:43,220 --> 00:53:46,199
I'll explain why
in a little bit.

1113
00:53:46,199 --> 00:53:46,990
It's crazy, I know.

1114
00:53:51,580 --> 00:53:54,845
I prefer the square because
it has less perimeter.

1115
00:54:00,010 --> 00:54:01,920
There are obviously many
things you might not

1116
00:54:01,920 --> 00:54:03,750
like about this strip folding.

1117
00:54:03,750 --> 00:54:06,740
It has high aspect
ratio, but in particular,

1118
00:54:06,740 --> 00:54:08,620
one way to characterize
why this is nasty,

1119
00:54:08,620 --> 00:54:10,630
it says high perimeter.

1120
00:54:10,630 --> 00:54:14,915
Perimeter is annoying
because-- for a few reasons.

1121
00:54:14,915 --> 00:54:16,706
Intuitively, if you
started with the sphere

1122
00:54:16,706 --> 00:54:18,890
and you wanted to
cut it open, then you

1123
00:54:18,890 --> 00:54:21,070
need a lot of cutting
because the perimeter is

1124
00:54:21,070 --> 00:54:22,860
the amount that you cut.

1125
00:54:22,860 --> 00:54:25,150
Or it's twice that.

1126
00:54:25,150 --> 00:54:29,382
So conversely, when you're
trying to wrap a surface,

1127
00:54:29,382 --> 00:54:30,840
if you have high
perimeter, there's

1128
00:54:30,840 --> 00:54:34,200
a lot to worry about, lots of
things that have to match up.

1129
00:54:34,200 --> 00:54:36,740
So you've got to be very
careful in your crinkling.

1130
00:54:36,740 --> 00:54:38,698
Whereas I really didn't
have to be very careful

1131
00:54:38,698 --> 00:54:42,140
wrapping with the square
because it has low perimeter.

1132
00:54:42,140 --> 00:54:44,310
Another sense in which
perimeter is annoying

1133
00:54:44,310 --> 00:54:46,870
is you're going to
have some error.

1134
00:54:46,870 --> 00:54:51,400
So in fact, you can't just
perfectly wrap a sphere.

1135
00:54:51,400 --> 00:54:53,740
Like, if I took a disk
of exactly this size,

1136
00:54:53,740 --> 00:54:56,100
I probably couldn't
wrap a sphere with it

1137
00:54:56,100 --> 00:54:58,762
because I'm going to
lose just some epsilon

1138
00:54:58,762 --> 00:55:00,220
and then there'll
be a little point

1139
00:55:00,220 --> 00:55:03,720
around the south
pole that's missing.

1140
00:55:03,720 --> 00:55:06,030
If you're really going to
build confectionery-- build

1141
00:55:06,030 --> 00:55:08,890
confectionery-- if you're really
going to wrap confectionery,

1142
00:55:08,890 --> 00:55:10,860
you probably make this
a little bit bigger.

1143
00:55:10,860 --> 00:55:14,510
How much area do
you lose when you

1144
00:55:14,510 --> 00:55:16,010
make your thing a
little bit bigger?

1145
00:55:16,010 --> 00:55:18,637
Well, it's roughly the
perimeter times epsilon.

1146
00:55:18,637 --> 00:55:21,220
It's a little bit different from
that, but basically perimeter

1147
00:55:21,220 --> 00:55:22,080
times epsilon.

1148
00:55:22,080 --> 00:55:24,050
So if you minimize
perimeter, then

1149
00:55:24,050 --> 00:55:27,944
your area wastage from
double coverage will go down.

1150
00:55:27,944 --> 00:55:29,360
So really what I
want is something

1151
00:55:29,360 --> 00:55:33,140
that's low area
and low perimeter.

1152
00:55:33,140 --> 00:55:35,090
That's our new goal.

1153
00:55:35,090 --> 00:55:37,820
And that's why I like
Furst better than Mirabell,

1154
00:55:37,820 --> 00:55:39,510
not just because
they taste better.

1155
00:55:39,510 --> 00:55:42,222
I really should verify
that they taste better.

1156
00:55:42,222 --> 00:55:45,010
Oh, yeah.

1157
00:55:45,010 --> 00:55:47,520
Now, they're very
different insides.

1158
00:55:47,520 --> 00:55:50,580
Oh, but it's so good.

1159
00:55:50,580 --> 00:55:52,230
Oh, much better chocolate.

1160
00:55:52,230 --> 00:55:52,740
All right.

1161
00:56:00,548 --> 00:56:04,160
Maybe I should have a
lottery for eating the rest.

1162
00:56:07,890 --> 00:56:08,570
Petal wrapping.

1163
00:56:12,790 --> 00:56:16,010
This is going to be
another way to put stretch

1164
00:56:16,010 --> 00:56:18,450
paths on the sphere and
get an unfolding-- get

1165
00:56:18,450 --> 00:56:21,410
a contractive
wrapping from that.

1166
00:56:21,410 --> 00:56:23,391
Which actually
corresponds to this toy.

1167
00:56:23,391 --> 00:56:25,140
I don't know if you
ever played with this.

1168
00:56:25,140 --> 00:56:26,460
The Frisbee that's also a ball.

1169
00:56:29,120 --> 00:56:30,940
These-- you see those cuts?

1170
00:56:33,839 --> 00:56:35,630
Those are the shortest
paths from the north

1171
00:56:35,630 --> 00:56:38,200
pole to the south pole.

1172
00:56:38,200 --> 00:56:41,030
But we're not--
before we were sort

1173
00:56:41,030 --> 00:56:44,650
of using-- kind of slicing
along but also preserving--

1174
00:56:44,650 --> 00:56:47,780
that's kind of confusing--
all of them, infinitely many.

1175
00:56:47,780 --> 00:56:49,280
Now I want to do
with finitely many.

1176
00:56:49,280 --> 00:56:51,240
Here I only have six.

1177
00:56:51,240 --> 00:56:52,800
Six?

1178
00:56:52,800 --> 00:56:53,450
Yes, six.

1179
00:56:56,772 --> 00:56:57,730
Doesn't have to be six.

1180
00:56:57,730 --> 00:57:00,260
Could be any number k.

1181
00:57:00,260 --> 00:57:07,070
So I'm going to look at k
stretch paths from north pole

1182
00:57:07,070 --> 00:57:07,850
to south pole.

1183
00:57:07,850 --> 00:57:11,190
So I have my sphere,
here is my north pole.

1184
00:57:11,190 --> 00:57:15,200
I do some constant number.

1185
00:57:15,200 --> 00:57:17,370
Here I've done five.

1186
00:57:17,370 --> 00:57:21,147
This is equally spaced
angular around the north pole,

1187
00:57:21,147 --> 00:57:22,980
they all go to the south
pole on the bottom.

1188
00:57:26,270 --> 00:57:27,924
Something like that.

1189
00:57:27,924 --> 00:57:29,340
It's going to start
getting messy.

1190
00:57:32,160 --> 00:57:33,790
These are going to
be stretch paths.

1191
00:57:33,790 --> 00:57:38,910
I have to fill these orange
slices in between them.

1192
00:57:38,910 --> 00:57:41,630
How should I fill them?

1193
00:57:41,630 --> 00:57:44,457
Here's an idea.

1194
00:57:44,457 --> 00:57:46,040
It's kind of like
the Voronoi diagram.

1195
00:57:46,040 --> 00:57:51,485
Voronoi diagram is you grow from
many objects simultaneously.

1196
00:57:51,485 --> 00:57:54,550
With this sphere wrapping
we grew from a single point

1197
00:57:54,550 --> 00:57:57,130
in all directions.

1198
00:57:57,130 --> 00:57:57,800
That was it.

1199
00:57:57,800 --> 00:57:59,380
Now, I've drawn these paths.

1200
00:57:59,380 --> 00:58:04,120
What if I grew from all of these
finite stretch paths at once?

1201
00:58:04,120 --> 00:58:07,299
What I would get, these
would at some point converge.

1202
00:58:07,299 --> 00:58:09,340
They would converge at
the angular bisector here.

1203
00:58:13,760 --> 00:58:15,560
Along the way,
what's happening is

1204
00:58:15,560 --> 00:58:20,330
each of these points is kind of
tracing along a straight line,

1205
00:58:20,330 --> 00:58:22,910
so to speak, a
great circular arc

1206
00:58:22,910 --> 00:58:25,640
until it reaches that point.

1207
00:58:25,640 --> 00:58:28,670
I'm going to make
these stretched paths,

1208
00:58:28,670 --> 00:58:31,794
and these are also stretched
pads, and basically cutting

1209
00:58:31,794 --> 00:58:32,710
along the dotted line.

1210
00:58:36,310 --> 00:58:36,810
OK.

1211
00:58:36,810 --> 00:58:38,800
I won't draw all of them.

1212
00:58:38,800 --> 00:58:39,860
But you get the idea.

1213
00:58:39,860 --> 00:58:43,015
When you unfold this thing, you
have these kind of wish bones.

1214
00:58:52,540 --> 00:58:55,430
These guys meet sort
of perpendicularly.

1215
00:58:55,430 --> 00:58:57,702
So when you unfold you
preserve those angles.

1216
00:58:57,702 --> 00:58:59,160
Remember these are
stretched paths.

1217
00:58:59,160 --> 00:59:01,430
These lengths are preserved.

1218
00:59:01,430 --> 00:59:04,780
The stuff in between,
it's really a curved thing

1219
00:59:04,780 --> 00:59:07,380
and it gets slightly
larger in this thing.

1220
00:59:07,380 --> 00:59:10,400
We call this thing a petal.

1221
00:59:10,400 --> 00:59:12,590
But this is an
unfolding of this sort

1222
00:59:12,590 --> 00:59:14,580
of one fifth of the sphere.

1223
00:59:14,580 --> 00:59:18,660
This is the center line and
these are the two sides.

1224
00:59:18,660 --> 00:59:23,540
So that's the idea
of petal wrapping.

1225
00:59:23,540 --> 00:59:26,950
Now, there's different ways
to join these petals together.

1226
00:59:26,950 --> 00:59:31,020
One idea is join them
all at the north pole.

1227
00:59:31,020 --> 00:59:33,390
So then what you get--
it's a little hard

1228
00:59:33,390 --> 00:59:36,020
to imagine that these things
don't intersect each other,

1229
00:59:36,020 --> 00:59:38,120
especially when I've
drawn them so fat.

1230
00:59:38,120 --> 00:59:40,245
For five, it's going to
look something like this.

1231
00:59:43,159 --> 00:59:44,200
I'm drawing it perfectly.

1232
00:59:46,659 --> 00:59:48,450
They won't actually
intersect in the center

1233
00:59:48,450 --> 00:59:50,533
because the total amount
of material at the center

1234
00:59:50,533 --> 00:59:52,660
is only 360 degrees.

1235
00:59:52,660 --> 00:59:56,250
And this will actually have--
the tangents will actually

1236
00:59:56,250 --> 00:59:57,090
match up here.

1237
00:59:57,090 --> 00:59:59,380
So they will perfectly
mean in the center

1238
00:59:59,380 --> 01:00:01,810
without self-intersection.

1239
01:00:01,810 --> 01:00:03,610
So it's only sort of
a vertex unfolding.

1240
01:00:03,610 --> 01:00:05,180
Because they're only
connected at the single point.

1241
01:00:05,180 --> 01:00:06,710
But that's maybe necessary.

1242
01:00:06,710 --> 01:00:07,702
I don't know.

1243
01:00:07,702 --> 01:00:09,410
And again, these are
the stretched paths.

1244
01:00:12,540 --> 01:00:15,450
It's a neat way to
think about unfolding.

1245
01:00:15,450 --> 01:00:20,160
You can draw these--
ignore the yellow for now.

1246
01:00:20,160 --> 01:00:22,734
That's if you might want
to actually cut these out.

1247
01:00:22,734 --> 01:00:24,650
So we have the three
petal unfolding, the four

1248
01:00:24,650 --> 01:00:26,770
petal unfolding, five
petal unfolding, six petal,

1249
01:00:26,770 --> 01:00:28,920
you can go arbitrarily high.

1250
01:00:28,920 --> 01:00:32,610
In the limit, the
area of the blue stuff

1251
01:00:32,610 --> 01:00:35,320
will converge to 4pi.

1252
01:00:35,320 --> 01:00:37,279
This is another way to
get arbitrarily close.

1253
01:00:37,279 --> 01:00:38,820
Because the smaller
these things are,

1254
01:00:38,820 --> 01:00:39,945
the less you're distorting.

1255
01:00:43,680 --> 01:00:45,530
So this is two
families of wrappings.

1256
01:00:45,530 --> 01:00:48,100
We call these-- the blue
stuff is the petal unfolding,

1257
01:00:48,100 --> 01:00:52,050
the yellow is the convex
hull of the petal unfoldings.

1258
01:00:52,050 --> 01:00:53,760
Not quite, actually.

1259
01:00:53,760 --> 01:01:00,090
So in the case of k equals 3,
we could think about a convex

1260
01:01:00,090 --> 01:01:02,110
hull, but what I've drawn
here is the smallest

1261
01:01:02,110 --> 01:01:04,880
enclosing equilateral triangle.

1262
01:01:04,880 --> 01:01:07,060
Now this is the first
class of unfoldings

1263
01:01:07,060 --> 01:01:10,690
we came up with after
the obvious sphere--

1264
01:01:10,690 --> 01:01:16,860
or circular one which
had pi cubed over 9pi.

1265
01:01:16,860 --> 01:01:21,090
One thing we wondered
is that triangle,

1266
01:01:21,090 --> 01:01:26,550
is it better than the
Mozart Kugel wrappings?

1267
01:01:26,550 --> 01:01:28,636
Or is it worse?

1268
01:01:28,636 --> 01:01:30,010
One way or the
other, presumably.

1269
01:01:30,010 --> 01:01:31,290
Or maybe it's the same again.

1270
01:01:31,290 --> 01:01:33,280
Who knows.

1271
01:01:33,280 --> 01:01:33,780
Let's see.

1272
01:01:36,601 --> 01:01:37,100
Yeah.

1273
01:01:37,100 --> 01:01:40,624
Let me tell you, at this point--
so I need more techniques

1274
01:01:40,624 --> 01:01:42,040
to cover both of
those unfoldings,

1275
01:01:42,040 --> 01:01:46,980
but that square is
the first wrapping.

1276
01:01:46,980 --> 01:01:48,274
This one.

1277
01:01:48,274 --> 01:01:50,440
I'll have to unfold another
one just so you can see.

1278
01:01:53,702 --> 01:01:54,640
All right.

1279
01:01:54,640 --> 01:01:57,510
So you see-- I mean,
I did a few times

1280
01:01:57,510 --> 01:01:59,000
so maybe you already know.

1281
01:01:59,000 --> 01:02:01,890
Here I have four corners,
four right angles.

1282
01:02:01,890 --> 01:02:04,110
They meet at the top.

1283
01:02:04,110 --> 01:02:07,320
So that is-- that's
where the four

1284
01:02:07,320 --> 01:02:08,820
corners of the
square which are also

1285
01:02:08,820 --> 01:02:11,980
the four corners of the
petal come together.

1286
01:02:11,980 --> 01:02:16,000
And then it's just
everything-- these lines

1287
01:02:16,000 --> 01:02:17,480
are stretched paths.

1288
01:02:17,480 --> 01:02:20,330
There's no distortion around
them, they just kind of unfold.

1289
01:02:20,330 --> 01:02:26,010
The stuff in the middle is--
it's like undressing here.

1290
01:02:26,010 --> 01:02:27,890
It's so tantalizing.

1291
01:02:27,890 --> 01:02:31,120
The stuff in the middle
is getting contracted.

1292
01:02:31,120 --> 01:02:33,860
So if-- I don't know if you
could see, maybe in the video

1293
01:02:33,860 --> 01:02:35,040
you'll be able to see.

1294
01:02:35,040 --> 01:02:36,900
Look closely at here.

1295
01:02:36,900 --> 01:02:40,950
Along this path there
isn't so much crinkling.

1296
01:02:40,950 --> 01:02:42,670
In between there's a
lot more crinkling.

1297
01:02:42,670 --> 01:02:43,990
It's like here there's
a lot of crinkling.

1298
01:02:43,990 --> 01:02:45,460
When I unfold it,
a lot more stuff

1299
01:02:45,460 --> 01:02:49,121
happens in between
the four top paths.

1300
01:02:49,121 --> 01:02:49,620
OK?

1301
01:02:49,620 --> 01:02:53,160
Now, they wanted to
cut very simple shapes.

1302
01:02:53,160 --> 01:02:55,320
In particular, shapes
that tiled the plane.

1303
01:02:55,320 --> 01:02:59,420
So they ended up using squares
instead of the sort of optimal

1304
01:02:59,420 --> 01:03:02,555
with that stretched paths
which are those four petals.

1305
01:03:05,122 --> 01:03:07,080
So they used the yellow,
whereas you could just

1306
01:03:07,080 --> 01:03:07,770
get away with the blue.

1307
01:03:07,770 --> 01:03:09,895
Of course, you want to add
a little bit of material

1308
01:03:09,895 --> 01:03:12,680
in the center so it
doesn't fall apart.

1309
01:03:12,680 --> 01:03:15,040
And you need to thicken
the edges a little bit.

1310
01:03:15,040 --> 01:03:19,740
So they care about that little
yellow minus blue savings?

1311
01:03:19,740 --> 01:03:21,056
I'm not sure.

1312
01:03:21,056 --> 01:03:24,330
But what about the triangle?

1313
01:03:24,330 --> 01:03:26,980
We spend a lot of time
computing these shapes.

1314
01:03:26,980 --> 01:03:28,800
It's some weird trigonometry.

1315
01:03:28,800 --> 01:03:30,360
It's in the notes.

1316
01:03:30,360 --> 01:03:33,540
Arc sine of sine over square
root of 1 over sine squared

1317
01:03:33,540 --> 01:03:36,250
minus cos squared of
various parameters

1318
01:03:36,250 --> 01:03:39,435
is the description of
one of these curves.

1319
01:03:39,435 --> 01:03:41,060
So we did all this
computation and then

1320
01:03:41,060 --> 01:03:44,680
we integrated it to compute
what these areas were and then--

1321
01:03:44,680 --> 01:03:47,690
or figure out exactly what these
shapes were, take the tangents,

1322
01:03:47,690 --> 01:03:50,780
say, well, is the smallest
enclosing triangle better

1323
01:03:50,780 --> 01:03:52,940
or worse than the square?

1324
01:03:52,940 --> 01:03:58,545
It turns out it's
better by 0.1%.

1325
01:04:02,620 --> 01:04:05,780
Now, I think-- millions
of Mozart Kugel

1326
01:04:05,780 --> 01:04:07,430
are made every year.

1327
01:04:07,430 --> 01:04:13,200
0.1% could equate
to huge savings.

1328
01:04:13,200 --> 01:04:14,790
Dollars.

1329
01:04:14,790 --> 01:04:16,710
But just think of
the material wastage

1330
01:04:16,710 --> 01:04:20,830
that Furst is committing
by using the square.

1331
01:04:20,830 --> 01:04:23,090
If they used the
triangle instead,

1332
01:04:23,090 --> 01:04:25,830
I mean, this could
solve global warming.

1333
01:04:25,830 --> 01:04:27,565
This could solve
chocolate melting.

1334
01:04:31,580 --> 01:04:34,230
We actually talk about global
warming in the paper as a joke.

1335
01:04:34,230 --> 01:04:36,830
It's a very funny paper.

1336
01:04:36,830 --> 01:04:39,440
Very silly paper, I should say.

1337
01:04:39,440 --> 01:04:39,940
Right.

1338
01:04:39,940 --> 01:04:41,315
Let me tell you
another wrapping.

1339
01:04:52,010 --> 01:04:53,430
Chalk.

1340
01:04:53,430 --> 01:04:54,440
The comb wrapping.

1341
01:05:05,010 --> 01:05:07,930
Comb wrapping, we're going
to use the same idea.

1342
01:05:07,930 --> 01:05:10,730
Again-- do I really
want to draw this again?

1343
01:05:14,825 --> 01:05:19,670
I'm going to draw those petals,
but instead of connecting them

1344
01:05:19,670 --> 01:05:21,610
at the north pole, we're
going to connect them

1345
01:05:21,610 --> 01:05:23,350
along the equator.

1346
01:05:23,350 --> 01:05:26,102
This is actually what
happens-- oh, no.

1347
01:05:26,102 --> 01:05:28,560
This is not-- this connects
them at the north pole I guess,

1348
01:05:28,560 --> 01:05:29,840
and it folds them in half.

1349
01:05:29,840 --> 01:05:31,350
Whatever.

1350
01:05:31,350 --> 01:05:35,060
But I have the equator
nicely denoted here.

1351
01:05:35,060 --> 01:05:39,080
So I want to keep the equator
as a new stretched path.

1352
01:05:39,080 --> 01:05:41,500
I want that to unfold straight.

1353
01:05:41,500 --> 01:05:43,830
But then attached to
it perpendicularly

1354
01:05:43,830 --> 01:05:46,690
are all these
petals which unfold.

1355
01:05:46,690 --> 01:05:52,400
So the result looks like this.

1356
01:05:52,400 --> 01:05:57,150
Here's the equatorial stretched
path and here are the petals.

1357
01:05:57,150 --> 01:05:59,820
Petals are exactly the
same shapes as before,

1358
01:05:59,820 --> 01:06:03,240
so we still have stretched
pads perpendicular like that.

1359
01:06:03,240 --> 01:06:04,950
But it's a new way
to cover every point

1360
01:06:04,950 --> 01:06:07,340
on the sphere with
stretched paths.

1361
01:06:07,340 --> 01:06:09,370
It leads to the blue unfoldings.

1362
01:06:09,370 --> 01:06:11,980
You can take the smallest
enclosing rectangle

1363
01:06:11,980 --> 01:06:13,076
and you get the yellow.

1364
01:06:17,024 --> 01:06:18,440
What do I want to
say about these?

1365
01:06:22,090 --> 01:06:24,570
They're the same
area, of course.

1366
01:06:24,570 --> 01:06:29,795
They look a lot like these
old cartograms, cartographs--

1367
01:06:29,795 --> 01:06:34,200
whatever you call them--
unfoldings of the earth.

1368
01:06:34,200 --> 01:06:35,410
I thought I brought an earth.

1369
01:06:35,410 --> 01:06:37,080
Maybe we didn't bring the earth.

1370
01:06:37,080 --> 01:06:40,349
A globe, that is.

1371
01:06:40,349 --> 01:06:41,890
You've seen pictures
maybe like this.

1372
01:06:41,890 --> 01:06:43,450
They're not unfolded
in the same way.

1373
01:06:43,450 --> 01:06:46,080
They don't have stretched
paths that are perpendicular.

1374
01:06:46,080 --> 01:06:48,250
They use some other
projection, so their petals

1375
01:06:48,250 --> 01:06:49,650
are a different shape.

1376
01:06:49,650 --> 01:06:51,810
And I'm not sure that
they're contractive--

1377
01:06:51,810 --> 01:06:54,320
or inverse contractive maps.

1378
01:06:54,320 --> 01:06:55,810
But same spirit anyway.

1379
01:06:58,540 --> 01:06:59,040
Let's see.

1380
01:07:02,130 --> 01:07:10,410
This one is the
Mirabell unfolding.

1381
01:07:10,410 --> 01:07:14,250
Should I unfold another
just to illustrate?

1382
01:07:14,250 --> 01:07:15,950
Going through chocolate
like crazy here.

1383
01:07:15,950 --> 01:07:19,180
This is the really
hard one to unfold.

1384
01:07:19,180 --> 01:07:22,587
So here's one corner--
a little hard to see.

1385
01:07:22,587 --> 01:07:23,920
Where's the equator going to be?

1386
01:07:23,920 --> 01:07:27,270
I think the equator
is around here.

1387
01:07:27,270 --> 01:07:28,690
That make sense?

1388
01:07:28,690 --> 01:07:29,190
No.

1389
01:07:29,190 --> 01:07:32,210
Actually it should be
perpendicular to that.

1390
01:07:32,210 --> 01:07:34,619
Around here.

1391
01:07:34,619 --> 01:07:35,910
This is a little harder to see.

1392
01:07:39,600 --> 01:07:41,350
Here's a corner, actually.

1393
01:07:41,350 --> 01:07:43,187
Tucked inside.

1394
01:07:43,187 --> 01:07:45,020
I've been really curious
what kind of robots

1395
01:07:45,020 --> 01:07:46,241
they have to do this.

1396
01:07:46,241 --> 01:07:46,740
All right.

1397
01:07:46,740 --> 01:07:47,531
Here's the equator.

1398
01:07:47,531 --> 01:07:48,570
Now I can see it.

1399
01:07:48,570 --> 01:07:50,420
So this-- the long
edge of the rectangle

1400
01:07:50,420 --> 01:07:52,400
is wrapped around the equator.

1401
01:07:52,400 --> 01:07:54,960
See it come off?

1402
01:07:54,960 --> 01:07:57,720
And then the corners
are wrapped up

1403
01:07:57,720 --> 01:08:02,130
against-- so there's
two-- well, yeah,

1404
01:08:02,130 --> 01:08:04,810
these guys go on one side,
these guys go on the other.

1405
01:08:04,810 --> 01:08:06,860
That's the short edge
of the rectangle.

1406
01:08:06,860 --> 01:08:09,580
Now we can actually compute how
big these things are, right?

1407
01:08:09,580 --> 01:08:14,780
So the-- where's the-- I
have a page about this.

1408
01:08:14,780 --> 01:08:17,149
Not much.

1409
01:08:17,149 --> 01:08:22,380
The vertical length,
that is one half equator.

1410
01:08:22,380 --> 01:08:24,160
So that's pi.

1411
01:08:24,160 --> 01:08:28,040
The horizontal length is 2pi.

1412
01:08:28,040 --> 01:08:36,990
So this is Mirabell
rectangle, pi by 2pi.

1413
01:08:36,990 --> 01:08:41,040
And we talked about
the Furst wrapping

1414
01:08:41,040 --> 01:08:52,600
which, let's see, this length--
so this goes like this.

1415
01:08:52,600 --> 01:08:55,090
So this is 2pi, right?

1416
01:08:55,090 --> 01:08:57,649
The diagonal is 2pi.

1417
01:08:57,649 --> 01:09:01,880
So that means this edge
length is root 2pi.

1418
01:09:01,880 --> 01:09:02,439
By root 2pi.

1419
01:09:05,359 --> 01:09:08,779
And magically-- I don't
really have a great reason why

1420
01:09:08,779 --> 01:09:10,939
this should be the case
but-- well, maybe I

1421
01:09:10,939 --> 01:09:13,760
do-- but these
have the same area.

1422
01:09:13,760 --> 01:09:14,790
Both 2pi squared.

1423
01:09:17,410 --> 01:09:19,479
Which is like 6pi.

1424
01:09:19,479 --> 01:09:23,689
So better than 9ish pi, but
still not quite as good as

1425
01:09:23,689 --> 01:09:26,250
4ish pi.

1426
01:09:26,250 --> 01:09:29,210
I have some exact numbers.

1427
01:09:29,210 --> 01:09:32,600
More exact numbers?

1428
01:09:32,600 --> 01:09:35,220
Oh, I have the perimeters.

1429
01:09:35,220 --> 01:09:39,170
So the first wrapping, the
perimeter is 5.7 times pi.

1430
01:09:39,170 --> 01:09:41,540
And the Mirabell
wrapping is 6 times pi.

1431
01:09:41,540 --> 01:09:43,415
So reasonable
savings in perimeter.

1432
01:09:46,430 --> 01:09:48,410
2pi squared.

1433
01:09:48,410 --> 01:09:52,359
The equilateral
triangle wrapping--

1434
01:09:52,359 --> 01:09:55,710
that one in the top left--
instead of 2pi squared,

1435
01:09:55,710 --> 01:10:00,810
it has area 1.9983pi squared.

1436
01:10:00,810 --> 01:10:06,430
That's the 0.1% improvement.

1437
01:10:06,430 --> 01:10:07,887
Amazing.

1438
01:10:07,887 --> 01:10:08,720
This was a surprise.

1439
01:10:08,720 --> 01:10:10,600
We thought it'd be a fair
amount better or worse

1440
01:10:10,600 --> 01:10:11,641
or the same or something.

1441
01:10:11,641 --> 01:10:13,140
But almost the same.

1442
01:10:13,140 --> 01:10:16,100
It's not what we expected.

1443
01:10:16,100 --> 01:10:18,680
Let's see.

1444
01:10:18,680 --> 01:10:22,290
I want to go to packing.

1445
01:10:22,290 --> 01:10:25,300
What if you want to do better
than those simple convex

1446
01:10:25,300 --> 01:10:25,800
shapes?

1447
01:10:25,800 --> 01:10:28,700
Yeah, equilateral triangles and
squares and rectangles pack.

1448
01:10:28,700 --> 01:10:31,800
But what if I tried to-- what
if I allowed non-convex shapes?

1449
01:10:31,800 --> 01:10:35,040
I still want to tile
the plane, so I'm

1450
01:10:35,040 --> 01:10:36,990
going to lose some
material for that.

1451
01:10:36,990 --> 01:10:39,320
I could add some of this
material to the blue guy, some

1452
01:10:39,320 --> 01:10:41,750
to the purple, some to the red.

1453
01:10:41,750 --> 01:10:43,980
You can compute the
area of this thing

1454
01:10:43,980 --> 01:10:46,780
and it does a lot better than
the equilateral triangle even.

1455
01:10:46,780 --> 01:10:48,190
So now we're getting down there.

1456
01:10:48,190 --> 01:10:51,640
The perimeter goes up a
little bit, but not by much.

1457
01:10:51,640 --> 01:10:57,690
Packing three petals gives
1.6 times pi squared.

1458
01:10:57,690 --> 01:10:58,360
Getting better.

1459
01:10:58,360 --> 01:11:01,240
By the way, 4pi-- because I'm
going to speak in pi squareds

1460
01:11:01,240 --> 01:11:08,050
now-- 4pi is about
1.27pi squared.

1461
01:11:08,050 --> 01:11:08,550
OK.

1462
01:11:08,550 --> 01:11:10,620
So what we want
is 1.27pi squared.

1463
01:11:10,620 --> 01:11:16,590
At this point, we're at
1.6033 times pi squared.

1464
01:11:16,590 --> 01:11:19,820
The comb does better.

1465
01:11:19,820 --> 01:11:23,530
In the limit, you don't
lose zero here in the limit,

1466
01:11:23,530 --> 01:11:26,310
but you lose quite little.

1467
01:11:26,310 --> 01:11:29,840
In the limit you get
1.333 times pi squared.

1468
01:11:29,840 --> 01:11:31,089
Versus 1.27.

1469
01:11:31,089 --> 01:11:31,880
That's pretty good.

1470
01:11:31,880 --> 01:11:35,830
I think this is-- this is
what we should manufacture.

1471
01:11:35,830 --> 01:11:36,330
Maybe.

1472
01:11:36,330 --> 01:11:39,940
Of course, the perimeter
goes to infinity also.

1473
01:11:39,940 --> 01:11:40,930
Slight problem.

1474
01:11:40,930 --> 01:11:42,247
AUDIENCE: [INAUDIBLE].

1475
01:11:42,247 --> 01:11:44,080
PROFESSOR: You don't
lose zero in the limit.

1476
01:11:44,080 --> 01:11:45,340
For awhile we thought we did.

1477
01:11:45,340 --> 01:11:46,590
You'll have to think about it.

1478
01:11:46,590 --> 01:11:47,298
It's not obvious.

1479
01:11:49,760 --> 01:11:51,630
Really, we have this
two dimensional space.

1480
01:11:51,630 --> 01:11:54,400
It's called the
Pareto curve or-- it's

1481
01:11:54,400 --> 01:11:57,670
a plot of perimeter on
the x-axis versus area

1482
01:11:57,670 --> 01:11:58,350
on the y-axis.

1483
01:11:58,350 --> 01:11:59,770
We want both to be small.

1484
01:12:02,470 --> 01:12:04,669
So two-source
wrapping, for example,

1485
01:12:04,669 --> 01:12:05,960
is pretty good in both metrics.

1486
01:12:05,960 --> 01:12:08,335
That's where you take the
Voronoi diagram, both the north

1487
01:12:08,335 --> 01:12:09,920
and south pole.

1488
01:12:09,920 --> 01:12:12,060
So you actually
have two disks-- one

1489
01:12:12,060 --> 01:12:14,870
for the southern hemisphere, one
for the northern hemisphere--

1490
01:12:14,870 --> 01:12:19,360
you get two disks and you
just attach them side by side.

1491
01:12:19,360 --> 01:12:22,690
Although-- yeah.

1492
01:12:22,690 --> 01:12:23,190
Let's see.

1493
01:12:23,190 --> 01:12:24,940
What's this one?

1494
01:12:24,940 --> 01:12:26,320
There's the Furst wrapping.

1495
01:12:26,320 --> 01:12:28,194
So they're really good
in terms of perimeter.

1496
01:12:28,194 --> 01:12:30,760
In fact, one of the
best perimeters we know.

1497
01:12:30,760 --> 01:12:33,570
It's an open problem
how far left you can go.

1498
01:12:33,570 --> 01:12:38,710
The stars are convex
hull of petal wrappings.

1499
01:12:38,710 --> 01:12:41,540
Because the square was a convex
hull of a four petal wrapping.

1500
01:12:41,540 --> 01:12:43,520
This is the convex of
the three petal wrapping.

1501
01:12:43,520 --> 01:12:45,880
Now, that doesn't
actually tile, but it

1502
01:12:45,880 --> 01:12:47,344
has the smallest
perimeter we now.

1503
01:12:47,344 --> 01:12:49,010
So it's where you
take the three petals,

1504
01:12:49,010 --> 01:12:50,801
but instead of completing
it to a triangle,

1505
01:12:50,801 --> 01:12:54,770
you actually make the sharp
turn around the petal.

1506
01:12:54,770 --> 01:13:02,790
So we have three
petals like this.

1507
01:13:02,790 --> 01:13:05,460
And so the convex hull
is some tangent here.

1508
01:13:05,460 --> 01:13:07,832
And then you follow
the petal for awhile.

1509
01:13:07,832 --> 01:13:09,540
And then you take a
tangent, and then you

1510
01:13:09,540 --> 01:13:12,550
follow the petal for a while,
and then you take a tangent.

1511
01:13:12,550 --> 01:13:18,030
So that's the smallest
perimeter unfolding of a sphere

1512
01:13:18,030 --> 01:13:19,450
that we know.

1513
01:13:19,450 --> 01:13:20,050
Open problem.

1514
01:13:20,050 --> 01:13:21,190
Can you do better?

1515
01:13:21,190 --> 01:13:23,170
Can you get arbitrarily small?

1516
01:13:23,170 --> 01:13:25,120
Probably not.

1517
01:13:25,120 --> 01:13:26,800
Some isoperimetry problem.

1518
01:13:26,800 --> 01:13:29,060
You can get arbitrarily
low area, arbitrarily

1519
01:13:29,060 --> 01:13:32,100
close to the optimal of 4pi.

1520
01:13:32,100 --> 01:13:35,770
That's like the strip wrapping.

1521
01:13:35,770 --> 01:13:37,230
Let's see.

1522
01:13:37,230 --> 01:13:38,560
So that's one limit.

1523
01:13:38,560 --> 01:13:41,500
That corresponds to the limit
of petal or comb wrappings.

1524
01:13:41,500 --> 01:13:43,750
Petal and comb wrappings,
if you don't take any hulls

1525
01:13:43,750 --> 01:13:47,640
or anything, they are the same
area and the same perimeter.

1526
01:13:47,640 --> 01:13:49,730
So I think out of
that you should

1527
01:13:49,730 --> 01:13:52,150
be able to see why, in the
special case of the square

1528
01:13:52,150 --> 01:13:53,900
and the rectangle, you
get the same stuff.

1529
01:13:53,900 --> 01:13:57,750
There's a bijection
between all the parts.

1530
01:13:57,750 --> 01:13:58,650
Not totally obvious.

1531
01:13:58,650 --> 01:14:01,860
You have to dissect
a little bit.

1532
01:14:01,860 --> 01:14:04,230
Mirabell wrappings,
they're not so good.

1533
01:14:04,230 --> 01:14:04,890
What are those?

1534
01:14:04,890 --> 01:14:07,880
The convex hull
of comb wrappings.

1535
01:14:07,880 --> 01:14:08,880
So they're way out here.

1536
01:14:08,880 --> 01:14:12,070
Clearly Furst is better.

1537
01:14:12,070 --> 01:14:14,520
Here's the equilateral
triangle, which you can't tell,

1538
01:14:14,520 --> 01:14:18,540
but it's slightly
below that dashed line.

1539
01:14:18,540 --> 01:14:21,712
And pretty good perimeter but
not quite as good as Furst.

1540
01:14:21,712 --> 01:14:24,170
So actually, probably, you
don't want to use the triangles.

1541
01:14:24,170 --> 01:14:25,990
You prefer the lower perimeter.

1542
01:14:25,990 --> 01:14:28,090
But this guy is interesting.

1543
01:14:28,090 --> 01:14:29,550
Anyway, this is a fun space.

1544
01:14:29,550 --> 01:14:31,735
What we kind of really
care about is-- this

1545
01:14:31,735 --> 01:14:32,860
is called the Pareto curve.

1546
01:14:32,860 --> 01:14:35,740
It's the, if I give you some
bound on perimeter, what's

1547
01:14:35,740 --> 01:14:37,760
the smallest area I can get?

1548
01:14:37,760 --> 01:14:39,540
This is the best
bound we know so far.

1549
01:14:39,540 --> 01:14:42,560
And we know the Pareto
curves below this thing,

1550
01:14:42,560 --> 01:14:44,480
but can you go lower?

1551
01:14:44,480 --> 01:14:45,560
We don't know.

1552
01:14:45,560 --> 01:14:48,550
So these are some fun examples,
but actually, the big questions

1553
01:14:48,550 --> 01:14:51,050
are still pretty
much unsolved here.

1554
01:14:51,050 --> 01:14:53,330
Even if you restrict
to-- I think

1555
01:14:53,330 --> 01:14:56,700
it's fair to restrict to
wrappings of the sphere

1556
01:14:56,700 --> 01:14:58,460
where every point
on the sphere gets

1557
01:14:58,460 --> 01:15:01,850
covered by a stretched path.

1558
01:15:01,850 --> 01:15:08,280
That seems like a nice,
natural class to think about.

1559
01:15:12,490 --> 01:15:15,640
But we have no idea what
that hull space looks like.

1560
01:15:15,640 --> 01:15:19,170
Can you do better than
these particular attempts?

1561
01:15:19,170 --> 01:15:21,140
Looks like we're
doing pretty good,

1562
01:15:21,140 --> 01:15:23,990
but we don't even know
how far left you can go.

1563
01:15:23,990 --> 01:15:25,410
So a pretty neat question.

1564
01:15:25,410 --> 01:15:26,790
And that is Mozart Kugel.

1565
01:15:26,790 --> 01:15:29,630
Any questions?

1566
01:15:29,630 --> 01:15:32,558
Other than, can I have some?

1567
01:15:32,558 --> 01:15:33,950
AUDIENCE: Where can we get some?

1568
01:15:33,950 --> 01:15:35,872
PROFESSOR: Where
can you get some.

1569
01:15:35,872 --> 01:15:36,830
That's a good question.

1570
01:15:36,830 --> 01:15:40,606
Most European airports
sell Mirabell.

1571
01:15:40,606 --> 01:15:42,980
So if you're flying through
Europe, it's the place to go.

1572
01:15:42,980 --> 01:15:46,420
Does anyone know if you
can get them locally?

1573
01:15:46,420 --> 01:15:48,487
I think maybe.

1574
01:15:48,487 --> 01:15:49,950
AUDIENCE: I think Cardullo's.

1575
01:15:49,950 --> 01:15:51,720
PROFESSOR: Cardullo's
that's a good place.

1576
01:15:51,720 --> 01:15:52,420
In Harvard Square.

1577
01:15:52,420 --> 01:15:53,850
They have a lot of
imported chocolates.

1578
01:15:53,850 --> 01:15:55,380
So, yeah, I think
they have Mirabell.

1579
01:15:55,380 --> 01:15:55,855
AUDIENCE: [INAUDIBLE].

1580
01:15:55,855 --> 01:15:58,230
There's a lot of imported
chocolates and confectioneries

1581
01:15:58,230 --> 01:15:59,760
in the Pru.

1582
01:15:59,760 --> 01:16:01,260
PROFESSOR: In the
Prudential Center.

1583
01:16:01,260 --> 01:16:02,200
All right.

1584
01:16:02,200 --> 01:16:04,600
These I think you have
to go to Salzburg.

1585
01:16:04,600 --> 01:16:06,590
Even in Austria
they're hard to get.

1586
01:16:06,590 --> 01:16:08,462
So these are very special.

1587
01:16:08,462 --> 01:16:12,191
I'll give Marty my second one.

1588
01:16:12,191 --> 01:16:12,690
All right.

1589
01:16:12,690 --> 01:16:14,117
That's all for today.

1590
01:16:14,117 --> 01:16:16,700
Hey, there's one thing I forgot
to talk about in lecture which

1591
01:16:16,700 --> 01:16:18,158
I think is really
cool so I'm going

1592
01:16:18,158 --> 01:16:22,100
to add a little bit to
what we were talking about.

1593
01:16:22,100 --> 01:16:23,840
We have all these
wrappings of the sphere

1594
01:16:23,840 --> 01:16:26,350
but I never proved to that
they're actually contractive.

1595
01:16:26,350 --> 01:16:31,790
How do you prove that all those
wrappings are contractive?

1596
01:16:31,790 --> 01:16:36,620
Well, there's a cool theorem we
proved just for this purpose.

1597
01:16:36,620 --> 01:16:45,055
I'll call it the Cauchy Arm
Lemma on a growing sphere.

1598
01:16:48,130 --> 01:16:50,442
It's a nice little connection
to Cauchy's Arm Lemma

1599
01:16:50,442 --> 01:16:55,930
which we talked about in the
context of Cauchy's rigidity

1600
01:16:55,930 --> 01:16:58,920
theorem for polyhedral.

1601
01:16:58,920 --> 01:17:00,630
Now, remember,
Cauchy's Arm Lemma

1602
01:17:00,630 --> 01:17:04,980
is about you have
some convex chain--

1603
01:17:04,980 --> 01:17:09,260
and actually it applies both
on the plane and the sphere.

1604
01:17:09,260 --> 01:17:10,820
So we're thinking
of an open chain

1605
01:17:10,820 --> 01:17:13,153
but we had the property that
even if you add the closing

1606
01:17:13,153 --> 01:17:14,970
edge it's convex.

1607
01:17:14,970 --> 01:17:17,520
And then we looked at
a flex of the chain

1608
01:17:17,520 --> 01:17:20,500
where all of the
angles increased

1609
01:17:20,500 --> 01:17:22,920
and we said, if all
the angles increase,

1610
01:17:22,920 --> 01:17:25,676
then this distance
increases, the endpoint--

1611
01:17:25,676 --> 01:17:27,550
the distance between
the endpoints increases.

1612
01:17:27,550 --> 01:17:30,800
That's regular
Cauchy's Arm Lemma.

1613
01:17:30,800 --> 01:17:34,430
But now I want to look at
not growing the angles,

1614
01:17:34,430 --> 01:17:38,590
but growing the sphere
that the linkage lives on.

1615
01:17:38,590 --> 01:17:40,640
So think of this in
the spherical case,

1616
01:17:40,640 --> 01:17:46,570
which we'll-- imagine this
thing is on a sphere and now I

1617
01:17:46,570 --> 01:17:52,030
transform it by taking a
larger sphere, larger radius,

1618
01:17:52,030 --> 01:17:55,900
and drawing exactly the same
edge lengths and angles.

1619
01:17:55,900 --> 01:17:57,600
Here, I'll draw it
a little bit curved

1620
01:17:57,600 --> 01:18:03,710
because in reality these
edges are great circular arcs.

1621
01:18:03,710 --> 01:18:06,800
So this edge length here
matches this length,

1622
01:18:06,800 --> 01:18:09,990
this angle matches this
angle, this edge length

1623
01:18:09,990 --> 01:18:12,350
matches this edge
length, and so on.

1624
01:18:12,350 --> 01:18:15,680
But I draw it on
a bigger sphere.

1625
01:18:15,680 --> 01:18:19,840
This theorem says, that the
distance between the endpoints

1626
01:18:19,840 --> 01:18:23,280
will increase also in that case.

1627
01:18:23,280 --> 01:18:25,500
So that's kind of neat.

1628
01:18:25,500 --> 01:18:30,640
And in the limit, if you take a
ginormous sphere, all the way--

1629
01:18:30,640 --> 01:18:35,170
if you take a really, really
big sphere, it becomes a plane.

1630
01:18:35,170 --> 01:18:36,980
So if I take
something on a sphere

1631
01:18:36,980 --> 01:18:42,910
and then I actually draw it on
the plane with the same angle--

1632
01:18:42,910 --> 01:18:46,690
so this planar angle
matches this spherical angle

1633
01:18:46,690 --> 01:18:50,270
and this length matches this
spherical length-- then,

1634
01:18:50,270 --> 01:18:53,310
in particular, this
distance will increase.

1635
01:18:53,310 --> 01:18:57,460
And in the reverse, if I
take a planar convex chain

1636
01:18:57,460 --> 01:19:01,620
and I redraw it on some
sphere, the distance

1637
01:19:01,620 --> 01:19:05,070
will go down provided
this thing is also convex.

1638
01:19:05,070 --> 01:19:07,250
So that's the Lemma.

1639
01:19:07,250 --> 01:19:09,882
Now, let me show you
how to use that to prove

1640
01:19:09,882 --> 01:19:11,590
that all those wrappings
are contractive.

1641
01:19:14,240 --> 01:19:16,073
It's really easy once
you have this Lemma.

1642
01:19:19,080 --> 01:19:20,945
Let's start with
the source wrapping.

1643
01:19:26,680 --> 01:19:29,570
So remember, we had a
sphere, the north pole,

1644
01:19:29,570 --> 01:19:33,340
and we took all these shortest
paths from the north pole

1645
01:19:33,340 --> 01:19:34,290
to the south pole.

1646
01:19:34,290 --> 01:19:39,340
All of those lines-- I want
to claim that-- and when

1647
01:19:39,340 --> 01:19:43,000
you unfold this you get a really
big disk and all those lines

1648
01:19:43,000 --> 01:19:45,430
become straight.

1649
01:19:45,430 --> 01:19:48,550
So what I need to show is that
between every pair of points,

1650
01:19:48,550 --> 01:19:53,502
the distance decreases when
I go from flat to sphere.

1651
01:19:53,502 --> 01:19:55,210
Which is exactly what
happened over here.

1652
01:19:55,210 --> 01:19:59,500
I went from flat to the sphere,
this distance decreased.

1653
01:19:59,500 --> 01:20:01,220
So that's how it's
going to be useful.

1654
01:20:01,220 --> 01:20:06,680
If I take two points, like,
say, this point and this point.

1655
01:20:06,680 --> 01:20:07,710
Any two points.

1656
01:20:07,710 --> 01:20:13,940
They live on one of these
spokes that goes to the center.

1657
01:20:13,940 --> 01:20:16,040
So draw those two
segments to the center

1658
01:20:16,040 --> 01:20:17,340
which was the north pole here.

1659
01:20:17,340 --> 01:20:20,740
So if I take the corresponding
two points up here,

1660
01:20:20,740 --> 01:20:24,690
I'm looking at that
path and that path.

1661
01:20:24,690 --> 01:20:28,146
Well, that's a convex chain
and the distance between them--

1662
01:20:28,146 --> 01:20:31,260
between those two points that
I started with, say, x and y,

1663
01:20:31,260 --> 01:20:36,270
is exactly the closing
distance of that convex chain.

1664
01:20:36,270 --> 01:20:38,650
So here's a convex chain,
it matches the angles

1665
01:20:38,650 --> 01:20:41,200
and the lengths of this
convex chain and the plane.

1666
01:20:41,200 --> 01:20:45,160
Therefore, this distance is
larger than this distance.

1667
01:20:45,160 --> 01:20:46,980
And so this is a
contractive map because it

1668
01:20:46,980 --> 01:20:49,240
works for any two
points x and y.

1669
01:20:49,240 --> 01:20:51,150
So that's the source wrapping.

1670
01:20:51,150 --> 01:20:54,470
With slightly more effort,
slightly more interesting

1671
01:20:54,470 --> 01:20:59,550
example, is the petal wrappings.

1672
01:20:59,550 --> 01:21:01,865
So think about one petal.

1673
01:21:04,410 --> 01:21:08,550
So let's say I just take-- in
fact, just think about half

1674
01:21:08,550 --> 01:21:10,420
of a petal.

1675
01:21:10,420 --> 01:21:14,240
So we had a petal look
something like that

1676
01:21:14,240 --> 01:21:16,960
going to the south pole here.

1677
01:21:16,960 --> 01:21:23,340
And so we bisected this thing
and then we took lots of ribs

1678
01:21:23,340 --> 01:21:24,270
like that.

1679
01:21:24,270 --> 01:21:25,580
So that's one half petal.

1680
01:21:29,610 --> 01:21:32,760
So when you unfold,
it looks like this.

1681
01:21:35,460 --> 01:21:37,890
Take any two points
on the half petal.

1682
01:21:37,890 --> 01:21:40,920
I'll show, first of all, that
the half petal is contractive.

1683
01:21:40,920 --> 01:21:44,710
So I take two points, say,
this one and this one,

1684
01:21:44,710 --> 01:21:47,170
call them x and y.

1685
01:21:47,170 --> 01:21:52,100
Now I take this rib,
this rib and this rib.

1686
01:21:52,100 --> 01:21:54,950
Those are the
stretched paths again.

1687
01:21:54,950 --> 01:21:57,120
They map to whatever up here.

1688
01:21:57,120 --> 01:21:59,880
Again, it's a
convex chain and so

1689
01:21:59,880 --> 01:22:02,760
this distance increases
when you go to the sphere

1690
01:22:02,760 --> 01:22:05,850
because it's convex
in both scenarios.

1691
01:22:05,850 --> 01:22:06,800
So that's half petals.

1692
01:22:06,800 --> 01:22:08,830
Now, when you worry
about a full petal

1693
01:22:08,830 --> 01:22:11,038
and you want to show the
whole wrapping is expansive,

1694
01:22:11,038 --> 01:22:14,940
you actually use a
different method.

1695
01:22:14,940 --> 01:22:17,850
So let's say we take
a whole petal here.

1696
01:22:20,510 --> 01:22:22,880
And I want to show not
only any pair of points

1697
01:22:22,880 --> 01:22:27,370
within a single half petal
contract but, in fact,

1698
01:22:27,370 --> 01:22:30,440
any two vertices, any
two points contract.

1699
01:22:30,440 --> 01:22:33,190
So I'm going to take x
and some other point.

1700
01:22:33,190 --> 01:22:35,000
Now I'll call it z.

1701
01:22:35,000 --> 01:22:38,360
If I did the-- just looked
at the stretched paths,

1702
01:22:38,360 --> 01:22:39,560
that's no longer convex.

1703
01:22:39,560 --> 01:22:41,100
I can't use the same trick.

1704
01:22:41,100 --> 01:22:43,400
But here's a new trick.

1705
01:22:43,400 --> 01:22:47,075
I take a straight line, a
shortest path from x to z,

1706
01:22:47,075 --> 01:22:50,570
which gives me a new
point y in the middle.

1707
01:22:50,570 --> 01:22:53,070
Now I know that this
distance contracts

1708
01:22:53,070 --> 01:22:55,700
because that's within
a single half petal.

1709
01:22:55,700 --> 01:22:58,090
So by the previous
argument, and by looking

1710
01:22:58,090 --> 01:23:01,820
at that little chain, I
know by this Arm Lemma

1711
01:23:01,820 --> 01:23:04,830
that distance contracts, I know
that this distance contracts.

1712
01:23:04,830 --> 01:23:06,840
So what about the
whole distance?

1713
01:23:06,840 --> 01:23:09,980
Well, if I look at the
distance from x to z--

1714
01:23:09,980 --> 01:23:15,790
that's what I care about--
let's say, in the plane,

1715
01:23:15,790 --> 01:23:18,460
this equals the sum
of those distances.

1716
01:23:18,460 --> 01:23:20,210
Because we drew a
straight line, it

1717
01:23:20,210 --> 01:23:23,400
equals the distance from x to y
plus the distance from y to z.

1718
01:23:26,360 --> 01:23:29,020
Now, I mapped these
onto the sphere,

1719
01:23:29,020 --> 01:23:32,540
call that d prime is
the spherical distance.

1720
01:23:32,540 --> 01:23:35,320
I know that when I look
at d prime I contract.

1721
01:23:35,320 --> 01:23:39,330
So d prime of xy is going
to be smaller for xy,

1722
01:23:39,330 --> 01:23:43,000
it's going to be smaller for yz.

1723
01:23:43,000 --> 01:23:44,930
What I really care
about is the distance

1724
01:23:44,930 --> 01:23:47,800
of the sphere between x and z.

1725
01:23:47,800 --> 01:23:50,840
But this is the
triangle inequality.

1726
01:23:50,840 --> 01:23:57,570
If you have a triangle xyz,
the distance from x to z

1727
01:23:57,570 --> 01:24:00,100
is always, at most,
the distance from x

1728
01:24:00,100 --> 01:24:01,980
to y plus the
distance from y to z.

1729
01:24:01,980 --> 01:24:03,841
That holds in any metric space.

1730
01:24:03,841 --> 01:24:05,590
So, in particular, it
holds on the sphere.

1731
01:24:05,590 --> 01:24:06,714
So this is sort of trivial.

1732
01:24:06,714 --> 01:24:08,550
This is triangle inequality.

1733
01:24:08,550 --> 01:24:12,950
And this was the contractiveness
that we already prove.

1734
01:24:12,950 --> 01:24:14,250
And this is just equal.

1735
01:24:14,250 --> 01:24:16,060
And so we get that
the new distance

1736
01:24:16,060 --> 01:24:18,550
is a contraction of
the old distance.

1737
01:24:18,550 --> 01:24:20,110
So that works on a full petal.

1738
01:24:20,110 --> 01:24:22,150
It actually-- this
exact same technique

1739
01:24:22,150 --> 01:24:25,040
works if you have a
full petal unfolding.

1740
01:24:25,040 --> 01:24:27,440
Maybe you have
something like this.

1741
01:24:27,440 --> 01:24:33,170
And you take-- finish it off--
you take two points anywhere

1742
01:24:33,170 --> 01:24:36,230
on any two petals,
like here and here, you

1743
01:24:36,230 --> 01:24:38,940
take the shortest
path between them

1744
01:24:38,940 --> 01:24:40,820
and again, you
could argue, well,

1745
01:24:40,820 --> 01:24:44,015
this is going to contract
because it is in a single half

1746
01:24:44,015 --> 01:24:45,890
petal, this is going to
contract because it's

1747
01:24:45,890 --> 01:24:47,150
in a single have petal.

1748
01:24:47,150 --> 01:24:49,770
Therefore, that sum
will also contract

1749
01:24:49,770 --> 01:24:51,630
by exactly the same argument.

1750
01:24:51,630 --> 01:24:53,960
And you can use that for
the comb wrapping too.

1751
01:24:53,960 --> 01:24:55,870
This is kind of neat
with this little Lemma

1752
01:24:55,870 --> 01:24:59,370
about growing spheres and
showing the distances increase,

1753
01:24:59,370 --> 01:25:01,880
you get to prove that all of
these wrappings are contractive

1754
01:25:01,880 --> 01:25:03,754
and therefore all these
things I was claiming

1755
01:25:03,754 --> 01:25:05,090
were working really do work.

1756
01:25:05,090 --> 01:25:09,170
You can formalize that
in a simple little way.