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PROFESSOR: We have a lot of
fun material to cover today,

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so let's get started.

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We start with a few basic
questions about-- well,

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first is a topology question.

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What is a handle?

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In general, a handle is really
a transformation on a surface.

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I thought I will demonstrate in
the context of glass blowing.

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So this is what it looks
like if you're making a cup.

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You take some hot glass.

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You attach it to your surface.

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You cut it off your hot pipe.

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And then you bring it around
and attach it to another point,

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and so that's a handle.

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You've probably used
handles before in real life.

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In general, or in the
mathematical setting,

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it's the same thing
but on a 2D surface.

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So imagine you have
some 2D surface.

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You take two disk-like
regions on the surface

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and then you attach on a
handle-- something like that.

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So it's an operation
you can do to a surface.

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And you can keep adding handles.

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I don't think there's
a clear way to say,

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oh, this is clearly a handle,
except to have added it

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as a handle.

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In general, the nice theorem
for two dimensional surfaces--

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so here's a coffee cup being
converted into a torus,

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because it has
genus 1, meaning it

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has essentially
one handle in it.

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In general, you take
any orientable surface

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without boundary-- this is
locally two dimensional-- then

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it will be a sphere plus some
non-negative number of handles.

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And that's a
classification theorem

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for orientable surfaces.

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It's only slightly
more complicated

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for non-orientable surfaces.

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And then 3D surfaces
are even harder

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and was only recently solved.

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But this is two
dimensional surfaces,

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which is easy and clean.

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There is a way to compute
genus and in some sense

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learn how many
handles are there.

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But there isn't a unique
thing of, oh, this part

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is clearly a handle.

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But when you draw it
this way, it kind of

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becomes clear this is a handle.

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This is a handle.

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But there isn't a
formal sense in which--

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I don't know-- this thing is not
a handle or some weird thing.

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I think that's
all about handles.

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Hopefully that answers things.

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Of course, things
get more complicated

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when you have a boundary,
which we usually call holes.

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But the next question is
about holes in the unfoldings.

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So I claim that convex
polyhedron, when

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you unfold them, never have
holes in the unfolded form.

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And this is to contrast this
example, which is not convex.

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But you can unfold it by
cutting these two edges.

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And you've got a little hole.

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So I showed this in lecture.

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And natural question
is, why isn't this

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possible for convex polyhedron?

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So I thought I would prove that.

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And the proof uses a cool
theorem, which we'll probably

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be seeing again, called
the Gauss-Bonnet theorem.

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And it says that if you
have some surface, which

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is homeomorphic to a
sphere-- so I don't

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want any handles
for this theorem.

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And of course, convex
polyhedra our sphere-like.

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They don't have handles.

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And I take a-- let me
take a better color--

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I take a closed curve
non-self-intersecting closed

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curve on that
surface-- that defines

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an interior and an exterior.

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Then what the
Gauss-Bonnet theorem

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says is that if you look
at the curvature that's

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enclosed by the curve-- the
total curvature by this closed

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curve on the surface-- and
you add on the total turn

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angle along the curve-- so here,
the curve is turning right.

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Here, it's turning
left, right, left.

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Left is positive.

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Right is negative.

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Then these always add
up to 360 degrees.

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We're not going to
prove this theorem,

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but we're going to use it.

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So this is a nice invariant
for sphere-like things.

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When you have handles,
this number changes.

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I think it's 0 for a torus.

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Well, I won't try to guess
it, because it's a little bit

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subtle to get right.

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One fun consequence of this--
let's just get warmed up--

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suppose you take a
sphere like object

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and you take a
closed curve-- this

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is going to be pretty
abstract-- in this direction,

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then it says, OK, the
total amount of curvature

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in here plus the total
turn angle equals 360.

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Now suppose that I
turn the curve around.

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So I just reverse the
direction-- like this.

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And so now, the interior
is this stuff out here.

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And then we get that the total
curvature outside the curve,

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plus the total turn angle of
the blue thing, equals 360.

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If I add those two
equations together,

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the total turn angle cancels.

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Because wherever red turns,
left blue turns right.

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And so this term disappears.

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And so what we get is
that the total curvature

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inside the curve, plus the total
curvature outside the curve,

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equals 720.

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So this is a nice
topological invariant

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of sphere-like polyhedra.

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You add up the total
curvature everywhere--

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it didn't really matter what the
curve was-- you always get 720.

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And so this is kind of neat.

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For convex polyhedra,
this means you

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have to somehow divvy
up this curvature.

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Because all curvatures
are positive.

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So 720 is, somehow,
spread around.

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But you have to have
exactly that much.

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For non-convex things, you could
have some negative curvature

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that balances out a lot
of positive curvature.

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So you can have a lot of both.

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But you have almost the
same amount of each.

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Just you have
exactly 720 excess.

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that's just a fun fact.

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We'll be using that in
the future, I think.

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OK, that was Gauss-Bonnet.

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Now let's use it
to prove that this

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can't happen for
convex polyhedron.

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So the idea is, suppose you have
an unfolding with a hole in it.

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So this is the surface out here.

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Then I'm going to take
a closed curve that

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walks around the hole but
stays inside the unfolding.

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And then, of course, I'm
visualizing that really

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on the polyhedron.

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So I want the
interior of the curve

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to enclose the hole-- or
what becomes the hole.

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Now, what we learned
from Gauss-Bonnet

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is that the curvature
enclosed by this thing,

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plus the total turn angle
of the curve, equals 360.

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The total turn angle
of the curve is 360.

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It's a planar walk here.

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So turn angle-- total
turn here is 360 degrees.

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Now, if we have a
convex polyhedron,

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then this curvature
has to be non-negative.

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Well, I mean, sorry.

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In any case here, this
curvature better equal 0.

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For convex polyhedra, this is
a sum of vertex curvatures.

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For convex polyhedra,
every vertex

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has strictly positive
curvature actually.

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I mean the zero curvature
vertices aren't vertices.

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They're just points
on the surface.

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So for convex
polyhedron, if you're

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going to have zero
total curvature,

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that means you actually
have no curvature.

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So you have no vertices
enclosed in here,

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which is a contradiction.

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That means there wasn't
a hole to open up.

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Technically, you could
do something weird

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like-- I don't know,
let's take a cube.

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You could say, OK, I'm going
to make a couple cuts here

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that do nothing.

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This is kind of a
weird situation.

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We're going to add those cuts.

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And then, of course, you could
draw this curve around it

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and say, oh yeah, look.

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I've got lots of zero
curvature vertices inside here.

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In general, whenever you
have zero curvature vertices,

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you could always just suture
them back up-- uncut them--

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and there'd be no
difference in the unfolding.

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So you have to assume
that you've already

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removed pointless cuts.

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Then there'll be no
vertices in there.

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And so then, in fact, there was
no hole for convex polyhedra.

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For non-convex polyhedra, you
can have a negative curvature

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vertex that balances out some
positive curvature vertices.

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And so the total curvature
here equals zero.

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But you have three vertices.

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And that just can't
happen for convex.

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Next quick question
is, when we're

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talking about the cut
locus and the ridge tree,

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we drew some pictures.

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The claim is that it
was a spanning tree

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of the polyhedron.

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And in particular, the
leaves of the tree--

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the degree 1
vertices-- are exactly

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the vertices of the polyhedron.

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And I hadn't actually
realized this,

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but in fact, it's not
really literally true.

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It's kind of spiritually true.

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The vertices, in fact, have
unique shortest paths to x.

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So remember, we have some
point, x, on the surface.

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And we're looking at
points, like this one,

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they have non-unique
shortest paths to x.

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This is, if you grow
the fire around x,

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where does the fire meet itself.

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And it will meet
itself along this edge,

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because you could
go around this way

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or go around this way and
it's equal length path.

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But at the vertex,
there's actually

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a unique way to go there.

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That's the black line.

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So technically, this point
is not on the ridge tree.

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But all of these points are.

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So this is kind of
like a limiting point

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of the ridge tree points.

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So we think of it as
being on the ridge tree.

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I mean, you could think
of as cut are not cut.

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It doesn't really matter.

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But you cut right next to it,
so effectively the same thing.

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But it is a neat
point-- a subtle point--

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that these guys have
unique shortest paths.

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Whereas, these do not.

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Still we cut all the
way up to corner.

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All right, that is that.

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So now we have a bunch of
newer and more exciting

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things-- or updates that
you haven't heard of.

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One question that we mentioned
was generalizing the star

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and source unfoldings.

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And there's a new
paper about this.

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This is with my PhD
advisor, Anna Lubiw,

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and this is just a
warm up to get started.

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So here, we have a box,
if we take a point x.

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And let's see, here we have
the source unfolding from x.

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And here, we have the star
unfolding, just for comparison.

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It's also kind of fun to
see them side by side.

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They're color coded.

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So where you cut is the
ridge tree, in this case.

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And you end up gluing
along the ridge tree

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in the star unfolding.

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Now, we're going to
generalize things

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a little bit in that we're
going to generalize the source

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unfolding, specifically.

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So source unfolding,
you have a point x.

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And you just sort of
shoot shortest paths all

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from x, and that's
what you keep.

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And you end up cutting
along the ridge tree, which

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is the void in our
diagram at this point.

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So I'm going to generalize
that a little bit

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and think of this as
a tiny little circle.

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And in general,
what I'm going to do

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is the source unfolding
outside the circle.

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And so when the circle is really
tiny, it is just the source

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unfolding.

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But in general, I'm going
to do the star unfolding

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inside the circle.

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OK.

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So in this case,
nothing changes.

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But next example is
going to be more general.

248
00:12:19,860 --> 00:12:21,770
Instead of being a
single point here,

249
00:12:21,770 --> 00:12:23,780
I'm going to take
a geodesic arc--

250
00:12:23,780 --> 00:12:26,030
a straight line on the surface.

251
00:12:26,030 --> 00:12:28,310
So here's an example of that.

252
00:12:28,310 --> 00:12:31,350
We have a square-based pyramid.

253
00:12:31,350 --> 00:12:34,470
We drew a straight
line on the surface.

254
00:12:34,470 --> 00:12:37,210
If you unfolded it,
it would be straight.

255
00:12:37,210 --> 00:12:40,296
We're thinking of
having a little--

256
00:12:40,296 --> 00:12:43,630
I think it's called a racetrack
curve in mathematics--

257
00:12:43,630 --> 00:12:46,229
around that straight line.

258
00:12:46,229 --> 00:12:48,270
And I'm going to do star
unfolding on the inside.

259
00:12:48,270 --> 00:12:49,940
In this case, there's no
vertices on the inside,

260
00:12:49,940 --> 00:12:51,002
so nothing happens.

261
00:12:51,002 --> 00:12:52,960
And then we do source
unfolding on the outside.

262
00:12:52,960 --> 00:12:54,334
This is actually
previously known

263
00:12:54,334 --> 00:12:57,290
to unfold by
O'Rourke and others.

264
00:12:57,290 --> 00:13:00,890
But we have a simpler
proof, essentially.

265
00:13:00,890 --> 00:13:03,000
And we're going to
generalize it more.

266
00:13:03,000 --> 00:13:05,490
But what we do is the source
unfolding on the outside.

267
00:13:05,490 --> 00:13:07,520
So you take shortest
paths-- from every point,

268
00:13:07,520 --> 00:13:12,300
you take its shortest
path to this geodesic.

269
00:13:12,300 --> 00:13:15,130
That's, some of these blue lines
show various shortest paths.

270
00:13:15,130 --> 00:13:16,810
That's what you keep.

271
00:13:16,810 --> 00:13:20,240
The ridge tree is if you
light fire simultaneously

272
00:13:20,240 --> 00:13:22,630
along this entire segment,
where does it burn out?

273
00:13:22,630 --> 00:13:24,520
And that's the purple stuff.

274
00:13:24,520 --> 00:13:27,740
And this is the
complementary diagram,

275
00:13:27,740 --> 00:13:32,100
where you do the reverse, which
would be the star unfolding

276
00:13:32,100 --> 00:13:34,230
on the outside, source
unfolding on the inside.

277
00:13:34,230 --> 00:13:36,400
So you glue along
the purple stuff

278
00:13:36,400 --> 00:13:39,870
instead of cutting
along the purple stuff.

279
00:13:39,870 --> 00:13:42,390
And this, we conjecture, doesn't
overlap but we don't know.

280
00:13:44,990 --> 00:13:46,750
All right, so fine.

281
00:13:46,750 --> 00:13:47,627
That looks easy.

282
00:13:47,627 --> 00:13:49,460
And you can prove that
this doesn't overlap.

283
00:13:49,460 --> 00:13:51,370
Before you proved
it, because you just

284
00:13:51,370 --> 00:13:54,630
had shortest paths emanating
in all directions around x.

285
00:13:54,630 --> 00:13:58,600
Now, you have shortest
paths emanating

286
00:13:58,600 --> 00:14:01,950
in 180 degrees of directions
around this endpoint.

287
00:14:01,950 --> 00:14:03,590
Then they are all just straight.

288
00:14:03,590 --> 00:14:05,660
And then they rotate
around 180 degrees here.

289
00:14:05,660 --> 00:14:07,140
And then they're
all just straight.

290
00:14:07,140 --> 00:14:08,700
And so there can't be
anything overlapping,

291
00:14:08,700 --> 00:14:10,300
because you're just
taking a continuum

292
00:14:10,300 --> 00:14:12,470
of these segments
of varying lengths.

293
00:14:12,470 --> 00:14:14,130
They don't overlap.

294
00:14:14,130 --> 00:14:15,540
Here, they're all parallel.

295
00:14:15,540 --> 00:14:16,660
Here, they sweep nicely.

296
00:14:16,660 --> 00:14:19,110
In general, if they
always turn clockwise

297
00:14:19,110 --> 00:14:22,471
as you walk along the
curve, you're fine.

298
00:14:22,471 --> 00:14:24,197
OK, here's a more general one.

299
00:14:24,197 --> 00:14:26,030
In general, what we can
prove is that if you

300
00:14:26,030 --> 00:14:29,820
have a convex curve
on the surface--

301
00:14:29,820 --> 00:14:33,172
so it always turns to the
left, I think technically.

302
00:14:33,172 --> 00:14:34,880
The angle on the left
hand side is always

303
00:14:34,880 --> 00:14:37,071
less than or equal
to 180 degrees.

304
00:14:37,071 --> 00:14:38,570
You have to be a
little careful what

305
00:14:38,570 --> 00:14:39,370
it means to turn to the left.

306
00:14:39,370 --> 00:14:41,660
When you hit a vertex, you've
got less than 360 total.

307
00:14:41,660 --> 00:14:43,120
So what does to the left mean?

308
00:14:43,120 --> 00:14:45,890
It just means you've got less
than 180 degrees of material

309
00:14:45,890 --> 00:14:47,430
on your left side.

310
00:14:47,430 --> 00:14:49,320
So this is an example
of a convex curve.

311
00:14:49,320 --> 00:14:50,840
It's got some circular arcs.

312
00:14:50,840 --> 00:14:53,442
We're no longer
tracking along some--

313
00:14:53,442 --> 00:14:55,400
we're no longer just
doubling along some curve.

314
00:14:55,400 --> 00:14:57,060
Here, we enclose a
vertex, which makes

315
00:14:57,060 --> 00:14:59,380
it a little more exciting,
because now what we're

316
00:14:59,380 --> 00:15:02,150
going to do star unfolding
on the inside, which remember

317
00:15:02,150 --> 00:15:05,760
was cutting along shortest
paths from every vertex

318
00:15:05,760 --> 00:15:06,610
to your thing.

319
00:15:06,610 --> 00:15:08,730
In this case, your thing
is no longer a point, x.

320
00:15:08,730 --> 00:15:10,690
It is now this convex curve.

321
00:15:10,690 --> 00:15:13,895
So let's say this is
the shortest path here.

322
00:15:13,895 --> 00:15:18,777
It might be than one choice, but
we're going to-- actually, no.

323
00:15:18,777 --> 00:15:19,860
This is the shortest path.

324
00:15:19,860 --> 00:15:21,050
This is a root 2 diagonal.

325
00:15:21,050 --> 00:15:22,896
This is length 1.

326
00:15:22,896 --> 00:15:24,270
So we're going to
come along here

327
00:15:24,270 --> 00:15:26,190
and that's this dashed line.

328
00:15:26,190 --> 00:15:29,140
So it got opened
up here, because we

329
00:15:29,140 --> 00:15:30,620
had too little material here.

330
00:15:30,620 --> 00:15:31,910
We open it up.

331
00:15:31,910 --> 00:15:33,410
But otherwise, it acts the same.

332
00:15:33,410 --> 00:15:36,050
In particular, outside
this red curve,

333
00:15:36,050 --> 00:15:37,290
it's just source unfolding.

334
00:15:37,290 --> 00:15:38,907
You've got all these
shortest paths.

335
00:15:38,907 --> 00:15:41,115
And again, what we argue--
and I'm not going to prove

336
00:15:41,115 --> 00:15:43,520
it here-- is that all
of these shortest paths

337
00:15:43,520 --> 00:15:47,800
keep turning clockwise and
do so exactly 360 degrees.

338
00:15:47,800 --> 00:15:49,950
There's some jumps when
you hit these gaps.

339
00:15:49,950 --> 00:15:51,770
And you have to kind
of jump over them.

340
00:15:51,770 --> 00:15:55,610
But still, you
have no collision.

341
00:15:55,610 --> 00:15:58,240
And drawn over here
is the reverse,

342
00:15:58,240 --> 00:16:03,616
where we would star unfold the
outside of a convex curve--

343
00:16:03,616 --> 00:16:05,740
because it's a convex curve,
the inside and outside

344
00:16:05,740 --> 00:16:08,930
are different-- and source
unfold on the inside.

345
00:16:08,930 --> 00:16:10,790
And this, we conjecture,
doesn't overlap.

346
00:16:10,790 --> 00:16:13,390
But we don't know
how to prove it.

347
00:16:13,390 --> 00:16:15,900
This thing is a generalization
of the source unfolding.

348
00:16:15,900 --> 00:16:18,590
Because when this curve is super
tiny, it is a source unfolding.

349
00:16:18,590 --> 00:16:20,090
This thing would
be a generalization

350
00:16:20,090 --> 00:16:23,641
of the star unfolding, but
we don't know whether works.

351
00:16:23,641 --> 00:16:25,840
I think one more example here.

352
00:16:25,840 --> 00:16:29,070
This is a way to
generate convex curves.

353
00:16:29,070 --> 00:16:31,320
You start from some point, x.

354
00:16:31,320 --> 00:16:34,130
And you design it-- you
choose a direction so

355
00:16:34,130 --> 00:16:36,980
that if you go straight--
so this curve goes straight

356
00:16:36,980 --> 00:16:38,230
everywhere except x.

357
00:16:38,230 --> 00:16:40,940
If you set it up right,
you come back to x.

358
00:16:40,940 --> 00:16:42,690
So this, in particular,
is a convex curve,

359
00:16:42,690 --> 00:16:44,440
because it's straight
everywhere except x.

360
00:16:44,440 --> 00:16:46,290
And at x, it's convex.

361
00:16:46,290 --> 00:16:49,730
And so, in this case, we
enclose a few vertices.

362
00:16:49,730 --> 00:16:53,440
On the inside, we've
got V3, V7, V6.

363
00:16:53,440 --> 00:16:59,950
So each of those ends up getting
cut here in these green lines.

364
00:16:59,950 --> 00:17:02,250
And so, like this one
is a very tiny cut.

365
00:17:02,250 --> 00:17:04,160
We just cut there at V3.

366
00:17:04,160 --> 00:17:05,960
The other ones
are little bigger.

367
00:17:05,960 --> 00:17:07,800
But they open up.

368
00:17:07,800 --> 00:17:12,359
And yeah, I guess,
this is the-- well,

369
00:17:12,359 --> 00:17:15,520
it's a few different
versions of the picture here.

370
00:17:15,520 --> 00:17:17,790
But again, you look
at the shortest paths.

371
00:17:17,790 --> 00:17:19,099
Here, they're all parallel.

372
00:17:19,099 --> 00:17:20,300
Here, they sweep.

373
00:17:20,300 --> 00:17:21,670
Here, they're all parallel.

374
00:17:21,670 --> 00:17:24,110
Here, we jump, but
it's just the same

375
00:17:24,110 --> 00:17:26,339
as sweeping-- it
doesn't hurt you.

376
00:17:26,339 --> 00:17:28,300
Then they're all parallel.

377
00:17:28,300 --> 00:17:28,890
Then we jump.

378
00:17:28,890 --> 00:17:31,139
Then they're all parallel,
in this particular example.

379
00:17:31,139 --> 00:17:33,030
But they will always
proceed clockwise

380
00:17:33,030 --> 00:17:37,600
around the curve, which
here is drawn red.

381
00:17:37,600 --> 00:17:39,390
Gets split up a
little bit, but you

382
00:17:39,390 --> 00:17:41,150
can show because
of that sweeping,

383
00:17:41,150 --> 00:17:42,640
they won't hit each other.

384
00:17:42,640 --> 00:17:46,290
And you have
non-overlapping unfolding.

385
00:17:46,290 --> 00:17:51,000
So we call this sun
unfolding, because of the rays

386
00:17:51,000 --> 00:17:52,642
that sweep around.

387
00:17:52,642 --> 00:17:54,100
And because we have
a convex curve,

388
00:17:54,100 --> 00:17:56,820
it's kind of like the sun.

389
00:17:56,820 --> 00:18:00,150
So that's a new unfolding.

390
00:18:00,150 --> 00:18:02,950
The obvious open
question is the reverse,

391
00:18:02,950 --> 00:18:04,600
when you glue around
the purple sides

392
00:18:04,600 --> 00:18:10,585
instead of the--
It's essentially,

393
00:18:10,585 --> 00:18:11,960
instead of having
a convex curve,

394
00:18:11,960 --> 00:18:14,780
you have a reflex curve,
exactly the opposite.

395
00:18:14,780 --> 00:18:16,106
What happens, we don't know.

396
00:18:16,106 --> 00:18:17,230
It's a lot harder to prove.

397
00:18:17,230 --> 00:18:18,490
Because in particular,
it's a lot harder

398
00:18:18,490 --> 00:18:20,531
to prove that the star
unfolding doesn't overlap.

399
00:18:20,531 --> 00:18:22,670
And you've got to include
at least that proof

400
00:18:22,670 --> 00:18:24,090
in any generalization of it.

401
00:18:26,610 --> 00:18:30,070
Next topic of unfolding
kind of related

402
00:18:30,070 --> 00:18:32,130
we call zipper unfolding.

403
00:18:32,130 --> 00:18:36,960
So these are some examples of
real felt models with a zipper.

404
00:18:36,960 --> 00:18:41,190
The goal is, I want an unfolding
that has a single zipper.

405
00:18:41,190 --> 00:18:44,410
And you just pull the zipper,
and it makes your polyhedron.

406
00:18:44,410 --> 00:18:46,340
So this is an example
of an octahedron.

407
00:18:46,340 --> 00:18:48,730
This one is actually
not even a polyhedron.

408
00:18:48,730 --> 00:18:51,650
It has two pyramidal
pockets in it.

409
00:18:51,650 --> 00:18:54,140
And it's actually closed
off in the middle.

410
00:18:54,140 --> 00:18:56,220
I think we have a
little video here

411
00:18:56,220 --> 00:19:00,180
of what it looks like
to open that octahedron.

412
00:19:00,180 --> 00:19:03,180
So what does it take
to have a single zipper

413
00:19:03,180 --> 00:19:08,445
line, that connects
everything together?

414
00:19:08,445 --> 00:19:09,820
Well, if you think
about it, that

415
00:19:09,820 --> 00:19:14,160
means that the cuts that you
make must follow a single path.

416
00:19:14,160 --> 00:19:18,380
So in general, the cuts form
a tree on a convex polyhedron.

417
00:19:18,380 --> 00:19:21,000
We want that tree
to be just a path.

418
00:19:21,000 --> 00:19:23,000
So it's like a Hamiltonian path.

419
00:19:23,000 --> 00:19:26,430
It's got to visit all the
vertices in some order.

420
00:19:26,430 --> 00:19:28,975
And you'd like it to
unfold without overlap.

421
00:19:28,975 --> 00:19:30,100
So is this always possible?

422
00:19:34,280 --> 00:19:39,780
This is a father, son,
mother, son, son paper.

423
00:19:39,780 --> 00:19:41,790
So this is my advisor
again and her two sons.

424
00:19:41,790 --> 00:19:44,164
Although we like to say this
a paper with her three sons,

425
00:19:44,164 --> 00:19:47,900
because I'm her academic
son-- or four, if you count

426
00:19:47,900 --> 00:19:48,560
Marty too.

427
00:19:48,560 --> 00:19:51,350
She was his advisor as well.

428
00:19:51,350 --> 00:19:53,800
So here are some examples
of good unfolding.

429
00:19:53,800 --> 00:19:55,200
So we have the platonic solids.

430
00:19:55,200 --> 00:19:57,170
These are just
typical unfoldings.

431
00:19:57,170 --> 00:19:59,220
These are all zipper unfoldings.

432
00:19:59,220 --> 00:20:02,425
So we're cutting along edges,
along a Hamiltonian path

433
00:20:02,425 --> 00:20:03,300
that doesn't overlap.

434
00:20:03,300 --> 00:20:06,210
They all have this kind
of nice snake like shape.

435
00:20:06,210 --> 00:20:07,640
And so all platonic
solids can be

436
00:20:07,640 --> 00:20:10,210
made by zipper edge unfoldings.

437
00:20:10,210 --> 00:20:11,740
Next, we did Archimedean solids.

438
00:20:11,740 --> 00:20:13,480
This is a lot more work.

439
00:20:13,480 --> 00:20:17,800
But again, you get these
nice S-like curves.

440
00:20:17,800 --> 00:20:19,224
And they're all
zipper unfolding,

441
00:20:19,224 --> 00:20:20,890
because they're all
possible and they'll

442
00:20:20,890 --> 00:20:27,060
have these Hamiltonian paths,
cuts, and avoid overlap.

443
00:20:27,060 --> 00:20:28,850
But you may notice,
there's one missing up

444
00:20:28,850 --> 00:20:30,890
there-- the great
Rhombi-Cosi-Dodecahedron,

445
00:20:30,890 --> 00:20:32,870
my favorite Archimedean solid.

446
00:20:32,870 --> 00:20:37,000
And it has a rather
different looking unfolding.

447
00:20:37,000 --> 00:20:39,430
As far as we can tell,
there's no S-shaped one--

448
00:20:39,430 --> 00:20:40,350
whatever that means.

449
00:20:40,350 --> 00:20:42,580
But you have to have a tree.

450
00:20:42,580 --> 00:20:45,310
These examples all
had path-like--

451
00:20:45,310 --> 00:20:47,470
The dual graph was
roughly a path.

452
00:20:47,470 --> 00:20:49,970
I guess it branches
a little bit here.

453
00:20:49,970 --> 00:20:53,810
Here, it's very
tree like, I guess.

454
00:20:53,810 --> 00:20:57,260
So all Archimedean
solids can be done.

455
00:20:57,260 --> 00:20:59,560
One open question,
next category up,

456
00:20:59,560 --> 00:21:01,890
is Johnson solids, which
are all polyhedron made

457
00:21:01,890 --> 00:21:04,570
with regular polygon faces--
convex polyhedron made

458
00:21:04,570 --> 00:21:06,484
by regular polygonal faces.

459
00:21:06,484 --> 00:21:08,150
Those we don't know,
whether they always

460
00:21:08,150 --> 00:21:10,040
have zipper unfoldings.

461
00:21:10,040 --> 00:21:13,160
But we do know there are
some convex polyhedra--

462
00:21:13,160 --> 00:21:19,970
like this rhombic dodecahedron--
that do not have zipper

463
00:21:19,970 --> 00:21:21,960
unfoldings if you
only cut along edges.

464
00:21:21,960 --> 00:21:24,710
Because this graph has
no Hamiltonian path.

465
00:21:24,710 --> 00:21:26,674
So never mind avoiding overlap.

466
00:21:26,674 --> 00:21:28,840
There's some polyhedra that
just aren't Hamiltonian.

467
00:21:28,840 --> 00:21:31,330
There's no path it visits
every vertex exactly

468
00:21:31,330 --> 00:21:33,890
once and only follows edges.

469
00:21:33,890 --> 00:21:36,180
So there's nothing
you could even

470
00:21:36,180 --> 00:21:39,100
hope to cut along
and avoid overlap.

471
00:21:39,100 --> 00:21:41,590
So that's bad news
for edge unfoldings.

472
00:21:41,590 --> 00:21:43,840
The big open question here
is for general unfoldings

473
00:21:43,840 --> 00:21:46,420
if you're allowed to cut
anywhere on a convex surface

474
00:21:46,420 --> 00:21:48,030
and do things like
the sun unfolding.

475
00:21:48,030 --> 00:21:49,676
I mean, edge
unfoldings, we don't

476
00:21:49,676 --> 00:21:50,800
know how to do them anyway.

477
00:21:50,800 --> 00:21:53,034
So if you allow
general unfoldings,

478
00:21:53,034 --> 00:21:55,450
we've got star unfolding,
source unfolding, sun unfolding,

479
00:21:55,450 --> 00:21:57,090
all sorts of things.

480
00:21:57,090 --> 00:21:59,740
But none of them are
zipper unfoldings.

481
00:21:59,740 --> 00:22:01,520
They all cut along trees.

482
00:22:01,520 --> 00:22:04,500
Like star unfolding
cuts along a star.

483
00:22:04,500 --> 00:22:06,460
The source unfolding cuts
along the ridge tree,

484
00:22:06,460 --> 00:22:08,910
which is going to be
very tree like thing.

485
00:22:08,910 --> 00:22:13,040
Can you always convert a
tree cut into a path cut?

486
00:22:13,040 --> 00:22:14,520
We've tried.

487
00:22:14,520 --> 00:22:17,550
It seems quite challenging.

488
00:22:17,550 --> 00:22:18,680
So that's the open problem.

489
00:22:18,680 --> 00:22:22,640
Does every convex polyhedron
have a general zipper

490
00:22:22,640 --> 00:22:26,420
unfolding, because edge
unfolding is always too much

491
00:22:26,420 --> 00:22:28,010
to hope for.

492
00:22:28,010 --> 00:22:32,542
So those are some fun
problems to think about.

493
00:22:32,542 --> 00:22:34,000
It's kind of like
zipper unfolding.

494
00:22:36,760 --> 00:22:38,870
So the next topic-- oh, right.

495
00:22:38,870 --> 00:22:43,240
One more thing to show, this is
a very fun talk that we gave.

496
00:22:43,240 --> 00:22:46,530
All five of us gave this
talk at Canadian Conference

497
00:22:46,530 --> 00:22:48,110
on Computational Geometry.

498
00:22:48,110 --> 00:22:51,270
And one of the props in the
talk was this cardboard box.

499
00:22:51,270 --> 00:22:53,844
And in the middle of the
talk, it starts jiggling.

500
00:22:53,844 --> 00:22:56,010
And actually, initially,
only four of the co-authors

501
00:22:56,010 --> 00:22:58,750
were giving the talk,
because the fifth one

502
00:22:58,750 --> 00:23:01,220
was hiding inside the box.

503
00:23:01,220 --> 00:23:03,170
And then in the
middle of the talk,

504
00:23:03,170 --> 00:23:05,750
he just jumps out
and then starts

505
00:23:05,750 --> 00:23:08,180
speaking, as if
nothing happened.

506
00:23:08,180 --> 00:23:09,720
It was a lot of fun.

507
00:23:09,720 --> 00:23:13,200
At this point, our son,
Jonah, is in the box.

508
00:23:13,200 --> 00:23:14,320
He was fairly small.

509
00:23:14,320 --> 00:23:17,090
He's grown up a lot
since, but at the time

510
00:23:17,090 --> 00:23:18,520
he fit nicely into
this cardboard

511
00:23:18,520 --> 00:23:21,930
box, which was maybe this big.

512
00:23:21,930 --> 00:23:23,870
I think we give him a
book and a flashlight

513
00:23:23,870 --> 00:23:26,640
and stuff to do while he
was sitting there, waiting

514
00:23:26,640 --> 00:23:30,440
for his slides, waiting
for the queue to come out.

515
00:23:30,440 --> 00:23:31,110
There we go.

516
00:23:36,610 --> 00:23:39,050
So those are unfolding of
the cube or a box in general.

517
00:23:41,910 --> 00:23:45,851
Next topic is going
back to edge unfolding.

518
00:23:45,851 --> 00:23:47,420
I like this comment.

519
00:23:47,420 --> 00:23:49,000
I thought they were
pretty obvious,

520
00:23:49,000 --> 00:23:50,820
but now you've
convinced me otherwise.

521
00:23:50,820 --> 00:23:54,450
And some of the evidence for
edge unfolding being difficult

522
00:23:54,450 --> 00:23:57,720
was this polyhedron, which we
proved has no edge unfolding.

523
00:23:57,720 --> 00:24:01,210
I didn't say it in lecture, but
we call it edge on unfoldable,

524
00:24:01,210 --> 00:24:04,350
because they cannot be unfolded.

525
00:24:04,350 --> 00:24:06,700
This is actually the first
example we came up with.

526
00:24:06,700 --> 00:24:10,092
I mention it only because
it has fewer faces.

527
00:24:10,092 --> 00:24:12,175
We usually show this one
because it's triangulated

528
00:24:12,175 --> 00:24:13,840
and that's kind of cooler.

529
00:24:13,840 --> 00:24:17,790
This one has convex faces still,
so still topologically convex.

530
00:24:17,790 --> 00:24:20,110
I mean, if I pushed
these points in,

531
00:24:20,110 --> 00:24:22,950
the polyhedron would be
convex but has fewer faces.

532
00:24:22,950 --> 00:24:34,000
It has I guess six faces per
hat, times 4 hats, so 24 faces.

533
00:24:34,000 --> 00:24:36,440
This was done in '99.

534
00:24:36,440 --> 00:24:38,740
It turns out at
exactly the same time,

535
00:24:38,740 --> 00:24:40,556
there was this paper
by Tarasov called,

536
00:24:40,556 --> 00:24:42,180
"Polyhedron with No
Natural Unfolding."

537
00:24:42,180 --> 00:24:44,799
The paper has no figures,
so I had to draw one

538
00:24:44,799 --> 00:24:46,090
to show you what it looks like.

539
00:24:46,090 --> 00:24:47,580
It's just a cube.

540
00:24:47,580 --> 00:24:50,100
And then at each corner of the
cube, you cut off the corner

541
00:24:50,100 --> 00:24:53,450
and then pull the point out.

542
00:24:53,450 --> 00:24:56,690
And they proved, by a pretty
similar argument-- I mean,

543
00:24:56,690 --> 00:24:58,730
essentially you treat
each of these as a hat.

544
00:24:58,730 --> 00:25:00,350
It's kind of a very simple hat.

545
00:25:00,350 --> 00:25:02,730
They happen to overlap;
they share edges.

546
00:25:02,730 --> 00:25:05,170
But again, because of these
negative curvature vertices,

547
00:25:05,170 --> 00:25:06,930
you have to cut through the hat.

548
00:25:06,930 --> 00:25:08,860
And that cut has to
keep going around.

549
00:25:08,860 --> 00:25:10,070
Eventually, it forms a cycle.

550
00:25:10,070 --> 00:25:15,480
So that's what Tarasov
proved in the same year, '99.

551
00:25:15,480 --> 00:25:22,830
And then Grunbaum-- he's a
famous geometer in Seattle--

552
00:25:22,830 --> 00:25:24,230
came up with some more examples.

553
00:25:24,230 --> 00:25:27,780
So he initially wanted to
make a star shaped example.

554
00:25:27,780 --> 00:25:30,680
Star shaped means that
there's a single point, namely

555
00:25:30,680 --> 00:25:33,250
the center, where you can
shine a light in all directions

556
00:25:33,250 --> 00:25:36,949
and the light reaches the entire
interior of the polyhedron.

557
00:25:36,949 --> 00:25:38,740
He didn't know about
our witch hat example.

558
00:25:38,740 --> 00:25:42,570
So he took the Tarasov example,
made it a dodecahedron,

559
00:25:42,570 --> 00:25:44,500
and then it is star shaped.

560
00:25:44,500 --> 00:25:47,200
I think ours are
already star shaped.

561
00:25:47,200 --> 00:25:48,130
So that's kind of fun.

562
00:25:48,130 --> 00:25:51,110
It looks like some scary
underwater creature.

563
00:25:51,110 --> 00:25:53,000
And then he learned
about our paper,

564
00:25:53,000 --> 00:25:54,541
and he said, all
right, I want to get

565
00:25:54,541 --> 00:25:56,000
as few faces as possible.

566
00:25:56,000 --> 00:25:59,630
And so we had 26, was it, 24?

567
00:25:59,630 --> 00:26:03,580
This one has only 13 faces.

568
00:26:03,580 --> 00:26:06,540
And it has tetrahedral--
well, that's not actually

569
00:26:06,540 --> 00:26:09,030
tetrahedrally asymmetric,
because this bottom spike is

570
00:26:09,030 --> 00:26:10,880
different from the others.

571
00:26:10,880 --> 00:26:13,540
But it's kind of like
Tarasov's example down

572
00:26:13,540 --> 00:26:17,780
to its very minimal
amounts, also star shaped.

573
00:26:17,780 --> 00:26:21,420
And his conjecture is that,
if you have 12 faces or fewer,

574
00:26:21,420 --> 00:26:24,480
there is no
un-unfoldable polyhedron.

575
00:26:24,480 --> 00:26:30,130
So he says polyhedra with 12
faces are un-un-unfoldable.

576
00:26:30,130 --> 00:26:33,259
He wanted to one up us.

577
00:26:33,259 --> 00:26:35,175
Open problem is to define
un-un-un-unfoldable.

578
00:26:37,934 --> 00:26:38,850
But we're up to three.

579
00:26:42,580 --> 00:26:45,040
So that's some
un-unfoldable polyhedra

580
00:26:45,040 --> 00:26:49,330
and some conjectured
un-un-unfoldable polyhedra.

581
00:26:49,330 --> 00:26:50,940
This is fun to say.

582
00:26:50,940 --> 00:26:55,430
The next result, which is
very recent is that-- this

583
00:26:55,430 --> 00:26:58,750
is with Zach Abel
and myself-- that it

584
00:26:58,750 --> 00:27:02,330
is NP-complete to decide,
given a polyhedron,

585
00:27:02,330 --> 00:27:05,460
is it unfoldable or
un-unfoldable-- by cutting

586
00:27:05,460 --> 00:27:06,250
along edges.

587
00:27:06,250 --> 00:27:07,850
These are all
cutting along edges.

588
00:27:07,850 --> 00:27:10,270
And we reduce from
this problem, which

589
00:27:10,270 --> 00:27:13,620
comes from parallel
computers actually--

590
00:27:13,620 --> 00:27:16,500
or geometry in general, you
want to pack a bunch of squares

591
00:27:16,500 --> 00:27:17,300
into a square.

592
00:27:17,300 --> 00:27:19,930
So you have squares
of different sizes.

593
00:27:19,930 --> 00:27:22,910
And you want to pack those
squares into a given square.

594
00:27:22,910 --> 00:27:23,640
Is it possible?

595
00:27:23,640 --> 00:27:24,490
Yes or no?

596
00:27:24,490 --> 00:27:27,030
Sometimes it is,
sometimes it isn't.

597
00:27:27,030 --> 00:27:32,259
This proof, by many
people-- Long, et al.--

598
00:27:32,259 --> 00:27:34,550
is from three partition,
which is a problem we've seen.

599
00:27:34,550 --> 00:27:35,466
I won't go through it.

600
00:27:35,466 --> 00:27:37,980
But essentially, it's kind
of like the disk packing

601
00:27:37,980 --> 00:27:40,460
proof, which I showed in
lecture some time ago related

602
00:27:40,460 --> 00:27:42,340
to tree maker.

603
00:27:42,340 --> 00:27:44,740
But you set up this tiny
space and you end up

604
00:27:44,740 --> 00:27:49,800
having to partition a bunch of
your squares into groups, each

605
00:27:49,800 --> 00:27:53,950
with the same sum
groups of size 3.

606
00:27:53,950 --> 00:27:57,290
So starting from this
problem of square packing,

607
00:27:57,290 --> 00:28:01,220
how do we convert it into
an unfolding problem?

608
00:28:01,220 --> 00:28:03,430
This is a rough idea
of the construction.

609
00:28:03,430 --> 00:28:05,780
So the big picture--
this is initially

610
00:28:05,780 --> 00:28:07,000
a polyhedron with boundary.

611
00:28:07,000 --> 00:28:08,680
Later on, we'll
remove the boundary.

612
00:28:08,680 --> 00:28:12,482
So just imagine a square, and
in the square, there's a tower.

613
00:28:12,482 --> 00:28:13,940
Things are not
drawn to scale here.

614
00:28:13,940 --> 00:28:17,270
The tower is super tall.

615
00:28:17,270 --> 00:28:22,010
This little pipe thing is very
narrow and also very long.

616
00:28:22,010 --> 00:28:27,130
I think even way longer than
anything in this picture.

617
00:28:27,130 --> 00:28:30,610
OK, so now along the
side of the tower--

618
00:28:30,610 --> 00:28:33,010
so this is an unfolding.

619
00:28:33,010 --> 00:28:36,510
Like if you cut along these
edges, you get a plus sign.

620
00:28:36,510 --> 00:28:38,370
This is the tower.

621
00:28:38,370 --> 00:28:40,770
Now, this grid
stuff is something

622
00:28:40,770 --> 00:28:42,100
that I haven't shown you yet.

623
00:28:42,100 --> 00:28:44,870
But basically, think
of it as water.

624
00:28:44,870 --> 00:28:45,960
It's very malleable.

625
00:28:45,960 --> 00:28:48,780
It can be cut open in
many different ways.

626
00:28:48,780 --> 00:28:51,520
And essentially, you don't have
to worry about it being there.

627
00:28:51,520 --> 00:28:54,010
It's like the glue that
holds everything together.

628
00:28:54,010 --> 00:28:58,100
But then, there's these
square faces, b1 up to bn.

629
00:28:58,100 --> 00:28:59,940
These are the things
you need to pack.

630
00:28:59,940 --> 00:29:02,340
So our goal is to set
things up that so, when

631
00:29:02,340 --> 00:29:04,550
you take this tower out,
you have a square hole.

632
00:29:04,550 --> 00:29:07,174
That is your target
square shape.

633
00:29:07,174 --> 00:29:08,590
And then on the
side of the tower,

634
00:29:08,590 --> 00:29:10,090
you've got all these squares.

635
00:29:10,090 --> 00:29:12,580
Basically those squares
have to fit into that hole.

636
00:29:12,580 --> 00:29:14,230
So it is square packing.

637
00:29:14,230 --> 00:29:15,200
That is our goal.

638
00:29:15,200 --> 00:29:19,560
Now, the challenge for that
is to make all of this stuff

639
00:29:19,560 --> 00:29:22,029
get out of the way and
also for these squares--

640
00:29:22,029 --> 00:29:24,070
normally when you unfold,
you're very constrained

641
00:29:24,070 --> 00:29:25,312
in how you lay out the faces.

642
00:29:25,312 --> 00:29:27,020
You can cut here or
cut here, but there's

643
00:29:27,020 --> 00:29:28,680
a discrete set of choices.

644
00:29:28,680 --> 00:29:31,670
Our goal is to design this stuff
so that these squares can just

645
00:29:31,670 --> 00:29:35,230
basically move willy-nilly
without hitting each other.

646
00:29:35,230 --> 00:29:37,610
It's not easy to do,
but that's the thing.

647
00:29:37,610 --> 00:29:40,949
This is one big face on the
outside-- this kind of L shape.

648
00:29:40,949 --> 00:29:43,490
We didn't want to go all the
way around for a couple reasons.

649
00:29:43,490 --> 00:29:45,930
One is we need a place
for things to get out.

650
00:29:45,930 --> 00:29:49,840
But also, we wanted this
to be topologically convex.

651
00:29:49,840 --> 00:29:53,870
So we didn't want to
face that is a donut.

652
00:29:53,870 --> 00:29:56,330
OK, so what's the next
part of the construction?

653
00:29:56,330 --> 00:29:59,280
Well, if we can
arrange for these guys

654
00:29:59,280 --> 00:30:03,150
to move willy-nilly-- that's
these very thin lines--

655
00:30:03,150 --> 00:30:05,510
we can just imagine that
if the squares are somehow

656
00:30:05,510 --> 00:30:09,780
packed, instead of having
things overlapping like this,

657
00:30:09,780 --> 00:30:14,687
you can route all of those
paths to avoid crossings.

658
00:30:14,687 --> 00:30:17,270
So as long as there's tiny gaps
between all the squares, which

659
00:30:17,270 --> 00:30:20,010
doesn't turn out to change the
square packing problem very

660
00:30:20,010 --> 00:30:24,080
much, you can do this.

661
00:30:24,080 --> 00:30:27,340
And then there's another issue,
which is there's a lot of stuff

662
00:30:27,340 --> 00:30:27,840
here.

663
00:30:27,840 --> 00:30:30,130
You've got to put it
somewhere, so you end up--

664
00:30:30,130 --> 00:30:32,620
it's very, very tiny--
so in some cases,

665
00:30:32,620 --> 00:30:34,760
you have to do this
kind of wiggling just

666
00:30:34,760 --> 00:30:39,510
to eat up length in
certain settings.

667
00:30:39,510 --> 00:30:41,040
It gets complicated.

668
00:30:41,040 --> 00:30:43,140
So how do we do
this wiggly stuff?

669
00:30:43,140 --> 00:30:47,980
Well, at the first
level, there's

670
00:30:47,980 --> 00:30:50,720
a very fine grid with
very small squares here.

671
00:30:53,370 --> 00:30:55,700
And we set it up so that
we can follow everything

672
00:30:55,700 --> 00:30:57,150
along a single path.

673
00:30:57,150 --> 00:30:59,960
We can visit all of this great
stuff along a single path.

674
00:30:59,960 --> 00:31:03,120
Occasionally, we will
encounter squares,

675
00:31:03,120 --> 00:31:05,500
but the path length
between any two squares

676
00:31:05,500 --> 00:31:08,640
is so big that these squares
have room to kind of stretch

677
00:31:08,640 --> 00:31:13,010
out as far away from each
other as they need to go.

678
00:31:13,010 --> 00:31:14,670
Now, these squares
are not squares.

679
00:31:14,670 --> 00:31:16,540
They are actually
this construction,

680
00:31:16,540 --> 00:31:18,440
which we call atoms.

681
00:31:18,440 --> 00:31:23,420
It looks like a weird set of
pyramids, or outside of towers.

682
00:31:23,420 --> 00:31:27,280
But in fact, there's many
different ways to unfold this.

683
00:31:27,280 --> 00:31:29,400
There's lots of different
ways to cut it open.

684
00:31:29,400 --> 00:31:30,660
And some of them go straight.

685
00:31:30,660 --> 00:31:31,618
Some of them turn left.

686
00:31:31,618 --> 00:31:32,910
Some of them turn right.

687
00:31:32,910 --> 00:31:34,830
These are three of the
unfoldings, I forget.

688
00:31:34,830 --> 00:31:37,080
There's a few dozen
unfoldings that we

689
00:31:37,080 --> 00:31:39,700
need that all have
the right parameters.

690
00:31:39,700 --> 00:31:41,340
This one's clearly turning left.

691
00:31:41,340 --> 00:31:43,677
This one's going straight
in a certain sense.

692
00:31:43,677 --> 00:31:46,010
This is coming from a left
turn and then going straight.

693
00:31:46,010 --> 00:31:48,060
So there's lots of
combinations here,

694
00:31:48,060 --> 00:31:50,350
slightly different
parities, and so on.

695
00:31:50,350 --> 00:31:54,410
But the end effect is that if
you have a big grid of these

696
00:31:54,410 --> 00:31:58,690
and you're following
them in a path,

697
00:31:58,690 --> 00:32:01,870
the path does some
turns on the surface.

698
00:32:01,870 --> 00:32:03,670
But you can force
it to do whatever

699
00:32:03,670 --> 00:32:05,320
turns you want in the unfolding.

700
00:32:05,320 --> 00:32:08,330
So you have complete freedom
to move the squares around.

701
00:32:08,330 --> 00:32:10,450
As I showed you, you
can avoid crossings,

702
00:32:10,450 --> 00:32:13,160
and it all works out.

703
00:32:13,160 --> 00:32:15,640
Maybe one detail,
which I didn't mention,

704
00:32:15,640 --> 00:32:19,370
is there's going to be
a ton of extra stuff.

705
00:32:19,370 --> 00:32:20,350
Where does it go?

706
00:32:25,630 --> 00:32:27,130
It's going to fill the pipe up.

707
00:32:30,240 --> 00:32:32,860
It's going to push
these guys up somewhat.

708
00:32:32,860 --> 00:32:34,780
And this thing is
so tall that there's

709
00:32:34,780 --> 00:32:39,790
room to put all the
excess stuff in here.

710
00:32:39,790 --> 00:32:41,710
And it's also so tall
that you can't just

711
00:32:41,710 --> 00:32:44,810
reach all the way out and put
all the squares on the outside.

712
00:32:44,810 --> 00:32:48,729
OK, so if you make that super,
super tall, none of the squares

713
00:32:48,729 --> 00:32:50,270
fit in here, because
it's too narrow.

714
00:32:50,270 --> 00:32:51,686
And none of the
squares can get up

715
00:32:51,686 --> 00:32:53,160
to the top, because
it's too long.

716
00:32:53,160 --> 00:32:55,880
So that's how it works.

717
00:32:55,880 --> 00:32:58,360
Now, that was a
polyhedron with boundary.

718
00:32:58,360 --> 00:33:00,070
Without boundary,
the construction

719
00:33:00,070 --> 00:33:01,690
is almost the same.

720
00:33:01,690 --> 00:33:04,930
You just add some extra
stuff on the outside

721
00:33:04,930 --> 00:33:08,180
to make it a regular polyhedron
homeomorphic to a sphere.

722
00:33:08,180 --> 00:33:10,530
But in the end, you have
basically the same construction

723
00:33:10,530 --> 00:33:11,770
of this nice square.

724
00:33:11,770 --> 00:33:14,490
The pipe and some other
stuff, you prove basically

725
00:33:14,490 --> 00:33:16,490
doesn't matter.

726
00:33:16,490 --> 00:33:20,940
So that is NP-completeness
of edge unfolding

727
00:33:20,940 --> 00:33:23,730
of topologically
convex polyhedra,

728
00:33:23,730 --> 00:33:25,590
even orthogonal
polyhedra, which is

729
00:33:25,590 --> 00:33:29,855
kind of nifty-- from last year.

730
00:33:32,455 --> 00:33:34,080
If you actually want
to unfold things--

731
00:33:34,080 --> 00:33:36,550
if you want to build
things out of paper,

732
00:33:36,550 --> 00:33:39,430
the current best heuristic
unfolder is called Pepakura.

733
00:33:39,430 --> 00:33:42,800
It's a free download online,
probably Windows only.

734
00:33:42,800 --> 00:33:44,840
You can take some
weird 3D model.

735
00:33:44,840 --> 00:33:48,330
And it uses multiple pieces, in
general, but you can add tabs

736
00:33:48,330 --> 00:33:50,550
and it's quite practical.

737
00:33:50,550 --> 00:33:54,050
And cut it out, either
manually or with your computer

738
00:33:54,050 --> 00:33:55,950
controlled sign
cutter or something

739
00:33:55,950 --> 00:34:00,020
and then fold up your pieces.

740
00:34:00,020 --> 00:34:02,270
When a one piece
unfolding is possible,

741
00:34:02,270 --> 00:34:03,480
it will typically find it.

742
00:34:03,480 --> 00:34:05,160
But it's not guaranteed.

743
00:34:05,160 --> 00:34:07,330
I don't know exactly
what algorithm it uses.

744
00:34:07,330 --> 00:34:08,889
But some combination
of brute force

745
00:34:08,889 --> 00:34:12,159
and just cutting into
multiple pieces when it fails.

746
00:34:18,620 --> 00:34:23,320
Next up, we have band unfolding.

747
00:34:23,320 --> 00:34:25,310
So I talked very briefly
about band unfolding.

748
00:34:25,310 --> 00:34:28,497
I thought I'd show you
some pictures about it.

749
00:34:28,497 --> 00:34:31,080
And so remember, we're talking
about volcano unfoldings, which

750
00:34:31,080 --> 00:34:32,830
is when you cut
along all the edges

751
00:34:32,830 --> 00:34:35,530
from a point or it's sort
of in the vertical thing.

752
00:34:35,530 --> 00:34:38,150
Band unfolding was when
you had some side faces,

753
00:34:38,150 --> 00:34:42,250
you kept those intact and
cut everything else away.

754
00:34:42,250 --> 00:34:44,150
So what do I have to show here?

755
00:34:44,150 --> 00:34:50,489
This is an example of a band
unfolding gone wrong-- I guess.

756
00:34:50,489 --> 00:34:52,139
It's a little hard to see.

757
00:34:52,139 --> 00:34:54,170
This is, in general,
a prismatoid.

758
00:34:54,170 --> 00:34:57,320
We had a top polygon, A,
here, and then below it,

759
00:34:57,320 --> 00:35:00,060
this is polygon, B. We
took the convex hull, which

760
00:35:00,060 --> 00:35:01,910
adds all these other edges.

761
00:35:01,910 --> 00:35:06,630
And this is an example of a
bad unfolding of a primatoid.

762
00:35:06,630 --> 00:35:08,530
Here's an example of
a bad-- this is really

763
00:35:08,530 --> 00:35:11,740
a band unfolding
that has gone wrong.

764
00:35:11,740 --> 00:35:15,620
So imagine here the top
polygon is this triangle.

765
00:35:15,620 --> 00:35:17,270
Bottom polygon is this triangle.

766
00:35:17,270 --> 00:35:18,005
That nicely nest.

767
00:35:18,005 --> 00:35:21,180
It's a very easy
example, seemingly.

768
00:35:21,180 --> 00:35:23,590
But for whatever reason,
we also added this cut.

769
00:35:23,590 --> 00:35:27,680
And you can actually force
that if I make this not quite--

770
00:35:27,680 --> 00:35:29,490
actually, maybe it's
not quite a triangle.

771
00:35:29,490 --> 00:35:30,470
It's a quadrilateral.

772
00:35:30,470 --> 00:35:32,930
Yeah, and this is
slightly a quadrilateral.

773
00:35:32,930 --> 00:35:35,200
So when you take a convex
hull, you get that edge.

774
00:35:35,200 --> 00:35:37,900
If you cut along that edge
and then cut along here

775
00:35:37,900 --> 00:35:41,010
and cut along here and
do the band unfolding

776
00:35:41,010 --> 00:35:44,080
thing-- we're not even drawing
the bottom polygon here-- just

777
00:35:44,080 --> 00:35:49,790
the band itself overlaps,
which is annoying.

778
00:35:49,790 --> 00:35:53,070
Nonetheless-- so this
is an old example--

779
00:35:53,070 --> 00:35:57,776
but nonetheless, we proved
that there's at least one place

780
00:35:57,776 --> 00:35:59,150
to cut the band--
like here would

781
00:35:59,150 --> 00:36:02,920
work-- that will avoid overlap.

782
00:36:02,920 --> 00:36:06,310
And this is some
parts the proof.

783
00:36:06,310 --> 00:36:10,370
In general, here we're
looking at the inner polygon.

784
00:36:10,370 --> 00:36:12,770
And we cut it somewhere.

785
00:36:12,770 --> 00:36:16,609
And then we argue about where
that vertex goes if we open up

786
00:36:16,609 --> 00:36:18,150
all the angles,
which is what happens

787
00:36:18,150 --> 00:36:20,440
when you squash it flat.

788
00:36:20,440 --> 00:36:23,200
And in particular,
we argue this vertex

789
00:36:23,200 --> 00:36:26,510
must stay in the gray
region, like in the picture

790
00:36:26,510 --> 00:36:30,850
on the right, so it
can't go up here.

791
00:36:30,850 --> 00:36:33,470
And so in general, if you look
at how these things unfold,

792
00:36:33,470 --> 00:36:36,280
if you're lucky, things will
be convex when you open it.

793
00:36:36,280 --> 00:36:38,130
This is always going to be good.

794
00:36:38,130 --> 00:36:40,400
There's this weakly
convex picture,

795
00:36:40,400 --> 00:36:42,880
where when you
close the ends, it's

796
00:36:42,880 --> 00:36:45,730
not quite convex but
only at one point.

797
00:36:48,840 --> 00:36:53,420
This is troublesome--
sometimes this works,

798
00:36:53,420 --> 00:36:54,510
sometimes it doesn't.

799
00:36:54,510 --> 00:36:56,550
If you look at this
example, I believe

800
00:36:56,550 --> 00:36:58,850
it is in the weakly
convex category.

801
00:36:58,850 --> 00:37:01,790
I haven't told you the
other one is a spiral.

802
00:37:01,790 --> 00:37:06,190
Here, if you draw a 90 degree
angle from this last bar,

803
00:37:06,190 --> 00:37:11,110
then this guy is on wrong
side of that 90 degree thing.

804
00:37:11,110 --> 00:37:13,740
This thing can't happen.

805
00:37:13,740 --> 00:37:17,190
So that's comforting.

806
00:37:17,190 --> 00:37:19,460
And it's related
to this property.

807
00:37:19,460 --> 00:37:21,390
But these things
can cause problems

808
00:37:21,390 --> 00:37:23,220
like in the previous picture.

809
00:37:23,220 --> 00:37:25,760
And so you have to argue
that there is a place

810
00:37:25,760 --> 00:37:29,600
to cut where you don't get
this or if when you get it,

811
00:37:29,600 --> 00:37:30,590
it's still OK.

812
00:37:30,590 --> 00:37:33,440
And that's a messy argument--
a lot of case analysis

813
00:37:33,440 --> 00:37:35,160
I won't go through here.

814
00:37:35,160 --> 00:37:39,970
I think I have one picture
of a nicely working example.

815
00:37:39,970 --> 00:37:45,300
This is, I guess, a
general primatoid.

816
00:37:45,300 --> 00:37:48,770
But here, we have a
nice cutting that works.

817
00:37:48,770 --> 00:37:53,510
The original question that
someone here posed was,

818
00:37:53,510 --> 00:37:54,970
what about prismoids?

819
00:37:54,970 --> 00:37:57,730
So we know the band
unfolds nicely.

820
00:37:57,730 --> 00:38:00,170
What we don't know is, can
you attach the top polygon

821
00:38:00,170 --> 00:38:03,067
and the bottom polygons to
the band and get an unfolding.

822
00:38:03,067 --> 00:38:05,400
We don't know, for example,
whether all prismatoids have

823
00:38:05,400 --> 00:38:07,890
edge unfoldings,
because we don't

824
00:38:07,890 --> 00:38:09,780
know how to place the
top and bottom things.

825
00:38:09,780 --> 00:38:11,730
Probably possible
but really hard

826
00:38:11,730 --> 00:38:14,820
to figure out where
they would go.

827
00:38:14,820 --> 00:38:20,150
A simpler problem is prismoids--
which this might actually

828
00:38:20,150 --> 00:38:20,802
be a prismoid.

829
00:38:20,802 --> 00:38:22,260
If you know that
all of these edges

830
00:38:22,260 --> 00:38:29,240
are parallel initially-- so it's
a very nice situation-- then

831
00:38:29,240 --> 00:38:31,170
that's a more special
case from prismatoids--

832
00:38:31,170 --> 00:38:33,590
then maybe you could attach
the interface and outerface.

833
00:38:33,590 --> 00:38:35,270
We don't know.

834
00:38:35,270 --> 00:38:37,770
We know volcano unfoldings
work for prismoids.

835
00:38:37,770 --> 00:38:39,270
We don't know about
band unfoldings.

836
00:38:39,270 --> 00:38:41,960
That could be an interesting
open problem to work on.

837
00:38:41,960 --> 00:38:43,890
Maybe it's easy with prismoids.

838
00:38:43,890 --> 00:38:48,330
They seem pretty clean,
but I don't know for sure.

839
00:38:48,330 --> 00:38:50,960
Oh, another fun thing, which
relates to our next topic,

840
00:38:50,960 --> 00:38:55,090
is that if you just
unfold the band part,

841
00:38:55,090 --> 00:38:58,720
you can do it by continuous
blooming-- meaning there's

842
00:38:58,720 --> 00:39:01,900
a continuous motion
from the folded thing

843
00:39:01,900 --> 00:39:04,130
to the unfolded
thing, or vice versa.

844
00:39:04,130 --> 00:39:06,220
And the easy way to see
that-- you kind of get

845
00:39:06,220 --> 00:39:09,860
a sense from this picture--
in general, when you squash it

846
00:39:09,860 --> 00:39:12,190
all the way flat, it opens.

847
00:39:12,190 --> 00:39:15,840
Like this initial grey diagram
is a projection of the original

848
00:39:15,840 --> 00:39:16,510
thing.

849
00:39:16,510 --> 00:39:18,670
If you-- instead of
opening it all the way--

850
00:39:18,670 --> 00:39:20,750
if you kind of squish
it from the top

851
00:39:20,750 --> 00:39:23,310
and just keep lowering
that top face, at the end,

852
00:39:23,310 --> 00:39:25,080
it's fully in the plane.

853
00:39:25,080 --> 00:39:27,050
But in the middle,
you have a motion.

854
00:39:27,050 --> 00:39:29,780
And by the same
argument, at all times,

855
00:39:29,780 --> 00:39:31,030
you are non-self intersecting.

856
00:39:31,030 --> 00:39:33,350
So that actually gives
you a continuous blooming

857
00:39:33,350 --> 00:39:37,385
of the band of a prismatoid,
which is kind of cool.

858
00:39:39,890 --> 00:39:42,690
So the next topic is blooming.

859
00:39:42,690 --> 00:39:44,350
This is the last topic also.

860
00:39:44,350 --> 00:39:46,400
A bunch of people asked
about continuous booming

861
00:39:46,400 --> 00:39:49,080
and so the obvious one
to learn more about.

862
00:39:49,080 --> 00:39:55,530
So I have some algorithms
and examples for you.

863
00:39:55,530 --> 00:39:57,840
This is the paper--
a bunch of authors,

864
00:39:57,840 --> 00:40:00,030
"Continuous Blooming
of Convex Polyhedra."

865
00:40:00,030 --> 00:40:01,880
This problem was
posed by Connolly,

866
00:40:01,880 --> 00:40:04,715
I believe, a bunch
of years prior.

867
00:40:10,380 --> 00:40:13,580
It's still not known whether
every unfolding continuously

868
00:40:13,580 --> 00:40:14,700
blooms.

869
00:40:14,700 --> 00:40:17,940
It could be that
every unfolding--

870
00:40:17,940 --> 00:40:21,020
every non-overlapping unfolding
of a convex polyhedron--

871
00:40:21,020 --> 00:40:22,160
continuously blooms.

872
00:40:22,160 --> 00:40:25,530
I would guess the answer is,
it doesn't but it's plausible.

873
00:40:25,530 --> 00:40:27,750
That was the original question.

874
00:40:27,750 --> 00:40:30,000
What we found are
two different ways

875
00:40:30,000 --> 00:40:33,060
to continuously bloom
any convex polyhedron,

876
00:40:33,060 --> 00:40:34,784
but we choose the unfolding.

877
00:40:34,784 --> 00:40:36,450
So we have a couple
different strategies

878
00:40:36,450 --> 00:40:38,920
for choosing unfoldings
that do continuously bloom.

879
00:40:38,920 --> 00:40:41,150
Whether every unfolding
continuously blooms,

880
00:40:41,150 --> 00:40:42,130
I don't know.

881
00:40:42,130 --> 00:40:44,730
There are definitely unfoldings
of non-convex polyhedra that

882
00:40:44,730 --> 00:40:48,390
do not continuously bloom,
based on knitting needles type

883
00:40:48,390 --> 00:40:50,880
examples, based on
bad linkage stuff.

884
00:40:50,880 --> 00:40:53,890
But for convex
polyhedra, I'm not sure.

885
00:40:56,930 --> 00:40:58,425
So what do we have first?

886
00:40:58,425 --> 00:41:02,800
First strategy actually starts
from any unfolding you have.

887
00:41:02,800 --> 00:41:06,370
So we start here from the
cross unfolding of the cube.

888
00:41:06,370 --> 00:41:07,920
And then it refines it.

889
00:41:07,920 --> 00:41:09,990
So a similar strategy
to hinged dissection,

890
00:41:09,990 --> 00:41:14,130
although I think this was
done before hinged dissection.

891
00:41:14,130 --> 00:41:16,670
Let's take some
hinged structure,

892
00:41:16,670 --> 00:41:18,665
add extra cuts and extra hinges.

893
00:41:18,665 --> 00:41:21,040
In this case, I mean the hinges
are going to be the same.

894
00:41:21,040 --> 00:41:22,770
It's just adding extra cuts.

895
00:41:22,770 --> 00:41:26,030
So we're going to cut along
the red thing and also

896
00:41:26,030 --> 00:41:27,160
this red dashed line.

897
00:41:27,160 --> 00:41:31,990
The red thing is a
spanning tree of the--

898
00:41:31,990 --> 00:41:33,780
or it's really the dual graph.

899
00:41:33,780 --> 00:41:35,640
So we have, the
dual graph is you

900
00:41:35,640 --> 00:41:37,580
have a vertex for every face.

901
00:41:37,580 --> 00:41:39,650
You connect them together
if they share an edge.

902
00:41:39,650 --> 00:41:41,710
In this case, when
they share an edge,

903
00:41:41,710 --> 00:41:43,070
we're going to cut along there.

904
00:41:43,070 --> 00:41:44,990
So we cut along
all this red stuff.

905
00:41:44,990 --> 00:41:48,760
And the cool thing is, if you
walk around the red structure,

906
00:41:48,760 --> 00:41:51,420
you get a cycle-- a
Hamiltonian cycle that

907
00:41:51,420 --> 00:41:53,520
visits all the
faces exactly once.

908
00:41:53,520 --> 00:41:57,100
We want to path, not a cycle,
so we add one more cut.

909
00:41:57,100 --> 00:42:00,230
So then this blue
dash thing is a path.

910
00:42:00,230 --> 00:42:02,720
And the claim is,
any path shaped

911
00:42:02,720 --> 00:42:05,980
unfolding-- where the faces are
connected together in a path--

912
00:42:05,980 --> 00:42:07,900
can be continuously bloomed.

913
00:42:07,900 --> 00:42:09,410
How do you do it?

914
00:42:09,410 --> 00:42:09,910
Roll.

915
00:42:13,210 --> 00:42:15,570
So it's a little
easier for me to think

916
00:42:15,570 --> 00:42:17,285
about unrolling
rather than rolling,

917
00:42:17,285 --> 00:42:18,410
but they're the same thing.

918
00:42:18,410 --> 00:42:21,580
So imagine, you
start with the cube.

919
00:42:21,580 --> 00:42:25,580
And then you just roll it--
you unroll one face at a time.

920
00:42:25,580 --> 00:42:28,490
So initially, there was a
90 degree fold angle here.

921
00:42:28,490 --> 00:42:31,812
Just unroll it, so that now
they're in the same plane.

922
00:42:31,812 --> 00:42:33,770
And there was initially
a 90 degree angle here.

923
00:42:33,770 --> 00:42:35,200
Unroll that.

924
00:42:35,200 --> 00:42:38,160
After three steps,
I think-- this guy's

925
00:42:38,160 --> 00:42:42,850
been flattened, two, three--
we have this picture.

926
00:42:42,850 --> 00:42:44,960
In general, I would
normally draw it rotated,

927
00:42:44,960 --> 00:42:50,140
so that the unfolded part
lives in the floor of xy-plane.

928
00:42:50,140 --> 00:42:52,930
And the polyhedron lives in z
greater than or equal to zero,

929
00:42:52,930 --> 00:42:55,340
so lives above that plane.

930
00:42:55,340 --> 00:42:58,800
What you know at all times
is that the unfolded part

931
00:42:58,800 --> 00:43:00,910
is a subset of the unfolding.

932
00:43:00,910 --> 00:43:03,010
It's a subset of this picture.

933
00:43:03,010 --> 00:43:04,970
So it doesn't overlap itself.

934
00:43:04,970 --> 00:43:06,910
If you start with a
non-overlapping unfolding,

935
00:43:06,910 --> 00:43:09,330
you do this cutting.

936
00:43:09,330 --> 00:43:11,390
And then you just take
some subset of the faces--

937
00:43:11,390 --> 00:43:13,710
a prefix of the
path-- then that will

938
00:43:13,710 --> 00:43:15,430
be a non-overlapping thing.

939
00:43:15,430 --> 00:43:17,390
On the other hand, the
polyhedron that remains,

940
00:43:17,390 --> 00:43:18,350
we haven't touched.

941
00:43:18,350 --> 00:43:19,940
I mean, it's rotating.

942
00:43:19,940 --> 00:43:22,680
You have to make sure it
stays above the plane here.

943
00:43:22,680 --> 00:43:26,390
But it is just a subset
of the faces of the cube.

944
00:43:26,390 --> 00:43:28,130
The cube is also not
self intersecting.

945
00:43:28,130 --> 00:43:29,780
So any subset is not
self intersecting.

946
00:43:29,780 --> 00:43:31,710
So this thing doesn't
self intersect.

947
00:43:31,710 --> 00:43:34,930
This thing doesn't
self intersect.

948
00:43:34,930 --> 00:43:39,700
Potentially, this cube will
be resting on like f1 here.

949
00:43:39,700 --> 00:43:43,470
It's possible for this thing
to unroll and just touch

950
00:43:43,470 --> 00:43:46,910
the plane-- xy-plane.

951
00:43:46,910 --> 00:43:49,790
But it won't penetrate it, just
because this thing's convex

952
00:43:49,790 --> 00:43:51,220
and this is a plane.

953
00:43:51,220 --> 00:43:53,530
So you can always keep this
thing above-- keep a convex

954
00:43:53,530 --> 00:43:56,650
shape above-- or what
was a convex shape--

955
00:43:56,650 --> 00:44:00,000
a partial version of a
convex shape-- above a plane.

956
00:44:00,000 --> 00:44:02,040
But they can be
potentially touching,

957
00:44:02,040 --> 00:44:04,900
even along two
dimensional surfaces.

958
00:44:04,900 --> 00:44:08,720
So if you allow touching, this
is the path unroll algorithm.

959
00:44:08,720 --> 00:44:09,990
It works fine.

960
00:44:09,990 --> 00:44:11,490
If you don't want
to allow touching,

961
00:44:11,490 --> 00:44:20,970
you need a slightly better
strategy, which I'll tell you.

962
00:44:20,970 --> 00:44:22,550
It's pretty simple.

963
00:44:22,550 --> 00:44:26,300
I won't argue that
it works, but it's

964
00:44:26,300 --> 00:44:28,170
called the 2-step unfolding.

965
00:44:28,170 --> 00:44:30,550
This is named after
the 2-step dance.

966
00:44:30,550 --> 00:44:33,180
So we alternate between
two kinds of steps.

967
00:44:33,180 --> 00:44:45,430
One is unfold of edge, ei, to
be almost coplanar-- sorry,

968
00:44:45,430 --> 00:44:57,710
this should be a
face-- face, fi,

969
00:44:57,710 --> 00:45:03,995
to be almost coplanar with
the previous one, fi minus 1.

970
00:45:03,995 --> 00:45:05,870
So that does correspond
to unfolding an edge,

971
00:45:05,870 --> 00:45:08,870
but I'm going to think
about numbering the faces.

972
00:45:08,870 --> 00:45:11,220
Almost means we
unfold it to epsilon

973
00:45:11,220 --> 00:45:14,980
within the angle of 180 degrees.

974
00:45:14,980 --> 00:45:26,230
Then we finish unfolding the
previous step-- fi minus 1.

975
00:45:26,230 --> 00:45:27,080
And then we repeat.

976
00:45:30,077 --> 00:45:31,035
This is a little funny.

977
00:45:33,830 --> 00:45:36,110
So we're worried that if
we unfold all the way flat,

978
00:45:36,110 --> 00:45:38,900
we'll actually live in the plane
and then we'll touch things.

979
00:45:38,900 --> 00:45:40,042
We don't want to do that.

980
00:45:40,042 --> 00:45:42,000
So we're going to unfold
it almost all the way.

981
00:45:42,000 --> 00:45:44,460
But when we're done,
we do want to be flat.

982
00:45:44,460 --> 00:45:47,729
So I don't want to just unfold
everything almost all the way.

983
00:45:47,729 --> 00:45:49,520
That might work, but
it's a little tricky--

984
00:45:49,520 --> 00:45:51,840
the almost might interact.

985
00:45:51,840 --> 00:45:55,400
So what I'm going to do is just
keep one guy almost unfolded.

986
00:45:55,400 --> 00:45:57,150
That will be the previous one.

987
00:45:57,150 --> 00:45:58,990
When I've almost
unfolded the next guy,

988
00:45:58,990 --> 00:46:01,516
I will finish unfolding
the previous guy.

989
00:46:01,516 --> 00:46:03,390
Then we will almost
unfold the next, next guy

990
00:46:03,390 --> 00:46:06,390
and then finish unfolding
the next guy, and so on.

991
00:46:06,390 --> 00:46:07,870
And this turns out to work.

992
00:46:07,870 --> 00:46:10,870
Essentially, you've got
your fully planar part,

993
00:46:10,870 --> 00:46:14,180
which is everything
before i minus 2.

994
00:46:14,180 --> 00:46:15,930
That would be completely flat.

995
00:46:15,930 --> 00:46:18,050
Then the next angle
will be almost flat.

996
00:46:18,050 --> 00:46:20,260
And then the rest will
be in its original state.

997
00:46:20,260 --> 00:46:22,610
And you can argue the
appropriate definitions

998
00:46:22,610 --> 00:46:23,650
of almost.

999
00:46:23,650 --> 00:46:26,190
There will be no--
in this case, there

1000
00:46:26,190 --> 00:46:28,300
will be no two
dimensional overlap.

1001
00:46:28,300 --> 00:46:31,890
You still can get an edge
resting on another thing--

1002
00:46:31,890 --> 00:46:33,210
still not quite perfect.

1003
00:46:33,210 --> 00:46:38,700
To make it completely
perfect, we need a waltz.

1004
00:46:38,700 --> 00:46:41,775
So the waltz is a
three step dance.

1005
00:46:45,960 --> 00:46:54,190
So the waltz, we have-- a
little tricky-- we unfold fi

1006
00:46:54,190 --> 00:47:02,480
to almost coplanar-- almost
180 degrees, with fi minus 1.

1007
00:47:02,480 --> 00:47:09,270
Then we unfold fi
plus 1 slightly.

1008
00:47:12,230 --> 00:47:23,690
And then we finish
unfolding fi minus 1.

1009
00:47:23,690 --> 00:47:26,440
So there's essentially
a potential interaction

1010
00:47:26,440 --> 00:47:28,850
between fi plus
1 and fi minus 1.

1011
00:47:28,850 --> 00:47:31,610
We do a little bit of
unfolding to prevent that

1012
00:47:31,610 --> 00:47:34,910
from being an issue here.

1013
00:47:34,910 --> 00:47:38,690
And so you end up with this
three step waltz thing.

1014
00:47:41,790 --> 00:47:45,980
Stefan Langerman is a dancer,
so he liked this terminology.

1015
00:47:45,980 --> 00:47:48,040
I won't argue here
that that works,

1016
00:47:48,040 --> 00:47:50,510
but now actually, you
avoid all touching.

1017
00:47:50,510 --> 00:47:54,110
Last picture I wanted
to show you is I

1018
00:47:54,110 --> 00:47:55,590
said there were
two ways to unfold.

1019
00:47:55,590 --> 00:47:58,419
One was you take any unfolding,
you make it Hamiltonian.

1020
00:47:58,419 --> 00:48:00,960
And then you unroll it, using
one of these three algorithms--

1021
00:48:00,960 --> 00:48:04,390
just regular unroll,
two step, or waltz.

1022
00:48:04,390 --> 00:48:08,660
But another strategy we know
works is the source unfolding.

1023
00:48:08,660 --> 00:48:10,710
Here, no extra cuts required.

1024
00:48:10,710 --> 00:48:13,290
Source unfolding is like the
cleanest, coolest unfolding.

1025
00:48:13,290 --> 00:48:15,490
It turns out it just unfolds.

1026
00:48:15,490 --> 00:48:17,080
No problem.

1027
00:48:17,080 --> 00:48:18,200
How do we unfold it?

1028
00:48:18,200 --> 00:48:22,020
We follow a post order
traversal of the tree of faces,

1029
00:48:22,020 --> 00:48:25,500
meaning we completely
do one subtree.

1030
00:48:25,500 --> 00:48:29,060
And now, here, we're in the
middle of doing this subtree.

1031
00:48:29,060 --> 00:48:31,920
So here, we just have a cube
with x being this point.

1032
00:48:31,920 --> 00:48:33,190
And so the reds are the cuts.

1033
00:48:33,190 --> 00:48:34,170
Pretty simple example.

1034
00:48:34,170 --> 00:48:36,450
You just have a
star of four prongs.

1035
00:48:36,450 --> 00:48:38,840
But here, we've completely
unfolded this subtree.

1036
00:48:38,840 --> 00:48:40,820
We've done one step
of this subtree.

1037
00:48:40,820 --> 00:48:42,800
The next thing we would
do is flip it open.

1038
00:48:42,800 --> 00:48:45,460
Then we do the next one,
then we do the next one.

1039
00:48:45,460 --> 00:48:48,550
Essentially what we argue
here, in one minute,

1040
00:48:48,550 --> 00:48:53,270
is as you unfold you
can think of-- well,

1041
00:48:53,270 --> 00:48:54,650
there's a couple
things going on.

1042
00:48:54,650 --> 00:48:57,650
One is just look
at a shortest path.

1043
00:48:57,650 --> 00:48:58,150
Right?

1044
00:48:58,150 --> 00:49:01,900
This source unfolding is made
by a star of shortest paths.

1045
00:49:01,900 --> 00:49:07,700
So just think of each shortest
path individually for now.

1046
00:49:07,700 --> 00:49:10,800
Individually, if you look
at a single shortest path,

1047
00:49:10,800 --> 00:49:12,850
it hits some sequence of faces.

1048
00:49:12,850 --> 00:49:17,190
Those faces form a
path-like unfolding.

1049
00:49:17,190 --> 00:49:19,360
So you just use path unroll.

1050
00:49:19,360 --> 00:49:21,590
And that will work.

1051
00:49:21,590 --> 00:49:24,090
The only issue is potential
interactions between the paths.

1052
00:49:24,090 --> 00:49:28,360
And that's where you have to get
into the post order traversal.

1053
00:49:28,360 --> 00:49:32,040
In general, you want this
shortest path to stay shortest.

1054
00:49:32,040 --> 00:49:34,400
What we're going to do
is imagine the polyhedron

1055
00:49:34,400 --> 00:49:37,220
in some sense as growing here.

1056
00:49:37,220 --> 00:49:42,139
So as we unfold-- like, as
we start unfolding this--

1057
00:49:42,139 --> 00:49:44,055
we can think of the
interior of the polyhedron

1058
00:49:44,055 --> 00:49:45,890
as just getting bigger.

1059
00:49:45,890 --> 00:49:47,570
And in that convex
polyhedron, it

1060
00:49:47,570 --> 00:49:50,500
turns out still these
paths remain shortest.

1061
00:49:50,500 --> 00:49:52,997
And by that, all the
variants to work out.

1062
00:49:52,997 --> 00:49:54,830
And you could argue
that the paths basically

1063
00:49:54,830 --> 00:49:56,560
don't interact with
each other, because

1064
00:49:56,560 --> 00:49:57,620
of post order traversal.

1065
00:49:57,620 --> 00:49:58,620
That was very hand wavy.

1066
00:49:58,620 --> 00:50:01,690
The actual proof is a
bit technical and more

1067
00:50:01,690 --> 00:50:05,900
than I can do in zero minutes.

1068
00:50:05,900 --> 00:50:09,320
That's a sketch of
continuous blooming

1069
00:50:09,320 --> 00:50:12,170
of source unfolding,
which is pretty cool.

1070
00:50:12,170 --> 00:50:14,110
Star unfolding is open.

1071
00:50:14,110 --> 00:50:18,800
Whether all unfoldings of
convex polyhedra work is open.

1072
00:50:18,800 --> 00:50:21,042
For non-convex
polyhedra, there are

1073
00:50:21,042 --> 00:50:23,500
bunch of non-convex polyhedra,
which we'll be talking about

1074
00:50:23,500 --> 00:50:27,150
in future lectures, that
we know have unfoldings.

1075
00:50:27,150 --> 00:50:28,600
Do they have
continuous blooming?

1076
00:50:28,600 --> 00:50:29,430
We have no idea.

1077
00:50:29,430 --> 00:50:32,610
This is the state of
the art for bloomings.

1078
00:50:32,610 --> 00:50:34,930
Still a lot of interesting
open questions.

1079
00:50:34,930 --> 00:50:38,952
Any other questions from you?

1080
00:50:38,952 --> 00:50:41,680
All right, that's it.