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PROFESSOR: Today we begin the
third major part of the class.

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We've done paper
folding, linkage folding,

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and now we're going to
do polyhedron folding.

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The very last topic we
did was hinged dissection,

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which is somewhere in the
middle of all these things.

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But with polyhedron
folding, we're

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thinking about a two
dimensional surface

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in 3D, something like
a cube, and we're

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interested in cutting along
the edges of that shape

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or somehow cutting along
the surface-- that's

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a good cutting-- and then
unfolding into some flat shape.

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So this is a standard cross
unfolding of the cube.

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This is the unfolding process,
and of course the reverse

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is the folding process, and
both of them are interesting.

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We're going to start
thinking about unfolding.

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That's one of the most
practical problems here.

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You really want to build some
3D shape out of sheet material.

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What shape do you
cut out in order

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to bend it into that surface?

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So you start with the surface.

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You need to figure out where
to cut so that when you unfold,

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you get something
that has no overlap.

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If you want to make
it from one sheet,

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it shouldn't have any overlap,
and ideally just one piece

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because then you have to do
less welding or edge joining

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to make that surface.

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So today and the
next couple lectures

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will be all about unfolding,
and then eventually we

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will turn to the reverse
problem, folding problem.

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And there are two
kinds of unfolding.

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The one that I just
drew is an edge

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unfolding because it
only cuts along edges.

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Edge unfolding only cut
along edges of the surface.

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This property is nice.

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If you're building
something, you

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don't want to have
visible seems.

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You have to be
bent at the edges,

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so it's not such a big
deal if you're also

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fusing along an edge.

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But in general, you could
imagine cutting anywhere

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on the surface, and
this is an example

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where the solid red
lines are cutting

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on what you see and the
dotted red lines were

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cutting on the backside.

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And when you unfold that thing,
you get this stoplight polygon,

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and this is what we call a
general unfolding, or just

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unfolding.

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There you can cut anywhere.

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In both cases, we want
one piece, no overlap,

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but you can change where
you're allowed to cut.

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So those are the
rules of the game,

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or two possible
rules of the game.

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And let me tell you what's
known about these kinds

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of unfoldings.

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I've mentioned it way
back in lecture one.

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so we can think
about edge unfolding,

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we can think about
general unfolding,

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and we can think about
them for convex polyhedra--

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simple things like the
cube have no dents--

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or general polyhedra,
non-convex polyhedra.

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And status is these
two corners are open.

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This one is solved.

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It can always be done.

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You can always generally
unfold a convex polyhedron,

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but we don't know about
general polyhedra.

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Edge unfolding of
convex polyhedra,

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we don't know whether
it's possible,

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but for non-convex
polyhedra, we know

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that's too much to hope for.

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Not always possible.

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Both of these questions
could go either way,

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but we know in various
generalizations, easy case

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of convex polyhedra general
unfolding, we can do it.

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The most restrictive,
hardest situation

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is edge unfolding
non-convex polyhedra.

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That's too much.

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But each of these
things, we don't know.

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And these are some of the
biggest open questions

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remaining in geometric folding
algorithms, this field.

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I'm going to talk mostly
about these three things

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today, so this open problem
and these two results.

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And then this topic will
essentially be next lecture.

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Obviously it's not solved, but
we still have a lot to say.

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There are various
partial results

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toward solving both
of those problems,

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I would say more on this one.

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This one is five centuries old.

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This one is one
decade old or so.

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So I need to set the stage
a little bit before I

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can talk about any
of these things,

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so let me start with a
little bit of terminology.

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We've in some sense talked
about curvature before.

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I don't think I gave
it this name, though,

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in the context of origami.

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And I use it almost
subconsciously

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so it's good for
me to define it.

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If you have some vertex
on your polyhedron.

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Let's say we're looking
at a corner of a cube,

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so this has three 90 degree
angles coming together.

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You sum those
angles, we get 270,

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and then you take
360 minus that sum,

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and that is your curvature.

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This is going to be 90.

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And what I really care about
is whether the curvature

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is positive, 0, or negative.

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Positive curvature, which
is what we got here--

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this is plus 90-- is
like a convex cone.

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So if you're thinking about
convex polyhedra, which

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is one of the situations
we care about,

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you'll always get positive or
0, maybe, curvature vertices,

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depending on what
you're thinking about.

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Zero, this is like
a piece of paper,

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so this is what you might call
flat, but it's locally flat.

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It could be drawn
flat or we know

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how to fold pieces of paper.

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You never change the curvature.

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The sum of the
instant face angles

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will always stay the same, no
matter how you fold this thing.

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Curvature is an invariant.

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If you just think locally and
forget about the actual folding

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of the thing, it's kind of flat.

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So you can think of this
as a piece of paper.

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We talked about
these two situations

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in the context of Kawasaki's
theorem way back when.

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Kawasaki's theorem
applied for both

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of these for the negative
curvature case, which

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is when you have lots
of material all joined

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at a single point,
something like this.

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Then Kawasaki's theorem
changed a little bit.

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There were some other
cases that could happen.

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So when you have
tons of material,

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this can only happen in
a non-convex polyhedron.

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That's sort of the point.

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If you're a convex
polyhedron, you

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know you have to have
only these two situations.

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Where do you get flat vertices?

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Well, if you think of
this point as a vertex

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and you add up all the angles
around it, well, that's 360.

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360 minus 360 is 0.

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So you have zero curvature on
all the faces and on the edges.

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Also, if you look here,
you have 180 and other 180.

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That adds to 360.

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And then at the actual
vertices, that's

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where you have positive
curvature for a convex

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polyhedron.

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So just some terminology.

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Get used to the idea of positive
curvature, zero curvature,

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and negative curvature.

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Then we have the
idea of a cutting.

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So cutting is just,
what edges do you cut?

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It's the red stuff
in this picture.

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What edges do you cut
in order to unfold?

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So this is an unfolding, the
mapping here, but the red part,

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I'll call it cutting.

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Sometimes it's also
called an unfolding,

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but that can be a
little confusing.

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The main point here is I want
to talk about what constraints

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cutting must satisfy in order
to be valid so we can just

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get a sense of what
is happening here.

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If you look at these pictures,
you can see two properties.

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One is that the cuttings
visit all the vertices

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of the polyhedron.

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So there's this red stuff.

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The red stuff is connected.

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It's acyclic, so it's actually
a tree in these pictures,

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and it's visiting
all the corners.

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Even back here, it's
visited on the back.

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Is that always true?

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Most of those properties
are mostly true,

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but it depends a little bit.

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One thing that is
sure is the cutting

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must span all non-zero
curvature vertices.

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But I do have to say
"non-zero curvature." "Spans"

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just means visits.

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You have to hit all the
vertices of non-zero curvature

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because anything of
non-zero curvature

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can't be flattened by itself.

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It needs to be cut.

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At that point, the cutting has
to have degree at least one.

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With zero curvature
vertices, even if they're not

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flat in three dimensions,
you could flatten them

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because we know locally,
it really is a flat thing.

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So there may not be any cuts
at zero curvature vertices.

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Most of the time, we won't
have to worry about them,

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but something to think about.

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Everybody else you have to span.

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You can say even more.

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If you have very
negative curvature,

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if the curvature of a vertex
is less than some integer,

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k times 360 with a minus
sign in front, then cutting

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must have degree strictly
more than k plus 1.

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So if you have
negative curvature,

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you know you have to
have at least two cuts

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for a negative curvature
vertex because if you

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had negative curvature--
one of these things--

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you made a single cut.

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Then all that material
is still there.

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If you then flattened it,
it's going to overlap itself

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because you're trying
to flatten it into 360.

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There's more than
360 material there.

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So negative curvature already,
you need two vertices.

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That's the plus 1 here.

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And as soon as you get to
smaller than negative 360,

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then you have to have at least
three cuts and so on, just

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to partition up into
at most 360 groups.

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This is useful.

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Negative curvature is basically
really tough to unfold

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and we'll use that to make
counter examples here.

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What else?

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your polyhedron has no handles,
then cutting has no cycles.

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A handle is something
of higher genus.

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If this is your polyhedron,
we call this a handle.

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You have sort of
a blob down here

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and you have a connection
from one place to the other.

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If you have a
handle, you can have

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a cycle of cuts like this
that does not disconnect

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the surface, but we
basically never think

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about that situation.

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Whenever we're thinking of
something of genus zero,

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like a convex polyhedron or a
regular, non-convex polyhedron,

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you really shouldn't have
a cycle in your cutting

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because if you
had a cycle, you'd

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disconnect your
surface into two parts

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normally, if you're like
a sphere or like a disk.

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That's the acyclic
condition and when it holds.

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Some other good things.

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If a polyhedron has no
boundary and no handles,

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and the unfolding has
no holes, then cutting

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is a spanning tree.

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Spanning tree, it's a concept
we've used a few times.

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It's just a tree.

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In this case, it's got
to visit all the vertices

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of non-zero curvature and
it's a tree, so it's connected

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and it's acyclic.

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The main new thing here
is that it's connected.

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We've already said
that it should

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be acyclic with no
handles, we've already

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said that it should
span everybody,

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so it's sort of a
summary theorem,

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but we have a whole
bunch of conditions here.

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In particular, this will hold
if your polyhedron is convex.

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So for convex polyhedra,
you have a spanning tree,

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and that's what's going
on in this picture even

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for general unfoldings.

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That's the interesting case.

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Here, it's maybe more
obvious, here it's

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a little less obvious.

248
00:16:20,160 --> 00:16:25,550
And to sort of see what could go
wrong here, I have an example.

249
00:16:25,550 --> 00:16:27,210
This is a non-convex polyhedron.

250
00:16:27,210 --> 00:16:29,270
There's a whole
bunch of views of it

251
00:16:29,270 --> 00:16:33,330
up here, and in
particular this top view.

252
00:16:33,330 --> 00:16:36,710
What we're doing is slicing
along just those two

253
00:16:36,710 --> 00:16:39,182
edges and some other
stuff around the outside,

254
00:16:39,182 --> 00:16:40,890
but in particular, we
cut those two edges

255
00:16:40,890 --> 00:16:46,060
and there aren't any
other cuts around there.

256
00:16:46,060 --> 00:16:48,610
A polyhedron is set up so
that when you open it up,

257
00:16:48,610 --> 00:16:50,100
this works out.

258
00:16:50,100 --> 00:16:51,170
It's nice and flat.

259
00:16:51,170 --> 00:16:52,280
This is extremely rare.

260
00:16:52,280 --> 00:16:54,750
If you perturbed this
example, it wouldn't hold.

261
00:16:54,750 --> 00:16:57,280
But you can set things up
so that, in fact, what's

262
00:16:57,280 --> 00:17:00,320
happening is that the total
curvature-- this vertex has

263
00:17:00,320 --> 00:17:03,150
positive curvature, this has
negative, this has positive,

264
00:17:03,150 --> 00:17:06,300
and the sum of those
three curvatures is zero,

265
00:17:06,300 --> 00:17:08,680
and that's what
allows this to be flat

266
00:17:08,680 --> 00:17:11,550
because the total curvature
in that little region is zero,

267
00:17:11,550 --> 00:17:13,750
and that's when things
are allowed to be flat.

268
00:17:13,750 --> 00:17:16,990
So you can cut here and make
a separate collection of cuts

269
00:17:16,990 --> 00:17:18,880
on the outside, but
it's disconnected,

270
00:17:18,880 --> 00:17:20,780
so this is kind of weird.

271
00:17:20,780 --> 00:17:25,020
Most of the time, things
will be connected,

272
00:17:25,020 --> 00:17:28,602
and as long as your
unfolding has no holes--

273
00:17:28,602 --> 00:17:30,060
and for an unfolding
to have holes,

274
00:17:30,060 --> 00:17:31,643
you'd have to have
this weird property

275
00:17:31,643 --> 00:17:34,270
that a bunch of curvatures sum
to zero like in that picture--

276
00:17:34,270 --> 00:17:36,910
as long as you don't
have that, you're OK.

277
00:17:42,434 --> 00:17:43,600
Let's get to these theorems.

278
00:17:46,820 --> 00:17:52,310
Actually, one thing
related to this problem,

279
00:17:52,310 --> 00:17:55,420
general unfolding of
arbitrary polyhedra,

280
00:17:55,420 --> 00:17:58,120
is you can't be too
general what you

281
00:17:58,120 --> 00:17:59,500
mean by non-convex polyhedra.

282
00:17:59,500 --> 00:18:01,670
So we have this example.

283
00:18:01,670 --> 00:18:05,330
It's the simplest nasty
polyhedron there is this.

284
00:18:05,330 --> 00:18:08,705
It has one vertex at the
top there with a big dot

285
00:18:08,705 --> 00:18:10,660
that has negative curvature.

286
00:18:10,660 --> 00:18:12,990
There's more than 360
degrees of material

287
00:18:12,990 --> 00:18:15,182
from all these triangles.

288
00:18:15,182 --> 00:18:16,640
And that's sort of
all that it has.

289
00:18:16,640 --> 00:18:17,930
This is the polyhedron.

290
00:18:17,930 --> 00:18:20,860
This is what we call
boundary of the polyhedron.

291
00:18:20,860 --> 00:18:23,360
So far, all the polyhedra
I've done haven't had that,

292
00:18:23,360 --> 00:18:25,430
and you may have
seen that I assumed

293
00:18:25,430 --> 00:18:29,608
that it wasn't there
for that last lemma.

294
00:18:29,608 --> 00:18:34,660
This polyhedron can't
be unfolded at all

295
00:18:34,660 --> 00:18:37,740
in one piece, no overlap.

296
00:18:37,740 --> 00:18:38,940
Why?

297
00:18:38,940 --> 00:18:40,309
Because it has one vertex.

298
00:18:40,309 --> 00:18:41,850
You could think of
these as vertices,

299
00:18:41,850 --> 00:18:42,984
but they're kind of flat.

300
00:18:42,984 --> 00:18:44,650
Everything's flat
except that one point.

301
00:18:44,650 --> 00:18:47,000
It has negative curvature.

302
00:18:47,000 --> 00:18:48,900
Negative curvature,
we said you have

303
00:18:48,900 --> 00:18:50,900
to have at least two
cuts coming in there.

304
00:18:50,900 --> 00:18:54,380
If you just made one cut, then
when you flatten this thing,

305
00:18:54,380 --> 00:18:56,450
it'll overlap itself locally.

306
00:18:56,450 --> 00:18:59,940
So there's got to be at least
two cuts coming in there.

307
00:18:59,940 --> 00:19:02,910
I don't think that
even matters much.

308
00:19:02,910 --> 00:19:04,850
I mean, there's at
least one, certainly.

309
00:19:04,850 --> 00:19:06,890
Where could those
guys go, those cuts?

310
00:19:06,890 --> 00:19:08,810
They have to wander
around the surface.

311
00:19:08,810 --> 00:19:11,220
If they just stop in
the middle of nowhere,

312
00:19:11,220 --> 00:19:14,429
then you're sort of doing
nothing because a cut that just

313
00:19:14,429 --> 00:19:15,970
stops in the middle
of a flat vertex,

314
00:19:15,970 --> 00:19:18,020
well, you might as well
not have done that cut,

315
00:19:18,020 --> 00:19:19,234
so you could erase it.

316
00:19:19,234 --> 00:19:20,400
So it's got to go somewhere.

317
00:19:20,400 --> 00:19:23,900
They could go to each other, in
which case you've made a cycle

318
00:19:23,900 --> 00:19:26,150
and then you've disconnected
your surface because this

319
00:19:26,150 --> 00:19:29,070
is like a disk, or they
could go to the boundary.

320
00:19:29,070 --> 00:19:30,650
And if they both
go to the boundary,

321
00:19:30,650 --> 00:19:33,640
again, you disconnect
your surface, two pieces.

322
00:19:33,640 --> 00:19:37,120
So this is kind of pathetic,
but you can't unfold this

323
00:19:37,120 --> 00:19:40,240
with one piece, no overlap.

324
00:19:40,240 --> 00:19:42,120
So when I say, in
this picture, and I

325
00:19:42,120 --> 00:19:44,800
say that it's open whether
non-convex polyhedra can

326
00:19:44,800 --> 00:19:48,020
be generally unfolded, I
mean non-convex polyhedra

327
00:19:48,020 --> 00:19:49,770
without boundary.

328
00:19:49,770 --> 00:19:51,955
I'll even give you
handles if you want.

329
00:19:51,955 --> 00:19:54,780
I'm not sure that
it matters too much,

330
00:19:54,780 --> 00:19:56,870
but boundary seems to
make a big difference.

331
00:19:56,870 --> 00:19:59,340
So what's wrong
is this polyhedron

332
00:19:59,340 --> 00:20:01,210
is kind of incomplete,
and as long

333
00:20:01,210 --> 00:20:04,380
as you close it up somehow
and don't have these boundary

334
00:20:04,380 --> 00:20:08,560
effects, then maybe you can
generally unfold everything.

335
00:20:08,560 --> 00:20:10,600
That's this question.

336
00:20:19,410 --> 00:20:21,250
That was a little bit on this.

337
00:20:21,250 --> 00:20:23,940
We'll come back to
it more next lecture.

338
00:20:23,940 --> 00:20:29,360
Next, I want to do this one,
general unfolding of convex

339
00:20:29,360 --> 00:20:29,870
polyhedra.

340
00:20:29,870 --> 00:20:33,110
This is really the most
positive news I could give you.

341
00:20:36,200 --> 00:20:39,540
All this unfolding stuff,
it's the one good result.

342
00:20:39,540 --> 00:20:42,257
We know several good results,
but in terms of that table,

343
00:20:42,257 --> 00:20:43,340
it's our only good result.

344
00:20:46,020 --> 00:20:50,530
And to do that, there's a bunch
of solutions to this problem.

345
00:20:50,530 --> 00:20:52,595
They all use the idea
of shortest paths.

346
00:21:11,540 --> 00:21:15,210
The shortest path is a
path that's shortest.

347
00:21:15,210 --> 00:21:19,140
So you have some surface,
you have some points.

348
00:21:19,140 --> 00:21:20,640
You do this all the
time when you're

349
00:21:20,640 --> 00:21:22,240
flying between two cities.

350
00:21:22,240 --> 00:21:25,870
You follow a shortest
path on a sphere.

351
00:21:25,870 --> 00:21:27,620
It's not straight
in three dimensions.

352
00:21:27,620 --> 00:21:30,890
It's the shortest thing subject
to lying on that surface, what

353
00:21:30,890 --> 00:21:32,310
is the shortest path you can do?

354
00:21:32,310 --> 00:21:33,717
So you're used to
it on a sphere.

355
00:21:33,717 --> 00:21:35,300
It's a little weirder
on a polyhedron,

356
00:21:35,300 --> 00:21:37,520
but it's just the same idea.

357
00:21:37,520 --> 00:21:40,570
Take all the possible paths you
could, find the shortest one.

358
00:21:40,570 --> 00:21:43,000
You fix the two
endpoints, x and y,

359
00:21:43,000 --> 00:21:44,260
and you get to optimize this.

360
00:21:44,260 --> 00:21:46,860
Whatever is shortest
is the right thing.

361
00:21:46,860 --> 00:21:52,660
I have some pictures of
what they look like here.

362
00:21:52,660 --> 00:21:56,240
So these are some
convex polyhedra.

363
00:21:56,240 --> 00:22:00,050
They're probably random
points on a sphere,

364
00:22:00,050 --> 00:22:01,650
take the convex hull.

365
00:22:01,650 --> 00:22:04,810
And then we pick some vertex,
or some point x in the corner

366
00:22:04,810 --> 00:22:06,810
there, and this is
computing the shortest paths

367
00:22:06,810 --> 00:22:10,360
from x to every other
vertex on the polyhedron.

368
00:22:10,360 --> 00:22:12,680
So they look very
straight in this case

369
00:22:12,680 --> 00:22:14,430
because the polyhedron
is almost a sphere.

370
00:22:14,430 --> 00:22:17,020
They look kind of like great
circular arcs on a sphere,

371
00:22:17,020 --> 00:22:18,130
but they're not quite.

372
00:22:18,130 --> 00:22:21,210
You can look carefully
at an edge here,

373
00:22:21,210 --> 00:22:23,470
this does bend a little bit.

374
00:22:23,470 --> 00:22:25,190
It's a bit subtle.

375
00:22:25,190 --> 00:22:28,030
It bends only a little bit
because the property you want

376
00:22:28,030 --> 00:22:31,130
is if you look at this
triangle and this triangle

377
00:22:31,130 --> 00:22:33,060
and you unfold
them-- so right now,

378
00:22:33,060 --> 00:22:36,110
they have some dihedral
angle between them,

379
00:22:36,110 --> 00:22:39,180
and if you open it out
so that they're flat,

380
00:22:39,180 --> 00:22:41,150
then this line
should be straight.

381
00:22:41,150 --> 00:22:44,260
Still the case that shortest
paths are straight lines,

382
00:22:44,260 --> 00:22:45,410
but only when you unfold.

383
00:22:45,410 --> 00:22:46,262
That's the idea.

384
00:22:46,262 --> 00:22:48,220
So if you look at all
the triangles are visited

385
00:22:48,220 --> 00:22:51,206
by a path, and you unfold them
to be a straight thing-- this

386
00:22:51,206 --> 00:22:54,670
is called developing, you
flatten it out-- then actually,

387
00:22:54,670 --> 00:22:56,420
the shortest path will
be a straight line,

388
00:22:56,420 --> 00:22:58,690
so that makes it really
easy to draw these things.

389
00:22:58,690 --> 00:23:01,250
There still might be multiple
candidates, like to get here,

390
00:23:01,250 --> 00:23:03,610
should I go this way or
the other way around,

391
00:23:03,610 --> 00:23:07,465
but each of those will
be locally straight,

392
00:23:07,465 --> 00:23:09,090
and this is a property
called geodesic.

393
00:23:13,630 --> 00:23:17,342
Shortest paths always
unfold straight.

394
00:23:17,342 --> 00:23:19,300
And in general, anything
that unfolds straight,

395
00:23:19,300 --> 00:23:21,700
it might not even be
shortest, is a geodesic.

396
00:23:26,110 --> 00:23:27,590
These are like locally shortest.

397
00:23:27,590 --> 00:23:28,910
Locally, you can't
make them any shorter,

398
00:23:28,910 --> 00:23:30,270
but they might have
made the wrong choice.

399
00:23:30,270 --> 00:23:32,020
They might have gone
around the wrong way.

400
00:23:35,400 --> 00:23:37,525
Geodesic means going
straight for a long time.

401
00:23:37,525 --> 00:23:39,150
You could actually
spiral around and do

402
00:23:39,150 --> 00:23:40,610
all sorts of crazy things.

403
00:23:40,610 --> 00:23:50,440
But shortest paths
never cross themselves

404
00:23:50,440 --> 00:23:52,966
because if you had the shortest
path that crossed itself,

405
00:23:52,966 --> 00:23:53,840
that wasn't shortest.

406
00:23:53,840 --> 00:23:56,980
You should have just
gotten rid of this part

407
00:23:56,980 --> 00:23:59,830
and gone straight
through the crossing.

408
00:23:59,830 --> 00:24:01,560
That's sort of trivial.

409
00:24:01,560 --> 00:24:03,875
And there's another
good, fun property

410
00:24:03,875 --> 00:24:07,330
that you may not have
noticed in those figures,

411
00:24:07,330 --> 00:24:14,865
but they never pass through
positive curvature vertex.

412
00:24:25,360 --> 00:24:28,060
So if you look at
these pictures,

413
00:24:28,060 --> 00:24:30,630
they might end at a
positive curvature vertex

414
00:24:30,630 --> 00:24:32,960
because that's where we told
the shortest path to go,

415
00:24:32,960 --> 00:24:35,050
but in the middle, they're
always crossing edges.

416
00:24:35,050 --> 00:24:37,920
This is a convex polyhedron so
everything's positive curvature

417
00:24:37,920 --> 00:24:38,810
or zero.

418
00:24:38,810 --> 00:24:41,390
In that case, the
paths can really only

419
00:24:41,390 --> 00:24:43,800
go through zero
curvature points.

420
00:24:43,800 --> 00:24:46,010
They might start
and end wherever,

421
00:24:46,010 --> 00:24:49,010
but in between, they
never hit a corner.

422
00:24:49,010 --> 00:24:49,980
Why?

423
00:24:49,980 --> 00:24:54,200
Because if you have a
positive curvature vertex--

424
00:24:54,200 --> 00:24:57,700
I'm just going to sketch this
idea-- if you went up here

425
00:24:57,700 --> 00:25:00,550
in order to go back down
somewhere like there,

426
00:25:00,550 --> 00:25:03,680
it's always better to
shortcut a little bit

427
00:25:03,680 --> 00:25:05,730
and not go through the vertex.

428
00:25:05,730 --> 00:25:08,276
It's better to go around
one way or the other.

429
00:25:08,276 --> 00:25:10,880
I'm just going to wave my
hands at that, but it's true.

430
00:25:13,550 --> 00:25:14,482
So what?

431
00:25:14,482 --> 00:25:16,690
Shortest paths are going to
be a really powerful tool

432
00:25:16,690 --> 00:25:17,648
for finding unfoldings.

433
00:25:25,530 --> 00:25:32,640
In particular, I want to
define the star, that picture,

434
00:25:32,640 --> 00:25:35,400
of all shortest
paths from one point.

435
00:25:50,630 --> 00:25:52,770
And do I want points here?

436
00:25:52,770 --> 00:25:53,960
I think I want vertices.

437
00:25:57,907 --> 00:25:59,240
That's the picture that we drew.

438
00:26:03,931 --> 00:26:06,430
The interesting thing about all
the shortest paths, they all

439
00:26:06,430 --> 00:26:10,304
start from the same point,
they never hit each other.

440
00:26:10,304 --> 00:26:11,970
So not only does a
shortest path not hit

441
00:26:11,970 --> 00:26:15,530
itself, but if you take many
shortest paths, or even two

442
00:26:15,530 --> 00:26:18,950
shortest paths from a
common starting point,

443
00:26:18,950 --> 00:26:21,800
they can't hit each other.

444
00:26:21,800 --> 00:26:24,790
It could be one is
a subset of another.

445
00:26:24,790 --> 00:26:27,859
For example, if I took
the shortest path to here

446
00:26:27,859 --> 00:26:29,900
and then I also took the
shortest path to a point

447
00:26:29,900 --> 00:26:32,080
just beyond it,
well, one's going

448
00:26:32,080 --> 00:26:33,380
to be a prefix of the other.

449
00:26:33,380 --> 00:26:36,860
Other than that, they will
never cross each other.

450
00:26:36,860 --> 00:26:39,930
I'm going to just assert
that, not prove it here.

451
00:26:39,930 --> 00:26:41,650
So really, it does
look like a star.

452
00:26:41,650 --> 00:26:43,960
In fact, you could fill
in more shortest paths.

453
00:26:43,960 --> 00:26:46,430
You could take the shortest
path to here, for example,

454
00:26:46,430 --> 00:26:51,210
and it'll fit in nicely, kind
of bisect that angle in there.

455
00:26:51,210 --> 00:26:53,150
It's a very simple
kind of structure.

456
00:26:53,150 --> 00:26:56,120
It's just from a point, you
have all this stuff going out.

457
00:26:56,120 --> 00:27:05,650
They never hit each
other except at something

458
00:27:05,650 --> 00:27:12,412
called the cut locus, also
called the ridge tree.

459
00:27:18,090 --> 00:27:27,675
And these are points with
non-unique shortest paths to x.

460
00:27:37,960 --> 00:27:39,915
So we're fixing some point, x.

461
00:27:45,810 --> 00:27:47,160
So here's a simpler polyhedron.

462
00:27:47,160 --> 00:27:51,110
It's just a
square-based pyramid.

463
00:27:51,110 --> 00:27:54,060
We're picking x to be in
the middle of this face,

464
00:27:54,060 --> 00:27:57,110
and drawn in these black
lines are the shortest paths

465
00:27:57,110 --> 00:28:00,180
from x to all the vertices.

466
00:28:00,180 --> 00:28:02,440
Here, that's a straight line.

467
00:28:02,440 --> 00:28:05,620
This one, if you unfolded
it, it would be straight,

468
00:28:05,620 --> 00:28:07,750
and there's some similar
ones on the back.

469
00:28:07,750 --> 00:28:10,060
And I think this is the back
in case you want to see.

470
00:28:12,730 --> 00:28:16,870
There's one back face that you
can't see at all, C, behind B

471
00:28:16,870 --> 00:28:20,977
here, and that's
what it looks like.

472
00:28:20,977 --> 00:28:22,310
There's no shortest paths there.

473
00:28:22,310 --> 00:28:23,810
Now, there's these dashed lines.

474
00:28:23,810 --> 00:28:25,640
That's the ridge tree.

475
00:28:25,640 --> 00:28:28,000
That's the cut locus.

476
00:28:28,000 --> 00:28:33,330
And these are points on the
backside with respect to x.

477
00:28:33,330 --> 00:28:37,260
If you're going from x,
you go around behind to C,

478
00:28:37,260 --> 00:28:42,330
or you could go from x around
behind to D and then to C.

479
00:28:42,330 --> 00:28:47,040
At some point, those things meet
and are still of equal length.

480
00:28:47,040 --> 00:28:49,970
So here, if you're
trying to go to x,

481
00:28:49,970 --> 00:28:52,800
you could go around this way or
you could go around this way.

482
00:28:52,800 --> 00:28:56,230
They will be the same
length, and all the points

483
00:28:56,230 --> 00:28:58,686
with that property
are the dashed lines.

484
00:28:58,686 --> 00:29:00,810
Now, if you're familiar
with [? Vornar ?] diagrams,

485
00:29:00,810 --> 00:29:05,700
this is the [? Vornar ?]
diagram of one point, x.

486
00:29:05,700 --> 00:29:10,160
Imagine you plant grass over
your polyhedron, you light

487
00:29:10,160 --> 00:29:15,250
a fire here, it burns at
uniform speed in all directions.

488
00:29:15,250 --> 00:29:17,710
Where the fire meets
itself on the backside,

489
00:29:17,710 --> 00:29:19,540
that is the ridge tree.

490
00:29:19,540 --> 00:29:22,030
That is the
[? Vornar ?] diagram.

491
00:29:22,030 --> 00:29:24,410
So it's kind of intuitive
that it's there.

492
00:29:24,410 --> 00:29:26,580
Maybe less obvious
is that it's a tree

493
00:29:26,580 --> 00:29:27,650
and it's a spanning tree.

494
00:29:27,650 --> 00:29:28,691
It hits all the vertices.

495
00:29:32,240 --> 00:29:35,390
So it's a natural cutting.

496
00:29:35,390 --> 00:29:38,330
That's why it's
called the cut locus,

497
00:29:38,330 --> 00:29:39,620
and it works really well.

498
00:29:58,359 --> 00:29:59,900
And it's called the
source unfolding.

499
00:30:11,690 --> 00:30:13,730
Source unfolding goes
back to the mid '80s.

500
00:30:13,730 --> 00:30:15,530
A bunch of people
discovered it for

501
00:30:15,530 --> 00:30:18,010
various computational
geometry applications.

502
00:30:18,010 --> 00:30:20,630
They didn't care about
unfolding at the time.

503
00:30:20,630 --> 00:30:25,640
And it's kind of obvious after
you think about it for a while,

504
00:30:25,640 --> 00:30:28,280
trying to solve this
general unfolding problem.

505
00:30:28,280 --> 00:30:36,970
Once you have this structure,
cut along the cut locus,

506
00:30:36,970 --> 00:30:43,105
unfold the star of
shortest paths from x.

507
00:30:46,770 --> 00:30:49,200
So let's do it for this example.

508
00:30:49,200 --> 00:30:50,860
It's the same
example on the left.

509
00:30:50,860 --> 00:30:52,150
This is the star unfolding.

510
00:30:52,150 --> 00:30:56,890
I've just splayed out
everything, so at the boundary,

511
00:30:56,890 --> 00:30:58,570
I'm cutting at the dashed part.

512
00:30:58,570 --> 00:31:00,820
That's the cut locus.

513
00:31:00,820 --> 00:31:02,320
Ignore these little
dashed lines.

514
00:31:02,320 --> 00:31:06,280
Those are just edges, unfolded.

515
00:31:06,280 --> 00:31:10,490
And what's happening is that
all the shortest paths are all

516
00:31:10,490 --> 00:31:12,200
here.

517
00:31:12,200 --> 00:31:15,490
If I look at any shortest path
from x on the left diagram,

518
00:31:15,490 --> 00:31:20,830
I can map it to a line
segment starting from x.

519
00:31:20,830 --> 00:31:22,100
x was a flat vertex.

520
00:31:22,100 --> 00:31:24,450
It has 360 degrees of
material around it.

521
00:31:24,450 --> 00:31:26,900
I just chose it to be somewhere
in the middle somewhere.

522
00:31:26,900 --> 00:31:28,590
Doesn't matter other than that.

523
00:31:28,590 --> 00:31:32,100
And now there's 360 degrees of
material on the flat unfolding.

524
00:31:32,100 --> 00:31:33,120
That's great.

525
00:31:33,120 --> 00:31:35,340
And you pick any
direction here, you

526
00:31:35,340 --> 00:31:37,620
can map it to a corresponding
direction over there

527
00:31:37,620 --> 00:31:40,780
on the surface, see where
that shortest path would

528
00:31:40,780 --> 00:31:43,850
go if you kept going until
you were no longer shortest.

529
00:31:43,850 --> 00:31:47,700
When you stop being shortest,
the edge of that place

530
00:31:47,700 --> 00:31:52,630
is the ridge tree, and
that's where you cut.

531
00:31:52,630 --> 00:31:54,584
You stop your segment there.

532
00:31:54,584 --> 00:31:57,000
So it's actually pretty obvious
this thing doesn't overlap

533
00:31:57,000 --> 00:31:58,620
because all of
these shortest paths

534
00:31:58,620 --> 00:32:01,940
are going in different
directions from x.

535
00:32:01,940 --> 00:32:05,450
What you have is what we call
a star shaped polygon around x.

536
00:32:05,450 --> 00:32:08,060
Every point on the
surface is visible from x

537
00:32:08,060 --> 00:32:12,090
because we just sort of
unrolled it to be right there.

538
00:32:12,090 --> 00:32:14,550
That's the source unfolding.

539
00:32:14,550 --> 00:32:16,350
It's a little hard to
see the 3D diagrams,

540
00:32:16,350 --> 00:32:20,585
but it's actually really easy
once you draw those shortest

541
00:32:20,585 --> 00:32:21,085
paths.

542
00:32:26,680 --> 00:32:29,984
So the source unfolding
is star shaped.

543
00:32:29,984 --> 00:32:32,150
We have another unfolding
called the star unfolding,

544
00:32:32,150 --> 00:32:33,661
which is not star shaped.

545
00:32:33,661 --> 00:32:34,285
Very confusing.

546
00:32:37,230 --> 00:32:40,710
I mean, what you call one
doesn't matter too much.

547
00:32:40,710 --> 00:32:44,120
This unfolding was
mentioned in 1948

548
00:32:44,120 --> 00:32:45,740
by Alexandrov, who
we'll be hearing

549
00:32:45,740 --> 00:32:49,270
about more in the
future, but wasn't

550
00:32:49,270 --> 00:32:52,770
improved to
non-overlap until '92.

551
00:32:52,770 --> 00:32:55,570
So this one is much less
obvious that it doesn't overlap,

552
00:32:55,570 --> 00:32:57,560
and I'm not going
to prove it here.

553
00:32:57,560 --> 00:33:00,470
But it goes back to
this idea of star.

554
00:33:00,470 --> 00:33:02,950
Say, OK, this cut locus
is nice, but let's

555
00:33:02,950 --> 00:33:05,050
focus on the star
of shortest paths

556
00:33:05,050 --> 00:33:08,750
from one point to all
the other vertices.

557
00:33:08,750 --> 00:33:14,600
Instead of keeping those paths
as the things that you unroll,

558
00:33:14,600 --> 00:33:17,700
what if we cut along them?

559
00:33:17,700 --> 00:33:20,430
It's another natural thing to
try and it turns out to work.

560
00:33:23,380 --> 00:33:27,650
So here, we cut along the star.

561
00:33:27,650 --> 00:33:30,930
We've already
proved this result,

562
00:33:30,930 --> 00:33:33,040
every convex polyhedron
is generally unfoldable.

563
00:33:33,040 --> 00:33:35,590
Just with that picture,
it's very easy.

564
00:33:35,590 --> 00:33:37,990
But hey, it's fun to have more.

565
00:33:37,990 --> 00:33:39,770
And this is what the
star unfolding looks

566
00:33:39,770 --> 00:33:41,990
like for the same example.

567
00:33:41,990 --> 00:33:49,075
This is a little less obvious,
but if you think about it,

568
00:33:49,075 --> 00:33:50,450
the star, the set
of all shortest

569
00:33:50,450 --> 00:33:53,500
paths to all the vertices,
is a spanning tree.

570
00:33:53,500 --> 00:33:56,280
It hits all the vertices
because we told it to,

571
00:33:56,280 --> 00:34:00,060
and it's a tree
because it's just

572
00:34:00,060 --> 00:34:02,960
all these edges coming
together at a point.

573
00:34:02,960 --> 00:34:05,320
So it's a natural cutting also.

574
00:34:05,320 --> 00:34:06,850
And magically it works.

575
00:34:06,850 --> 00:34:10,260
How it works is a
little less obvious.

576
00:34:10,260 --> 00:34:14,650
You see here the ridge tree
drawn on the unfolding,

577
00:34:14,650 --> 00:34:17,310
and you can see all the
parts and map the letter E,

578
00:34:17,310 --> 00:34:20,409
this is the bottom square,
and stays connected.

579
00:34:20,409 --> 00:34:22,510
It's just like you
reattach everything

580
00:34:22,510 --> 00:34:26,510
around the ridge tree
and it doesn't overlap.

581
00:34:30,710 --> 00:34:33,639
It's quite difficult, I think,
to give intuition why this

582
00:34:33,639 --> 00:34:37,024
doesn't overlap, but it doesn't.

583
00:34:43,178 --> 00:34:44,719
I'll wave my hands
at some point when

584
00:34:44,719 --> 00:34:47,690
we have the necessary
tools to prove it,

585
00:34:47,690 --> 00:34:49,574
I can mention how it's done.

586
00:34:49,574 --> 00:34:50,699
But we don't have them yet.

587
00:34:50,699 --> 00:34:52,698
We'll get them in a couple
of lectures, I think.

588
00:34:56,150 --> 00:35:02,260
These results can be extended
also in various directions.

589
00:35:02,260 --> 00:35:04,985
Let me tell you briefly
about some extensions.

590
00:35:10,850 --> 00:35:13,370
For a long time, these
were the two ways

591
00:35:13,370 --> 00:35:15,380
to solve that problem
and that was sort of it.

592
00:35:17,564 --> 00:35:19,980
It would be nice, for example,
to have some general family

593
00:35:19,980 --> 00:35:22,521
of unfoldings that includes the
star unfolding and the source

594
00:35:22,521 --> 00:35:24,489
unfolding and there's
something in the middle,

595
00:35:24,489 --> 00:35:26,280
but we don't necessarily
know what that is.

596
00:35:26,280 --> 00:35:27,292
Do you have a question?

597
00:35:27,292 --> 00:35:28,208
AUDIENCE: [INAUDIBLE]?

598
00:35:33,307 --> 00:35:35,640
PROFESSOR: How do you find
the creases is your question?

599
00:35:40,002 --> 00:35:42,210
You find the creases because
you know for every point

600
00:35:42,210 --> 00:35:44,370
here where it was
on the surface,

601
00:35:44,370 --> 00:35:47,700
and if it was on an edge,
then it's a crease point.

602
00:35:47,700 --> 00:35:48,950
That's the easy way to do it.

603
00:35:50,845 --> 00:35:52,720
I don't know if that's
a satisfactory answer,

604
00:35:52,720 --> 00:35:55,061
but it can be done.

605
00:35:55,061 --> 00:35:57,770
AUDIENCE: Does it matter
where you choose x to be?

606
00:35:57,770 --> 00:36:00,660
PROFESSOR: Does it matter
where you choose x to be?

607
00:36:00,660 --> 00:36:02,560
It will change the unfolding.

608
00:36:02,560 --> 00:36:06,411
It will never overlap,
but in that sense

609
00:36:06,411 --> 00:36:08,160
there's a whole family
of star unfoldings,

610
00:36:08,160 --> 00:36:09,270
depending on where you move x.

611
00:36:09,270 --> 00:36:10,800
There's a whole family
of source unfoldings

612
00:36:10,800 --> 00:36:12,049
depending on where you move x.

613
00:36:12,049 --> 00:36:15,740
AUDIENCE: Does it make it
more efficient, any choice?

614
00:36:15,740 --> 00:36:18,340
PROFESSOR: I mean,
some of the unfoldings

615
00:36:18,340 --> 00:36:21,134
might have more cuts,
some might have less cuts.

616
00:36:21,134 --> 00:36:22,550
You're going to
get more cuts when

617
00:36:22,550 --> 00:36:24,760
you have-- depends
how you count.

618
00:36:24,760 --> 00:36:29,760
In some sense, there's only
n cuts, n shortest paths.

619
00:36:29,760 --> 00:36:33,960
But a shortest path
might cut over many faces

620
00:36:33,960 --> 00:36:37,140
or it might just
cut over one face.

621
00:36:37,140 --> 00:36:37,947
Depends where x is.

622
00:36:37,947 --> 00:36:39,530
If you choose x to
be in a nice place,

623
00:36:39,530 --> 00:36:41,450
maybe you could get
away with fewer cuts

624
00:36:41,450 --> 00:36:46,540
and have to do less welding, but
there's no theory about that.

625
00:36:46,540 --> 00:36:48,222
More questions.

626
00:36:48,222 --> 00:36:51,020
AUDIENCE: So it doesn't
matter where you put x.

627
00:36:51,020 --> 00:36:54,966
If it works at one spot, it
basically works at every spot.

628
00:36:54,966 --> 00:36:56,122
[INAUDIBLE]?

629
00:36:56,122 --> 00:36:57,580
PROFESSOR: For
convex polyhedra, it

630
00:36:57,580 --> 00:36:59,150
works no matter where you put x.

631
00:36:59,150 --> 00:37:02,730
For non-convex polyhedra, some
x's might work, some might not.

632
00:37:06,330 --> 00:37:08,874
Here, let me show you.

633
00:37:08,874 --> 00:37:10,040
Here's more star unfoldings.

634
00:37:10,040 --> 00:37:11,030
Cool.

635
00:37:11,030 --> 00:37:15,320
They're really crazy looking
and not really star shaped.

636
00:37:15,320 --> 00:37:18,410
There isn't one point
that can see everything.

637
00:37:18,410 --> 00:37:21,620
They look kind of spiky like a
star, but it's quite different.

638
00:37:21,620 --> 00:37:23,790
These are random
points on a sphere,

639
00:37:23,790 --> 00:37:26,736
like take 42 random points on
a sphere, take the convex hull,

640
00:37:26,736 --> 00:37:27,860
then take a star unfolding.

641
00:37:30,620 --> 00:37:35,060
Here's an example with
a non-convex polyhedron.

642
00:37:35,060 --> 00:37:39,170
So one thing we hoped for
for a little while briefly

643
00:37:39,170 --> 00:37:41,940
was that if you only had one
vertex of negative curvature,

644
00:37:41,940 --> 00:37:46,574
which is this one, maybe if
you choose x to be right there

645
00:37:46,574 --> 00:37:48,240
and did the star
unfolding, because that

646
00:37:48,240 --> 00:37:51,860
cuts x into lots of
little tiny pieces, then

647
00:37:51,860 --> 00:37:55,710
maybe it would unfold about
overlap, but it doesn't work.

648
00:37:55,710 --> 00:37:56,805
We were destroyed.

649
00:37:59,649 --> 00:38:01,190
In fact, you can
show neither of them

650
00:38:01,190 --> 00:38:03,700
will work in general for
a non-convex polyhedra,

651
00:38:03,700 --> 00:38:06,020
so there's no hope of
solving this problem

652
00:38:06,020 --> 00:38:10,430
with these techniques, at least
with these exact algorithms.

653
00:38:18,550 --> 00:38:20,500
You might get lucky,
but most of the time,

654
00:38:20,500 --> 00:38:21,880
I think they won't work.

655
00:38:30,690 --> 00:38:32,410
Currently, x is a point.

656
00:38:32,410 --> 00:38:36,371
You can actually
let x be-- I guess

657
00:38:36,371 --> 00:38:37,870
I'm not going to
be safe-- I'm going

658
00:38:37,870 --> 00:38:39,750
to call it a geodesic path.

659
00:38:39,750 --> 00:38:43,370
There's some restrictions
on when this is allowed,

660
00:38:43,370 --> 00:38:45,930
but the idea is you
have your surface,

661
00:38:45,930 --> 00:38:49,067
you take some straight
path on the surface,

662
00:38:49,067 --> 00:38:50,900
and then you take
shortest paths from there.

663
00:38:54,080 --> 00:38:58,550
And if you just think
about the source unfolding

664
00:38:58,550 --> 00:39:02,304
where you keep x intact
and unfold from there,

665
00:39:02,304 --> 00:39:03,720
the picture is
going to look like,

666
00:39:03,720 --> 00:39:04,840
well, you have
this straight line

667
00:39:04,840 --> 00:39:06,660
when you unfold that
thing, and then you

668
00:39:06,660 --> 00:39:10,560
have this nice star
of stuff around it.

669
00:39:10,560 --> 00:39:14,050
And under some
simple conditions,

670
00:39:14,050 --> 00:39:16,430
that will work without overlap.

671
00:39:16,430 --> 00:39:18,530
What's more impressive is
that the star unfolding

672
00:39:18,530 --> 00:39:20,984
works from a source like that.

673
00:39:20,984 --> 00:39:22,400
Again, there are
some restrictions

674
00:39:22,400 --> 00:39:24,249
on x that I'm not
going to define here,

675
00:39:24,249 --> 00:39:25,040
but it can be done.

676
00:39:25,040 --> 00:39:29,120
These are two very recent
papers by O'Rourke and Itoh

677
00:39:29,120 --> 00:39:33,209
and Vilku from the
last two years.

678
00:39:33,209 --> 00:39:34,000
So that's exciting.

679
00:39:34,000 --> 00:39:37,811
We now have four general methods
for unfolding convex polyhedra.

680
00:39:37,811 --> 00:39:39,060
Again, these are big families.

681
00:39:39,060 --> 00:39:40,450
You can choose
any geodesic path.

682
00:39:40,450 --> 00:39:41,940
Maybe some are
nicer than others.

683
00:39:45,390 --> 00:39:46,860
You can do higher dimensions.

684
00:39:55,275 --> 00:39:57,405
Star unfolding doesn't
really make sense

685
00:39:57,405 --> 00:39:58,280
in higher dimensions.

686
00:39:58,280 --> 00:39:59,988
I don't think I've
thought about it much,

687
00:39:59,988 --> 00:40:02,300
but source unfolding
makes sense.

688
00:40:02,300 --> 00:40:05,420
So you have some four polytopes,
a little hard to imagine.

689
00:40:05,420 --> 00:40:09,150
You take some point and
just radiate out from there,

690
00:40:09,150 --> 00:40:12,440
unfold like that until you hit
yourself, and then you stop,

691
00:40:12,440 --> 00:40:14,510
and that works in any dimension.

692
00:40:14,510 --> 00:40:17,240
Source unfolding works
in any dimension.

693
00:40:17,240 --> 00:40:22,870
That's fairly recent,
2003, Miller and Pak.

694
00:40:22,870 --> 00:40:27,765
Another thing you can do
is continuous blooming.

695
00:40:32,110 --> 00:40:34,810
This is an idea posed by
Connolly several years

696
00:40:34,810 --> 00:40:39,017
back and then solved
last year, I guess.

697
00:40:39,017 --> 00:40:40,100
It was presented in Japan.

698
00:40:42,660 --> 00:40:46,242
This is about folded states
versus folding motions,

699
00:40:46,242 --> 00:40:48,450
an issue we have thought
about many times in origami.

700
00:40:48,450 --> 00:40:51,050
It was easy for any
polygonal piece of paper.

701
00:40:51,050 --> 00:40:54,120
We could go from
unfolded to folded state

702
00:40:54,120 --> 00:40:56,450
by a continuous motion
without self intersection.

703
00:40:56,450 --> 00:40:58,400
For linkages, it
was the big deal.

704
00:40:58,400 --> 00:41:02,030
It was all about, can
we get from a to b?

705
00:41:02,030 --> 00:41:06,590
For convex polyhedra, you can
do it, continuous blooming.

706
00:41:06,590 --> 00:41:10,670
This is in the middle of an
algorithm of continuously

707
00:41:10,670 --> 00:41:12,970
blooming the source unfolding.

708
00:41:12,970 --> 00:41:14,380
So here's our point x.

709
00:41:14,380 --> 00:41:16,930
This is the cube, and
the source unfolding

710
00:41:16,930 --> 00:41:19,990
of a cube for this
point x in the center

711
00:41:19,990 --> 00:41:23,530
of a face, these four vertical
lines and then a little x

712
00:41:23,530 --> 00:41:26,370
on the top side.

713
00:41:26,370 --> 00:41:28,500
The algorithm unfolds
one edge at a time

714
00:41:28,500 --> 00:41:30,040
but in a very specific order.

715
00:41:30,040 --> 00:41:34,124
So it ends up unfolding this
entire house shape first,

716
00:41:34,124 --> 00:41:35,790
and then it's done
one edge of this one,

717
00:41:35,790 --> 00:41:38,081
and then it's going to fold
the other one down and then

718
00:41:38,081 --> 00:41:41,180
unfold one, two for the
backside, and then one,

719
00:41:41,180 --> 00:41:42,670
two for the left side.

720
00:41:42,670 --> 00:41:44,670
And you can show that
will not self intersect

721
00:41:44,670 --> 00:41:47,060
as long as you had
a convex polyhedron.

722
00:41:47,060 --> 00:41:50,564
It's not obvious but it's true.

723
00:41:50,564 --> 00:41:52,230
There are some other
results that if you

724
00:41:52,230 --> 00:41:57,280
have any unfolding that doesn't
self intersect at the end,

725
00:41:57,280 --> 00:42:01,805
you can add some cuts and make
it actually continuously bloom.

726
00:42:01,805 --> 00:42:03,365
It's kind of like
hinged dissections.

727
00:42:03,365 --> 00:42:05,220
It may not work by
itself, but you'd

728
00:42:05,220 --> 00:42:09,450
cut the pieces into smaller
pieces and then it will work.

729
00:42:09,450 --> 00:42:11,750
That's pretty good news,
and source unfolding just

730
00:42:11,750 --> 00:42:12,620
works as is.

731
00:42:16,070 --> 00:42:18,420
I might talk about that
some future lecture.

732
00:42:23,610 --> 00:42:26,370
Not known, for example, whether
the star unfolding continuously

733
00:42:26,370 --> 00:42:27,305
blooms.

734
00:42:27,305 --> 00:42:30,750
That sounds a
little scary to me.

735
00:42:30,750 --> 00:42:33,490
We don't actually have
an unfolding of a convex

736
00:42:33,490 --> 00:42:36,940
polyhedron that does
not continuously bloom.

737
00:42:36,940 --> 00:42:40,460
I think there should be one,
but that's an open problem.

738
00:42:40,460 --> 00:42:43,560
Is this always possible, or
is there some crazy collection

739
00:42:43,560 --> 00:42:45,650
of cuts that you cannot escape?

740
00:42:49,470 --> 00:42:54,050
That's that, general
unfolding of convex polyhedra.

741
00:42:54,050 --> 00:42:58,900
Let's turn to the--
it's not the last topic.

742
00:42:58,900 --> 00:43:00,710
We still want to
cover both of these.

743
00:43:00,710 --> 00:43:05,320
Next topic is edge unfolding
of convex polyhedra.

744
00:43:05,320 --> 00:43:07,260
Now, this problem,
as I said, goes

745
00:43:07,260 --> 00:43:13,090
back to 1525,
implicitly at least,

746
00:43:13,090 --> 00:43:20,445
by this guy, Albrecht
Durer, who was a cool guy.

747
00:43:20,445 --> 00:43:22,240
This is Renaissance time.

748
00:43:22,240 --> 00:43:23,730
He did many different things.

749
00:43:23,730 --> 00:43:26,159
I guess painter is maybe
his primary profession,

750
00:43:26,159 --> 00:43:27,700
but he did a lot of
different things.

751
00:43:27,700 --> 00:43:32,300
He studied early perspective,
all that good stuff.

752
00:43:32,300 --> 00:43:33,700
This is one of
his famous prints.

753
00:43:33,700 --> 00:43:36,860
It was actually on display
at the MFA last year,

754
00:43:36,860 --> 00:43:39,580
I got to see it, and it
has a little polyhedron

755
00:43:39,580 --> 00:43:42,260
thrown in there.

756
00:43:42,260 --> 00:43:43,700
It's a nice polyhedron.

757
00:43:43,700 --> 00:43:44,600
How did he draw them?

758
00:43:44,600 --> 00:43:48,151
Well, he probably built
models out of some material.

759
00:43:48,151 --> 00:43:49,650
And he was just
generally interested

760
00:43:49,650 --> 00:43:51,720
in the third dimension
and understanding

761
00:43:51,720 --> 00:43:55,640
how all these things worked, and
so he made a lot of unfoldings

762
00:43:55,640 --> 00:43:57,080
in this book.

763
00:43:57,080 --> 00:43:59,290
This is the original title.

764
00:43:59,290 --> 00:44:00,900
Here's a translation.

765
00:44:00,900 --> 00:44:03,140
Titles were a lot
longer in those days.

766
00:44:03,140 --> 00:44:04,967
The Painter's Manual
is what I usually

767
00:44:04,967 --> 00:44:07,050
call it, "A Manual of
Measurement of Lines, Areas,

768
00:44:07,050 --> 00:44:08,716
and Solids by Means
of Compass and Ruler

769
00:44:08,716 --> 00:44:11,280
Assembled by Albrecht Durer for
the Use of all Lovers of Art

770
00:44:11,280 --> 00:44:12,905
with Appropriate
Illustrations Arranged

771
00:44:12,905 --> 00:44:16,320
to be Printed in
the Year MDXXV."

772
00:44:16,320 --> 00:44:20,360
And I don't even know what the
subtitle is, but there you go.

773
00:44:20,360 --> 00:44:22,210
If you read German,
I'm told this

774
00:44:22,210 --> 00:44:26,240
is a challenge because
of the old script.

775
00:44:26,240 --> 00:44:29,620
So unfoldings like this.

776
00:44:29,620 --> 00:44:32,410
He did mostly Archimedian
solids in this book.

777
00:44:32,410 --> 00:44:34,840
This book is several
hundred pages long.

778
00:44:34,840 --> 00:44:36,790
I have a copy.

779
00:44:36,790 --> 00:44:41,380
This is a
cuboctahedron unfolded.

780
00:44:41,380 --> 00:44:45,290
He is only cutting along edges,
and he liked this, I presume,

781
00:44:45,290 --> 00:44:47,290
for building models.

782
00:44:47,290 --> 00:44:48,896
Here's a fancy
one, the snub cube.

783
00:44:48,896 --> 00:44:50,520
This one, I think,
might have an error.

784
00:44:50,520 --> 00:44:55,380
A couple of them have very small
errors, but for the most part,

785
00:44:55,380 --> 00:44:58,030
he was the inventor
of edge unfolding.

786
00:44:58,030 --> 00:45:03,230
Now, he didn't ask, is this
always possible, but we did.

787
00:45:03,230 --> 00:45:15,370
Mathematicians did in
1976, I believe, or '85.

788
00:45:15,370 --> 00:45:22,230
'75, G. C. Shephard, geometer.

789
00:45:22,230 --> 00:45:24,830
He was the first one
to write it in a paper,

790
00:45:24,830 --> 00:45:27,390
does every convex polyhedron
have an edge unfolding?

791
00:45:31,910 --> 00:45:36,000
So what can I tell you
about this problem?

792
00:45:36,000 --> 00:45:37,460
It's hard.

793
00:45:37,460 --> 00:45:41,035
Many people have
thought about it.

794
00:45:41,035 --> 00:45:42,660
I'm not really sure
that it's possible.

795
00:45:48,420 --> 00:45:49,770
A lot of people do, though.

796
00:46:05,240 --> 00:46:08,780
It's possible for
a lot of polyhedra.

797
00:46:08,780 --> 00:46:11,020
In fact, every polyhedron
we've tried to unfold we

798
00:46:11,020 --> 00:46:14,100
have eventually unfolded--
convex polyhedron.

799
00:46:14,100 --> 00:46:17,680
Simple set of examples are
the Archimedian solids.

800
00:46:17,680 --> 00:46:21,440
Doesn't take that much effort to
find unfoldings of all of them.

801
00:46:21,440 --> 00:46:25,310
You can do this with exhaustive
search or just playing around.

802
00:46:25,310 --> 00:46:28,020
A lot of people have
done this over the years.

803
00:46:28,020 --> 00:46:30,550
This is all the
unfoldings of those guys.

804
00:46:30,550 --> 00:46:33,370
There's a lot of
heuristic software

805
00:46:33,370 --> 00:46:35,680
for actually doing this.

806
00:46:35,680 --> 00:46:39,020
I think the coolest one
right now is Pepakura.

807
00:46:39,020 --> 00:46:41,020
If you want to build a
paper model of something,

808
00:46:41,020 --> 00:46:43,319
you have a 3D model
thrown into Pepakura

809
00:46:43,319 --> 00:46:45,610
and it'll probably give you
a one piece unfolding, even

810
00:46:45,610 --> 00:46:46,690
for non-convex shapes.

811
00:46:46,690 --> 00:46:48,740
Even though it's
not always possible,

812
00:46:48,740 --> 00:46:50,480
it'll do edge
cuttings, it'll try

813
00:46:50,480 --> 00:46:53,540
to do some exhaustive search,
and usually does pretty well.

814
00:46:53,540 --> 00:46:55,310
Sometimes it'll use
multiple pieces.

815
00:46:55,310 --> 00:46:58,350
That's the catch.

816
00:46:58,350 --> 00:47:00,670
There's a link in
the lecture notes.

817
00:47:00,670 --> 00:47:03,710
The most thorough
mathematical search

818
00:47:03,710 --> 00:47:09,310
is by this guy, Wolfram
Schlickenrieder,

819
00:47:09,310 --> 00:47:13,040
who wrote the equivalent
of a master's thesis

820
00:47:13,040 --> 00:47:19,556
in Berlin 13 years ago.

821
00:47:19,556 --> 00:47:21,180
I just wrote to him
about it yesterday.

822
00:47:21,180 --> 00:47:23,434
I was like, hey,
this is cool stuff.

823
00:47:23,434 --> 00:47:24,600
Can I show it in my lecture?

824
00:47:24,600 --> 00:47:25,900
He was like, yeah, it lives on.

825
00:47:29,600 --> 00:47:33,170
He had a class of 10
different algorithms,

826
00:47:33,170 --> 00:47:36,750
around that many,
possible algorithms

827
00:47:36,750 --> 00:47:38,640
for unfolding all
convex polyhedra,

828
00:47:38,640 --> 00:47:41,110
and then he came up with
a dozen different families

829
00:47:41,110 --> 00:47:41,750
of polyhedra.

830
00:47:41,750 --> 00:47:44,208
I don't have the polyhedra
drawn here, just the unfoldings.

831
00:47:47,420 --> 00:47:51,160
They're generators of
big classes of polyhedra.

832
00:47:51,160 --> 00:47:54,440
He applied every
algorithm to every class.

833
00:47:54,440 --> 00:47:56,900
Ideally, you get an algorithm
that works for all classes

834
00:47:56,900 --> 00:48:00,620
or you find a class that
foils all algorithms.

835
00:48:00,620 --> 00:48:03,080
Sadly, he found neither.

836
00:48:03,080 --> 00:48:07,190
Every algorithm was
foiled by some example,

837
00:48:07,190 --> 00:48:09,935
yet every example was
foiled by some algorithm.

838
00:48:13,060 --> 00:48:16,960
It's really annoying, but here's
some examples where they fail.

839
00:48:16,960 --> 00:48:20,710
They just barely fail, just a
couple triangles are messed up.

840
00:48:20,710 --> 00:48:22,280
Little squares are messed up.

841
00:48:22,280 --> 00:48:24,320
I think these are called
the turtle polyhedra.

842
00:48:24,320 --> 00:48:29,080
It's like a big, flat thing
and then a dome on top.

843
00:48:29,080 --> 00:48:31,960
So inconclusive, I
guess, is the answer.

844
00:48:31,960 --> 00:48:34,970
There was one algorithm that
had a degree of freedom.

845
00:48:34,970 --> 00:48:36,830
You had to choose
a direction, then

846
00:48:36,830 --> 00:48:39,600
you cut along all the edges
that are most in that direction.

847
00:48:39,600 --> 00:48:41,980
And there was a conjecture
on the table from this thesis

848
00:48:41,980 --> 00:48:44,570
that maybe for every
polyhedron, there

849
00:48:44,570 --> 00:48:46,520
is a direction that
works, but then

850
00:48:46,520 --> 00:48:49,950
that was destroyed
four years ago.

851
00:48:49,950 --> 00:48:53,514
Brendan Lucier from Waterloo
proved that that's not true.

852
00:48:53,514 --> 00:48:54,930
He found a polyhedron
that doesn't

853
00:48:54,930 --> 00:48:56,777
work from any direction.

854
00:48:56,777 --> 00:48:58,860
So at the moment, we have
no candidate algorithms,

855
00:48:58,860 --> 00:49:01,440
which makes me worry whether
this could possibly be true.

856
00:49:04,200 --> 00:49:06,750
We have some more bad examples.

857
00:49:06,750 --> 00:49:11,086
Here are some annoying things,
like a cube with a corner

858
00:49:11,086 --> 00:49:11,585
cut off.

859
00:49:14,370 --> 00:49:17,030
It's a very local thing, but
you can mess up really easily.

860
00:49:17,030 --> 00:49:19,674
It's actually quite common
to mess up, I would say,

861
00:49:19,674 --> 00:49:21,340
even though for things
like Archimedian,

862
00:49:21,340 --> 00:49:23,470
it's much harder to mess up.

863
00:49:23,470 --> 00:49:28,340
Simplest polyhedron that messes
up is this sliver tetrahedron.

864
00:49:28,340 --> 00:49:30,800
So the polyhedron
is drawn at the top.

865
00:49:30,800 --> 00:49:39,650
It's a little hard to
see, but on the backside,

866
00:49:39,650 --> 00:49:47,250
there's an edge like
that, if you can imagine.

867
00:49:47,250 --> 00:49:50,050
So it's almost
flat, so that's why

868
00:49:50,050 --> 00:49:53,330
it's drawn flat, and
the four corners,

869
00:49:53,330 --> 00:49:55,574
like a tetrahedron
should have, four faces,

870
00:49:55,574 --> 00:49:57,240
the front two triangles
and the back two

871
00:49:57,240 --> 00:49:59,580
triangles with the
line that I drew.

872
00:49:59,580 --> 00:50:03,085
And if you unfold it in a simple
way, you just cut from a to b,

873
00:50:03,085 --> 00:50:05,240
it unfolds like
this, no problem.

874
00:50:05,240 --> 00:50:07,340
But if you unfold
the wrong way, which

875
00:50:07,340 --> 00:50:09,820
is to cut both of
the diagonal lines,

876
00:50:09,820 --> 00:50:12,990
then you end up with the
spears that cross each other,

877
00:50:12,990 --> 00:50:15,590
so you really have
to be careful.

878
00:50:15,590 --> 00:50:21,660
In fact, if you take
a random polyhedron--

879
00:50:21,660 --> 00:50:25,370
I choose 80 points on a
sphere, I take the convex hull.

880
00:50:25,370 --> 00:50:28,600
It's a very nice, round
sphere-like polyhedron.

881
00:50:28,600 --> 00:50:30,954
And I look at all
the unfoldings--

882
00:50:30,954 --> 00:50:32,620
this is probably not
all the unfoldings,

883
00:50:32,620 --> 00:50:37,750
but I randomly generate
unfoldings, and I evaluate,

884
00:50:37,750 --> 00:50:42,870
what is the observed
probability that I get overlap?

885
00:50:42,870 --> 00:50:46,700
And it's very close to 100%.

886
00:50:46,700 --> 00:50:48,650
The conjecture is that
as n goes to infinity,

887
00:50:48,650 --> 00:50:51,560
the probability of
overlap goes to 1.

888
00:50:51,560 --> 00:50:52,810
We don't have a proof of that.

889
00:50:52,810 --> 00:50:55,456
That would be nice to prove.

890
00:50:55,456 --> 00:50:57,580
There's good reason to
believe that's easy to prove

891
00:50:57,580 --> 00:51:00,291
or that it's true, let's
say, that there is a proof.

892
00:51:00,291 --> 00:51:02,040
I don't know how to
actually formalize it.

893
00:51:05,310 --> 00:51:07,450
This is some work
from the late '80s.

894
00:51:17,570 --> 00:51:19,380
Just because most
unfoldings fail

895
00:51:19,380 --> 00:51:22,280
doesn't mean there isn't
one unfolding that works.

896
00:51:22,280 --> 00:51:25,280
If I were to try to prove that
there is an unfolding that

897
00:51:25,280 --> 00:51:28,520
works, here's my best hope,
and it ties into some things

898
00:51:28,520 --> 00:51:31,420
that we've seen
with tensegrities.

899
00:51:31,420 --> 00:51:35,840
If you take a really big
polyhedron, which is the case

900
00:51:35,840 --> 00:51:40,150
you worry about, and you look
at a small portion of it,

901
00:51:40,150 --> 00:51:43,810
that portion will be
almost completely flat.

902
00:51:43,810 --> 00:51:48,450
Locally, this thing
is mostly flat.

903
00:51:48,450 --> 00:51:50,350
Instead of thinking
about the big polyhedron,

904
00:51:50,350 --> 00:51:54,350
if we at least wanted to
get it to work in any patch,

905
00:51:54,350 --> 00:51:59,820
think about the case of
an almost flat polyhedron.

906
00:51:59,820 --> 00:52:01,050
So it's like a little dome.

907
00:52:01,050 --> 00:52:02,462
It has boundary now.

908
00:52:02,462 --> 00:52:04,170
Makes life a little
harder, but let's say

909
00:52:04,170 --> 00:52:05,430
it's a nice, convex boundary.

910
00:52:05,430 --> 00:52:07,309
This is a convex dome.

911
00:52:07,309 --> 00:52:08,350
It's actually polyhedral.

912
00:52:12,560 --> 00:52:15,170
Well, let's make it
super, super flat.

913
00:52:15,170 --> 00:52:17,480
So we have this z-coordinate.

914
00:52:17,480 --> 00:52:20,864
Scale z to squash it
down into the plane.

915
00:52:20,864 --> 00:52:22,280
What's nice about
this is then you

916
00:52:22,280 --> 00:52:24,240
can think of your
convex polyhedron

917
00:52:24,240 --> 00:52:28,830
just as some drawing
in the plane.

918
00:52:28,830 --> 00:52:31,220
So here maybe is my
convex polyhedron.

919
00:52:31,220 --> 00:52:32,740
Now, in fact, each
of these vertices

920
00:52:32,740 --> 00:52:35,180
is lifted a tiny amount,
some infinitesimal amount,

921
00:52:35,180 --> 00:52:38,570
but you can think
of it in the plane.

922
00:52:38,570 --> 00:52:40,157
These are convex faces.

923
00:52:40,157 --> 00:52:42,490
We know this thing can be
lifted to a convex polyhedron.

924
00:52:42,490 --> 00:52:45,840
That lifting is
a positive stress

925
00:52:45,840 --> 00:52:48,380
on all the edges except
the boundary edges.

926
00:52:48,380 --> 00:52:51,650
Maybe that stress gives
you some useful structure.

927
00:52:51,650 --> 00:52:53,430
If there's any hope
of this working,

928
00:52:53,430 --> 00:52:56,680
that structure better be useful,
but I don't know how to use it.

929
00:52:56,680 --> 00:52:58,640
But in particular,
open question.

930
00:52:58,640 --> 00:53:01,430
If I give you a super
shallow, arbitrarily

931
00:53:01,430 --> 00:53:06,540
shallow, convex polyhedral
dome, can it be unfolded?

932
00:53:06,540 --> 00:53:07,270
I don't know.

933
00:53:11,290 --> 00:53:13,670
Maybe we'll work on it
in the problem session.

934
00:53:16,426 --> 00:53:17,925
Haven't thought
about it for awhile.

935
00:53:20,890 --> 00:53:23,840
That's if you want
to solve everything.

936
00:53:23,840 --> 00:53:27,580
What if you just want to
solve some special cases?

937
00:53:27,580 --> 00:53:31,900
That is a special case
motivated by the general case.

938
00:53:31,900 --> 00:53:34,920
Well, you can solve the
case of a polyhedron

939
00:53:34,920 --> 00:53:36,130
with at most six vertices.

940
00:53:36,130 --> 00:53:38,950
That's as far as we've gotten.

941
00:53:38,950 --> 00:53:42,210
You can solve some simple
examples like pyramids

942
00:53:42,210 --> 00:53:46,380
if you take any convex
polygon, doesn't

943
00:53:46,380 --> 00:53:47,909
have to be irregular
or anything,

944
00:53:47,909 --> 00:53:50,200
and then you take a point
and you take the convex hull.

945
00:53:52,780 --> 00:53:55,360
So that's a polyhedron
and it unfolds.

946
00:53:55,360 --> 00:53:56,580
How does it unfold?

947
00:53:56,580 --> 00:53:57,850
Any suggestions where to cut?

948
00:54:03,670 --> 00:54:05,954
How?

949
00:54:05,954 --> 00:54:07,430
AUDIENCE: From the top?

950
00:54:07,430 --> 00:54:08,870
PROFESSOR: From the top.

951
00:54:08,870 --> 00:54:10,500
Yeah, just cut here.

952
00:54:10,500 --> 00:54:12,680
Here, we're only allowed
to cut along edges.

953
00:54:12,680 --> 00:54:17,021
This is called the
volcano unfolding.

954
00:54:17,021 --> 00:54:18,010
What do we have?

955
00:54:18,010 --> 00:54:23,310
Like that, and then there's
just a bunch of triangles.

956
00:54:23,310 --> 00:54:25,340
What's nice about
the volcano unfolding

957
00:54:25,340 --> 00:54:31,370
is if you look at these
perpendicular strips here,

958
00:54:31,370 --> 00:54:34,600
those triangles will fit inside
those perpendicular strips,

959
00:54:34,600 --> 00:54:37,370
and therefore they won't
intersect each other

960
00:54:37,370 --> 00:54:39,370
because these strips don't
intersect each other.

961
00:54:39,370 --> 00:54:41,120
You could probably
even continuously bloom

962
00:54:41,120 --> 00:54:41,940
this, no problem.

963
00:54:41,940 --> 00:54:45,997
So a very simple example,
easy to do with volcanoes.

964
00:54:45,997 --> 00:54:47,205
That's the so-called pyramid.

965
00:54:51,030 --> 00:54:56,240
Prism, a little bit more
interesting, but not by much.

966
00:54:56,240 --> 00:54:59,791
You take some polygon.

967
00:54:59,791 --> 00:55:02,010
Again, doesn't
have to be regular,

968
00:55:02,010 --> 00:55:05,304
but you take two copies of
it vertically offset, take

969
00:55:05,304 --> 00:55:05,970
the convex hull.

970
00:55:08,600 --> 00:55:14,510
This one you can
cut in two ways.

971
00:55:14,510 --> 00:55:15,920
You could do a
volcano-like thing

972
00:55:15,920 --> 00:55:20,350
where you cut all
the vertical edges

973
00:55:20,350 --> 00:55:24,860
and then you cut all but
one of the bottom edges.

974
00:55:24,860 --> 00:55:30,280
Then you'll have basically a
volcano with little rectangles

975
00:55:30,280 --> 00:55:33,420
hanging off all the sides.

976
00:55:33,420 --> 00:55:37,654
And then there's another copy,
the bottom copy of this face,

977
00:55:37,654 --> 00:55:39,320
and it just hangs out
over here, and you

978
00:55:39,320 --> 00:55:41,090
could prove that that
won't hit anybody

979
00:55:41,090 --> 00:55:43,250
in this very simple situation.

980
00:55:43,250 --> 00:55:46,260
There's another unfolding,
though, that I mention

981
00:55:46,260 --> 00:55:50,080
because it's useful
for other things,

982
00:55:50,080 --> 00:55:51,905
and it's so-called
band unfolding.

983
00:55:51,905 --> 00:55:58,220
Band unfolding, I
want to keep intact

984
00:55:58,220 --> 00:56:04,266
the radial band around the
thing, maybe like this.

985
00:56:04,266 --> 00:56:07,600
What do I want to do
to keep that edge?

986
00:56:07,600 --> 00:56:09,450
Cut, cut.

987
00:56:09,450 --> 00:56:11,600
So I cut on all the
top edges except one,

988
00:56:11,600 --> 00:56:13,700
I cut on all the bottom
edges except the same one,

989
00:56:13,700 --> 00:56:17,110
and then I also cut
this vertical edge.

990
00:56:17,110 --> 00:56:21,560
So what I should get
is I get my top face,

991
00:56:21,560 --> 00:56:24,020
then I get a
rectangle, then I get

992
00:56:24,020 --> 00:56:27,020
the band that goes
around the outside,

993
00:56:27,020 --> 00:56:32,486
and then I get another copy
of my shape, the bottom side.

994
00:56:32,486 --> 00:56:33,985
This is what we
call band unfolding.

995
00:56:36,890 --> 00:56:40,100
There's some nice theorems about
band unfoldings working out.

996
00:56:44,440 --> 00:56:46,690
Let me get to more
interesting polyhedra.

997
00:56:49,780 --> 00:56:51,650
These are some
pretty simple cases,

998
00:56:51,650 --> 00:56:55,180
but we don't know very
much on the positive side.

999
00:57:05,231 --> 00:57:05,730
Prismoid.

1000
00:57:13,000 --> 00:57:20,480
Suppose you take a polygon,
and you basically inset it.

1001
00:57:20,480 --> 00:57:24,400
So I want to make a new
version of the polygon

1002
00:57:24,400 --> 00:57:32,100
where all of the angles match,
so these edges are parallel

1003
00:57:32,100 --> 00:57:32,760
to each other.

1004
00:57:32,760 --> 00:57:36,170
I think, actually, the
lengths can change.

1005
00:57:36,170 --> 00:57:37,935
And then you take
the convex hull.

1006
00:57:40,790 --> 00:57:43,165
These unfold with the
volcano unfolding.

1007
00:57:46,024 --> 00:57:47,440
It's a little more
subtle to prove

1008
00:57:47,440 --> 00:57:50,180
but it's proved in the textbook.

1009
00:57:54,720 --> 00:57:57,070
Edges have to stay parallel.

1010
00:57:57,070 --> 00:57:59,576
I think that might
force something.

1011
00:57:59,576 --> 00:58:01,200
I'm not quite sure,
but the constraints

1012
00:58:01,200 --> 00:58:02,300
are that the edges
have to be parallel

1013
00:58:02,300 --> 00:58:03,633
and the angles have to be equal.

1014
00:58:08,370 --> 00:58:09,660
Dome.

1015
00:58:09,660 --> 00:58:13,160
Dome is actually in
some ways simpler.

1016
00:58:13,160 --> 00:58:20,720
You take some base, and then
I want a whole bunch of faces

1017
00:58:20,720 --> 00:58:22,500
that all touch the base.

1018
00:58:22,500 --> 00:58:26,610
So in general, it's going to be
like a tree of faces, something

1019
00:58:26,610 --> 00:58:28,710
like this.

1020
00:58:28,710 --> 00:58:31,840
Every face has to touch
an edge of the base.

1021
00:58:31,840 --> 00:58:34,405
These also unfold, and
also with a volcano.

1022
00:58:37,060 --> 00:58:40,490
This is proved also
in the textbook.

1023
00:58:40,490 --> 00:58:42,456
Both results are
by Joe O'Rourke.

1024
00:58:45,650 --> 00:58:47,380
Prismatoid.

1025
00:58:47,380 --> 00:58:49,870
This is the coolest
special case I'm

1026
00:58:49,870 --> 00:58:53,380
going to talk about,
the most interesting.

1027
00:58:53,380 --> 00:58:59,340
I take some convex polygon,
and then in a parallel offset,

1028
00:58:59,340 --> 00:59:06,337
I take some other
convex polygon--

1029
00:59:06,337 --> 00:59:08,212
no relation to each
other except that they're

1030
00:59:08,212 --> 00:59:09,950
in parallel planes--
and then I take

1031
00:59:09,950 --> 00:59:15,640
the convex hull,
something like that.

1032
00:59:22,160 --> 00:59:23,595
This one, sadly, is open.

1033
00:59:27,250 --> 00:59:29,110
Now, what we do know,
and this is related

1034
00:59:29,110 --> 00:59:32,660
to the band
unfolding, if you look

1035
00:59:32,660 --> 00:59:38,650
at the band around the
sides of this prismatoid

1036
00:59:38,650 --> 00:59:41,460
and just unfold
that by itself, that

1037
00:59:41,460 --> 00:59:45,160
will not self intersect,
and that's quite nontrivial,

1038
00:59:45,160 --> 00:59:45,820
the proof.

1039
00:59:45,820 --> 00:59:49,040
But if you just
unroll the side faces,

1040
00:59:49,040 --> 00:59:53,041
they will wander around but
they won't hit themselves.

1041
00:59:53,041 --> 00:59:54,540
The only remaining
problem is, where

1042
00:59:54,540 --> 00:59:57,960
does the top and
the bottom face go?

1043
00:59:57,960 --> 01:00:01,920
I think I have an example
where that's a little dicey.

1044
01:00:01,920 --> 01:00:04,060
Prism.

1045
01:00:04,060 --> 01:00:04,860
Prismoid.

1046
01:00:04,860 --> 01:00:07,120
That gives you an
answer to your offset.

1047
01:00:07,120 --> 01:00:10,070
So they can be sheared
away from each other,

1048
01:00:10,070 --> 01:00:12,840
but this is a prismoid.

1049
01:00:12,840 --> 01:00:17,420
All of the sides are parallel
and the angles match, a and b.

1050
01:00:17,420 --> 01:00:22,900
Here is the volcano
unfolding of the prismoid,

1051
01:00:22,900 --> 01:00:26,580
and here's showing that it's
not so obvious that it always

1052
01:00:26,580 --> 01:00:27,080
works.

1053
01:00:27,080 --> 01:00:30,330
If you're not careful where
you put the top face-- so this

1054
01:00:30,330 --> 01:00:34,230
is a on top of b
in plan-- you could

1055
01:00:34,230 --> 01:00:36,860
attach a to any of these faces.

1056
01:00:36,860 --> 01:00:38,250
You have to choose
one that's not

1057
01:00:38,250 --> 01:00:41,420
so inside because
it's overlapping here.

1058
01:00:41,420 --> 01:00:42,920
But if you choose
the outermost one,

1059
01:00:42,920 --> 01:00:45,890
it works and you can prove that.

1060
01:00:45,890 --> 01:00:51,490
This is the dome.

1061
01:00:51,490 --> 01:00:53,900
You can even have
overhang and they

1062
01:00:53,900 --> 01:00:57,190
will unfold, volcano style.

1063
01:00:57,190 --> 01:01:00,260
You no longer have those
nice perpendicular wedges

1064
01:01:00,260 --> 01:01:01,650
to say everything is disjoined.

1065
01:01:01,650 --> 01:01:02,719
But it still turns out.

1066
01:01:02,719 --> 01:01:03,510
Nothing intersects.

1067
01:01:06,380 --> 01:01:09,290
Here's the prismatoid
I wanted to show you.

1068
01:01:09,290 --> 01:01:11,820
This one's actually almost flat.

1069
01:01:11,820 --> 01:01:16,290
So a is this triangle right on
top of b, the triangle beneath.

1070
01:01:16,290 --> 01:01:17,790
You take the convex
hull and you get

1071
01:01:17,790 --> 01:01:20,970
all those edges on the outside.

1072
01:01:20,970 --> 01:01:23,710
And if you're not careful--
this is more volcano

1073
01:01:23,710 --> 01:01:26,745
style unfolding-- but
you can get overlap.

1074
01:01:32,020 --> 01:01:36,910
One thing we do know,
this is kind of weird.

1075
01:01:36,910 --> 01:01:41,380
If you try to go more general,
and instead of taking a convex

1076
01:01:41,380 --> 01:01:43,970
polygon on the top and a
convex polygon on the bottom,

1077
01:01:43,970 --> 01:01:46,290
instead you take a
smooth convex curve

1078
01:01:46,290 --> 01:01:47,890
on the top and a
smooth convex curve

1079
01:01:47,890 --> 01:01:50,594
on the bottom, which is getting
weird because now there's

1080
01:01:50,594 --> 01:01:52,010
infinitely many
vertices and going

1081
01:01:52,010 --> 01:01:54,200
to be infinitely many cuts.

1082
01:01:54,200 --> 01:01:58,440
But there is a natural
notion of unfolding

1083
01:01:58,440 --> 01:01:59,990
which sort of takes
the continuum.

1084
01:01:59,990 --> 01:02:02,815
You take all those lines
that go from one side

1085
01:02:02,815 --> 01:02:07,630
to the other, the rule
lines of the convex shape,

1086
01:02:07,630 --> 01:02:09,647
and you just unfold them.

1087
01:02:09,647 --> 01:02:11,730
You don't actually preserve
area when you do this.

1088
01:02:11,730 --> 01:02:14,580
It's not a valid unfolding
in the usual sense,

1089
01:02:14,580 --> 01:02:16,430
but it's a natural
generalization

1090
01:02:16,430 --> 01:02:18,756
of unfolding to smooth shapes.

1091
01:02:18,756 --> 01:02:21,380
You can prove, well, that part's
going to work just fine if you

1092
01:02:21,380 --> 01:02:23,410
just volcano it, and
then you can actually

1093
01:02:23,410 --> 01:02:26,420
find a place for the top face.

1094
01:02:33,920 --> 01:02:35,420
Those are the special
cases we know,

1095
01:02:35,420 --> 01:02:37,780
and even some pretty simple
cases that we don't know,

1096
01:02:37,780 --> 01:02:40,480
although it's almost there.

1097
01:02:55,359 --> 01:02:56,275
One more open problem.

1098
01:03:21,716 --> 01:03:23,590
Just mention this because
it's a fun problem.

1099
01:03:26,430 --> 01:03:38,380
So again, I want to edge
unfold all convex polyhedra,

1100
01:03:38,380 --> 01:03:41,709
but I allow multiple
pieces to make

1101
01:03:41,709 --> 01:03:42,875
the problem a little easier.

1102
01:03:46,140 --> 01:03:53,690
So let's say I have a
polyhedron with f faces.

1103
01:03:53,690 --> 01:03:56,740
I want to know how few
pieces could I get away with?

1104
01:03:56,740 --> 01:03:59,836
The big open problem is, can
I get away with one piece?

1105
01:03:59,836 --> 01:04:00,960
What if you make it easier?

1106
01:04:00,960 --> 01:04:05,760
What if I just want, say,
little o of f pieces?

1107
01:04:05,760 --> 01:04:08,410
Smaller than any
constant times f.

1108
01:04:08,410 --> 01:04:09,290
This is open.

1109
01:04:13,210 --> 01:04:15,390
What we do know is
some constant times

1110
01:04:15,390 --> 01:04:19,710
f where the constant
is less than 1.

1111
01:04:19,710 --> 01:04:23,230
Best constant I have
written here is 1/2.

1112
01:04:23,230 --> 01:04:24,900
There is a better
bound, but I think

1113
01:04:24,900 --> 01:04:27,557
it's not so easy to
summarize in this form.

1114
01:04:27,557 --> 01:04:29,140
So you can get a
little less than half

1115
01:04:29,140 --> 01:04:32,380
the faces' number of pieces,
but that's pretty pathetic.

1116
01:04:35,507 --> 01:04:37,840
It's a problem that seemed
like it would be a good idea,

1117
01:04:37,840 --> 01:04:40,450
but so far it hasn't seemed to
make the problem much easier.

1118
01:04:44,670 --> 01:04:46,810
What do I mean by pieces?

1119
01:04:46,810 --> 01:04:49,580
Well, we're all about
one piece unfolding,

1120
01:04:49,580 --> 01:04:53,170
so now your cutting
can have cycles in it

1121
01:04:53,170 --> 01:04:55,315
and disconnect the surface
into multiple parts.

1122
01:04:55,315 --> 01:04:56,231
AUDIENCE: [INAUDIBLE].

1123
01:04:58,752 --> 01:05:00,460
PROFESSOR: Multiple
connected components.

1124
01:05:15,426 --> 01:05:17,050
The tricky part is
to pair up the faces

1125
01:05:17,050 --> 01:05:20,050
so that everybody has a mate.

1126
01:05:20,050 --> 01:05:22,510
That's not always possible.

1127
01:05:22,510 --> 01:05:24,134
AUDIENCE: [INAUDIBLE]?

1128
01:05:24,134 --> 01:05:24,800
PROFESSOR: Yeah.

1129
01:05:24,800 --> 01:05:28,880
You would hope that one
third, but-- I mean,

1130
01:05:28,880 --> 01:05:30,700
you can do lots of
little local arguments

1131
01:05:30,700 --> 01:05:33,799
and prove this constant,
but the big question

1132
01:05:33,799 --> 01:05:35,340
is, can you get less
than a constant?

1133
01:05:44,730 --> 01:05:50,615
So last topic for today is edge
unfolding non-convex polyhedra.

1134
01:05:56,970 --> 01:06:00,210
Sort of did that
problem addressed,

1135
01:06:00,210 --> 01:06:03,330
and now this is not
always possible.

1136
01:06:03,330 --> 01:06:06,200
So I want to give you a
polyhedron where you cannot

1137
01:06:06,200 --> 01:06:06,910
edge unfold it.

1138
01:06:06,910 --> 01:06:10,510
This is actually pretty easy,
and we did it way back in '98

1139
01:06:10,510 --> 01:06:12,340
when we started
working on folding,

1140
01:06:12,340 --> 01:06:14,810
but it's kind of cheating.

1141
01:06:14,810 --> 01:06:18,630
This is a box on a box,
and the only edges here

1142
01:06:18,630 --> 01:06:20,790
are the edges of the two boxes.

1143
01:06:20,790 --> 01:06:23,460
There's no edges connecting
the outside of this face

1144
01:06:23,460 --> 01:06:24,670
to the inside of that face.

1145
01:06:24,670 --> 01:06:28,930
That face here is a donut.

1146
01:06:28,930 --> 01:06:29,840
It's a square donut.

1147
01:06:33,080 --> 01:06:37,590
So if you're only cutting along
edges, that face is intact.

1148
01:06:37,590 --> 01:06:43,100
Now I ask you, where
does the top box go?

1149
01:06:43,100 --> 01:06:45,660
The top box has five
square faces, not six.

1150
01:06:45,660 --> 01:06:47,490
There's nothing on the bottom.

1151
01:06:47,490 --> 01:06:49,920
And somehow, it has to
be attached to the rest

1152
01:06:49,920 --> 01:06:50,810
if I want one piece.

1153
01:06:50,810 --> 01:06:54,170
That means that all
five faces fit in here,

1154
01:06:54,170 --> 01:06:56,990
but there's only room
for one in terms of area

1155
01:06:56,990 --> 01:06:59,050
so you're screwed.

1156
01:06:59,050 --> 01:07:03,290
But this is cheating because
what I really wanted to do

1157
01:07:03,290 --> 01:07:05,920
was generalize this
problem, edge unfolding

1158
01:07:05,920 --> 01:07:08,080
of convex polyhedra.

1159
01:07:08,080 --> 01:07:10,000
Now, I know they're not
going to look convex,

1160
01:07:10,000 --> 01:07:13,240
but that thing is
really not convex.

1161
01:07:13,240 --> 01:07:14,740
It's really not
convex in the sense

1162
01:07:14,740 --> 01:07:18,500
that you have this face that is
not even topologically a disk.

1163
01:07:18,500 --> 01:07:19,750
It has a hole in it.

1164
01:07:19,750 --> 01:07:22,160
Convex polyhedra
don't have that.

1165
01:07:22,160 --> 01:07:25,210
So in some sense, this
is a topological problem,

1166
01:07:25,210 --> 01:07:27,500
you might say.

1167
01:07:27,500 --> 01:07:30,800
And you might hope that if
I looked at polyhedra that

1168
01:07:30,800 --> 01:07:34,000
are topologically
convex, maybe those

1169
01:07:34,000 --> 01:07:37,020
would unfold by edge cuts.

1170
01:07:37,020 --> 01:07:39,920
"Topologically convex"
means that if you just

1171
01:07:39,920 --> 01:07:43,570
move the vertices around but
preserve the edge structure,

1172
01:07:43,570 --> 01:07:46,509
then it becomes a
convex polyhedron.

1173
01:07:46,509 --> 01:07:48,550
In other words, I take a
convex polyhedron, which

1174
01:07:48,550 --> 01:07:50,664
we think maybe has
an edge unfolding,

1175
01:07:50,664 --> 01:07:52,330
and then I just pull
the vertices around

1176
01:07:52,330 --> 01:07:55,150
but preserve all the faces.

1177
01:07:55,150 --> 01:07:57,070
Then can you edge unfold those?

1178
01:07:57,070 --> 01:07:59,470
And the answer is no.

1179
01:07:59,470 --> 01:08:03,400
I have one more example
before we get there.

1180
01:08:03,400 --> 01:08:06,160
This polyhedron, at least
all the faces are disks.

1181
01:08:06,160 --> 01:08:07,600
There's no holes.

1182
01:08:07,600 --> 01:08:09,560
So it's a cube with
little bites taken out

1183
01:08:09,560 --> 01:08:12,320
of all the edges, same paper.

1184
01:08:12,320 --> 01:08:13,820
And this thing also
does not unfold.

1185
01:08:13,820 --> 01:08:14,945
It's a little less obvious.

1186
01:08:17,622 --> 01:08:19,330
What's cheating about
this example is you

1187
01:08:19,330 --> 01:08:22,090
have two faces, like this
purple one and the yellow one,

1188
01:08:22,090 --> 01:08:24,859
that share two different edges.

1189
01:08:24,859 --> 01:08:26,890
And for convex polyhedra,
two faces either

1190
01:08:26,890 --> 01:08:29,649
share an edge or a
vertex or nothing,

1191
01:08:29,649 --> 01:08:34,300
but they can't reach
around and do to joins.

1192
01:08:34,300 --> 01:08:40,390
Again, this is not topologically
convex, but this is.

1193
01:08:40,390 --> 01:08:43,300
So this is two views
of the same thing,

1194
01:08:43,300 --> 01:08:46,640
and if I take these spikes and
I just push that vertex down

1195
01:08:46,640 --> 01:08:50,396
to be really close to this
triangle, this will be convex.

1196
01:08:50,396 --> 01:08:52,020
So it's just a convex
polyhedron that I

1197
01:08:52,020 --> 01:08:55,060
pull on four of the points.

1198
01:08:55,060 --> 01:08:56,580
Same facial structure.

1199
01:08:56,580 --> 01:09:00,899
This has no edge unfolding
and we're going to prove that.

1200
01:09:12,830 --> 01:09:14,990
This example is even
stronger in that

1201
01:09:14,990 --> 01:09:16,200
all the faces are triangles.

1202
01:09:25,560 --> 01:09:30,177
So what we're going to
do is take this thing.

1203
01:09:30,177 --> 01:09:31,885
It's kind of appropriate
for this season.

1204
01:10:00,680 --> 01:10:03,690
So I've got a
tetrahedral spike on top.

1205
01:10:03,690 --> 01:10:05,830
Think of that as going
out of the board.

1206
01:10:05,830 --> 01:10:10,041
And then in the plane of
the board is this triangle,

1207
01:10:10,041 --> 01:10:11,540
and then I just add
in all the edges

1208
01:10:11,540 --> 01:10:13,480
to make it a triangulation.

1209
01:10:13,480 --> 01:10:15,242
Adding edges is a
worry because that's

1210
01:10:15,242 --> 01:10:16,950
where we're allowed
to cut, so you really

1211
01:10:16,950 --> 01:10:18,796
have to worry about
all those edges we add.

1212
01:10:18,796 --> 01:10:21,720
If we add them in
that way, I claim,

1213
01:10:21,720 --> 01:10:24,950
you take this-- we call
it the witch's hat--

1214
01:10:24,950 --> 01:10:29,690
then you multiply, in some
sense, by a tetrahedron,

1215
01:10:29,690 --> 01:10:32,370
meaning I take four
copies of this hat,

1216
01:10:32,370 --> 01:10:36,390
I put one on each face
of the tetrahedron,

1217
01:10:36,390 --> 01:10:39,470
and you get that example.

1218
01:10:39,470 --> 01:10:43,050
Here's one witch's hat and
they're just joined along edges

1219
01:10:43,050 --> 01:10:45,244
to make the tetrahedron.

1220
01:10:45,244 --> 01:10:46,910
You could do this
with bigger polyhedra,

1221
01:10:46,910 --> 01:10:49,800
too, like octahedron, anything
with equilateral triangles,

1222
01:10:49,800 --> 01:10:52,410
but tetrahedron is the
smallest where it works.

1223
01:10:52,410 --> 01:10:55,160
Just two of them glued face
to face would not work,

1224
01:10:55,160 --> 01:10:56,760
but this way does work.

1225
01:10:56,760 --> 01:10:57,850
Is that true?

1226
01:11:02,350 --> 01:11:05,392
Now I've got to think about two
of them joined face to face.

1227
01:11:05,392 --> 01:11:06,350
I don't think it works.

1228
01:11:06,350 --> 01:11:07,808
Otherwise, we would
have done that.

1229
01:11:09,910 --> 01:11:12,000
Oh yes, I think I see why.

1230
01:11:19,380 --> 01:11:21,780
Why doesn't this work?

1231
01:11:21,780 --> 01:11:25,540
I should say it doesn't work
if the spikes are really tall

1232
01:11:25,540 --> 01:11:28,804
and the base is really flat.

1233
01:11:28,804 --> 01:11:30,470
I'm going to define
what "really" means,

1234
01:11:30,470 --> 01:11:32,130
but we'll get there.

1235
01:11:36,790 --> 01:11:38,890
So here's our witch's hat.

1236
01:11:43,160 --> 01:11:47,430
When this spike is really
tall, these angles, alpha,

1237
01:11:47,430 --> 01:11:50,600
are very close to 90,
a little bit under 90.

1238
01:11:50,600 --> 01:11:54,880
This angle in the floor here--
well, this angle in the floor

1239
01:11:54,880 --> 01:11:57,850
is about 60, probably
actually is exactly 60.

1240
01:11:57,850 --> 01:11:59,900
This is an equilateral triangle.

1241
01:11:59,900 --> 01:12:04,720
So if this thing is very flat,
this angle will be almost 300

1242
01:12:04,720 --> 01:12:07,250
because it's 360 minus 60.

1243
01:12:07,250 --> 01:12:15,360
It'll be a little bit less
than 300 but almost 300.

1244
01:12:15,360 --> 01:12:17,950
These matter.

1245
01:12:17,950 --> 01:12:21,060
In particular, the
total sum of angles

1246
01:12:21,060 --> 01:12:26,210
here is 300 plus twice
90, which is big.

1247
01:12:26,210 --> 01:12:28,170
It's bigger than 360.

1248
01:12:28,170 --> 01:12:28,960
That's the point.

1249
01:12:28,960 --> 01:12:33,420
In fact, 300 plus one
of these angles, 90,

1250
01:12:33,420 --> 01:12:38,260
would be almost 390,
which is way above 360.

1251
01:12:38,260 --> 01:12:40,991
So this has negative curvature,
and even if you cut out

1252
01:12:40,991 --> 01:12:42,740
one of the spike
triangles, it would still

1253
01:12:42,740 --> 01:12:44,480
have negative curvature.

1254
01:12:44,480 --> 01:12:49,032
That's going to be bad
news because let's just

1255
01:12:49,032 --> 01:12:49,990
imagine for the moment.

1256
01:12:49,990 --> 01:12:51,605
We know unfolding
things with boundary

1257
01:12:51,605 --> 01:12:54,960
is hard, but let's pretend
for now that you wanted

1258
01:12:54,960 --> 01:12:57,340
to unfold a hat in isolation.

1259
01:12:57,340 --> 01:13:01,610
You wanted to unfold it into
one piece without overlap.

1260
01:13:01,610 --> 01:13:04,480
Think about what you could do.

1261
01:13:04,480 --> 01:13:09,110
I have to cut with
a spanning forest,

1262
01:13:09,110 --> 01:13:12,260
I guess, meaning
it's acyclic, it's

1263
01:13:12,260 --> 01:13:14,030
got to hit all the vertices.

1264
01:13:14,030 --> 01:13:16,410
There's only four vertices here.

1265
01:13:16,410 --> 01:13:19,500
How could I hit all the
vertices with a forest?

1266
01:13:19,500 --> 01:13:22,420
You may recall from way back
when that trees have leaves.

1267
01:13:22,420 --> 01:13:25,460
Every tree has at
least two leaves.

1268
01:13:25,460 --> 01:13:28,337
Here I might have multiple
trees, maybe, but unlikely,

1269
01:13:28,337 --> 01:13:29,920
and I have to have
at least to leaves.

1270
01:13:29,920 --> 01:13:32,150
Where could those two leaves be?

1271
01:13:32,150 --> 01:13:35,500
Could they be at any leaves
or vertices at degree one?

1272
01:13:35,500 --> 01:13:38,472
Could they be at any of
these three vertices?

1273
01:13:38,472 --> 01:13:40,180
No, because they have
negative curvature.

1274
01:13:40,180 --> 01:13:41,905
We know a vertex with
negative curvature

1275
01:13:41,905 --> 01:13:43,780
has to have at least
two cuts incident to it.

1276
01:13:43,780 --> 01:13:46,960
You can't stop at
these three vertices.

1277
01:13:46,960 --> 01:13:52,030
That only leaves one
vertex and the boundary.

1278
01:13:52,030 --> 01:13:53,510
So the only thing
you can do if you

1279
01:13:53,510 --> 01:13:56,050
want to visit all the
vertices and start somewhere

1280
01:13:56,050 --> 01:13:58,640
on the boundary and get
to the x, get to the peak.

1281
01:14:05,280 --> 01:14:06,350
That's all you can do.

1282
01:14:06,350 --> 01:14:08,600
You couldn't have multiple
connections to the boundary

1283
01:14:08,600 --> 01:14:10,780
because then you
would disconnect.

1284
01:14:10,780 --> 01:14:12,340
Then there's really
only two choices

1285
01:14:12,340 --> 01:14:16,480
and they're
reflectionally symmetric.

1286
01:14:16,480 --> 01:14:18,310
And we're only allowed
to go along edges.

1287
01:14:18,310 --> 01:14:19,760
It's super constrained.

1288
01:14:19,760 --> 01:14:21,870
You can go here, walk
around, and go up,

1289
01:14:21,870 --> 01:14:24,309
or you can walk around
the other way and go up.

1290
01:14:24,309 --> 01:14:26,350
Those are actually slightly
different because you

1291
01:14:26,350 --> 01:14:29,280
have two choices of which
edge to follow here.

1292
01:14:29,280 --> 01:14:33,240
They're both screwed
because if you

1293
01:14:33,240 --> 01:14:36,540
look at this point, the
white dot, or the white dot

1294
01:14:36,540 --> 01:14:40,610
up there, what remains
here on the outside

1295
01:14:40,610 --> 01:14:44,230
is almost 300 degrees
of material on the base

1296
01:14:44,230 --> 01:14:46,840
plus one of the 90 degree
faces, the back one

1297
01:14:46,840 --> 01:14:47,910
that you can't see.

1298
01:14:47,910 --> 01:14:49,320
It's easier to see here.

1299
01:14:49,320 --> 01:14:51,200
You have the 300
degrees on the bottom

1300
01:14:51,200 --> 01:14:54,630
and then the white face
is still attached to it,

1301
01:14:54,630 --> 01:14:57,415
and that's almost 390.

1302
01:14:57,415 --> 01:14:58,790
When you flatten
that thing, it's

1303
01:14:58,790 --> 01:15:01,650
going to overlap
itself at that point.

1304
01:15:01,650 --> 01:15:03,020
Bad news.

1305
01:15:03,020 --> 01:15:05,810
So this just says, if I
look at a hat in isolation,

1306
01:15:05,810 --> 01:15:08,510
it can't unfold, but that's
not what I care about.

1307
01:15:08,510 --> 01:15:10,720
I care about four of
them joined together.

1308
01:15:17,050 --> 01:15:19,550
Suppose you had some
unfolding of the whole thing,

1309
01:15:19,550 --> 01:15:21,290
and then I look
at, well, what cuts

1310
01:15:21,290 --> 01:15:27,090
happen within the witch's hat?

1311
01:15:27,090 --> 01:15:29,364
I know the witch's hat
cannot remain in one piece.

1312
01:15:29,364 --> 01:15:30,780
Therefore, it must
be disconnected

1313
01:15:30,780 --> 01:15:35,100
into multiple pieces,
something like this.

1314
01:15:35,100 --> 01:15:45,050
Again, the cuts have to visit
all the vertices somehow,

1315
01:15:45,050 --> 01:15:48,320
but we know from the
perspective of a single hat,

1316
01:15:48,320 --> 01:15:51,590
that hat must split
into two parts.

1317
01:15:51,590 --> 01:15:54,430
There are lots of things
you could consider.

1318
01:15:54,430 --> 01:15:57,300
Let's suppose this is
possible, not even worry

1319
01:15:57,300 --> 01:15:59,540
about these kinds of pictures.

1320
01:15:59,540 --> 01:16:02,255
Well, I claim we have a problem.

1321
01:16:27,521 --> 01:16:28,395
Here's a tetrahedron.

1322
01:16:31,410 --> 01:16:34,394
These vertices are all
the same point, actually.

1323
01:16:34,394 --> 01:16:35,810
I've just unfolded
the tetrahedron

1324
01:16:35,810 --> 01:16:37,893
because it's way easier
to draw in two dimensions.

1325
01:16:41,720 --> 01:16:46,670
So if I look at the hat
that is on this triangle,

1326
01:16:46,670 --> 01:16:49,070
this hat gets disconnected
into two parts.

1327
01:16:49,070 --> 01:16:51,090
There's only three
connection points

1328
01:16:51,090 --> 01:16:52,300
to the rest of the world.

1329
01:16:52,300 --> 01:16:54,890
So what these pictures
have to look like is they

1330
01:16:54,890 --> 01:16:59,266
connect two of the vertices
by a collection of cuts.

1331
01:16:59,266 --> 01:17:00,890
If you're going to
cut into two halves,

1332
01:17:00,890 --> 01:17:03,380
you've got to have
a path across,

1333
01:17:03,380 --> 01:17:08,150
and there's only three
vertices to visit.

1334
01:17:08,150 --> 01:17:10,026
There's some collection
of cuts that go from,

1335
01:17:10,026 --> 01:17:11,608
let's say, this
vertex to this vertex.

1336
01:17:11,608 --> 01:17:13,440
At this point,
everything's symmetric.

1337
01:17:13,440 --> 01:17:16,689
Maybe it would be more obvious
if I started the center.

1338
01:17:16,689 --> 01:17:18,230
It could be from
here to there, could

1339
01:17:18,230 --> 01:17:20,771
be from here to there or from
here to there, but at least one

1340
01:17:20,771 --> 01:17:23,880
of those things exist,
and by rotational symmetry

1341
01:17:23,880 --> 01:17:27,990
of this diagram,
say it's that one.

1342
01:17:27,990 --> 01:17:30,540
Well, what happens to this face?

1343
01:17:30,540 --> 01:17:32,780
Could there be
something like this?

1344
01:17:32,780 --> 01:17:36,170
No, because then this
would be disconnected

1345
01:17:36,170 --> 01:17:38,480
from the rest of the world.

1346
01:17:38,480 --> 01:17:41,720
So suddenly, this
hat is constrained.

1347
01:17:41,720 --> 01:17:43,890
Maybe it could look like this.

1348
01:17:43,890 --> 01:17:44,640
You have a choice.

1349
01:17:44,640 --> 01:17:46,392
It could look like
that or like that,

1350
01:17:46,392 --> 01:17:48,100
but by reflectional
symmetry, same thing.

1351
01:17:48,100 --> 01:17:49,740
So let's say this one.

1352
01:17:49,740 --> 01:17:53,100
Remember, x is the
same everywhere.

1353
01:17:53,100 --> 01:17:56,280
Well, that means
this is impossible

1354
01:17:56,280 --> 01:17:59,420
because this edge is
actually glued to this edge,

1355
01:17:59,420 --> 01:18:01,440
and so then this thing
would be disconnected

1356
01:18:01,440 --> 01:18:03,220
from the rest of the world.

1357
01:18:03,220 --> 01:18:05,280
So there's a couple
of possibilities.

1358
01:18:05,280 --> 01:18:08,880
It could look like this,
or it could look like this.

1359
01:18:12,417 --> 01:18:14,000
It was one of those
two for this face.

1360
01:18:14,000 --> 01:18:15,000
We've got one face left.

1361
01:18:18,120 --> 01:18:23,190
This is impossible because
this edge glues to this one,

1362
01:18:23,190 --> 01:18:26,070
so imagine this thing being
picked up and moved over here.

1363
01:18:26,070 --> 01:18:28,615
So we have a wiggly line
here and then this stuff,

1364
01:18:28,615 --> 01:18:29,990
and so that would
be disconnected

1365
01:18:29,990 --> 01:18:33,660
from the rest of the world,
so that can't happen.

1366
01:18:33,660 --> 01:18:36,660
What about this one?

1367
01:18:36,660 --> 01:18:38,340
Is that the harder one?

1368
01:18:38,340 --> 01:18:41,322
I don't know.

1369
01:18:41,322 --> 01:18:43,280
It's been a while since
I've used the argument.

1370
01:18:43,280 --> 01:18:45,200
If we have this
together with this,

1371
01:18:45,200 --> 01:18:48,010
that's clearly bad because
that forms a cycle.

1372
01:18:48,010 --> 01:18:50,290
No good.

1373
01:18:50,290 --> 01:18:52,430
But what if I have this
together with this one?

1374
01:18:57,377 --> 01:18:58,210
It gets hard to see.

1375
01:19:02,130 --> 01:19:03,110
It's bad.

1376
01:19:03,110 --> 01:19:03,990
It's a cycle.

1377
01:19:03,990 --> 01:19:08,370
It starts and ends at x,
and if you fold it up right,

1378
01:19:08,370 --> 01:19:10,470
you'll see there's really
two sides to that cycle.

1379
01:19:10,470 --> 01:19:13,550
It's actually forced,
so that's bad.

1380
01:19:13,550 --> 01:19:19,380
All right, one more
choice, this one.

1381
01:19:19,380 --> 01:19:22,394
If I do this together
with this one,

1382
01:19:22,394 --> 01:19:24,310
that's going to be bad
because that's a cycle.

1383
01:19:24,310 --> 01:19:27,360
We start and end at x.

1384
01:19:27,360 --> 01:19:29,930
What if I do this
one and that one?

1385
01:19:29,930 --> 01:19:31,430
Well, that's also
bad because here's

1386
01:19:31,430 --> 01:19:33,880
a cycle that starts
at x, ends at x.

1387
01:19:33,880 --> 01:19:36,540
This and this form the
inner of the cycle.

1388
01:19:36,540 --> 01:19:42,595
All cases are bad,
so no edge unfolding.

1389
01:19:46,960 --> 01:19:49,910
Tragic.

1390
01:19:49,910 --> 01:19:52,680
We can think briefly about
the case-- do I have time?

1391
01:19:52,680 --> 01:19:54,720
I have 10 seconds.

1392
01:19:54,720 --> 01:19:56,960
About the case where
there are two triangles.

1393
01:19:56,960 --> 01:20:01,960
I guess I should
draw them like this.

1394
01:20:01,960 --> 01:20:05,276
So this point is the
same as this point.

1395
01:20:05,276 --> 01:20:07,650
So if I try to simplify this,
instead of using four hats,

1396
01:20:07,650 --> 01:20:10,220
I just use two hats,
put one here, one here.

1397
01:20:10,220 --> 01:20:16,060
Then I could do something
like this, I think,

1398
01:20:16,060 --> 01:20:20,530
and maybe that's OK if
there's no cycle formed there.

1399
01:20:20,530 --> 01:20:23,240
So we really needed the
tetrahedron somehow.

1400
01:20:23,240 --> 01:20:26,080
I think it does work for
octahedron and larger also,

1401
01:20:26,080 --> 01:20:29,610
but just two triangles
is not enough.

1402
01:20:29,610 --> 01:20:32,875
And that is unfolding.

1403
01:20:32,875 --> 01:20:37,080
We did not always edge foldable
for general polyhedra, even

1404
01:20:37,080 --> 01:20:39,127
topologically convex polyhedra.

1405
01:20:39,127 --> 01:20:41,210
Next time, we'll talk more
about general unfolding

1406
01:20:41,210 --> 01:20:43,440
of arbitrary polyhedra.