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PROFESSOR: All right. lecture
14 was about two main topics,

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I guess.

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We had slender adorned chains,
the sort of fatter linkages,

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and then hinged dissection.

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Most of our time was actually
spent with the slender

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adornments and proving
that that works.

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But most of our questions today
are about hinged dissections

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because that's kind
of the most fun,

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and there's a lot more
to say about them.

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So first question is,
is there any software

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for hinged dissections?

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And the short answer
is no, surprisingly.

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So this would definitely be
a cool project possibility.

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There are a bunch
of examples-- let

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me switch to these on the web--
just sort of random examples,

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cool dissections people
thought were so neat

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that they wanted
to animate them.

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And so they
basically constructed

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where the coordinates were over
time in Mathematica, and then

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put it on the web as a
illustration of that.

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So this is a
equilateral triangle

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to a hexagon-- a
regular hexagon.

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It's a hinged dissection
by Greg Frederickson,

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and then is drawn
by have Rick Mabry.

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Here's another one for
an equilateral triangle

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to a pentagon.

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Pretty cool.

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And they're hinged in
a tree-like fashion

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even, which is kind of unusual.

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Greg Frederickson is one of the
masters of hinged dissections,

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and dissections in general.

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He's probably the master
of dissections in general.

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And he has three books of
different kinds of dissections.

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This is actually hinged the
section on the cover here.

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The purple and pink
pieces hinge like this

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into a smaller star from
the outline of a big star

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to the interior
of a smaller star.

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And then this star
fits nicely inside.

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This is another
hinged dissection.

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This book is entirely
about hinged dissections,

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although not just
the kinds we've seen.

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Another kind called
twist hinging,

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which I think this
is a twist hinge.

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The piece flips
around the other side.

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And then there's the third
book about a different kind

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of hinged dissection that's more
of a surface hinged dissection

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where you've got two--
you've got the front

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and back of this surface
and you fold them

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with like piano hinges
with hinges in the plane.

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All are very cool books.

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You should check
them out if you want

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to know more about dissections.

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They're more about here are
cool examples, some design

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techniques for how to make them.

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I'll show you one such design
technique later on today.

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but not a ton of theory here, in
particular because there wasn't

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a ton of theory when
these books were written.

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So that's some
hinged dissections.

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And as I said,
cool project would

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be to make a general tool for
animating hinged dissections.

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There's only a
handful out there.

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Greg has digital files of lots
of his hinged dissections.

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He'd probably be
willing to share them,

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though I haven't
talked to him about it.

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If there was a good
engine for animating them

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I think it would be cool.

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Even cooler would
be to implement

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the slender adorn
chain business.

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Take one of these hinged
dissections, maybe they just

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hinge but sometimes
there's collisions.

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But we already
know if you refine

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these guys to be
slender, which you

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can do-- if they are
triangulated you can do it

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with only losing a factor of
three in the number of pieces.

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It would be cool
to implement that.

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And then you can do the
slender adorned folding

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via CDR, which I have
an implementation of,

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or it's not that hard to build
one if you have a LP solver.

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So various project
possibilities.

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Another cool project
would be to just design

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more hinged dissections.

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There's still
interesting questions.

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Either use fewer pieces or
just make elegant designs.

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Related to the implementation
idea, a particular family

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of hinged dissections that
could be fun to implement

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are embodied by this alphabet.

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I showed this in lecture.

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You can take the letter six
and convert it into a square,

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and convert it into an eight,
and convert it into a four,

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and convert it into a
nine via these 128 pieces.

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I didn't talk much about
this theorem though,

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so I thought I'd give you
a little sketch of how

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this works.

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It's actually very simple to
construct the folded states

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of these hinged
dissections, and it

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could be an interesting
thing to implement.

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And it's also just kind of fun.

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This is way earlier,
1999, way before

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we knew that everything
was possible.

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We could at least do all
polyominoes of a given size.

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So let's just think about
polyominoes, about polyhexes,

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polyiamonds, where you
have equilateral triangles.

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And these are called
polyabolos for silly reasons,

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basically, by analogy
to a diablo, which

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is a juggling device.

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You can hinge dissect
any of them here.

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You take each square and you
cut it into two half-squares,

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and then you hinge them
together like this.

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This Is 1, 2, 3, 4, 5, 6, 7, 8.

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So this will make any
four-square object,

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any tetris piece.

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And generally, you
take two end pieces

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and you can make any anomina.

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And the way you prove
that that is universal,

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that it can fold into
anything-- it's not

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so clear from this picture,
but it's actually really easy

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to prove by induction.

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So the first thing to do
in this inductive proof

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is to check that you can
do it for n equals 1.

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OK.

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That may sound trivial
but this is actually core.

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The key property you need
in a hinged dissection

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of a single square into
your general family

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is that there's a hinge visible
on every edge of the object.

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So here, this hinge kind
of covers this edge,

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it covers that edge.

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So both of these edges
have hinges on them,

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and the other two edges
have a hinge on them.

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They happen to be shared
hinges but that's OK.

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And each of these, that's true.

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The triangle, it's a
little more awkward.

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You actually need two hinges
to cover the three sides.

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But you only need
these two pieces.

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One of them is non-convex.

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It may be hard to
fold continuously

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but you'd refine
it if you wanted

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to do slender adornments.

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So let's not worry about
continuous motion yet.

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So that's the base
case of the induction.

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How do I do it for n equals 1?

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Now inductively, if
I have some shape

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I want to build I'll
take what I call

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the dual graph of that shape.

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So make a vertex
for every square,

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connect them together
if they share

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an edge-- the squares
share an edge--

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and then look at a spanning
tree of that shape.

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So just cut some of
these edges until you

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have tree connectivity
among those squares.

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Every tree has at least two
leaves, except in the fall,

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but every mathematical tree
has at least two leaves.

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Like this is a leaf, if I cut
here this would also be a leaf.

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Leaf is a degree one vertex.

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So that's a square that
only shares one side.

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So pluck off that leaf,
remove that square.

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The resulting n minus 1
square is, by assumption,

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can be made by this hinged
dissection with two times

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n minus 1 pieces.

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So now we just have
to attach this guy on.

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And here's a figure for
that down at the bottom.

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This is the same
thing for triangles,

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and polyabolos are
in the upper right.

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So you have some existing
hinged dissection,

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you don't really
know what it's like,

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and you want to add
this leaf back on so it

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shares one edge with one guy.

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Now this guy could
be oriented this way,

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or it could be
oriented this way,

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but it's the same by reflection.

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So let's say it's
oriented this way.

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We know the square is made up by
two half-squares, by induction,

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and so we know that
there's a hinge here.

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Now this hinge connects to
some things, in this case,

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to some t prime.

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It could be here,
it could be up here.

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And all we do is
stick s on here.

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Now s can rotate.

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We have our solution
for one guy,

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and there's two different
orientations for him.

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We're going to choose
this orientation

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because it puts this
hinge right there.

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And so once we do that,
normally this would be a cycle,

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and this thing would be
a cycle through here,

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but we just redo
the hinges in here

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so that the cycle gets bigger.

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And the important
thing to verify

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is that the orientations
to the triangles

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are the same, just like the
hinged dissection picture

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I showed.

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We always go from the base
edge to the next base edge,

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to the next base edge of these
right isosceles triangles,

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and all the triangles are
on the outside of the cycle.

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So we actually construct a
cyclic hinged dissection.

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Then at the end you could
break it and make it a path.

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And this one is even slender.

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Remember, right triangles
are slender, barely.

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You can look at all
the inward normals.

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They hit the base edge.

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So this will even
move continuously

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if it's an open chain.

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For closed chains we don't know.

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So that's polyominoes.

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Polyaimonds are similar.

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Pretty much the same thing.

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You just-- in this
case, you might

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have hinges on both sides, but
you rotate this thing so one

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of the hinges lines up, and
you just reconnect the hinges.

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And it's not hard to
show you can always

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do that, the hinges
will never cross.

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And this proves that
these folded states exist,

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and then we use the slender
stuff to do continuous motions.

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Actually, when this
paper was written

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we didn't have slender
adornments back in '99,

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even in 2005 when the
journal version appeared.

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So it's only now that we know
that motions are possible by,

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in this case directly, in this
case with some refinement.

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So I thought that would
just be fun to see.

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You can do some
other crazy things.

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So this is a hinged dissection
from any four-iamond.

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So this is four equilateral
triangles joined together

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to any four amino.

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This is a tetris piece.

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It's essentially a
superposition of this idea

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with-- you see in here,
these four lines make

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the hinged dissection
of Dudeney,

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from 1902, from an equilateral
triangle to a square.

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And with some extra
stuff added in-- this

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is maybe a foreshadowing
of the idea of refinement,

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although we didn't really
realize it at the time.

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We want to add some
hinges so that we

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have hinges on the midpoints
of the edges instead

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of the corners.

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That turns out to be a bit
more efficient in this case.

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So we add some hinging,
still hingeable individually,

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but now we have
hinges at the corners

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and so-- at the
midpoints, and we'll

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have the same
property over here.

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And it allows you to
hinge these together.

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Actually, here it looks like
some of them are at the corners

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not the midpoints.

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So it's a bit messy.

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In general, we can prove
if you have any shape

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and you want to make
poly that shape-- so

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let's call this
shape x, you want

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to make polyexes-- you can do it
as long as the copies of the x

246
00:10:49,025 --> 00:10:54,310
are only rotated and they're
joined at corresponding edges.

247
00:10:54,310 --> 00:10:57,350
So if you check, this guy's
just been rotated 180 degrees.

248
00:10:57,350 --> 00:10:59,290
In general you can join
these things together

249
00:10:59,290 --> 00:11:01,980
at matching edges.

250
00:11:01,980 --> 00:11:04,237
And the basic technique
is just subdivide

251
00:11:04,237 --> 00:11:06,820
the thing, triangulate, draw in
the dual of the triangulation,

252
00:11:06,820 --> 00:11:09,610
and then connect to the
midpoints of the edges.

253
00:11:09,610 --> 00:11:11,960
And you can show,
basically, instead

254
00:11:11,960 --> 00:11:13,530
of the hinged
dissection going around

255
00:11:13,530 --> 00:11:16,090
like this you can just
make it go around like this

256
00:11:16,090 --> 00:11:17,460
and come back this way.

257
00:11:17,460 --> 00:11:19,850
And if you check the sequence
of pieces they could visit,

258
00:11:19,850 --> 00:11:22,250
it's identical if you
go around this way

259
00:11:22,250 --> 00:11:24,360
or if you go around this way.

260
00:11:24,360 --> 00:11:30,484
And that's enough to show
that any folded state is valid

261
00:11:30,484 --> 00:11:31,900
With the triangles
and the squares

262
00:11:31,900 --> 00:11:34,316
we're essentially exploiting
the symmetry of these pieces.

263
00:11:34,316 --> 00:11:36,507
So you can rotate them
to make them compatible.

264
00:11:36,507 --> 00:11:38,840
Here they're forced to be
compatible by assuming we only

265
00:11:38,840 --> 00:11:42,360
join matching edges.

266
00:11:42,360 --> 00:11:46,082
So that was the
2D polyform paper.

267
00:11:46,082 --> 00:11:49,680
You can see Frederickson
was one of the authors.

268
00:11:49,680 --> 00:11:54,159
In 3D, here's easy way
to generalize that.

269
00:11:54,159 --> 00:11:55,825
If you take, for
example, a tetrahedron,

270
00:11:55,825 --> 00:12:01,120
a regular tetrahedron, you take
the centroid and cut everything

271
00:12:01,120 --> 00:12:01,999
to the centroid.

272
00:12:01,999 --> 00:12:03,665
And you end up cutting
your tetrahedron,

273
00:12:03,665 --> 00:12:10,050
it has four sides, into four of
these more slender tetrahedra.

274
00:12:10,050 --> 00:12:12,770
And then you take four of
them and join them together

275
00:12:12,770 --> 00:12:13,556
in this way.

276
00:12:13,556 --> 00:12:15,055
You do have to be
careful in the way

277
00:12:15,055 --> 00:12:16,471
that you join them
because, again,

278
00:12:16,471 --> 00:12:20,510
on every face we want to
have an incident hinge.

279
00:12:20,510 --> 00:12:22,700
So we've got to
take care in the way

280
00:12:22,700 --> 00:12:25,720
that you hinge together to make
sure that that is the case.

281
00:12:25,720 --> 00:12:27,110
But it's also cyclically hinged.

282
00:12:27,110 --> 00:12:28,490
This gets joined to that.

283
00:12:28,490 --> 00:12:31,100
And basically, the same
inductive proof works.

284
00:12:31,100 --> 00:12:35,370
You just pluck off a leaf, show
that you can turn the thing so

285
00:12:35,370 --> 00:12:37,980
that one of the hinges
aligns with the inductive

286
00:12:37,980 --> 00:12:40,460
construction, and then
just join the hinges

287
00:12:40,460 --> 00:12:43,830
across instead of
within the cycles.

288
00:12:43,830 --> 00:12:44,920
So pretty easy.

289
00:12:47,860 --> 00:12:49,360
What are we talking about?

290
00:12:49,360 --> 00:12:51,120
Hinged dissections
software, I guess.

291
00:12:51,120 --> 00:12:52,960
Those would be fun
things to implement.

292
00:12:52,960 --> 00:12:55,070
They've never been
implemented, and especially,

293
00:12:55,070 --> 00:12:55,904
to see them folding.

294
00:12:55,904 --> 00:12:57,361
I thought I'd show
you a little bit

295
00:12:57,361 --> 00:12:59,100
about hinged
dissection hardware,

296
00:12:59,100 --> 00:13:01,780
different ways you could
make them physically real.

297
00:13:01,780 --> 00:13:04,610
This is kind of
mesoscale I'd call it.

298
00:13:04,610 --> 00:13:08,230
This is at one centimeter
bar, so not super tiny,

299
00:13:08,230 --> 00:13:10,410
but I think this could
scale down quite a bit.

300
00:13:10,410 --> 00:13:13,495
We have a Petri dish here
with some liquid in it,

301
00:13:13,495 --> 00:13:15,460
if you could read up there.

302
00:13:15,460 --> 00:13:17,510
Maybe this is the
coolest example.

303
00:13:17,510 --> 00:13:20,780
We have a square made
up of four pieces,

304
00:13:20,780 --> 00:13:23,780
and you add a little bit
of salt to that liquid

305
00:13:23,780 --> 00:13:27,840
and it pops into the equilateral
triangle configuration.

306
00:13:27,840 --> 00:13:31,470
So it's sort of spontaneously
folding, hinging.

307
00:13:31,470 --> 00:13:33,890
Essentially these pieces
are slanted a little bit

308
00:13:33,890 --> 00:13:37,260
and they prefer-- one weighting
causes them to fold one way

309
00:13:37,260 --> 00:13:39,900
but when you add the salt they
end up flopping the other way.

310
00:13:39,900 --> 00:13:42,460
You could see they're a little
bit inexact because of that,

311
00:13:42,460 --> 00:13:45,490
but pretty awesome the
kinds of hinged dissections.

312
00:13:45,490 --> 00:13:48,470
You can get them all to
actuate even without much room

313
00:13:48,470 --> 00:13:50,520
to do so.

314
00:13:50,520 --> 00:13:57,387
This is done at Harvard, George
Whiteside's group, chemistry.

315
00:13:57,387 --> 00:13:59,595
Kind of related, it's not
exactly hinged dissections,

316
00:13:59,595 --> 00:14:01,700
but I feel like it's
the same spirit,

317
00:14:01,700 --> 00:14:04,910
is this idea of DNA
origami, it's called,

318
00:14:04,910 --> 00:14:07,740
where you take one
big strand of DNA

319
00:14:07,740 --> 00:14:10,600
and you force it to fold
into a particular shape.

320
00:14:10,600 --> 00:14:13,340
Here we're folding
it into a happy face.

321
00:14:13,340 --> 00:14:15,310
The way that's done
is you add in a bunch

322
00:14:15,310 --> 00:14:17,140
of little pieces of DNA.

323
00:14:17,140 --> 00:14:20,200
So this string, basically,
has a-- this DNA strand

324
00:14:20,200 --> 00:14:22,780
has a random string
written on it basically,

325
00:14:22,780 --> 00:14:25,850
and you identify, oh, I want
these guys to glue together.

326
00:14:25,850 --> 00:14:27,890
So you take this piece
of the random string,

327
00:14:27,890 --> 00:14:28,690
and this piece of
the random string,

328
00:14:28,690 --> 00:14:31,023
and you construct a piece of
DNA that has both of those,

329
00:14:31,023 --> 00:14:33,600
like a little zipper to
cause those to zip up.

330
00:14:33,600 --> 00:14:34,940
You do that all over the place.

331
00:14:34,940 --> 00:14:36,550
And there's now automatic
tools to do this,

332
00:14:36,550 --> 00:14:38,091
it's really easy to
make DNA origami.

333
00:14:38,091 --> 00:14:39,962
It, basically, always works.

334
00:14:39,962 --> 00:14:41,670
There's a limit to
how big this thing can

335
00:14:41,670 --> 00:14:46,060
be because the main strand
here is a single piece of DNA,

336
00:14:46,060 --> 00:14:49,350
and those are hard to make
super big, at least currently.

337
00:14:49,350 --> 00:14:51,440
But you get some really
nice happy faces and mass

338
00:14:51,440 --> 00:14:52,620
produce them.

339
00:14:52,620 --> 00:14:54,332
Hundred-nanometer scale.

340
00:14:54,332 --> 00:14:55,790
It's kind of like
hinged dissection

341
00:14:55,790 --> 00:14:58,780
because that strand
of DNA is moving,

342
00:14:58,780 --> 00:15:00,440
it's actually more
like a fixed angle

343
00:15:00,440 --> 00:15:03,060
chain, kind of like
a hinged dissection.

344
00:15:03,060 --> 00:15:05,760
And we're essentially
using here universality

345
00:15:05,760 --> 00:15:07,210
of hinged dissections
of something

346
00:15:07,210 --> 00:15:09,400
like polyominoes,
though the shapes

347
00:15:09,400 --> 00:15:10,800
are a little bit more awkward.

348
00:15:10,800 --> 00:15:12,410
And they've made a
maps of the world.

349
00:15:12,410 --> 00:15:16,480
You could do two-color patterns,
make snowflakes, the word DNA,

350
00:15:16,480 --> 00:15:19,750
and crazy stuff.

351
00:15:19,750 --> 00:15:22,190
So it was started by Paul
Rothman, though a lot of people

352
00:15:22,190 --> 00:15:23,820
do DNA origami these days.

353
00:15:26,810 --> 00:15:29,350
Cool.

354
00:15:29,350 --> 00:15:34,510
Next paper I wanted to show
you-- this is fairly recent--

355
00:15:34,510 --> 00:15:38,890
and it's about getting
continuous motions,

356
00:15:38,890 --> 00:15:42,000
in particular, in 3D, of
hinged dissection-like things.

357
00:15:42,000 --> 00:15:44,510
So here we have
a chain of balls.

358
00:15:44,510 --> 00:15:46,387
These are more like
ball and socket joints.

359
00:15:46,387 --> 00:15:47,970
So you can maybe see
them better here.

360
00:15:47,970 --> 00:15:51,830
There's a member going
in from the green guy

361
00:15:51,830 --> 00:15:55,100
into the center of the red
guy, and there's a slot,

362
00:15:55,100 --> 00:15:58,030
and the red guy can fold
around the-- or the blue guy

363
00:15:58,030 --> 00:16:00,420
can fold around the red guy.

364
00:16:00,420 --> 00:16:03,366
And the question is,
OK, this is great.

365
00:16:03,366 --> 00:16:05,490
You can prove universality,
you can make any shape.

366
00:16:05,490 --> 00:16:08,250
You just subdivide your
dog, or whatever, into two

367
00:16:08,250 --> 00:16:10,149
by two by two
squarelets and then

368
00:16:10,149 --> 00:16:11,690
we know how to
connect those together

369
00:16:11,690 --> 00:16:14,940
to make a nice Hamiltonian
cycle that visits everything.

370
00:16:14,940 --> 00:16:18,150
But can you actually fold
a chain of balls like this

371
00:16:18,150 --> 00:16:20,530
into that dog?

372
00:16:20,530 --> 00:16:22,830
And the answer is always yes.

373
00:16:22,830 --> 00:16:28,310
Essentially, you feed a
big string of these balls

374
00:16:28,310 --> 00:16:30,990
into-- that's actually what's
happening in this animation

375
00:16:30,990 --> 00:16:33,520
here, although it's a little
hard to tell-- you're feeding

376
00:16:33,520 --> 00:16:37,060
in, say, at one of the legs,
one of the extreme points

377
00:16:37,060 --> 00:16:40,170
in some direction,
this chain of balls.

378
00:16:40,170 --> 00:16:43,619
And as they go in they just
start tracking along the path.

379
00:16:43,619 --> 00:16:46,160
And you just need to check that
you can track along the path.

380
00:16:46,160 --> 00:16:48,410
As this guy goes into
a corner, for example,

381
00:16:48,410 --> 00:16:51,241
you can actually navigate the
corner while, at all times,

382
00:16:51,241 --> 00:16:52,240
staying within the tube.

383
00:16:52,240 --> 00:16:53,890
If you can stay
within the tube you

384
00:16:53,890 --> 00:16:55,931
know you won't collide
with the rest of the chain

385
00:16:55,931 --> 00:16:58,740
because this tube is
non self-intersecting.

386
00:16:58,740 --> 00:17:00,980
And so the 2D version
is fairly easy.

387
00:17:00,980 --> 00:17:03,607
This is just circles.

388
00:17:03,607 --> 00:17:05,940
A little trickier to check
that it actually is possible,

389
00:17:05,940 --> 00:17:09,422
with just one turn, with a
U-turn, and with a kind of-- I

390
00:17:09,422 --> 00:17:11,470
don't know what you call
this, not a U-turn--

391
00:17:11,470 --> 00:17:13,089
where you change
in two directions--

392
00:17:13,089 --> 00:17:14,609
two dimensions all at once.

393
00:17:14,609 --> 00:17:17,514
All of these are possible with
this particular mechanism,

394
00:17:17,514 --> 00:17:18,680
whatever mechanism you have.

395
00:17:18,680 --> 00:17:21,369
If it can do this then
you can make anything.

396
00:17:21,369 --> 00:17:23,880
So that's another
way to prove motions

397
00:17:23,880 --> 00:17:26,950
exist for this kind of
polyform special case.

398
00:17:26,950 --> 00:17:28,069
Why do we care about this?

399
00:17:28,069 --> 00:17:29,950
For building robots.

400
00:17:29,950 --> 00:17:32,160
So these are somewhat
different mechanisms,

401
00:17:32,160 --> 00:17:35,020
but I have two
examples built here

402
00:17:35,020 --> 00:17:36,950
at the MIT Center
for Bits and Atoms

403
00:17:36,950 --> 00:17:40,780
over in the Media Lab
with Neil Gershenfeld

404
00:17:40,780 --> 00:17:44,750
and many, many people.

405
00:17:44,750 --> 00:17:48,540
So you get some idea--
this is a fairly small guy.

406
00:17:48,540 --> 00:17:51,907
I mean, the actual
size is about this big.

407
00:17:51,907 --> 00:17:53,365
You see some feet
in the background

408
00:17:53,365 --> 00:17:56,380
to give you some sense of scale.

409
00:17:56,380 --> 00:17:59,696
It's not very many pieces, but
if you made a really long chain

410
00:17:59,696 --> 00:18:01,570
it would really be able
to fold into anything

411
00:18:01,570 --> 00:18:04,640
you want, just servos
to make the turns here.

412
00:18:04,640 --> 00:18:06,590
This is a much larger one.

413
00:18:06,590 --> 00:18:09,200
The right version is folding.

414
00:18:09,200 --> 00:18:11,280
And you get some
idea of scale here,

415
00:18:11,280 --> 00:18:15,460
this is, when it's
fully extended,

416
00:18:15,460 --> 00:18:21,560
144-- should that
be feet or inches?

417
00:18:21,560 --> 00:18:24,250
It's really big.

418
00:18:24,250 --> 00:18:26,770
So a little bit
slower, of course,

419
00:18:26,770 --> 00:18:28,240
because it has to
move a lot more,

420
00:18:28,240 --> 00:18:31,700
and it's also
quite a bit longer.

421
00:18:31,700 --> 00:18:35,970
This is built, in particular,
by Skylar Tibbits here.

422
00:18:35,970 --> 00:18:38,340
So that's the idea of robots.

423
00:18:38,340 --> 00:18:39,757
In general, we
like to make robots

424
00:18:39,757 --> 00:18:40,923
that can change their shape.

425
00:18:40,923 --> 00:18:42,730
We've seen sheet folding
robots, but these

426
00:18:42,730 --> 00:18:47,170
are more chain folding robots
inspired by proteins, and DNA,

427
00:18:47,170 --> 00:18:50,652
and things like that, sort of
big versions of DNA origami.

428
00:18:50,652 --> 00:18:52,110
What's cool about
them is that they

429
00:18:52,110 --> 00:18:54,040
stay connected
throughout the motion.

430
00:18:54,040 --> 00:18:57,750
You can keep your wiring, and
you can keep your batteries,

431
00:18:57,750 --> 00:19:00,930
and whatnot, and your
communication channels

432
00:19:00,930 --> 00:19:03,200
connected in this
kind of scenario.

433
00:19:03,200 --> 00:19:06,370
This is, by contrast, to
more common approaches

434
00:19:06,370 --> 00:19:08,210
to reconfigurable robots.

435
00:19:08,210 --> 00:19:10,422
You have individual
units and they can attach

436
00:19:10,422 --> 00:19:11,630
and detached from each other.

437
00:19:11,630 --> 00:19:13,937
You could see like these
guys picking up blocks,

438
00:19:13,937 --> 00:19:14,770
moving stuff around.

439
00:19:14,770 --> 00:19:17,649
It's definitely cool,
but in practice it's

440
00:19:17,649 --> 00:19:19,440
a lot harder to build
these kinds of robots

441
00:19:19,440 --> 00:19:21,752
because the attach
detach mechanism,

442
00:19:21,752 --> 00:19:23,460
it's hard to get them
to align perfectly,

443
00:19:23,460 --> 00:19:25,640
it's hard to get the
electrical connectivity.

444
00:19:25,640 --> 00:19:27,400
Every piece has to
have a battery instead

445
00:19:27,400 --> 00:19:29,740
of like every 10th
piece, or one battery

446
00:19:29,740 --> 00:19:33,450
to drive everything, or
tethering, or whatever.

447
00:19:33,450 --> 00:19:36,634
You can do some very cool things
and there's a lot of algorithms

448
00:19:36,634 --> 00:19:37,550
around for doing this.

449
00:19:37,550 --> 00:19:39,540
Daniella Rus, here at
MIT, built this robot,

450
00:19:39,540 --> 00:19:41,600
and a bunch of others.

451
00:19:41,600 --> 00:19:43,660
There's also a very
cool theory about these.

452
00:19:43,660 --> 00:19:44,760
I've worked on them.

453
00:19:44,760 --> 00:19:49,670
You can prove, for example,
that all of these models

454
00:19:49,670 --> 00:19:53,020
can simulate each other up
to constant factors in scale.

455
00:19:53,020 --> 00:19:56,190
So you can take your
favorite robot in a molecube

456
00:19:56,190 --> 00:19:59,469
and simulate a crystalline
robot, or vice versa.

457
00:19:59,469 --> 00:20:01,010
And then there's
efficient algorithms

458
00:20:01,010 --> 00:20:02,970
to-- these crystalline
robots, they

459
00:20:02,970 --> 00:20:07,500
can just expand and contract
and detection and attach.

460
00:20:07,500 --> 00:20:10,720
And you can prove that
given two configurations

461
00:20:10,720 --> 00:20:14,180
you can change it from one
to the other up to some scale

462
00:20:14,180 --> 00:20:14,770
factor.

463
00:20:14,770 --> 00:20:17,790
You can even do it
extremely fast in log n time

464
00:20:17,790 --> 00:20:20,079
if all the robots are
actuating all at once.

465
00:20:20,079 --> 00:20:22,370
Anyway, there's cool stuff
about reconfigurable robots,

466
00:20:22,370 --> 00:20:24,420
but the hinged dissections
offers an alternative

467
00:20:24,420 --> 00:20:28,540
where everything stays
connected at all times,

468
00:20:28,540 --> 00:20:30,890
but closely related.

469
00:20:30,890 --> 00:20:34,730
I think that was
the hardware story.

470
00:20:34,730 --> 00:20:38,720
So we go back to our proof
of hinged dissections

471
00:20:38,720 --> 00:20:42,060
and why it works.

472
00:20:42,060 --> 00:20:45,530
And one of the-- I was kind of
surprised I didn't show this

473
00:20:45,530 --> 00:20:50,640
in lecture, but I don't
remember why I didn't.

474
00:20:50,640 --> 00:20:52,200
One missing piece
with-- how do you

475
00:20:52,200 --> 00:20:55,180
go from a rectangle of one
size to a rectangle of another?

476
00:20:55,180 --> 00:21:02,030
You may recall,
we had a triangle,

477
00:21:02,030 --> 00:21:04,660
we triangulated our
polygons so we ended up

478
00:21:04,660 --> 00:21:06,380
with some arbitrary triangles.

479
00:21:06,380 --> 00:21:10,432
Then we cut parallel
to the base halfway up.

480
00:21:10,432 --> 00:21:12,390
You can put this over
here, put this over here,

481
00:21:12,390 --> 00:21:15,750
you get a rectangle of
some unknown height.

482
00:21:15,750 --> 00:21:19,200
And then to make it universal
we wanted to convert everything

483
00:21:19,200 --> 00:21:21,200
into a rectangle
of height epsilon

484
00:21:21,200 --> 00:21:24,280
so that then we could just
string them together--

485
00:21:24,280 --> 00:21:26,219
obviously, the area has
to be preserved here.

486
00:21:26,219 --> 00:21:28,510
If we string together all
the epsilon height rectangles

487
00:21:28,510 --> 00:21:33,500
we've got one super long
epsilon height rectangle.

488
00:21:33,500 --> 00:21:36,410
And then we overlay
the two dissections.

489
00:21:36,410 --> 00:21:37,870
This is how we did dissections.

490
00:21:37,870 --> 00:21:43,220
But how do you do this step
from one rectangle to another?

491
00:21:43,220 --> 00:21:47,560
This is a very old
dissection, at least 1778.

492
00:21:47,560 --> 00:21:49,980
It wasn't published
by Montucla but he's

493
00:21:49,980 --> 00:21:52,640
credited in this
publication, and this

494
00:21:52,640 --> 00:21:55,450
is Frederickson's diagram of it.

495
00:21:55,450 --> 00:21:59,330
So you take the fatter
rectangle and then

496
00:21:59,330 --> 00:22:02,750
you take the longer rectangle
and you-- first, you

497
00:22:02,750 --> 00:22:06,840
make multiple copies of the fat
rectangle, just sort of tile

498
00:22:06,840 --> 00:22:09,420
strip of the plane to the right.

499
00:22:09,420 --> 00:22:13,625
And then you angle the
thin rectangle, slightly.

500
00:22:13,625 --> 00:22:15,250
First of all, you
line up these corners

501
00:22:15,250 --> 00:22:17,300
so the top left corners
line up, and then we

502
00:22:17,300 --> 00:22:20,040
want the top right corner
of the thin rectangle

503
00:22:20,040 --> 00:22:23,550
to lie on this bottom line.

504
00:22:23,550 --> 00:22:25,300
Turns out this always works.

505
00:22:25,300 --> 00:22:28,430
It's not totally obvious but,
essentially, these copies

506
00:22:28,430 --> 00:22:31,474
of the rectangle you can
kind of fold them up.

507
00:22:31,474 --> 00:22:33,140
And when you go off
the right edge here,

508
00:22:33,140 --> 00:22:35,990
you're essentially coming
back on the left edge here.

509
00:22:35,990 --> 00:22:39,480
And then you're going this
way, and you're going this way,

510
00:22:39,480 --> 00:22:44,460
and this little piece is exactly
the same as this little piece.

511
00:22:44,460 --> 00:22:48,320
And from that you
get a dissection.

512
00:22:48,320 --> 00:22:54,050
It's not hinged, but you can
see that this big rectangle has

513
00:22:54,050 --> 00:22:56,420
the tiny piece here, which
conveniently fits right

514
00:22:56,420 --> 00:22:56,920
over there.

515
00:22:56,920 --> 00:23:00,820
It's like a wrap around
in the other direction.

516
00:23:00,820 --> 00:23:04,749
And then this piece-- well,
everything matches up here.

517
00:23:04,749 --> 00:23:06,540
The only other weird
thing is this bottom--

518
00:23:06,540 --> 00:23:11,390
when you go below the bottom
you also wrap around to the top.

519
00:23:11,390 --> 00:23:13,360
And just check all
the pieces match up,

520
00:23:13,360 --> 00:23:15,956
and you've got your dissection.

521
00:23:15,956 --> 00:23:16,850
It's kind of crazy.

522
00:23:16,850 --> 00:23:21,570
You have to check this works
for all parameters, but it does.

523
00:23:21,570 --> 00:23:24,070
And in general, of course, if
you have a very long rectangle

524
00:23:24,070 --> 00:23:26,760
you need many pieces,
relative to the fat one,

525
00:23:26,760 --> 00:23:30,640
but that's essentially optimal.

526
00:23:30,640 --> 00:23:31,540
OK.

527
00:23:31,540 --> 00:23:33,200
For fun-- this is
a general technique

528
00:23:33,200 --> 00:23:35,300
called the piece lie
technique, or superposing

529
00:23:35,300 --> 00:23:41,450
two tessellation of your shape.

530
00:23:41,450 --> 00:23:43,400
You can use that same
technique, for example,

531
00:23:43,400 --> 00:23:47,930
to get the hinged dissection
from a regular square

532
00:23:47,930 --> 00:23:50,060
to the equilateral triangle.

533
00:23:50,060 --> 00:23:52,840
You just angle it right
so that, for example,

534
00:23:52,840 --> 00:23:55,400
this midpoint hits
this midpoint,

535
00:23:55,400 --> 00:23:57,140
and various other
alignments happen,

536
00:23:57,140 --> 00:24:00,380
like this midpoint
falls on that edge.

537
00:24:00,380 --> 00:24:03,420
And if you look at it
right these cuts give you

538
00:24:03,420 --> 00:24:07,190
the four pieces for the square
to-- I guess you can see it

539
00:24:07,190 --> 00:24:10,200
right here, here are the
four pieces of the square.

540
00:24:10,200 --> 00:24:12,130
And if you check,
everything matches up.

541
00:24:12,130 --> 00:24:13,807
You can also make
equilateral triangle.

542
00:24:13,807 --> 00:24:15,390
In this case, it
happens to be hinged.

543
00:24:15,390 --> 00:24:17,210
That doesn't always happen.

544
00:24:17,210 --> 00:24:19,270
It's a little tricky
to tell, maybe,

545
00:24:19,270 --> 00:24:22,420
but with practice
you can see it.

546
00:24:22,420 --> 00:24:25,690
I mentioned, at some point, that
you could take this and turn it

547
00:24:25,690 --> 00:24:31,580
into a table that either has
four sides or has three sides.

548
00:24:31,580 --> 00:24:33,480
One of the annoying
things about the table

549
00:24:33,480 --> 00:24:36,015
is that you need legs
on each of the pieces.

550
00:24:36,015 --> 00:24:38,140
So Frederickson was playing
around with this fairly

551
00:24:38,140 --> 00:24:40,920
recently, in 2008,
and he came up

552
00:24:40,920 --> 00:24:43,660
with this alternative way of--
essentially the same technique,

553
00:24:43,660 --> 00:24:46,240
but you end up
with one big piece

554
00:24:46,240 --> 00:24:48,460
and lots of smaller pieces.

555
00:24:48,460 --> 00:24:51,490
So the idea is you just have
a big leg, or a bunch of legs,

556
00:24:51,490 --> 00:24:54,740
under one piece of the table.

557
00:24:54,740 --> 00:24:58,390
And so this is what the
dissection looks like.

558
00:24:58,390 --> 00:25:00,440
Unfortunately,
it's not hingeable.

559
00:25:00,440 --> 00:25:02,300
But if you add in
a couple pieces

560
00:25:02,300 --> 00:25:04,060
you can make it hingeable.

561
00:25:04,060 --> 00:25:06,631
So at this point, the
universality result

562
00:25:06,631 --> 00:25:07,380
was probably none.

563
00:25:07,380 --> 00:25:10,570
This is actually a lot
easier than the way we do it,

564
00:25:10,570 --> 00:25:12,560
specialized to this
kind of scenario.

565
00:25:12,560 --> 00:25:15,930
This hinges, I think, something
like this-- maybe even

566
00:25:15,930 --> 00:25:16,730
an animation of it?

567
00:25:16,730 --> 00:25:17,920
Yeah.

568
00:25:17,920 --> 00:25:20,627
Drawn by Frederickson.

569
00:25:20,627 --> 00:25:22,710
So you could see a careful
orchestration here just

570
00:25:22,710 --> 00:25:26,870
to make sure that, indeed,
you can avoid collision.

571
00:25:26,870 --> 00:25:29,030
And so that's the
proposed table.

572
00:25:29,030 --> 00:25:29,940
No one has built it.

573
00:25:29,940 --> 00:25:32,590
Another project would be to
build some hinged dissections,

574
00:25:32,590 --> 00:25:34,685
for example this one,
as real furniture.

575
00:25:34,685 --> 00:25:36,670
It would be pretty neat.

576
00:25:36,670 --> 00:25:41,490
I have a couple examples
here of real furniture built.

577
00:25:41,490 --> 00:25:45,139
This is the Dudeney dissection,
a four-piece kind of a cabinet.

578
00:25:45,139 --> 00:25:46,180
It's got lots of shelves.

579
00:25:46,180 --> 00:25:48,580
It looks really practical.

580
00:25:48,580 --> 00:25:50,060
And I don't know the bottom.

581
00:25:50,060 --> 00:25:52,220
It looks like there's a
bunch of wheels down there.

582
00:25:52,220 --> 00:25:53,720
Definitely, you
have to have a bunch

583
00:25:53,720 --> 00:25:55,639
of table legs in this case.

584
00:25:55,639 --> 00:25:57,180
But you can really
reconfiguration it

585
00:25:57,180 --> 00:25:59,360
in all sorts of ways.

586
00:25:59,360 --> 00:26:00,160
The close up.

587
00:26:02,962 --> 00:26:03,920
That looks pretty cool.

588
00:26:03,920 --> 00:26:06,030
It's made by D Haus Company.

589
00:26:10,130 --> 00:26:11,190
Any German speakers?

590
00:26:11,190 --> 00:26:12,900
Anyone know what "haus" means?

591
00:26:12,900 --> 00:26:15,420
Same in English, house.

592
00:26:15,420 --> 00:26:17,160
So they actually built a house.

593
00:26:17,160 --> 00:26:21,240
And I can't tell whether this
is a real building or a very

594
00:26:21,240 --> 00:26:22,655
good computer rendering.

595
00:26:22,655 --> 00:26:23,720
It may be real.

596
00:26:23,720 --> 00:26:24,940
AUDIENCE: [INAUDIBLE]

597
00:26:24,940 --> 00:26:25,540
PROFESSOR: What's that?

598
00:26:25,540 --> 00:26:26,420
AUDIENCE: It looks
like a rendering.

599
00:26:26,420 --> 00:26:27,380
PROFESSOR: It looks
like a rendering.

600
00:26:27,380 --> 00:26:27,710
Yeah.

601
00:26:27,710 --> 00:26:29,710
At some point later they
have people walking by,

602
00:26:29,710 --> 00:26:31,190
but it could be a composition.

603
00:26:31,190 --> 00:26:34,380
Anyway, it's an idea of
having a house for any season.

604
00:26:34,380 --> 00:26:38,300
You can reconfigure it
dynamically with these tracks.

605
00:26:38,300 --> 00:26:39,680
It's a pretty cool idea.

606
00:26:39,680 --> 00:26:42,090
It would be neat
to experiment with.

607
00:26:45,060 --> 00:26:50,090
Anyway, hinged
dissections in practice.

608
00:26:50,090 --> 00:26:51,557
It's funny to take
a 2D dissection,

609
00:26:51,557 --> 00:26:53,140
but, I think, in
architectural setting

610
00:26:53,140 --> 00:26:54,670
you can't change
where the floor is.

611
00:26:54,670 --> 00:26:57,430
So probably, 2D
dissection makes sense.

612
00:26:57,430 --> 00:26:59,940
There's the real,
maybe real version?

613
00:26:59,940 --> 00:27:03,230
I don't know.

614
00:27:03,230 --> 00:27:06,210
So that was
rectangular rectangle.

615
00:27:06,210 --> 00:27:07,500
OK.

616
00:27:07,500 --> 00:27:09,742
I'm cheating a little bit.

617
00:27:09,742 --> 00:27:10,450
Another question.

618
00:27:10,450 --> 00:27:12,140
This is a very
specific question,

619
00:27:12,140 --> 00:27:15,300
but for step three,
which is where

620
00:27:15,300 --> 00:27:18,610
we did all the action of
rehinging stuff, I said,

621
00:27:18,610 --> 00:27:20,030
number of pieces
roughly doubles.

622
00:27:20,030 --> 00:27:23,544
I meant to say at
least roughly doubles.

623
00:27:23,544 --> 00:27:24,960
So in the worst
case, the point is

624
00:27:24,960 --> 00:27:26,610
that can be at
least exponential.

625
00:27:26,610 --> 00:27:30,220
It definitely can be more
because, in general-- remember,

626
00:27:30,220 --> 00:27:32,425
it looks something
like this-- The point

627
00:27:32,425 --> 00:27:34,940
is, you need at least
two triangles per edge

628
00:27:34,940 --> 00:27:37,100
here because they
need to fit together

629
00:27:37,100 --> 00:27:41,200
to make these little kites.

630
00:27:41,200 --> 00:27:44,364
So you at least double, for
every edge that you visit.

631
00:27:44,364 --> 00:27:46,030
In the worst case,
you visit the whole--

632
00:27:46,030 --> 00:27:48,290
all the edges of the polygon.

633
00:27:48,290 --> 00:27:52,210
So you end up
doubling everything.

634
00:27:52,210 --> 00:27:54,002
But it can be worse
because sometimes,

635
00:27:54,002 --> 00:27:55,960
if you don't have a lot
of room in this corner,

636
00:27:55,960 --> 00:27:58,256
you've got to divide into
lots of very tiny triangles.

637
00:27:58,256 --> 00:28:00,630
I think that probably only
happens towards the beginning.

638
00:28:00,630 --> 00:28:02,650
After you've cut
them small, you won't

639
00:28:02,650 --> 00:28:04,990
have to cut them
even, even smaller.

640
00:28:04,990 --> 00:28:07,280
But I don't know for sure.

641
00:28:07,280 --> 00:28:09,800
The point is, it's
at least exponential.

642
00:28:09,800 --> 00:28:13,060
And this is the more
complicated diagram.

643
00:28:13,060 --> 00:28:17,640
But I claim that you could
get a pseudopolynomial bound.

644
00:28:17,640 --> 00:28:18,470
How do you do that?

645
00:28:18,470 --> 00:28:20,290
This is a little
[? trickable, ?]

646
00:28:20,290 --> 00:28:24,510
and still have time though.

647
00:28:24,510 --> 00:28:29,330
So let me go over the rough
idea, also what the claim is.

648
00:28:29,330 --> 00:28:30,710
So pseudopolynomial bound.

649
00:28:40,000 --> 00:28:42,170
I'm not going to claim this
for arbitrary polygons,

650
00:28:42,170 --> 00:28:43,700
although I think
it's probably true.

651
00:28:43,700 --> 00:28:47,155
What we argue in the paper
is that if the vertices

652
00:28:47,155 --> 00:28:51,360
of the polygon lie on
our grid, then we're OK.

653
00:28:55,581 --> 00:28:57,705
It's just a little hard to
keep track of otherwise.

654
00:29:04,250 --> 00:29:09,200
I will scale things to
make this the integer grid.

655
00:29:09,200 --> 00:29:15,910
And then the claim is
the number of pieces

656
00:29:15,910 --> 00:29:21,335
is polynomial in the number
of vertices, n and r--

657
00:29:21,335 --> 00:29:22,944
r is usually some
ratio of the longest

658
00:29:22,944 --> 00:29:24,360
distance to the
smallest distance.

659
00:29:24,360 --> 00:29:28,680
In this case r is the grid
size, like an r by r grid.

660
00:29:28,680 --> 00:29:31,470
That's like the size of
the overall grid divided

661
00:29:31,470 --> 00:29:33,060
by the size of a grid cell.

662
00:29:33,060 --> 00:29:34,350
So, basically, the same thing.

663
00:29:37,100 --> 00:29:40,570
So, how do we prove this?

664
00:29:40,570 --> 00:29:49,000
The general idea-- so we have
these messy constructions,

665
00:29:49,000 --> 00:29:50,460
and essentially,
we're inducting.

666
00:29:50,460 --> 00:29:52,905
We're moving one hinge, and
then moving the next hinge,

667
00:29:52,905 --> 00:29:54,000
and moving the next hinge.

668
00:29:54,000 --> 00:29:55,740
And essentially, all
of those inductions

669
00:29:55,740 --> 00:29:58,230
are nested inside each other.

670
00:29:58,230 --> 00:30:00,300
You completely refine
to do one thing then

671
00:30:00,300 --> 00:30:03,020
you have to refine to do
the next one in the existing

672
00:30:03,020 --> 00:30:03,520
refinement.

673
00:30:03,520 --> 00:30:05,050
So we have a very
deep recursion.

674
00:30:05,050 --> 00:30:06,820
It's one way to think of it.

675
00:30:06,820 --> 00:30:13,010
Order n depth recursion, so we
end up with exponential in n.

676
00:30:13,010 --> 00:30:20,220
But instead, what we can do is
only recurse to constant depth.

677
00:30:20,220 --> 00:30:25,967
And if you're just more careful
in the overall construction

678
00:30:25,967 --> 00:30:26,675
this is possible.

679
00:30:30,161 --> 00:30:30,660
How?

680
00:30:33,180 --> 00:30:34,895
Let me give you
some of the steps.

681
00:30:38,324 --> 00:30:42,444
You need more gadgets and you
need to follow-- So before,

682
00:30:42,444 --> 00:30:44,360
I said, oh, there's some
dissection out there,

683
00:30:44,360 --> 00:30:45,840
it's known.

684
00:30:45,840 --> 00:30:49,310
You triangulate, you convert
triangle to square, triangle

685
00:30:49,310 --> 00:30:53,112
to rectangle, rectangle to
rectangle, then superpose.

686
00:30:53,112 --> 00:30:55,320
It does the dissection, then
we hinge it arbitrarily,

687
00:30:55,320 --> 00:30:56,861
then we fix the
hinges one at a time.

688
00:30:56,861 --> 00:30:59,340
Here, I want to actually
follow those steps

689
00:30:59,340 --> 00:31:02,613
and keep it hinged dissection
as much as possible.

690
00:31:05,259 --> 00:31:07,800
So we're going to triangulate
the polygons, but in this case,

691
00:31:07,800 --> 00:31:11,170
we're going to subdivide
further and also triangulated

692
00:31:11,170 --> 00:31:14,050
with all the grid
points as vertices.

693
00:31:17,610 --> 00:31:24,700
It's little hard to
draw, but here's a grid.

694
00:31:24,700 --> 00:31:26,960
Let's draw a polygon.

695
00:31:29,880 --> 00:31:32,530
Hard to make a very exciting
polygon, so few vertices,

696
00:31:32,530 --> 00:31:36,721
but maybe something like that.

697
00:31:36,721 --> 00:31:37,220
OK.

698
00:31:37,220 --> 00:31:42,439
If I triangulate this thing
and all the interior points--

699
00:31:42,439 --> 00:31:44,730
there aren't very many interior
points in this example.

700
00:31:44,730 --> 00:31:46,438
Maybe I'll make a
slightly different one.

701
00:31:50,984 --> 00:31:52,150
There's two interior points.

702
00:31:52,150 --> 00:31:53,525
I want to triangulate,
with those

703
00:31:53,525 --> 00:31:54,750
as vertices of the triangle.

704
00:31:54,750 --> 00:31:57,370
So maybe I'll do
something like this.

705
00:32:04,050 --> 00:32:06,070
A couple different
shapes of triangles here,

706
00:32:06,070 --> 00:32:08,410
but they all have the same area.

707
00:32:08,410 --> 00:32:11,290
This is called Pick's theorem,
special case of Pick's theorem.

708
00:32:11,290 --> 00:32:13,840
So here, they're
all a half-square.

709
00:32:13,840 --> 00:32:16,570
Even though this one
spans a weird shape,

710
00:32:16,570 --> 00:32:18,450
it's one-half square of area.

711
00:32:18,450 --> 00:32:20,530
So the nice thing is if
I do this in polygon a

712
00:32:20,530 --> 00:32:23,320
and in polygon be
the triangles--

713
00:32:23,320 --> 00:32:25,577
there's equal number of
triangles of the same size

714
00:32:25,577 --> 00:32:27,410
because they have
matching areas originally.

715
00:32:36,880 --> 00:32:39,559
There's probably a way to do
this for general polygons.

716
00:32:39,559 --> 00:32:41,600
I think this is the only
step that requires grids

717
00:32:41,600 --> 00:32:46,610
except it's also a lot easier to
analyze, this bound with grids.

718
00:32:46,610 --> 00:32:49,792
So it's, I guess, an
open problem to work out

719
00:32:49,792 --> 00:32:50,375
without grids.

720
00:32:52,730 --> 00:32:53,230
OK.

721
00:32:53,230 --> 00:32:55,563
The next thing is we'd really
like a chain of triangles.

722
00:32:55,563 --> 00:32:59,010
Right now we just have
a blob of triangles.

723
00:32:59,010 --> 00:33:06,060
And we can chainify
the triangles.

724
00:33:06,060 --> 00:33:08,790
This is a step
that was-- I don't

725
00:33:08,790 --> 00:33:12,100
know if I showed the
figure last time.

726
00:33:12,100 --> 00:33:15,430
This is what we do to
slenderfy everything.

727
00:33:15,430 --> 00:33:17,157
We have some general
hinged dissection.

728
00:33:17,157 --> 00:33:18,490
I don't know what it looks like.

729
00:33:18,490 --> 00:33:19,948
We just take each
of the triangles,

730
00:33:19,948 --> 00:33:22,650
subdivide at their
in center, cut,

731
00:33:22,650 --> 00:33:25,420
and then you hinge
around the outside.

732
00:33:25,420 --> 00:33:27,340
And you'll get
one-- in this case,

733
00:33:27,340 --> 00:33:30,150
one cycle of slender triangles.

734
00:33:30,150 --> 00:33:34,012
In this case, all we care
about is that it's a chain.

735
00:33:34,012 --> 00:33:35,470
So we have some
general thing here.

736
00:33:35,470 --> 00:33:38,500
We subdivide each
of them like this,

737
00:33:38,500 --> 00:33:39,770
and then you hinge around.

738
00:33:39,770 --> 00:33:42,892
And so now I've got a hinged
collection of triangles for a,

739
00:33:42,892 --> 00:33:44,850
and I've got his collection
of triangles for b.

740
00:33:44,850 --> 00:33:46,520
I'm just going to
do a to b here.

741
00:33:46,520 --> 00:33:48,620
I should probably say that.

742
00:33:48,620 --> 00:33:49,495
Two shapes.

743
00:33:53,056 --> 00:33:54,430
And conveniently,
these triangles

744
00:33:54,430 --> 00:33:56,540
will still have matching areas.

745
00:33:56,540 --> 00:34:00,490
They're all now 1/6,
if we do it right.

746
00:34:00,490 --> 00:34:11,040
So we get a chain of
area 1/6 triangles.

747
00:34:11,040 --> 00:34:14,420
And I have the same
number for a and for b.

748
00:34:14,420 --> 00:34:15,300
So this kind of cool.

749
00:34:15,300 --> 00:34:17,570
Of course, the triangles
could be different shapes,

750
00:34:17,570 --> 00:34:21,639
but I basically have a
chain of various triangles.

751
00:34:21,639 --> 00:34:25,920
They're all the same area-- a
little hard to draw-- for a.

752
00:34:25,920 --> 00:34:30,360
I have a similar chain for b.

753
00:34:30,360 --> 00:34:32,310
And I just need to
convert, basically,

754
00:34:32,310 --> 00:34:34,699
triangle per
triangle from a to b.

755
00:34:34,699 --> 00:34:36,230
So now my problem
is a lot easier.

756
00:34:36,230 --> 00:34:38,104
I have these hinges
which I need to preserve.

757
00:34:38,104 --> 00:34:39,889
That's a little trickier.

758
00:34:39,889 --> 00:34:43,340
This is actually an idea
suggested by Epstein

759
00:34:43,340 --> 00:34:44,889
before the universality result.

760
00:34:44,889 --> 00:34:46,880
It's like, all we need
to do is do triangle

761
00:34:46,880 --> 00:34:48,940
to triangle while
preserving two hinges.

762
00:34:48,940 --> 00:34:51,030
Then we could do
anything to anything.

763
00:34:51,030 --> 00:34:53,139
So we're following that plan.

764
00:34:53,139 --> 00:34:56,387
And now we're going to use
all the fancy gadgets we have

765
00:34:56,387 --> 00:34:58,720
to do triangle to triangle
while preserving these hinges

766
00:34:58,720 --> 00:35:03,152
and not blowing up the
number of pieces too much.

767
00:35:03,152 --> 00:35:04,110
But definitely simpler.

768
00:35:04,110 --> 00:35:06,420
We're down to
triangle to triangle.

769
00:35:06,420 --> 00:35:07,170
Next step.

770
00:35:09,930 --> 00:35:10,430
OK.

771
00:35:10,430 --> 00:35:13,110
Next problem Yeah.

772
00:35:13,110 --> 00:35:14,590
This is slightly annoying.

773
00:35:14,590 --> 00:35:18,150
I said, oh great, these
triangles are matching up.

774
00:35:18,150 --> 00:35:21,810
But I'm not going to be able
to do triangle to triangle

775
00:35:21,810 --> 00:35:24,307
and get exactly the hinges
I want where I want them,

776
00:35:24,307 --> 00:35:26,140
so I'm going to have
to end up, for example,

777
00:35:26,140 --> 00:35:28,690
moving this hinge
to another corner.

778
00:35:28,690 --> 00:35:32,540
So we're going to use
a new gadget, actually,

779
00:35:32,540 --> 00:35:44,390
for fixing which vertices
connect to which triangles.

780
00:35:44,390 --> 00:35:47,770
This is, maybe, not obvious yet
that we need this, but we will.

781
00:35:47,770 --> 00:35:50,970
And we're going to use a
slightly, a somewhat more

782
00:35:50,970 --> 00:35:54,830
efficient version of,
essentially, the same idea.

783
00:35:54,830 --> 00:35:57,400
So we've got a hinge
here, in the middle.

784
00:35:57,400 --> 00:35:59,909
And basically, can't control
where the hinge goes, but it's

785
00:35:59,909 --> 00:36:01,450
supposed to go to
one of the corners.

786
00:36:01,450 --> 00:36:04,180
So we're going to
reconfigure in this way.

787
00:36:04,180 --> 00:36:06,960
So we assume we have
some way of doing it.

788
00:36:06,960 --> 00:36:11,400
And here's the thing, we assume
that maybe this has already

789
00:36:11,400 --> 00:36:12,630
happened to a.

790
00:36:12,630 --> 00:36:16,710
We don't want to recurse
into a because then we

791
00:36:16,710 --> 00:36:18,550
get exponential blow up.

792
00:36:18,550 --> 00:36:21,220
I'm going to have to do this
for every single triangle here.

793
00:36:21,220 --> 00:36:21,720
There's n of them.

794
00:36:21,720 --> 00:36:22,330
That's a lot.

795
00:36:22,330 --> 00:36:23,850
I don't want to
get deep recursion,

796
00:36:23,850 --> 00:36:26,775
I don't want to get
depth n recursion.

797
00:36:26,775 --> 00:36:31,975
But if I cut up in this way, in
fact, I only need to cut up b.

798
00:36:31,975 --> 00:36:34,860
And if b hasn't been
touched yet this is OK.

799
00:36:34,860 --> 00:36:36,340
And then I'll do
it the next way,

800
00:36:36,340 --> 00:36:38,120
and the next triangle,
next triangle,

801
00:36:38,120 --> 00:36:39,480
and they won't interact.

802
00:36:39,480 --> 00:36:41,760
That's the good news.

803
00:36:41,760 --> 00:36:42,790
So how do we do it?

804
00:36:42,790 --> 00:36:48,830
Well, we cut up a little, oh,
what do we call it, kite fan,

805
00:36:48,830 --> 00:36:50,130
I believe, here.

806
00:36:50,130 --> 00:36:53,040
Here there's two kites,
and we get these triangles

807
00:36:53,040 --> 00:36:55,360
to match these two, these
triangles to match these two.

808
00:36:55,360 --> 00:36:57,610
We cut up this little
piece along the side.

809
00:36:57,610 --> 00:37:00,220
And either the green
stays in here-- green

810
00:37:00,220 --> 00:37:04,140
is attached to the
pink or magenta.

811
00:37:04,140 --> 00:37:07,470
So if we keep the green in
here, the triangle stays there.

812
00:37:07,470 --> 00:37:10,220
If we pull everything out--
and there's a little hole

813
00:37:10,220 --> 00:37:12,340
made here to make
that more plausible.

814
00:37:12,340 --> 00:37:17,230
But in reality, we have to
subdivide to get slender.

815
00:37:17,230 --> 00:37:20,780
So if we instead reconfigure
the green to lie along the edge,

816
00:37:20,780 --> 00:37:25,047
and the blue can turn
around here and fit inside

817
00:37:25,047 --> 00:37:26,630
because it has exactly
the same shape,

818
00:37:26,630 --> 00:37:30,150
these two chains are identical,
it can also fit in here.

819
00:37:30,150 --> 00:37:33,920
And then we've moved the
magenta over to that side.

820
00:37:33,920 --> 00:37:34,950
So that's cool.

821
00:37:34,950 --> 00:37:37,560
That works, and it
doesn't touch a.

822
00:37:37,560 --> 00:37:41,150
So it's a slight variation
of what we had before.

823
00:37:41,150 --> 00:37:43,810
And it's good.

824
00:37:43,810 --> 00:37:46,966
So, that's psudeopolynomial,
and they don't interact.

825
00:37:46,966 --> 00:37:48,590
And so we can move
these things however

826
00:37:48,590 --> 00:37:52,726
we need to according to what
step four produces for us.

827
00:37:52,726 --> 00:37:54,557
So this maybe
slightly out of order.

828
00:37:54,557 --> 00:37:56,890
I could have called that step
four, and this step three.

829
00:37:59,430 --> 00:38:01,145
Get to the more exciting part.

830
00:38:01,145 --> 00:38:03,740
Finally, we do
triangle to triangle.

831
00:38:11,090 --> 00:38:12,030
This a little crazy.

832
00:38:12,030 --> 00:38:16,010
I'm going to give you
three constructions that

833
00:38:16,010 --> 00:38:17,432
give us what we want.

834
00:38:17,432 --> 00:38:19,390
And then I'm going to
claim I can overlay them.

835
00:38:19,390 --> 00:38:21,680
This is what we can't do
with hinged dissections,

836
00:38:21,680 --> 00:38:22,970
but I'm going to do it anyway.

837
00:38:22,970 --> 00:38:24,370
So bear with me.

838
00:38:24,370 --> 00:38:28,110
The final gadget will
say how to overlay them.

839
00:38:28,110 --> 00:38:30,160
But let's start with the
relatively simple goal

840
00:38:30,160 --> 00:38:33,740
of triangle to rectangle.

841
00:38:33,740 --> 00:38:35,590
This I already showed you.

842
00:38:35,590 --> 00:38:37,870
And the nice thing about
triangle to rectangle,

843
00:38:37,870 --> 00:38:40,910
this three-piece dissection, is
you can hinge it here and here

844
00:38:40,910 --> 00:38:42,160
and it works just fine.

845
00:38:46,337 --> 00:38:47,920
So that's already a
hinged dissection.

846
00:38:47,920 --> 00:38:48,850
That's the easy step.

847
00:38:51,880 --> 00:38:54,920
Then we want to
take that rectangle

848
00:38:54,920 --> 00:39:01,340
and convert it into a tiny--
or not tiny, same area,

849
00:39:01,340 --> 00:39:04,264
but an epsilon-height rectangle.

850
00:39:04,264 --> 00:39:05,930
Because remember, we
have two triangles,

851
00:39:05,930 --> 00:39:08,130
they're different shapes so
they have different heights.

852
00:39:08,130 --> 00:39:09,921
This one will end up
being half the height,

853
00:39:09,921 --> 00:39:12,130
but it won't match what
we'd get for this triangle.

854
00:39:12,130 --> 00:39:15,670
So I'm going to do steps a and
b for each of the triangles.

855
00:39:15,670 --> 00:39:19,990
And then I have two
epsilon-height rectangles.

856
00:39:19,990 --> 00:39:22,870
And then the challenge is to
convert one into the other.

857
00:39:22,870 --> 00:39:26,830
This is a challenge because
they have hinges on them.

858
00:39:26,830 --> 00:39:30,600
So with dissections you just
overlay these two cut ups.

859
00:39:30,600 --> 00:39:33,410
But hinged dissections, there's
hinges you have to preserve,

860
00:39:33,410 --> 00:39:35,220
we can't do that.

861
00:39:35,220 --> 00:39:35,720
OK.

862
00:39:39,920 --> 00:39:46,630
First part is step b,
which I showed you already,

863
00:39:46,630 --> 00:39:48,380
going from one
rectangle to another.

864
00:39:48,380 --> 00:39:49,820
Here's another diagram of it.

865
00:39:49,820 --> 00:39:52,940
It turns out it's almost hinged.

866
00:39:52,940 --> 00:39:55,910
You can, essentially,
just flop back and forth

867
00:39:55,910 --> 00:39:58,120
and back and forth,
except at the end

868
00:39:58,120 --> 00:40:00,460
you might be in trouble.

869
00:40:00,460 --> 00:40:03,410
So there's one step here,
and depending on parity

870
00:40:03,410 --> 00:40:07,210
exactly this piece of the
rectangle is hinged here.

871
00:40:07,210 --> 00:40:09,170
But I really want
to be hinged here,

872
00:40:09,170 --> 00:40:11,020
so I'm just going to
move it over here.

873
00:40:11,020 --> 00:40:12,882
I have tools for
moving hinges around.

874
00:40:12,882 --> 00:40:15,090
So it turns out, you have
to check that this is safe.

875
00:40:15,090 --> 00:40:20,000
But you just do one hinge
moving, and then you're OK.

876
00:40:20,000 --> 00:40:22,250
So in this case-- this should
actually go a little bit

877
00:40:22,250 --> 00:40:25,310
deeper-- the bottom
figure shows when

878
00:40:25,310 --> 00:40:28,340
you go too deep you can
cut, cut-- and this is just

879
00:40:28,340 --> 00:40:31,250
like the previous diagram
of triangle to rectangle.

880
00:40:31,250 --> 00:40:33,500
You do that at the
bottom you'll be fine.

881
00:40:33,500 --> 00:40:35,750
There's a couple different
cases in exactly the parity

882
00:40:35,750 --> 00:40:37,030
and how you end up.

883
00:40:37,030 --> 00:40:39,174
Three cases I guess.

884
00:40:39,174 --> 00:40:40,840
But in all cases the
rest can be hinged.

885
00:40:40,840 --> 00:40:48,054
You just need this one step
in the middle to fix it.

886
00:40:48,054 --> 00:40:49,970
So most of it is just
swinging back and forth.

887
00:40:49,970 --> 00:40:51,780
So it's almost hinged,
which is good news

888
00:40:51,780 --> 00:40:55,320
because we have tools to make
almost hinged things actually

889
00:40:55,320 --> 00:40:57,360
hinged.

890
00:40:57,360 --> 00:40:59,350
So that's cool.

891
00:40:59,350 --> 00:41:03,460
So basically, we've covered
a and b at this point.

892
00:41:03,460 --> 00:41:07,540
But the last part
is c, or how do

893
00:41:07,540 --> 00:41:10,030
we superpose all these things?

894
00:41:10,030 --> 00:41:14,980
And this is using another
gadget called pseudocuts.

895
00:41:14,980 --> 00:41:19,600
And essentially, you have some
nice hinged dissection already,

896
00:41:19,600 --> 00:41:22,280
and you want to add
a cut and a hinge.

897
00:41:22,280 --> 00:41:26,370
So just imagine cutting
all the way through here

898
00:41:26,370 --> 00:41:29,280
and adding a hinge, I guess,
on the yellow side here.

899
00:41:29,280 --> 00:41:31,360
And somehow, I want
this thing to fold

900
00:41:31,360 --> 00:41:33,250
in all the ways it used
to be able to fold.

901
00:41:33,250 --> 00:41:34,880
So it could fold into a.

902
00:41:34,880 --> 00:41:38,170
But then I also want it to be
able to fold at this hinge,

903
00:41:38,170 --> 00:41:40,930
and eventually fold into b.

904
00:41:40,930 --> 00:41:44,370
And it's complicated,
but again, the same idea.

905
00:41:44,370 --> 00:41:49,090
So we've got these yellow guys,
which normally live in here,

906
00:41:49,090 --> 00:41:52,560
and so yellows is yellow.

907
00:41:52,560 --> 00:41:53,860
These are triangles.

908
00:41:53,860 --> 00:41:55,470
These are triangles
minus triangles,

909
00:41:55,470 --> 00:41:57,330
so they're like little quads.

910
00:41:57,330 --> 00:41:59,680
They have holes just the
right size for the yellow.

911
00:41:59,680 --> 00:42:02,330
These guys have holes
just the right side--

912
00:42:02,330 --> 00:42:05,720
I'm sorry, how does it go?

913
00:42:05,720 --> 00:42:06,220
OK.

914
00:42:06,220 --> 00:42:06,720
I see.

915
00:42:06,720 --> 00:42:11,496
It's purple, then blue,
then yellow, I believe.

916
00:42:11,496 --> 00:42:13,860
So the yellow fits
into the blue-- anyway.

917
00:42:13,860 --> 00:42:16,870
Whatever works.

918
00:42:16,870 --> 00:42:18,370
These guys nest together.

919
00:42:18,370 --> 00:42:21,970
And when they nest together
they fill these little holes.

920
00:42:21,970 --> 00:42:24,830
And then there's matching
patterns out here.

921
00:42:24,830 --> 00:42:27,670
So they all fit.

922
00:42:27,670 --> 00:42:28,360
How does it go?

923
00:42:28,360 --> 00:42:30,380
Actually, sorry, I think
they're all triangles.

924
00:42:30,380 --> 00:42:32,830
This just looks multicolored.

925
00:42:32,830 --> 00:42:34,590
So it looks like
purple here is going

926
00:42:34,590 --> 00:42:37,530
into the cyan one
at the next level.

927
00:42:37,530 --> 00:42:40,381
The yellow guys are
going into the purple.

928
00:42:40,381 --> 00:42:40,880
I see.

929
00:42:40,880 --> 00:42:43,130
So there's a triangle
and a quad here.

930
00:42:43,130 --> 00:42:44,610
Lovely.

931
00:42:44,610 --> 00:42:47,785
And then these guys
stretch across.

932
00:42:50,940 --> 00:42:54,110
Definitely a little
more complicated.

933
00:42:54,110 --> 00:42:56,890
And you lose a factor
of two, or whatever,

934
00:42:56,890 --> 00:43:03,687
but if you apply these
pseudocuts in the right order--

935
00:43:03,687 --> 00:43:06,020
and these are fairly simple
cuttings that we have to do.

936
00:43:06,020 --> 00:43:10,550
We know that these cuts
are mostly a striping.

937
00:43:10,550 --> 00:43:14,405
So if you just apply them in
order you don't get blow up.

938
00:43:14,405 --> 00:43:16,603
I'll just wave my hands at that.

939
00:43:16,603 --> 00:43:19,750
It's a little hard to draw
the picture, obviously,

940
00:43:19,750 --> 00:43:21,605
but that's how it goes.

941
00:43:21,605 --> 00:43:24,522
And that's pseudopolynomial
hinged dissection.

942
00:43:24,522 --> 00:43:26,605
This is why-- it was
intentional I didn't cover it

943
00:43:26,605 --> 00:43:30,110
in lecture because it's
pretty complicated.

944
00:43:30,110 --> 00:43:32,302
There wasn't time.

945
00:43:32,302 --> 00:43:32,885
Any questions?

946
00:43:35,550 --> 00:43:40,190
Last topic is higher dimensions.

947
00:43:40,190 --> 00:43:45,140
Can we get a brief
overview of 3D dissections?

948
00:43:45,140 --> 00:43:47,307
So this is more a dissection
question than a hinging

949
00:43:47,307 --> 00:43:49,139
question, although, of
course you could ask,

950
00:43:49,139 --> 00:43:51,110
does all this work for
hinged dissections?

951
00:43:51,110 --> 00:43:54,480
Pseudopolynomial, we
don't necessarily know.

952
00:43:54,480 --> 00:43:58,930
For straight up proving that
hinged dissections exist,

953
00:43:58,930 --> 00:44:02,640
the claim is-- it hasn't been
written up formally yet--

954
00:44:02,640 --> 00:44:04,880
the same techniques work.

955
00:44:04,880 --> 00:44:07,280
You can take any
dissection and convert it

956
00:44:07,280 --> 00:44:09,130
into a hinged dissection.

957
00:44:09,130 --> 00:44:11,880
But in 3D, it turns out,
dissections, by themselves,

958
00:44:11,880 --> 00:44:15,710
are not so simple, as
a lot of open problems.

959
00:44:15,710 --> 00:44:17,590
Some nice things are known.

960
00:44:17,590 --> 00:44:23,580
So let me tell you
about 3D dissection.

961
00:44:23,580 --> 00:44:25,300
If I want to convert
one polyhedron

962
00:44:25,300 --> 00:44:30,300
p into another polyhedron
q, obviously, the volumes

963
00:44:30,300 --> 00:44:41,290
must be the same assuming we're
doing a reasonable cutting

964
00:44:41,290 --> 00:44:44,350
and not some crazy
axiom of choice thing.

965
00:44:44,350 --> 00:44:47,737
So volumes have to match, just
like for polygons the areas

966
00:44:47,737 --> 00:44:48,320
have to match.

967
00:44:48,320 --> 00:44:50,590
But that turns out
to be not enough.

968
00:44:50,590 --> 00:44:54,170
And this goes back
to a Hilbert problem.

969
00:44:54,170 --> 00:44:56,510
So you may have heard
of David Hilbert.

970
00:44:56,510 --> 00:45:01,510
He wrote this paper of like
23 open problems at the turn

971
00:45:01,510 --> 00:45:04,630
the previous century, 1900.

972
00:45:04,630 --> 00:45:05,735
This is problem three.

973
00:45:09,230 --> 00:45:11,520
It wasn't directly about
hinged dissections,

974
00:45:11,520 --> 00:45:13,010
or about dissections rather.

975
00:45:13,010 --> 00:45:16,600
A little bit convoluted--
it's about some certain axioms

976
00:45:16,600 --> 00:45:18,230
and proving certain things.

977
00:45:18,230 --> 00:45:19,850
But in particular,
he was asking,

978
00:45:19,850 --> 00:45:23,490
are there two tetrahedra
of equal base and altitude,

979
00:45:23,490 --> 00:45:25,510
so equal volume,
which can in no way

980
00:45:25,510 --> 00:45:27,640
be split up into
congruent tetrahedra?

981
00:45:27,640 --> 00:45:30,380
So there's no way to
dissect one into the other.

982
00:45:30,380 --> 00:45:32,340
If that's true it would
show the certain axioms

983
00:45:32,340 --> 00:45:34,710
are necessary and
certain proofs.

984
00:45:34,710 --> 00:45:35,900
And it turns out it is true.

985
00:45:35,900 --> 00:45:39,060
There are tetrahedral of equal
volume where you cannot do

986
00:45:39,060 --> 00:45:40,600
this.

987
00:45:40,600 --> 00:45:42,400
And that-- I don't
have a slide for it--

988
00:45:42,400 --> 00:45:47,650
but this was proved
by a guy named Dehn.

989
00:45:47,650 --> 00:45:50,460
And he came up with something
called the-- well, that we now

990
00:45:50,460 --> 00:45:51,540
call the Dehn invariant.

991
00:45:51,540 --> 00:45:54,270
He didn't call it that himself.

992
00:45:54,270 --> 00:45:56,810
And these things
must also match.

993
00:45:56,810 --> 00:45:58,810
It's called invariant
meaning that no matter how

994
00:45:58,810 --> 00:46:01,530
you cut the things up and
reassemble the Dehn invariant

995
00:46:01,530 --> 00:46:02,770
doesn't change.

996
00:46:02,770 --> 00:46:05,140
And so if you have any
hope of going from p to q,

997
00:46:05,140 --> 00:46:07,130
those two things must match.

998
00:46:07,130 --> 00:46:14,480
And then, Sidler--
so this was 1901,

999
00:46:14,480 --> 00:46:17,160
Dehn proved that this was
a necessary condition.

1000
00:46:17,160 --> 00:46:20,200
So like a year
after that appeared.

1001
00:46:20,200 --> 00:46:23,760
In 1965, a little
bit later, Sidler

1002
00:46:23,760 --> 00:46:26,400
proved that this is
all that's necessary.

1003
00:46:26,400 --> 00:46:28,060
So these are
sufficient conditions.

1004
00:46:28,060 --> 00:46:30,770
If p and q have the same volume
and the same Dehn invariant,

1005
00:46:30,770 --> 00:46:32,874
then there is
actually a dissection.

1006
00:46:32,874 --> 00:46:34,540
And he proved it
somewhat algebraically,

1007
00:46:34,540 --> 00:46:36,730
somewhat constructively,
I'm not sure exactly.

1008
00:46:36,730 --> 00:46:41,360
There's a simpler proof
by [? Jephson ?] in 1968.

1009
00:46:41,360 --> 00:46:44,100
And he proved that, actually,
in 4D the same is true.

1010
00:46:44,100 --> 00:46:45,750
In 4D you need the
volumes to match

1011
00:46:45,750 --> 00:46:47,940
and the Dehn invariants to
match, and that's enough.

1012
00:46:47,940 --> 00:46:52,680
In 5D and higher no one knows
what it takes for a dissection.

1013
00:46:52,680 --> 00:46:53,800
Pretty weird.

1014
00:46:53,800 --> 00:46:57,480
It could be interesting to
study these more carefully.

1015
00:46:57,480 --> 00:47:00,760
Let me tell you briefly
about Dehn invariants.

1016
00:47:00,760 --> 00:47:04,430
A little awkward unless you're
familiar with tensor product

1017
00:47:04,430 --> 00:47:05,200
space.

1018
00:47:05,200 --> 00:47:07,910
How many people know about
tensor product space?

1019
00:47:07,910 --> 00:47:08,410
A few.

1020
00:47:08,410 --> 00:47:08,710
OK.

1021
00:47:08,710 --> 00:47:11,210
If you've done quantum stuff,
I guess, it's more common.

1022
00:47:11,210 --> 00:47:16,040
I'm not familiar with tensor
product space, but here we go.

1023
00:47:16,040 --> 00:47:19,800
Tensor product space.

1024
00:47:24,580 --> 00:47:29,040
But I can read Wikipedia
with the best of them.

1025
00:47:29,040 --> 00:47:31,190
It's a fairly simple
notion, it just

1026
00:47:31,190 --> 00:47:32,490
has somewhat weird notation.

1027
00:47:32,490 --> 00:47:34,880
You can do things
like take something x

1028
00:47:34,880 --> 00:47:37,630
and write tensor product with y.

1029
00:47:37,630 --> 00:47:39,660
And what this means
is, basically,

1030
00:47:39,660 --> 00:47:40,960
don't mess with this product.

1031
00:47:40,960 --> 00:47:41,810
OK.

1032
00:47:41,810 --> 00:47:42,900
It's a product.

1033
00:47:42,900 --> 00:47:45,370
Really, this is two
things, x and y.

1034
00:47:45,370 --> 00:47:46,870
They're not
interchangeable, they're

1035
00:47:46,870 --> 00:47:49,490
in completely different worlds,
different units, whatever.

1036
00:47:49,490 --> 00:47:51,190
You can't like multiply them.

1037
00:47:51,190 --> 00:47:52,800
They just hang out side by side.

1038
00:47:52,800 --> 00:47:54,320
You also can't flip them around.

1039
00:47:54,320 --> 00:47:56,020
It's not commutative.

1040
00:47:56,020 --> 00:47:57,430
OK.

1041
00:47:57,430 --> 00:47:58,690
Fine.

1042
00:47:58,690 --> 00:47:59,930
But some things hold.

1043
00:47:59,930 --> 00:48:05,250
Like if you take, I don't know,
z and add it to this product,

1044
00:48:05,250 --> 00:48:07,310
you do have
distributivity, so you

1045
00:48:07,310 --> 00:48:13,370
can get x-- is this-- no, this
doesn't look very correct.

1046
00:48:13,370 --> 00:48:17,060
If I have this then you
can multiply that out.

1047
00:48:17,060 --> 00:48:24,330
So you get x tensored
with y plus x tensored z.

1048
00:48:24,330 --> 00:48:26,860
So that holds.

1049
00:48:26,860 --> 00:48:29,360
It also holds on the left.

1050
00:48:29,360 --> 00:48:31,850
And the other thing is
that constants come out.

1051
00:48:31,850 --> 00:48:37,190
So if we have c times
x tensored with y,

1052
00:48:37,190 --> 00:48:44,350
this is the same thing as
c times x tensored with y.

1053
00:48:44,350 --> 00:48:46,860
So in the end I'm
going to have a bunch

1054
00:48:46,860 --> 00:48:48,640
of these pairs,
these tensor pairs.

1055
00:48:48,640 --> 00:48:50,540
And I'm also able to
add them together.

1056
00:48:50,540 --> 00:48:52,130
And nothing happens when
you add them together,

1057
00:48:52,130 --> 00:48:52,930
they just hang out.

1058
00:48:52,930 --> 00:48:55,221
So in general-- you could
also have a constant factor--

1059
00:48:55,221 --> 00:48:59,225
so you have a linear
combination of pairs, basically.

1060
00:48:59,225 --> 00:49:00,740
Why am I doing this?

1061
00:49:00,740 --> 00:49:04,440
Because here's the
Dehn invariant.

1062
00:49:04,440 --> 00:49:06,450
Dehn invariant says,
look, with polyhedra

1063
00:49:06,450 --> 00:49:09,100
you've got two things-- it's
going to be the x and the y

1064
00:49:09,100 --> 00:49:11,140
over there-- you've
got edge links

1065
00:49:11,140 --> 00:49:13,410
and you've got dihedral angles.

1066
00:49:13,410 --> 00:49:15,660
So look at every edge.

1067
00:49:15,660 --> 00:49:20,390
Here's an edge of
my polyhedron here.

1068
00:49:20,390 --> 00:49:25,020
It has some length,
which I'll call l of e.

1069
00:49:25,020 --> 00:49:29,250
And there's some angle here,
which I'll call theta of e.

1070
00:49:29,250 --> 00:49:30,800
Add those up over every edge.

1071
00:49:30,800 --> 00:49:32,300
So the Dehn invariant
is going to be

1072
00:49:32,300 --> 00:49:40,148
the sum over all edges of the
length tensored with the angle.

1073
00:49:40,148 --> 00:49:44,530
AUDIENCE: Isn't an angle a
function of two [? of these? ?]

1074
00:49:44,530 --> 00:49:47,500
PROFESSOR: Angle is the angle
between these two planes.

1075
00:49:47,500 --> 00:49:49,540
So that's a dihedral angle.

1076
00:49:49,540 --> 00:49:50,660
Yep.

1077
00:49:50,660 --> 00:49:53,857
So for every edge there's
one dihedral angle.

1078
00:49:53,857 --> 00:49:55,690
Just sort of the interiors
of all that angle

1079
00:49:55,690 --> 00:49:58,860
there at the edge.

1080
00:49:58,860 --> 00:50:00,980
So this is kind of
what's going on.

1081
00:50:00,980 --> 00:50:02,480
And these things have to match.

1082
00:50:02,480 --> 00:50:04,690
Now it's a little
more complicated.

1083
00:50:04,690 --> 00:50:09,410
Sorry, it's not
really just the angle.

1084
00:50:09,410 --> 00:50:12,300
Essentially, if you add
rational multiples of pi nothing

1085
00:50:12,300 --> 00:50:13,100
happens.

1086
00:50:13,100 --> 00:50:20,180
So you actually take this weird
group, all rationals times

1087
00:50:20,180 --> 00:50:22,210
pi-- All this means
is if you have

1088
00:50:22,210 --> 00:50:25,320
two angles and their difference,
that you subtract them

1089
00:50:25,320 --> 00:50:27,410
and you get a rational
multiple of pi,

1090
00:50:27,410 --> 00:50:29,715
then those two angles
are considered the same.

1091
00:50:29,715 --> 00:50:31,340
So what this is really
saying is I only

1092
00:50:31,340 --> 00:50:34,020
care about the irrational
part of pi, roughly.

1093
00:50:34,020 --> 00:50:36,920
You add pi over 2, that
doesn't change anything.

1094
00:50:36,920 --> 00:50:38,260
Why this thing?

1095
00:50:38,260 --> 00:50:40,330
Well if I take an edge,
and for example, I

1096
00:50:40,330 --> 00:50:42,680
cut it in half
anywhere, I could cut it

1097
00:50:42,680 --> 00:50:45,220
at an irrational
fraction, or whatever,

1098
00:50:45,220 --> 00:50:46,920
I will get two
lengths but they'll

1099
00:50:46,920 --> 00:50:48,280
be tensored with the same angle.

1100
00:50:48,280 --> 00:50:49,630
I didn't change the angle.

1101
00:50:49,630 --> 00:50:53,460
And so by
distributivity-- once you

1102
00:50:53,460 --> 00:50:55,330
get things inside
the same place.

1103
00:50:55,330 --> 00:50:58,770
So in this case, we'll get
two lengths that add up.

1104
00:50:58,770 --> 00:50:59,890
They match.

1105
00:50:59,890 --> 00:51:00,390
OK.

1106
00:51:00,390 --> 00:51:02,500
So as long as you
have matching angles

1107
00:51:02,500 --> 00:51:04,330
you can add the
lengths together.

1108
00:51:04,330 --> 00:51:05,940
That's what
distributivity tells you.

1109
00:51:05,940 --> 00:51:09,240
Similarly, if I tried to cut
this angle in some piece,

1110
00:51:09,240 --> 00:51:12,910
it could be an irrational
ratio between the two pieces,

1111
00:51:12,910 --> 00:51:15,129
they will have the
same edge length.

1112
00:51:15,129 --> 00:51:16,670
And when I have
matching edge lengths

1113
00:51:16,670 --> 00:51:19,630
I can use distributivity and
add the angles back together.

1114
00:51:19,630 --> 00:51:24,270
So basically, when you dissect,
this thing will not change.

1115
00:51:24,270 --> 00:51:27,104
It's a little more
awkward when I cut here

1116
00:51:27,104 --> 00:51:28,520
because this was
originally a pie,

1117
00:51:28,520 --> 00:51:29,978
and then I cut it
into some pieces.

1118
00:51:29,978 --> 00:51:32,300
And this is where you need
the rational multiples of pi

1119
00:51:32,300 --> 00:51:33,710
not mattering.

1120
00:51:33,710 --> 00:51:36,460
But eventually you can prove
Dehn invariant is invariant.

1121
00:51:36,460 --> 00:51:39,170
The harder proof, you can
prove that it's also sufficient

1122
00:51:39,170 --> 00:51:41,250
if you have the
matching volumes.

1123
00:51:41,250 --> 00:51:44,750
As recently proved
like a few years ago,

1124
00:51:44,750 --> 00:51:49,280
2008, that whether the Dehn
invariant of one polyhedron

1125
00:51:49,280 --> 00:51:52,150
and another match is decidable.

1126
00:51:52,150 --> 00:51:54,710
So there is an algorithm to
tell whether two polyhedron have

1127
00:51:54,710 --> 00:51:57,090
the same this thing.

1128
00:51:57,090 --> 00:51:58,711
Decidable is a pretty
weak statement.

1129
00:51:58,711 --> 00:52:00,085
Natural open
problem is, is there

1130
00:52:00,085 --> 00:52:01,520
a good algorithm to do it?

1131
00:52:01,520 --> 00:52:02,470
We don't know.

1132
00:52:02,470 --> 00:52:04,310
If it does match, is
there a good algorithm

1133
00:52:04,310 --> 00:52:06,110
to find the dissection?

1134
00:52:06,110 --> 00:52:07,180
We don't know.

1135
00:52:07,180 --> 00:52:09,970
These may be easy if you really
understand the proofs deeply.

1136
00:52:09,970 --> 00:52:12,410
But at the time no one
cared about algorithms.

1137
00:52:12,410 --> 00:52:15,830
At this point, we need to go
back and really understand

1138
00:52:15,830 --> 00:52:18,350
how to actually do 3D
dissections so that we could

1139
00:52:18,350 --> 00:52:20,040
then do a 3D hinged dissections.

1140
00:52:22,840 --> 00:52:24,190
That's it.

1141
00:52:26,720 --> 00:52:29,350
Don't forget, orgami
convention is on Saturday.

1142
00:52:29,350 --> 00:52:31,200
Should be fun.