1
00:00:50,565 --> 00:00:51,440
PROFESSOR: All right.

2
00:00:51,440 --> 00:00:53,170
Let's get started.

3
00:00:53,170 --> 00:00:57,130
So we are continuing the
theme of unfolding polyhedra,

4
00:00:57,130 --> 00:01:01,290
and the general picture we are
thinking about in terms of edge

5
00:01:01,290 --> 00:01:03,670
unfolding versus
general unfolding which

6
00:01:03,670 --> 00:01:05,440
are these two pictures.

7
00:01:05,440 --> 00:01:07,030
Top is an edge
unfolding of the cube.

8
00:01:07,030 --> 00:01:10,680
Bottom is a general
unfolding of the cube.

9
00:01:10,680 --> 00:01:12,640
If you have a
complex polyhedron,

10
00:01:12,640 --> 00:01:14,930
we found general unfoldings.

11
00:01:14,930 --> 00:01:16,241
There are like four of them.

12
00:01:16,241 --> 00:01:16,740
They work.

13
00:01:16,740 --> 00:01:18,799
We proved one of them.

14
00:01:18,799 --> 00:01:20,340
If you want an edge
unfolding, that's

15
00:01:20,340 --> 00:01:22,760
the centuries-old open problem.

16
00:01:22,760 --> 00:01:24,352
For non-convex
polyhedra, we know

17
00:01:24,352 --> 00:01:25,560
this is too much to hope for.

18
00:01:25,560 --> 00:01:27,879
Even if it's
topologically convex,

19
00:01:27,879 --> 00:01:29,420
there's not always
an edge unfolding.

20
00:01:29,420 --> 00:01:33,410
That was the
tetrahedral Witch's Hat.

21
00:01:33,410 --> 00:01:37,470
But for general
unfolding, we don't know.

22
00:01:37,470 --> 00:01:40,320
So today's lecture is actually
mostly about these two

23
00:01:40,320 --> 00:01:45,910
open problems and
different variations of it

24
00:01:45,910 --> 00:01:49,070
that we know how to
solve, special cases,

25
00:01:49,070 --> 00:01:51,160
and changes in the model.

26
00:01:54,280 --> 00:01:56,280
At the end, there'll
also be some stuff

27
00:01:56,280 --> 00:01:58,270
about the reverse
direction folding,

28
00:01:58,270 --> 00:02:02,190
but mostly, it will
be about unfolding.

29
00:02:02,190 --> 00:02:05,300
So there's a third
kind of unfolding

30
00:02:05,300 --> 00:02:10,330
which we call vertex
unfolding, and it's

31
00:02:10,330 --> 00:02:13,900
kind of like a
hinged dissection.

32
00:02:13,900 --> 00:02:16,430
So instead of normally
in unfolding, something

33
00:02:16,430 --> 00:02:23,450
like a cross, it's a
nice, connected polygon.

34
00:02:23,450 --> 00:02:26,430
The faces are
joined along edges.

35
00:02:26,430 --> 00:02:30,770
But what if I allowed
disconnecting along edges,

36
00:02:30,770 --> 00:02:37,240
but I just wanted things to
stay connected along vertices,

37
00:02:37,240 --> 00:02:38,610
like at a hinge.

38
00:02:38,610 --> 00:02:41,390
So it still should be
one piece in the sense

39
00:02:41,390 --> 00:02:45,840
that this is still
connected, and in this case,

40
00:02:45,840 --> 00:02:48,710
I still want an edge unfolding.

41
00:02:48,710 --> 00:02:50,280
You're only allowed
to cut on edges.

42
00:02:53,480 --> 00:02:55,500
And in fact, we will
cut on all the edges

43
00:02:55,500 --> 00:02:57,190
because that'll be
the most flexible.

44
00:02:57,190 --> 00:02:59,510
Every edge gets cut,
but you're going

45
00:02:59,510 --> 00:03:09,830
to leave intact certain
vertices to make one, quote,

46
00:03:09,830 --> 00:03:14,970
"piece" at vertices.

47
00:03:18,635 --> 00:03:20,240
So we still want one piece.

48
00:03:23,310 --> 00:03:28,869
So this is vertex unfolding,
and it's always possible.

49
00:03:28,869 --> 00:03:30,660
We've even implemented
this algorithm here.

50
00:03:30,660 --> 00:03:32,560
A bunch of random
points on a sphere.

51
00:03:32,560 --> 00:03:37,030
Take the convex hull, and
then, take a vertex unfolding.

52
00:03:37,030 --> 00:03:40,330
So you get this nice
chain of triangles,

53
00:03:40,330 --> 00:03:42,450
don't intersect each other,
and this will fold up

54
00:03:42,450 --> 00:03:45,000
into that 3D
polyhedron on the left.

55
00:03:45,000 --> 00:03:47,591
Here's some bigger examples,
hundreds of vertices.

56
00:03:47,591 --> 00:03:48,090
Amazing.

57
00:03:52,880 --> 00:03:53,380
All right.

58
00:03:53,380 --> 00:03:57,179
So how do we do this?

59
00:03:57,179 --> 00:03:57,678
All right.

60
00:03:57,678 --> 00:04:01,330
Let me state a theorem.

61
00:04:01,330 --> 00:04:14,050
Every connected
triangulated manifold--

62
00:04:14,050 --> 00:04:15,300
this is a very general result.

63
00:04:15,300 --> 00:04:17,536
It actually holds
in any dimension.

64
00:04:17,536 --> 00:04:19,910
We're going to think about
two dimensional sources in 3D,

65
00:04:19,910 --> 00:04:21,493
but it could be D
dimensional surfaces

66
00:04:21,493 --> 00:04:23,963
in D plus 1 dimensions.

67
00:04:23,963 --> 00:04:26,740
--has a vertex unfolding.

68
00:04:34,640 --> 00:04:38,420
So this works both for
convex and for non-convex.

69
00:04:38,420 --> 00:04:42,740
The only catch is that every
face has to be a triangle.

70
00:04:42,740 --> 00:04:44,660
It's an open problem
for something

71
00:04:44,660 --> 00:04:48,955
like a cube where you actually
have quadrilateral faces.

72
00:04:55,260 --> 00:05:00,720
So the way we prove
this is to construct

73
00:05:00,720 --> 00:05:03,125
what we call a facet-path.

74
00:05:08,640 --> 00:05:14,330
This is a path that
alternates between visiting

75
00:05:14,330 --> 00:05:22,910
faces, triangles, and vertices.

76
00:05:28,490 --> 00:05:31,960
So the idea is you
have some-- say

77
00:05:31,960 --> 00:05:34,690
we're doing a
tetrahedron or something.

78
00:05:34,690 --> 00:05:36,020
You start at a facet.

79
00:05:36,020 --> 00:05:37,520
Then, you go to one
of its vertices.

80
00:05:37,520 --> 00:05:41,340
Then, you go to one of the
facets that shares that vertex.

81
00:05:41,340 --> 00:05:42,840
Then, you go to one
of the vertices.

82
00:05:42,840 --> 00:05:44,259
Then, you go to a facet.

83
00:05:44,259 --> 00:05:45,300
Then, you go to a vertex.

84
00:05:45,300 --> 00:05:46,490
Then, you go to a facet.

85
00:05:46,490 --> 00:05:48,380
This is a facet-path.

86
00:05:48,380 --> 00:05:55,680
It should visit every facet,
every triangle, exactly once.

87
00:05:59,730 --> 00:06:01,900
Vertices you can
visit multiple times,

88
00:06:01,900 --> 00:06:04,790
although I think it's a bad idea
to visit the same vertex twice

89
00:06:04,790 --> 00:06:06,460
in a row.

90
00:06:06,460 --> 00:06:08,810
Other than that, you can
visit a vertex more than once.

91
00:06:08,810 --> 00:06:10,964
Maybe I would visit it
again coming up here.

92
00:06:10,964 --> 00:06:12,630
If there was another
triangle like this,

93
00:06:12,630 --> 00:06:18,050
I could go up to this vertex
again and over to the triangle.

94
00:06:18,050 --> 00:06:19,850
So the claim is
facet-paths always

95
00:06:19,850 --> 00:06:25,390
exist for any triangulated,
connected surface.

96
00:06:25,390 --> 00:06:28,690
Once you have this
facet-path, you're

97
00:06:28,690 --> 00:06:36,810
basically golden because you
can lay it out without overlap.

98
00:06:36,810 --> 00:06:40,970
And that's the next picture.

99
00:06:40,970 --> 00:06:46,980
So if you have a triangle and
you have some corners of it

100
00:06:46,980 --> 00:06:49,120
that are hinged to
adjacent triangles,

101
00:06:49,120 --> 00:06:50,930
you can rotate
that triangle 'til

102
00:06:50,930 --> 00:06:52,910
it fits in a vertical slab.

103
00:06:52,910 --> 00:06:55,650
And the hinges are on
the ends of the slab.

104
00:06:55,650 --> 00:06:58,150
And so each triangle
lives in its own slab.

105
00:06:58,150 --> 00:06:59,499
Slabs don't intersect.

106
00:06:59,499 --> 00:07:00,165
No intersection.

107
00:07:13,970 --> 00:07:15,720
The hard part is really
getting this path.

108
00:07:15,720 --> 00:07:18,190
Once you have that, you
can just lay it out.

109
00:07:18,190 --> 00:07:20,540
It's a little bit nontrivial
with obtuse triangles.

110
00:07:20,540 --> 00:07:23,680
They don't necessarily just
lie along the horizontal line.

111
00:07:23,680 --> 00:07:25,940
You have to set things
up so that those guys are

112
00:07:25,940 --> 00:07:28,420
on the boundary,
but you can always

113
00:07:28,420 --> 00:07:29,965
rotate it so that
that's possible.

114
00:07:34,260 --> 00:07:44,920
So the real question is, how
do we construct a facet-path?

115
00:07:53,820 --> 00:07:54,800
All right.

116
00:07:54,800 --> 00:07:56,740
So I have a bunch of triangles.

117
00:07:56,740 --> 00:07:59,360
In general, they
make some surface.

118
00:07:59,360 --> 00:08:04,340
And maybe I should draw a
running example over here.

119
00:08:04,340 --> 00:08:08,960
Let's think about a triangulated
cube, something simple.

120
00:08:13,220 --> 00:08:15,499
Imagine there's more
triangles in the back.

121
00:08:15,499 --> 00:08:17,540
It's a little hard to
think about the 3D picture.

122
00:08:17,540 --> 00:08:20,600
I would really like to
two dimensionalify it,

123
00:08:20,600 --> 00:08:22,740
and because this
theorem works, not only

124
00:08:22,740 --> 00:08:27,620
for a polyhedron-- it works
for any manifold, any sort

125
00:08:27,620 --> 00:08:30,880
of locally, two dimensional
thing-- it would be nice

126
00:08:30,880 --> 00:08:32,919
if I could just sort
of cut this open

127
00:08:32,919 --> 00:08:36,730
and think about a disc instead
of a sphere, topologically.

128
00:08:36,730 --> 00:08:37,436
And I can.

129
00:08:37,436 --> 00:08:39,470
I mean this theorem's supposed
to work for discs just as well.

130
00:08:39,470 --> 00:08:41,110
It should work for
anything that's connected.

131
00:08:41,110 --> 00:08:43,651
So ideally, I cut it all apart
into lots of little triangles,

132
00:08:43,651 --> 00:08:46,100
but it has to stay connected.

133
00:08:46,100 --> 00:08:49,240
So I'll unfold it.

134
00:08:49,240 --> 00:08:49,920
Why not?

135
00:08:49,920 --> 00:08:53,420
So I can add some
cuts until I get down

136
00:08:53,420 --> 00:09:02,160
to a spanning tree of the
faces, a spanning tree

137
00:09:02,160 --> 00:09:03,921
of the dual graph.

138
00:09:03,921 --> 00:09:04,420
OK.

139
00:09:04,420 --> 00:09:09,084
For whatever reason, this is
the triangulation I chose,

140
00:09:09,084 --> 00:09:10,500
and this is the
unfolding I chose.

141
00:09:10,500 --> 00:09:13,230
But just keep cutting
edges until cutting an edge

142
00:09:13,230 --> 00:09:14,990
would cause it to disconnect.

143
00:09:14,990 --> 00:09:17,740
So the maximal set of cuts,
there are many ways to do it.

144
00:09:17,740 --> 00:09:19,512
When I unfold, it might
overlap because we

145
00:09:19,512 --> 00:09:21,220
don't know whether
edge unfoldings exist,

146
00:09:21,220 --> 00:09:22,590
but that's fine.

147
00:09:22,590 --> 00:09:24,810
You can think of it as a
two dimensional picture,

148
00:09:24,810 --> 00:09:26,877
but it might need a few layers.

149
00:09:26,877 --> 00:09:28,710
We're not going to
actually fold this thing,

150
00:09:28,710 --> 00:09:31,500
but we're going to cut
this up into a facet-path.

151
00:09:31,500 --> 00:09:32,910
This could actually
be our input.

152
00:09:32,910 --> 00:09:34,780
Instead of being
given a polyhedron,

153
00:09:34,780 --> 00:09:37,270
we're given a disc like
this, a triangulated disc,

154
00:09:37,270 --> 00:09:38,729
and we have to deal with it.

155
00:09:38,729 --> 00:09:40,770
Somehow we've got to
construct a facet-path here,

156
00:09:40,770 --> 00:09:43,020
visit every triangle
exactly once,

157
00:09:43,020 --> 00:09:44,825
and passing through vertices.

158
00:09:44,825 --> 00:09:46,200
AUDIENCE: So if
you have overlap,

159
00:09:46,200 --> 00:09:47,682
it doesn't really matter.

160
00:09:47,682 --> 00:09:49,640
PROFESSOR: Yeah, overlap
doesn't really matter.

161
00:09:49,640 --> 00:09:50,980
We're going to cut it
up, and then, we're

162
00:09:50,980 --> 00:09:52,413
going to splay
out the triangles.

163
00:09:52,413 --> 00:09:53,788
AUDIENCE: Distances
don't matter.

164
00:09:53,788 --> 00:09:55,150
Right?

165
00:09:55,150 --> 00:09:56,620
PROFESSOR: No, uh.

166
00:09:56,620 --> 00:09:58,620
AUDIENCE: If you're trying
to just find a path--

167
00:09:58,620 --> 00:09:58,712
PROFESSOR: Right.

168
00:09:58,712 --> 00:10:00,410
From the perspective
of facet-path,

169
00:10:00,410 --> 00:10:01,440
distances don't matter.

170
00:10:01,440 --> 00:10:02,360
It's just topology.

171
00:10:02,360 --> 00:10:02,859
Yeah.

172
00:10:02,859 --> 00:10:08,430
You could think of it as a
circle with some decomposition

173
00:10:08,430 --> 00:10:12,000
into triangles, if you like,
but that's maybe harder

174
00:10:12,000 --> 00:10:13,850
to think about.

175
00:10:13,850 --> 00:10:15,630
I got to think
about it this way.

176
00:10:15,630 --> 00:10:17,970
It's useful because then,
we'll get to Mickey Mouses,

177
00:10:17,970 --> 00:10:21,000
as we'll see.

178
00:10:21,000 --> 00:10:30,310
So cut edges until
you can't anymore,

179
00:10:30,310 --> 00:10:31,925
so until cutting
would disconnect.

180
00:10:38,520 --> 00:10:40,730
So this means what
you're left with

181
00:10:40,730 --> 00:10:43,356
will be sort of a tree of faces.

182
00:10:43,356 --> 00:10:45,480
There'll be no cycles
because if there was a cycle,

183
00:10:45,480 --> 00:10:49,770
you could cut one of the edges,
and it wouldn't fall apart.

184
00:10:49,770 --> 00:10:52,570
So obviously,
there's a tree here.

185
00:10:52,570 --> 00:10:54,980
Now, trees have leaves.

186
00:10:54,980 --> 00:10:57,260
That's our favorite
lemma lately.

187
00:10:57,260 --> 00:10:59,480
They have at least two leaves.

188
00:10:59,480 --> 00:11:01,140
In the case of a
triangulated polygon,

189
00:11:01,140 --> 00:11:02,330
we usually call them ears.

190
00:11:02,330 --> 00:11:03,580
So here's a fun term.

191
00:11:06,700 --> 00:11:11,740
So let's say color the ears.

192
00:11:11,740 --> 00:11:19,105
These are leaves and in that
tree which call the dual tree.

193
00:11:22,190 --> 00:11:25,524
So in reality, we're thinking
about a dual graph which

194
00:11:25,524 --> 00:11:27,190
looks something like
this, where there's

195
00:11:27,190 --> 00:11:31,160
a vertex for every triangle and
edges for every edge connecting

196
00:11:31,160 --> 00:11:33,590
triangles, and there's
leaves of that tree.

197
00:11:33,590 --> 00:11:36,820
But in the original thing,
we think of this triangle

198
00:11:36,820 --> 00:11:41,180
as being an ear, and
this triangle is an ear.

199
00:11:41,180 --> 00:11:46,850
And this triangle is an ear,
and this triangle is an ear.

200
00:11:46,850 --> 00:11:49,150
I like ears because they're
kind of on the boundary,

201
00:11:49,150 --> 00:11:51,450
on the surface, so
they're just triangles

202
00:11:51,450 --> 00:11:54,650
that are adjacent to
only one other triangle.

203
00:11:54,650 --> 00:11:57,770
Now, the next step
is to color what

204
00:11:57,770 --> 00:12:00,340
are called the
second-level ears.

205
00:12:00,340 --> 00:12:02,020
I'll call them the
remaining ears,

206
00:12:02,020 --> 00:12:04,970
if you remove those ears, what
would, then, become an ear.

207
00:12:09,300 --> 00:12:09,800
All right.

208
00:12:09,800 --> 00:12:12,531
Now, I wish I had another color.

209
00:12:12,531 --> 00:12:14,260
Yeah, that's kind of yellowish.

210
00:12:14,260 --> 00:12:16,060
So this would become an ear.

211
00:12:16,060 --> 00:12:17,170
This would become an ear.

212
00:12:17,170 --> 00:12:19,742
This would become an ear,
and this would become an ear.

213
00:12:22,340 --> 00:12:26,280
And I'm going to stop
there, just two levels.

214
00:12:26,280 --> 00:12:29,609
What could I get
in this process?

215
00:12:29,609 --> 00:12:31,150
I mean what it looks
like I'm getting

216
00:12:31,150 --> 00:12:33,370
is I get an ear and
then, a second-level ear.

217
00:12:33,370 --> 00:12:35,540
It could be a little
more general than that.

218
00:12:35,540 --> 00:12:38,490
Maybe, for example, if I
had a triangle like this,

219
00:12:38,490 --> 00:12:40,190
both of these would
be first-level ears,

220
00:12:40,190 --> 00:12:42,600
and then, this would
become a second-level ear.

221
00:12:42,600 --> 00:12:45,080
Turns out that's
all that can happen.

222
00:12:45,080 --> 00:12:48,852
Because these are
triangles, second-level ear,

223
00:12:48,852 --> 00:12:51,060
at this point, it's only
adjacent to one other thing.

224
00:12:51,060 --> 00:12:52,760
So what was it
adjacent to before?

225
00:12:52,760 --> 00:12:54,580
Well, maybe, one
ear, certainly not

226
00:12:54,580 --> 00:12:57,000
zero because it wasn't
a first-level ear.

227
00:12:57,000 --> 00:12:59,270
So at least one
ear, maybe two ears.

228
00:12:59,270 --> 00:13:00,157
That's it.

229
00:13:00,157 --> 00:13:02,115
AUDIENCE: Unless there's
only four pieces left.

230
00:13:02,115 --> 00:13:03,062
Right?

231
00:13:03,062 --> 00:13:04,520
PROFESSOR: Sorry,
what do you mean?

232
00:13:04,520 --> 00:13:06,477
AUDIENCE: You could
have three ears.

233
00:13:06,477 --> 00:13:08,060
PROFESSOR: You could
have three years.

234
00:13:08,060 --> 00:13:09,180
Yeah, so here I have two.

235
00:13:09,180 --> 00:13:11,096
I mean the first-level
ear and a second-level,

236
00:13:11,096 --> 00:13:14,020
or it could be a first-level--
Oh, you're saying at the end.

237
00:13:14,020 --> 00:13:14,740
Right.

238
00:13:14,740 --> 00:13:15,970
It could be like this.

239
00:13:15,970 --> 00:13:18,220
That would only happen
at the very end.

240
00:13:18,220 --> 00:13:20,230
But yeah, this could
be first-level ears,

241
00:13:20,230 --> 00:13:21,605
and this is the
second-level ear.

242
00:13:21,605 --> 00:13:22,810
Good point.

243
00:13:22,810 --> 00:13:34,040
Most of the time, we
will either get something

244
00:13:34,040 --> 00:13:37,470
like this, the rest of
the polygons over here--

245
00:13:37,470 --> 00:13:38,780
so this is a first-level ear.

246
00:13:38,780 --> 00:13:41,020
This is a second-level ear.

247
00:13:41,020 --> 00:13:53,110
--or I get-- this is the
Mickey Mouse picture.

248
00:13:53,110 --> 00:13:55,435
Probably not allowed
to say Mickey Mouse.

249
00:13:55,435 --> 00:14:00,650
It's under copyright,
but there you go.

250
00:14:00,650 --> 00:14:02,820
So two first-level
ears, second-level ear.

251
00:14:02,820 --> 00:14:03,677
Boom.

252
00:14:03,677 --> 00:14:05,260
This is, most of the
time, what you'll

253
00:14:05,260 --> 00:14:07,670
get in the base case
of this induction.

254
00:14:07,670 --> 00:14:09,054
I'm going to pluck
off these ears

255
00:14:09,054 --> 00:14:10,470
and keep making
the thing smaller.

256
00:14:10,470 --> 00:14:13,630
At the end, there are a
few cases to think about.

257
00:14:13,630 --> 00:14:17,175
I have them drawn here
somewhere, base cases.

258
00:14:20,190 --> 00:14:21,230
Nothing.

259
00:14:21,230 --> 00:14:21,900
Yeah, well.

260
00:14:30,500 --> 00:14:32,830
This won't work out if
you have one triangle,

261
00:14:32,830 --> 00:14:37,670
if you have zero triangles,
which is this empty picture,

262
00:14:37,670 --> 00:14:39,875
or if you just have two maybe.

263
00:14:39,875 --> 00:14:41,437
Well, I guess--
Yeah, because then,

264
00:14:41,437 --> 00:14:42,770
these are both first-level ears.

265
00:14:42,770 --> 00:14:44,780
It doesn't look quite the same.

266
00:14:44,780 --> 00:14:50,273
Or maybe just a Mickey Mouse
because those are all-- Well,

267
00:14:50,273 --> 00:14:51,590
it probably works.

268
00:14:51,590 --> 00:14:57,082
Anyway, these are the sort
of cases you worry about.

269
00:14:57,082 --> 00:14:58,790
But for these cases,
I just need to check

270
00:14:58,790 --> 00:15:00,140
that I can find a facet-path.

271
00:15:00,140 --> 00:15:02,930
So for example, this one,
I just visit the triangle.

272
00:15:02,930 --> 00:15:06,820
This one-- I don't
know-- I do it like that.

273
00:15:06,820 --> 00:15:11,660
In fact, I can make it a
cycle if I want to go crazy.

274
00:15:11,660 --> 00:15:15,166
This one-- I don't know--
something like this.

275
00:15:15,166 --> 00:15:17,290
That'd be one way to do it.

276
00:15:17,290 --> 00:15:18,290
(SINGING) Doo, doo, doo.

277
00:15:21,400 --> 00:15:24,350
Hm.

278
00:15:24,350 --> 00:15:26,500
Got to be a little careful.

279
00:15:26,500 --> 00:15:27,000
[INAUDIBLE]

280
00:15:30,980 --> 00:15:32,350
Something like that.

281
00:15:32,350 --> 00:15:33,100
All right.

282
00:15:33,100 --> 00:15:37,340
But really, I care
about these two cases.

283
00:15:37,340 --> 00:15:39,856
So what I'm going to
do for each of those--

284
00:15:39,856 --> 00:15:48,430
I guess this is
still step four--

285
00:15:48,430 --> 00:15:53,706
is I actually want to make
cycles for these guys.

286
00:15:53,706 --> 00:15:55,005
I care about that.

287
00:16:03,440 --> 00:16:05,220
Am I doing something wrong?

288
00:16:05,220 --> 00:16:05,720
Oh, no.

289
00:16:05,720 --> 00:16:06,630
Over there.

290
00:16:06,630 --> 00:16:07,130
Good.

291
00:16:09,906 --> 00:16:11,780
So in both of these
cases, I can make cycles.

292
00:16:11,780 --> 00:16:13,720
In this one, in the
base cases, not always.

293
00:16:13,720 --> 00:16:17,770
Like this guy, hard
to make a cycle.

294
00:16:17,770 --> 00:16:19,870
But for the two
general cases, if I

295
00:16:19,870 --> 00:16:22,370
find two ears or three ears,
I can just draw that in.

296
00:16:22,370 --> 00:16:24,756
So I'll do that for this
example, though, by now,

297
00:16:24,756 --> 00:16:26,756
it's gotten a little
messy, so let me redraw it.

298
00:16:30,930 --> 00:16:43,630
So we have-- All right.

299
00:16:43,630 --> 00:16:49,390
So I'm going to connect these
guys and connect these guys.

300
00:16:49,390 --> 00:16:51,080
This case, I only
get the two-ear case,

301
00:16:51,080 --> 00:16:53,220
but I think we'll get
another case shortly.

302
00:17:01,370 --> 00:17:02,180
OK?

303
00:17:02,180 --> 00:17:05,400
So obviously, I haven't finished
it, and these are disconnected.

304
00:17:05,400 --> 00:17:07,750
There's lots of
things left to do.

305
00:17:07,750 --> 00:17:11,170
But then, the idea is repeat.

306
00:17:13,690 --> 00:17:15,880
So imagine those
guys as being done.

307
00:17:15,880 --> 00:17:19,349
I'm left with these
four triangles,

308
00:17:19,349 --> 00:17:22,640
actually, a little
boring because I

309
00:17:22,640 --> 00:17:25,089
don't get the Mickey Mouse case.

310
00:17:25,089 --> 00:17:26,720
But then, this will be an ear.

311
00:17:26,720 --> 00:17:28,410
This will be a second-level ear.

312
00:17:28,410 --> 00:17:32,380
And so I'll end up doing this.

313
00:17:32,380 --> 00:17:34,540
And this will be
an ear, and this

314
00:17:34,540 --> 00:17:37,030
will be an ear,
second-level here.

315
00:17:37,030 --> 00:17:40,369
I mean this is actually
the base case, if you will.

316
00:17:40,369 --> 00:17:42,660
So in general, I just pluck
off two or three triangles,

317
00:17:42,660 --> 00:17:45,040
repeat until I get
one of the base cases.

318
00:17:45,040 --> 00:17:48,110
Now, what I have
is not connected,

319
00:17:48,110 --> 00:17:50,310
but it's a bunch of cycles.

320
00:17:50,310 --> 00:17:54,545
But there's one cycle here, one
cycle there, one cycle there.

321
00:17:54,545 --> 00:17:56,170
You could actually
connect some of them

322
00:17:56,170 --> 00:17:57,211
together, like these two.

323
00:17:57,211 --> 00:17:59,820
You could go around like
this and then, go like that,

324
00:17:59,820 --> 00:18:01,780
so that could be a bigger cycle.

325
00:18:01,780 --> 00:18:04,670
But these cycles are not even
attached to these cycles,

326
00:18:04,670 --> 00:18:06,017
so it's kind of a problem.

327
00:18:06,017 --> 00:18:08,350
That's step five is we're
going to fix all the problems.

328
00:18:11,020 --> 00:18:13,080
That'd be a great title.

329
00:18:13,080 --> 00:18:17,195
Connect cycles together.

330
00:18:20,800 --> 00:18:22,800
We're going to do that
by local switches,

331
00:18:22,800 --> 00:18:24,870
so here's the general picture.

332
00:18:24,870 --> 00:18:28,660
Suppose you have two adjacent
triangles and two completely

333
00:18:28,660 --> 00:18:31,940
separate paths.

334
00:18:31,940 --> 00:18:36,180
So it's a cycle, so
some cycle over here.

335
00:18:36,180 --> 00:18:38,170
And if this guy shared
a vertex, then I

336
00:18:38,170 --> 00:18:39,320
could actually
connect them together.

337
00:18:39,320 --> 00:18:40,861
So suppose it doesn't
share a vertex.

338
00:18:40,861 --> 00:18:42,640
That means it has
to use these two,

339
00:18:42,640 --> 00:18:46,510
and then, it--
something like that.

340
00:18:46,510 --> 00:18:50,940
What I want to do is remove
this edge and remove this edge

341
00:18:50,940 --> 00:18:55,060
and, instead, add this
edge and that edge.

342
00:18:55,060 --> 00:18:58,620
So that's a local change, and
now, it will be one big cycle.

343
00:18:58,620 --> 00:19:00,640
We've probably seen
this trick once or twice

344
00:19:00,640 --> 00:19:05,840
before, I think in the Mountain
Valley assignment stuff.

345
00:19:05,840 --> 00:19:09,940
So here I have, for example,
these two triangles,

346
00:19:09,940 --> 00:19:12,760
which are adjacent, but
the paths don't meet.

347
00:19:12,760 --> 00:19:15,390
So I'm just going to erase
this edge, put in that one,

348
00:19:15,390 --> 00:19:17,295
erase this edge,
put in that one.

349
00:19:17,295 --> 00:19:24,010
Lo and behold, I have a
bigger cycle now, like that.

350
00:19:24,010 --> 00:19:26,410
Still not everything, so
like these two triangles

351
00:19:26,410 --> 00:19:27,510
don't touch.

352
00:19:27,510 --> 00:19:28,400
So erase that edge.

353
00:19:28,400 --> 00:19:29,444
Erase that edge.

354
00:19:29,444 --> 00:19:30,110
Put in that one.

355
00:19:30,110 --> 00:19:31,620
Put in that one.

356
00:19:31,620 --> 00:19:34,140
I'm preserving, at all
times, that I'm a facet path,

357
00:19:34,140 --> 00:19:36,270
and I'm merging components.

358
00:19:36,270 --> 00:19:40,365
So by the end, I'll
have one big component.

359
00:19:40,365 --> 00:19:42,740
Now, I haven't necessarily
described this as a big cycle,

360
00:19:42,740 --> 00:19:44,910
but as I've kind of been
hinting, what I do now

361
00:19:44,910 --> 00:19:46,920
is take an Euler tour
around this thing.

362
00:19:46,920 --> 00:19:48,870
We've seen Euler tours.

363
00:19:48,870 --> 00:19:52,350
All the vertices in this
graph will have even degree.

364
00:19:52,350 --> 00:19:55,960
And so I can take an
Euler tour, and really, I

365
00:19:55,960 --> 00:19:59,310
should take what's called
a non-crossing Euler tour.

366
00:20:01,879 --> 00:20:03,920
Point is you've got a
bunch of cycles that touch.

367
00:20:03,920 --> 00:20:06,540
You just walk
around the outside.

368
00:20:06,540 --> 00:20:08,482
That will visit all the cycles.

369
00:20:08,482 --> 00:20:10,190
It'll visit all these
edges exactly once,

370
00:20:10,190 --> 00:20:11,850
and that will actually
be a facet-path.

371
00:20:18,340 --> 00:20:20,778
And we're done.

372
00:20:20,778 --> 00:20:24,210
Now, I have a facet-path
for a triangulated cube.

373
00:20:24,210 --> 00:20:26,110
So I can splay out
those triangles,

374
00:20:26,110 --> 00:20:28,940
get a non-overlapping
chain like this.

375
00:20:28,940 --> 00:20:33,890
Actually, it's probably the
top one, something like that,

376
00:20:33,890 --> 00:20:34,390
might be.

377
00:20:34,390 --> 00:20:36,640
I may not have matched exactly
what's in the textbook.

378
00:20:40,360 --> 00:20:41,500
Cool.

379
00:20:41,500 --> 00:20:44,480
One slight detail is I was
sort of waving my hands,

380
00:20:44,480 --> 00:20:47,430
assuming there was
actually a cycle here.

381
00:20:47,430 --> 00:20:48,900
I really only need a path.

382
00:20:48,900 --> 00:20:52,270
In fact, there won't always be
a cycle because at the very end,

383
00:20:52,270 --> 00:20:53,930
you're not able to
construct a cycle.

384
00:20:53,930 --> 00:20:56,135
So that will actually
cause you to make--

385
00:20:56,135 --> 00:20:59,690
if I do something like
this, so at least it

386
00:20:59,690 --> 00:21:02,510
can connect to other things.

387
00:21:02,510 --> 00:21:05,702
That will cause you make
two vertices of odd degree.

388
00:21:05,702 --> 00:21:07,160
But that's OK
because there's still

389
00:21:07,160 --> 00:21:10,850
an Euler path that starts at one
of the vertices of odd degree,

390
00:21:10,850 --> 00:21:12,600
visits all the other
edges, and then,

391
00:21:12,600 --> 00:21:15,460
comes to the other
vertex of odd degree.

392
00:21:15,460 --> 00:21:16,925
And I just need a path.

393
00:21:16,925 --> 00:21:19,050
You can actually characterize
when you get a cycle.

394
00:21:19,050 --> 00:21:24,860
It's when the original thing
is not too colorable, I think.

395
00:21:24,860 --> 00:21:28,960
Anyway, that's vertex unfolding.

396
00:21:28,960 --> 00:21:29,710
It's kind of easy.

397
00:21:32,470 --> 00:21:34,837
I think we solved most of
it in like an afternoon.

398
00:21:34,837 --> 00:21:35,545
We had this idea.

399
00:21:35,545 --> 00:21:39,785
It was cool, and then, we
solved it kind of quickly.

400
00:21:39,785 --> 00:21:41,090
That's how it often goes.

401
00:21:41,090 --> 00:21:44,734
Once you have a cool
problem, it falls quickly.

402
00:21:44,734 --> 00:21:47,400
There's lots of interesting open
problems remaining about vertex

403
00:21:47,400 --> 00:21:49,900
unfolding, and they
seem a lot harder.

404
00:21:49,900 --> 00:21:53,920
So for example, what if I have
non-triangulated polyhedra?

405
00:21:53,920 --> 00:21:56,710
And natural version is to think
about what we were originally

406
00:21:56,710 --> 00:21:58,382
trying to attack,
convex polyhedra.

407
00:21:58,382 --> 00:22:00,090
This turned out to
not require convexity.

408
00:22:00,090 --> 00:22:03,290
But what about convex
polyhedra, not triangulated?

409
00:22:03,290 --> 00:22:05,330
Is there always a
vertex unfolding?

410
00:22:05,330 --> 00:22:07,490
We don't know about
edge unfoldings.

411
00:22:07,490 --> 00:22:11,620
And the answer is no.

412
00:22:11,620 --> 00:22:13,710
Well, no that's not right.

413
00:22:13,710 --> 00:22:15,050
Sorry.

414
00:22:15,050 --> 00:22:17,030
The answer is we don't know.

415
00:22:17,030 --> 00:22:18,970
What's annoying
about this example

416
00:22:18,970 --> 00:22:20,610
is that there's no facet-path.

417
00:22:20,610 --> 00:22:22,620
So there are two things
that could go wrong.

418
00:22:22,620 --> 00:22:24,350
One is that the
facet-path doesn't exist,

419
00:22:24,350 --> 00:22:26,690
and that can happen in
this more general scenario.

420
00:22:26,690 --> 00:22:29,400
And the other is that when
you lay it out, it overlaps.

421
00:22:29,400 --> 00:22:31,770
Both of these things
could go wrong

422
00:22:31,770 --> 00:22:33,990
in the general convex case.

423
00:22:33,990 --> 00:22:36,640
So in this example, this
is a truncated cube.

424
00:22:36,640 --> 00:22:41,360
There's eight triangles
and only six octagons.

425
00:22:41,360 --> 00:22:43,847
And if you look, once
you're at a triangle,

426
00:22:43,847 --> 00:22:45,680
you have to go to an
octagon because there's

427
00:22:45,680 --> 00:22:47,380
no adjacent triangles.

428
00:22:47,380 --> 00:22:50,400
So at best, you could alternate
 triangle-octagon-triangle-octagon

429
00:22:50,400 --> 00:22:52,870
if you want to pack all those
triangles in, but you run out

430
00:22:52,870 --> 00:22:54,310
of octagons to get there.

431
00:22:54,310 --> 00:22:58,714
So there's no facet-path,
but maybe, you

432
00:22:58,714 --> 00:22:59,630
don't need facet-path.

433
00:22:59,630 --> 00:23:03,960
You could make a tree,
just a little trickier.

434
00:23:03,960 --> 00:23:06,450
Obviously, if you triangulate
that surface, you're done.

435
00:23:06,450 --> 00:23:11,030
So the analog of general
unfolding with vertex unfolding

436
00:23:11,030 --> 00:23:11,830
is trivial.

437
00:23:11,830 --> 00:23:15,130
You just triangulate, and
then, you use the edge case.

438
00:23:15,130 --> 00:23:17,110
But if you really only
want to cut along edges,

439
00:23:17,110 --> 00:23:18,510
this example can be done.

440
00:23:18,510 --> 00:23:21,450
I mean it has an edge unfolding,
so if it has an edge unfolding,

441
00:23:21,450 --> 00:23:23,150
it definitely has
a vertex unfolding,

442
00:23:23,150 --> 00:23:25,010
but we don't know
how to prove it.

443
00:23:25,010 --> 00:23:27,176
The other thing that could
go wrong is once you have

444
00:23:27,176 --> 00:23:33,690
octagons, even if you could
find a way to lay them out--

445
00:23:33,690 --> 00:23:37,630
that's not a very good
octagon-- you can't-- let's say

446
00:23:37,630 --> 00:23:39,650
if your two hinges
were here and here,

447
00:23:39,650 --> 00:23:42,110
if that's where you attach
two adjacent pieces,

448
00:23:42,110 --> 00:23:44,076
you can't fit that
in a vertical strip.

449
00:23:44,076 --> 00:23:46,450
There's no way to turn it so
it fits in a vertical strip.

450
00:23:46,450 --> 00:23:49,180
So also this layout
problem doesn't

451
00:23:49,180 --> 00:23:52,044
work if you have something
more than triangles.

452
00:23:52,044 --> 00:23:53,585
So some pretty
fascinating questions.

453
00:23:53,585 --> 00:23:57,210
It's even open for non-convex.

454
00:23:57,210 --> 00:24:03,460
If I take non-convex polyhedra,
and I want a vertex unfolding--

455
00:24:03,460 --> 00:24:06,400
let's say all the
faces are convex.

456
00:24:06,400 --> 00:24:10,635
At the very least, you need
to forbid faces having holes.

457
00:24:10,635 --> 00:24:17,480
If you remember this
example, the box on a box,

458
00:24:17,480 --> 00:24:19,720
this also doesn't have
a vertex unfolding

459
00:24:19,720 --> 00:24:23,220
because still, this guy has to
fit in that little square hole.

460
00:24:25,840 --> 00:24:27,930
But as long as the
faces don't have holes--

461
00:24:27,930 --> 00:24:31,310
let's say the faces are convex--
would be a nice version,

462
00:24:31,310 --> 00:24:36,550
like the Witch's Hat,
the spiked tetrahedron.

463
00:24:36,550 --> 00:24:38,280
That probably has
a vertex unfolding.

464
00:24:38,280 --> 00:24:40,902
AUDIENCE: If you triangulate
that, [INAUDIBLE].

465
00:24:40,902 --> 00:24:42,360
PROFESSOR: If you
triangulate this,

466
00:24:42,360 --> 00:24:44,490
it'll definitely have a--
if you triangulate anything,

467
00:24:44,490 --> 00:24:45,740
it'll have a vertex unfolding.

468
00:24:45,740 --> 00:24:47,380
Yeah.

469
00:24:47,380 --> 00:24:47,880
All right.

470
00:24:47,880 --> 00:24:50,110
That's all I want to say
about vertex unfolding,

471
00:24:50,110 --> 00:24:53,060
and that's sort of addressing
edge unfolding of convex

472
00:24:53,060 --> 00:24:57,060
polyhedra, or actually,
both of these.

473
00:24:57,060 --> 00:24:59,485
Now, I'd like to go to
sort of the real problem.

474
00:24:59,485 --> 00:25:01,360
This is one of my favorite
problems, I think.

475
00:25:01,360 --> 00:25:04,520
A bunch of us posed it in '98,
which was right when I was--

476
00:25:04,520 --> 00:25:08,220
or '99, right when
I was starting out.

477
00:25:08,220 --> 00:25:11,020
See, it's tantalizing, in
some ways more natural,

478
00:25:11,020 --> 00:25:13,566
because it's nice to
allow cuts anywhere.

479
00:25:13,566 --> 00:25:15,440
And it could potentially
work for everything.

480
00:25:15,440 --> 00:25:19,850
I conjecture every non-convex
polyhedron without boundary

481
00:25:19,850 --> 00:25:22,220
can be generally
unfolded, and there's

482
00:25:22,220 --> 00:25:26,220
some really good evidence for
this as of a couple years ago.

483
00:25:26,220 --> 00:25:50,720
That's what I want
to talk about,

484
00:25:50,720 --> 00:25:52,045
which is orthogonal polyhedra.

485
00:25:58,810 --> 00:26:02,120
This is one of my favorite
unfolding results.

486
00:26:02,120 --> 00:26:08,830
It's by Mirela Damian, Robin
Flatland, and Joe O'Rourke.

487
00:26:08,830 --> 00:26:11,530
They've done a lot of work
in this unfolding area.

488
00:26:11,530 --> 00:26:14,880
An orthogonal polyhedron is
one where all the faces are

489
00:26:14,880 --> 00:26:17,640
perpendicular to one of
the three coordinate axes.

490
00:26:17,640 --> 00:26:21,160
So for example, here is
a orthogonal polyhedron

491
00:26:21,160 --> 00:26:22,726
I drew this morning.

492
00:26:22,726 --> 00:26:24,470
If you want to draw
orthogonal polyhedra,

493
00:26:24,470 --> 00:26:26,880
Google SketchUp
makes it really easy,

494
00:26:26,880 --> 00:26:30,900
and you can add
shadows and texture.

495
00:26:30,900 --> 00:26:32,970
So there are three
kinds of faces.

496
00:26:32,970 --> 00:26:37,230
There's the ones perpendicular
to x, like these guys.

497
00:26:37,230 --> 00:26:40,470
There's the ones
perpendicular to y.

498
00:26:40,470 --> 00:26:42,494
That's all the yellow faces.

499
00:26:42,494 --> 00:26:44,160
And there's the ones
perpendicular to z.

500
00:26:44,160 --> 00:26:45,870
That's these top guys.

501
00:26:45,870 --> 00:26:47,710
So we call them x-faces,
y-faces, z-faces.

502
00:26:51,480 --> 00:26:54,060
What we're going to do--
so the theorem is these

503
00:26:54,060 --> 00:26:56,540
have general unfoldings.

504
00:26:56,540 --> 00:26:59,520
That's pretty awesome
because every polyhedron

505
00:26:59,520 --> 00:27:03,780
is approximately orthogonal
if you can voxelize it.

506
00:27:03,780 --> 00:27:05,469
So this is really
a lot of stuff.

507
00:27:05,469 --> 00:27:08,010
I would love to generalize this
approach arbitrary polyhedra,

508
00:27:08,010 --> 00:27:11,900
but that's the
big open question.

509
00:27:11,900 --> 00:27:13,380
So what do we do?

510
00:27:13,380 --> 00:27:16,270
Well, we're going to single
out, from this color coding,

511
00:27:16,270 --> 00:27:18,430
the y-faces.

512
00:27:18,430 --> 00:27:20,290
Just color them yellow.

513
00:27:20,290 --> 00:27:21,740
Then, there's all
the other faces.

514
00:27:21,740 --> 00:27:25,090
Well, they form bands.

515
00:27:25,090 --> 00:27:26,470
They're cycles.

516
00:27:26,470 --> 00:27:29,174
They go around in a loop.

517
00:27:29,174 --> 00:27:32,820
A lot of them here, I've just
drawn as rectangular loops,

518
00:27:32,820 --> 00:27:36,280
but in general, all those
wooden faces, the x and z-faces,

519
00:27:36,280 --> 00:27:39,290
will form a bunch of loops.

520
00:27:39,290 --> 00:27:41,120
And then, there's
the y-faces which

521
00:27:41,120 --> 00:27:43,495
we're going to have to deal
with, but sort of ignore them

522
00:27:43,495 --> 00:27:44,374
for a while.

523
00:27:44,374 --> 00:27:46,040
I should have drawn
it with them erased.

524
00:27:46,040 --> 00:27:47,230
It would look cool.

525
00:27:47,230 --> 00:27:50,250
But if you think about
just those bands,

526
00:27:50,250 --> 00:27:53,270
this is sort of how they're
connected together, this tree.

527
00:27:53,270 --> 00:27:57,270
There's a big one out here,
and then, it has two children.

528
00:27:57,270 --> 00:27:59,405
There's this one and this one.

529
00:27:59,405 --> 00:28:02,450
And then, this child has
one child hanging off of it,

530
00:28:02,450 --> 00:28:05,120
and this one has two
children hanging off of it.

531
00:28:05,120 --> 00:28:08,810
So that's these two guys.

532
00:28:08,810 --> 00:28:12,014
In general, this guy might
have some children hanging off

533
00:28:12,014 --> 00:28:12,930
the back side as well.

534
00:28:12,930 --> 00:28:14,880
I'm not going to try to
represent that in this picture.

535
00:28:14,880 --> 00:28:16,350
There's just a
bunch of children.

536
00:28:16,350 --> 00:28:17,725
There will be some
front children

537
00:28:17,725 --> 00:28:18,975
and some back children.

538
00:28:18,975 --> 00:28:20,850
You pick some root
arbitrarily, and then, you

539
00:28:20,850 --> 00:28:22,370
have children
going off of there.

540
00:28:22,370 --> 00:28:26,040
Now, if you're orthogonal
polyhedron has genus 0--

541
00:28:26,040 --> 00:28:29,100
it's topologically a
sphere-- this will be a tree.

542
00:28:29,100 --> 00:28:32,730
If it's like a doughnut,
it will have a cycle.

543
00:28:32,730 --> 00:28:35,420
So this theorem only
applies for our genius zero.

544
00:28:39,380 --> 00:28:44,370
So we're going to exploit
that that dual drawing, how

545
00:28:44,370 --> 00:28:47,540
the bands are connected
together, is like a tree.

546
00:28:47,540 --> 00:28:52,960
So why don't I
write down, a band

547
00:28:52,960 --> 00:29:09,640
is a cycle of x and z-faces,
and they are connected together

548
00:29:09,640 --> 00:29:12,610
in a tree.

549
00:29:12,610 --> 00:29:14,550
Now, the rough
idea is we're going

550
00:29:14,550 --> 00:29:17,180
to take a depth-first
traversal of this tree.

551
00:29:17,180 --> 00:29:18,790
We're going to
start at the root,

552
00:29:18,790 --> 00:29:21,720
work our way down and come
back, work our way down,

553
00:29:21,720 --> 00:29:23,710
and something like that.

554
00:29:23,710 --> 00:29:25,360
It's not going to
be quite so simple.

555
00:29:28,710 --> 00:29:31,910
The challenge, I guess you
could say, is avoiding overlap.

556
00:29:31,910 --> 00:29:32,410
OK?

557
00:29:32,410 --> 00:29:34,451
If you wanted to unfold
a band, obviously, a band

558
00:29:34,451 --> 00:29:36,010
can just unfold straight.

559
00:29:36,010 --> 00:29:38,630
It's like a nice, long strip.

560
00:29:38,630 --> 00:29:40,220
So each band,
individually, is fine.

561
00:29:40,220 --> 00:29:42,386
It's how do you piece those
bands together and then,

562
00:29:42,386 --> 00:29:45,100
have room for the
yellow faces to attach

563
00:29:45,100 --> 00:29:49,366
on the sides, no overlap?

564
00:29:49,366 --> 00:29:53,690
But it can be done with
the awesome, crazy idea

565
00:29:53,690 --> 00:29:54,815
that we'll get to shortly.

566
00:29:57,452 --> 00:29:59,160
It's going to start
out kind of innocent,

567
00:29:59,160 --> 00:30:11,450
but the general
approach is always

568
00:30:11,450 --> 00:30:18,640
proceed rightward
in the unfolding.

569
00:30:21,870 --> 00:30:27,495
So the unfolding will
look something like this.

570
00:30:36,910 --> 00:30:38,587
OK, whatever.

571
00:30:38,587 --> 00:30:39,670
Always going to the right.

572
00:30:39,670 --> 00:30:42,540
We start here, and we
might go up and down,

573
00:30:42,540 --> 00:30:46,080
but we never go left.

574
00:30:46,080 --> 00:30:49,100
And then, that's going
to be all the band faces.

575
00:30:49,100 --> 00:30:51,102
All the band stuff will
be connected like that,

576
00:30:51,102 --> 00:30:53,060
and then, there's going
to be yellow faces that

577
00:30:53,060 --> 00:30:54,226
can just hang off the sides.

578
00:30:56,809 --> 00:30:57,850
So these are the y-faces.

579
00:31:01,140 --> 00:31:03,130
As long as I get
the band to do this,

580
00:31:03,130 --> 00:31:05,260
y-faces can hang up and down.

581
00:31:05,260 --> 00:31:07,600
It's not going to
intersect anybody.

582
00:31:07,600 --> 00:31:08,580
Pretty clear?

583
00:31:08,580 --> 00:31:10,060
So it's clear if
we could do this,

584
00:31:10,060 --> 00:31:12,060
we can get non-selfintersection.

585
00:31:12,060 --> 00:31:15,050
The amazing thing is
that this is possible.

586
00:31:15,050 --> 00:31:16,900
What this is essentially
a limitation on

587
00:31:16,900 --> 00:31:19,900
is how far you could
turn your bearing.

588
00:31:19,900 --> 00:31:21,502
So you start going right.

589
00:31:21,502 --> 00:31:23,710
You can turn right, but I
could not turn right again.

590
00:31:23,710 --> 00:31:25,371
I have to turn left next.

591
00:31:25,371 --> 00:31:27,120
I can actually do two
left turns in a row,

592
00:31:27,120 --> 00:31:29,145
as long as I was
initially going down.

593
00:31:29,145 --> 00:31:31,770
I can turn left twice, then,
I could alternate left-right.

594
00:31:31,770 --> 00:31:32,650
That's always good.

595
00:31:32,650 --> 00:31:37,940
Or I could turn right twice, but
only if I'm initially going up.

596
00:31:37,940 --> 00:31:38,780
That's the rules.

597
00:31:38,780 --> 00:31:41,500
As long as I adhere to
those rules, I'm fine.

598
00:31:41,500 --> 00:31:43,950
Now, we're going
to heavily exploit

599
00:31:43,950 --> 00:31:45,760
that we can do
general unfolding,

600
00:31:45,760 --> 00:31:48,260
that we can subdivide
those strips

601
00:31:48,260 --> 00:31:50,210
into lots of little pieces.

602
00:31:50,210 --> 00:31:52,600
We're going to subdivide
into a lot of little pieces,

603
00:31:52,600 --> 00:32:00,100
an exponential number of pieces,
so this is kind of hard core.

604
00:32:00,100 --> 00:32:02,210
So here is one example.

605
00:32:05,430 --> 00:32:07,190
This is a leaf.

606
00:32:07,190 --> 00:32:09,360
So trees have leaves,
and at the end,

607
00:32:09,360 --> 00:32:10,860
we're going to have
to visit a leaf.

608
00:32:10,860 --> 00:32:13,140
So this is one box.

609
00:32:13,140 --> 00:32:15,250
There's this funny
view, so you can

610
00:32:15,250 --> 00:32:19,490
see like a mirror on the bottom
and a mirror on the right

611
00:32:19,490 --> 00:32:21,780
and on the side.

612
00:32:21,780 --> 00:32:27,850
So this is if you had one
box, here's what you would do.

613
00:32:27,850 --> 00:32:29,800
And we're actually
assuming-- notice

614
00:32:29,800 --> 00:32:34,710
this side does not get covered.

615
00:32:34,710 --> 00:32:36,210
The idea is that
site doesn't exist.

616
00:32:36,210 --> 00:32:37,920
That's attached to the
rest of the polyhedron,

617
00:32:37,920 --> 00:32:39,410
so that's where
our parent lives.

618
00:32:39,410 --> 00:32:42,600
And our parent tells us
you have to start at s,

619
00:32:42,600 --> 00:32:45,280
and it says you
better finish at t.

620
00:32:45,280 --> 00:32:49,180
And I want the property
that if initially,

621
00:32:49,180 --> 00:32:51,720
I think, initially,
you're going right.

622
00:32:51,720 --> 00:32:55,900
No, it looks like
initially, you're going up.

623
00:32:55,900 --> 00:32:57,680
It matters.

624
00:32:57,680 --> 00:32:59,910
You can do stuff,
and then, at the end,

625
00:32:59,910 --> 00:33:02,060
you should still be facing up.

626
00:33:02,060 --> 00:33:06,395
So you have to visit all these
faces, but not turn in total.

627
00:33:06,395 --> 00:33:08,020
And normally, that
would be hard to do.

628
00:33:08,020 --> 00:33:09,936
If you just tried to
visit one face at a time,

629
00:33:09,936 --> 00:33:12,640
you can't do that,
but if you visit

630
00:33:12,640 --> 00:33:15,660
faces multiple times and kind
of weave around in a clever way,

631
00:33:15,660 --> 00:33:16,880
you can do it.

632
00:33:16,880 --> 00:33:24,061
So maybe I'll point
with this thing.

633
00:33:24,061 --> 00:33:24,560
Yeah.

634
00:33:24,560 --> 00:33:25,910
So I start at s.

635
00:33:25,910 --> 00:33:27,060
I go up.

636
00:33:27,060 --> 00:33:28,670
I turn right.

637
00:33:28,670 --> 00:33:29,700
Now, I better turn left.

638
00:33:29,700 --> 00:33:32,840
I go down over here, up there.

639
00:33:32,840 --> 00:33:34,140
I turn left.

640
00:33:34,140 --> 00:33:36,560
Then, I turn right.

641
00:33:36,560 --> 00:33:40,420
So if you follow along here,
I just turn right here.

642
00:33:40,420 --> 00:33:41,890
So now, I go down here.

643
00:33:41,890 --> 00:33:44,780
And that is a left
turn because it's

644
00:33:44,780 --> 00:33:46,990
on the bottom, a little
hard to think about.

645
00:33:46,990 --> 00:33:52,330
So I turn left here, and
then, I go, turn right.

646
00:33:52,330 --> 00:33:58,900
And then, I go down, turn
left and then, right.

647
00:33:58,900 --> 00:34:00,560
This is confusing
that I'm upside down.

648
00:34:00,560 --> 00:34:06,310
And I come to t, and lo and
behold, I'm facing up again.

649
00:34:06,310 --> 00:34:08,520
In fact, I basically
just zigzagged.

650
00:34:08,520 --> 00:34:11,231
This would also work if
it was rotated 90 degrees.

651
00:34:11,231 --> 00:34:13,480
I'd initially be going right
and, then, down and right

652
00:34:13,480 --> 00:34:14,659
and down and right and down.

653
00:34:14,659 --> 00:34:16,090
So this can kind
of go in a couple

654
00:34:16,090 --> 00:34:17,726
of different orientations.

655
00:34:17,726 --> 00:34:18,600
It's really powerful.

656
00:34:18,600 --> 00:34:20,474
It also works if t is
on the other side of s.

657
00:34:20,474 --> 00:34:22,560
You could do sort of the
mirror image traversal.

658
00:34:22,560 --> 00:34:24,949
Now, obviously, I didn't
cover the entire surface.

659
00:34:24,949 --> 00:34:27,940
I'm leaving room for
later, but if this

660
00:34:27,940 --> 00:34:32,710
was all I was going to do, I
would actually sort of fill out

661
00:34:32,710 --> 00:34:35,360
all those strips, just
kind of extend them.

662
00:34:35,360 --> 00:34:37,469
It just makes this
kind of fatter.

663
00:34:37,469 --> 00:34:39,070
So this got a little bigger.

664
00:34:39,070 --> 00:34:42,290
I've got the first half
and then, the second half.

665
00:34:42,290 --> 00:34:44,929
This is really glued up there.

666
00:34:44,929 --> 00:34:47,489
But you can imagine once you
have these sort of paths that

667
00:34:47,489 --> 00:34:49,889
visit everything,
you just fatten out,

668
00:34:49,889 --> 00:34:51,889
and it's no big deal.

669
00:34:51,889 --> 00:34:53,770
And also, in this
case here, we're

670
00:34:53,770 --> 00:34:56,870
imagining-- oh, this is
actually two of them.

671
00:34:56,870 --> 00:34:58,040
Fun.

672
00:34:58,040 --> 00:35:00,770
Two of these strips
joined together.

673
00:35:00,770 --> 00:35:02,670
And so there's a few side faces.

674
00:35:02,670 --> 00:35:04,415
They just attach up and down.

675
00:35:04,415 --> 00:35:06,040
They're not going to
intersect anything

676
00:35:06,040 --> 00:35:07,770
because this is not
actually below this.

677
00:35:07,770 --> 00:35:08,686
This is way over here.

678
00:35:12,330 --> 00:35:16,610
So that's the
leaves, and I still

679
00:35:16,610 --> 00:35:18,110
haven't gotten to
the exciting part.

680
00:35:20,940 --> 00:35:23,010
So imagine you have a band.

681
00:35:23,010 --> 00:35:25,889
Just going to represent
that by this big rectangle,

682
00:35:25,889 --> 00:35:27,180
and it has a bunch of children.

683
00:35:27,180 --> 00:35:29,030
Remember, it can
have front children.

684
00:35:29,030 --> 00:35:31,370
This is down in
the y-coordinate.

685
00:35:31,370 --> 00:35:37,640
And it could have back children
up in the y-coordinate.

686
00:35:37,640 --> 00:35:41,070
And suppose I'm actually
attached to some parent,

687
00:35:41,070 --> 00:35:45,011
let's say, as a down neighbor,
and I start at some s.

688
00:35:45,011 --> 00:35:46,760
Let's say I'm told you
have to start here.

689
00:35:46,760 --> 00:35:49,340
You have to finish here, and
you should preserve orientation

690
00:35:49,340 --> 00:35:52,519
because I don't want to have to
think about whether it turned,

691
00:35:52,519 --> 00:35:54,060
don't want to have
to depend on that.

692
00:35:54,060 --> 00:35:56,127
So you can do it.

693
00:35:56,127 --> 00:35:58,460
Initially, you must be facing
up because immediately I'm

694
00:35:58,460 --> 00:35:59,892
going to make two right turns.

695
00:35:59,892 --> 00:36:01,350
And I could handle
two right turns,

696
00:36:01,350 --> 00:36:03,141
as long as the next
thing I did was a left.

697
00:36:03,141 --> 00:36:05,210
So I come into
this thing saying,

698
00:36:05,210 --> 00:36:07,940
look, you're facing--
this is facing up.

699
00:36:07,940 --> 00:36:09,430
Now, you're facing down.

700
00:36:09,430 --> 00:36:11,870
You better turn left
next, and by the end,

701
00:36:11,870 --> 00:36:13,590
I still want to be facing down.

702
00:36:13,590 --> 00:36:14,090
OK.

703
00:36:14,090 --> 00:36:15,409
Now, I make two left turns.

704
00:36:15,409 --> 00:36:16,450
Now, I'm facing up again.

705
00:36:16,450 --> 00:36:20,230
I tell this guy, you better
turn right next, and preserve.

706
00:36:20,230 --> 00:36:22,084
At the end, you should
be facing up again.

707
00:36:22,084 --> 00:36:23,250
Now, I make two right turns.

708
00:36:23,250 --> 00:36:25,160
Now, I'm facing down,
and so on and so on.

709
00:36:25,160 --> 00:36:28,380
And here, I'm wrapping
around to here

710
00:36:28,380 --> 00:36:30,829
because this is actually
a band that cycles around.

711
00:36:30,829 --> 00:36:31,620
Then, I go in here.

712
00:36:31,620 --> 00:36:32,520
It's the same thing.

713
00:36:32,520 --> 00:36:34,820
It's just a little
hard to see because I'm

714
00:36:34,820 --> 00:36:37,970
drawing it on a flat surface.

715
00:36:37,970 --> 00:36:40,170
But if it was on a ring,
it would be much clearer.

716
00:36:40,170 --> 00:36:42,190
Just going left and
right and left and right,

717
00:36:42,190 --> 00:36:46,570
alternating direction, so that
I preserve my orientation.

718
00:36:46,570 --> 00:36:50,160
At some point, I get to here.

719
00:36:50,160 --> 00:36:51,090
I loop around.

720
00:36:51,090 --> 00:36:53,530
I make a little
wiggle at some point,

721
00:36:53,530 --> 00:36:55,429
and then, I can visit
all the top neighbors.

722
00:36:55,429 --> 00:36:57,970
You just have to slightly switch
your orientation, but again,

723
00:36:57,970 --> 00:37:00,310
preserve that you're doing
left-left-right-right,

724
00:37:00,310 --> 00:37:02,890
left-left-right-right,
left-left-right-right.

725
00:37:02,890 --> 00:37:05,030
Then, you will preserve
your orientation.

726
00:37:05,030 --> 00:37:07,363
You tell each of the children
which way you're initially

727
00:37:07,363 --> 00:37:10,070
going, and they
can deal with it.

728
00:37:10,070 --> 00:37:11,969
It's basically telling
you whether s and t

729
00:37:11,969 --> 00:37:13,510
is like this or the
other way around.

730
00:37:16,380 --> 00:37:17,350
OK?

731
00:37:17,350 --> 00:37:20,000
So now, we end up
way up there where

732
00:37:20,000 --> 00:37:24,100
the lavender edge is at t10.

733
00:37:24,100 --> 00:37:25,620
Now what?

734
00:37:25,620 --> 00:37:29,040
We want to come back
here, and I'm not

735
00:37:29,040 --> 00:37:30,587
allowed to sort of
intersect myself.

736
00:37:30,587 --> 00:37:32,420
That would be the paper
going into two parts

737
00:37:32,420 --> 00:37:34,620
of this unfolding,
so that's not good.

738
00:37:34,620 --> 00:37:37,700
But I have all this
space, so natural thing

739
00:37:37,700 --> 00:37:40,670
is to just wander from
there back down to here,

740
00:37:40,670 --> 00:37:42,450
using up the space.

741
00:37:42,450 --> 00:37:45,650
So it's going to look like this.

742
00:37:45,650 --> 00:37:49,090
Everything that you
did, you just undo.

743
00:37:49,090 --> 00:37:51,310
Now, where this is
painful is not only

744
00:37:51,310 --> 00:37:53,030
do I have to undo
it in this diagram,

745
00:37:53,030 --> 00:37:54,924
but I have to recursively
undo everything

746
00:37:54,924 --> 00:37:57,340
I did in here, everything I
didn't in here, and everything

747
00:37:57,340 --> 00:37:58,400
I did in here.

748
00:37:58,400 --> 00:38:01,930
So this recursive thing
from this structure

749
00:38:01,930 --> 00:38:03,360
ends up getting doubled.

750
00:38:03,360 --> 00:38:05,700
At the parent structure,
it will also get doubled.

751
00:38:05,700 --> 00:38:07,620
At every level of
the tree, you're

752
00:38:07,620 --> 00:38:10,144
going to double
what was below you,

753
00:38:10,144 --> 00:38:12,060
so that's why you get
exponential, in general.

754
00:38:12,060 --> 00:38:21,941
If your tree is ugly
like something like this,

755
00:38:21,941 --> 00:38:23,690
you'll start with
something nice and small

756
00:38:23,690 --> 00:38:26,100
down here, maybe
constant number of terms.

757
00:38:26,100 --> 00:38:26,970
Then, you'll double.

758
00:38:26,970 --> 00:38:27,730
Then, you'll double again.

759
00:38:27,730 --> 00:38:28,490
Then, you'll double again.

760
00:38:28,490 --> 00:38:30,910
Then, you'll double again,
and it will be exponential.

761
00:38:30,910 --> 00:38:34,230
So if this is n, the number
of things you're doing here

762
00:38:34,230 --> 00:38:37,055
is 2 theta n.

763
00:38:37,055 --> 00:38:38,680
On the other hand,
if your tree happens

764
00:38:38,680 --> 00:38:43,930
to be nice and balanced,
doubling is not so bad

765
00:38:43,930 --> 00:38:46,270
because here you'll
have constant.

766
00:38:46,270 --> 00:38:48,507
This will double
everything below.

767
00:38:48,507 --> 00:38:51,090
This a double everything below,
but there's only log n levels.

768
00:38:54,610 --> 00:38:57,486
So is that linear?

769
00:38:57,486 --> 00:38:59,610
It should be about linear.

770
00:38:59,610 --> 00:39:03,140
It's certainly 2
to the theta log n,

771
00:39:03,140 --> 00:39:04,800
and it matters what
this constant is.

772
00:39:04,800 --> 00:39:09,790
I think it's n or maybe n
squared, but not too bad.

773
00:39:09,790 --> 00:39:12,330
So if you're lucky and the just
the structure of your bands

774
00:39:12,330 --> 00:39:14,220
is balanced, it's good.

775
00:39:14,220 --> 00:39:16,600
In general, though, it's
going to be exponential.

776
00:39:16,600 --> 00:39:18,420
Open problem.

777
00:39:18,420 --> 00:39:21,100
Can you deal with these
situations and sort of balance

778
00:39:21,100 --> 00:39:24,670
them and make them only make
a polynomial number of cuts.

779
00:39:24,670 --> 00:39:26,320
Certainly be nice.

780
00:39:26,320 --> 00:39:31,960
Exponential number cuts
is a lot, but it works.

781
00:39:31,960 --> 00:39:34,380
You can unfold every
orthogonal polyhedron this way.

782
00:39:34,380 --> 00:39:35,921
I would love to see
an implementation

783
00:39:35,921 --> 00:39:37,610
of this algorithm.

784
00:39:37,610 --> 00:39:39,069
You could only do
it in a computer

785
00:39:39,069 --> 00:39:41,360
because you'd be splicing
into all these little things,

786
00:39:41,360 --> 00:39:42,960
and it would fall apart.

787
00:39:42,960 --> 00:39:44,400
Jason?

788
00:39:44,400 --> 00:39:47,476
AUDIENCE: You've been
making these, I guess,

789
00:39:47,476 --> 00:39:55,305
gadgets [INAUDIBLE] voxel
would attach just by a side.

790
00:39:55,305 --> 00:39:59,091
You could imagine it attaching
at a corner attaching

791
00:39:59,091 --> 00:40:01,550
to multiple sides.

792
00:40:01,550 --> 00:40:04,655
PROFESSOR: So you're worried
about-- not quite sure.

793
00:40:07,210 --> 00:40:10,840
What we think about is one
band, which looks something

794
00:40:10,840 --> 00:40:15,090
like this, attaching
to another band.

795
00:40:15,090 --> 00:40:17,310
That will always
happen on a face.

796
00:40:20,582 --> 00:40:22,290
I'm not quite sure
what you're imagining.

797
00:40:22,290 --> 00:40:29,260
Maybe something like that where
they share a partial face here?

798
00:40:29,260 --> 00:40:30,665
AUDIENCE: Yeah,
but it could also

799
00:40:30,665 --> 00:40:31,980
be inset into the [INAUDIBLE].

800
00:40:31,980 --> 00:40:35,320
PROFESSOR: If it's inset, I'm
cutting with every-- I maybe

801
00:40:35,320 --> 00:40:37,940
didn't mention that--
through every vertex,

802
00:40:37,940 --> 00:40:40,230
I'm going to slice
with a y-plane.

803
00:40:40,230 --> 00:40:42,252
So that will cut into
lots of little strips,

804
00:40:42,252 --> 00:40:44,460
and then, there's no sort
of overlap with the strips.

805
00:40:44,460 --> 00:40:47,299
I'm subdividing into
little substrips.

806
00:40:47,299 --> 00:40:48,840
So that sort of
deals with that issue

807
00:40:48,840 --> 00:40:51,760
if this was moved that way.

808
00:40:51,760 --> 00:40:53,592
Then, there will be
three strips, one

809
00:40:53,592 --> 00:40:55,300
over here, one where
they're overlapping,

810
00:40:55,300 --> 00:40:57,020
and one on the right.

811
00:40:57,020 --> 00:40:57,520
Yeah.

812
00:40:57,520 --> 00:40:58,103
Good question.

813
00:40:58,103 --> 00:41:02,190
I forgot to mention the
subdivision at the beginning.

814
00:41:02,190 --> 00:41:05,520
Speaking of subdivision
at the beginning,

815
00:41:05,520 --> 00:41:09,245
this leads to another notion
which we call grid unfolding.

816
00:41:37,010 --> 00:41:41,610
So the grid of an
orthogonal polyhedron

817
00:41:41,610 --> 00:41:45,850
is what you get
when you subdivide

818
00:41:45,850 --> 00:42:01,730
by extending every
face into a plane

819
00:42:01,730 --> 00:42:03,455
and cutting with that plane.

820
00:42:03,455 --> 00:42:05,080
So that's sort of
what I was doing here

821
00:42:05,080 --> 00:42:06,121
when they're overlapping.

822
00:42:06,121 --> 00:42:10,439
Even in this picture,
there's like a face here,

823
00:42:10,439 --> 00:42:11,980
this vertical face,
and so I'd end up

824
00:42:11,980 --> 00:42:15,755
slicing through this thing
with a band in that direction.

825
00:42:19,590 --> 00:42:21,927
And I don't know,
this vertex you'll

826
00:42:21,927 --> 00:42:23,260
end up slicing through this guy.

827
00:42:26,690 --> 00:42:28,800
So hopefully, you
can imagine that.

828
00:42:28,800 --> 00:42:31,400
With every vertex,
slice x, y, and z.

829
00:42:31,400 --> 00:42:32,620
That's another way to do it.

830
00:42:32,620 --> 00:42:35,840
And that subdivides
in sort of a nice grid

831
00:42:35,840 --> 00:42:38,370
where every face will
now be a rectangle,

832
00:42:38,370 --> 00:42:41,730
and rectangles always meet
whole edge to whole edge.

833
00:42:41,730 --> 00:42:43,400
So it's a nice simplification.

834
00:42:43,400 --> 00:42:46,870
What I'm proposing is add
those edges to your polyhedron.

835
00:42:46,870 --> 00:42:49,670
It's kind of like assuming that
you started with unit cubes

836
00:42:49,670 --> 00:42:54,710
and build something but kept
all the edges of all the cubes.

837
00:42:54,710 --> 00:42:57,700
I want an edge
unfolding of that.

838
00:42:57,700 --> 00:42:58,720
Do those exist?

839
00:42:58,720 --> 00:43:00,545
These are what we
call grid unfoldings.

840
00:43:07,270 --> 00:43:11,787
So grid unfolding is an
edge unfolding of the grid.

841
00:43:11,787 --> 00:43:13,745
This only makes sense
for orthogonal polyhedra.

842
00:43:20,460 --> 00:43:21,730
Open question.

843
00:43:21,730 --> 00:43:24,300
Do grid unfoldings always exist?

844
00:43:24,300 --> 00:43:28,170
It's essentially the
orthogonal analog

845
00:43:28,170 --> 00:43:33,587
of this question, edge
unfolding a convex polyhedra.

846
00:43:33,587 --> 00:43:35,170
If you want to go
to orthogonal, which

847
00:43:35,170 --> 00:43:38,832
is like in between convex
and general non-convex,

848
00:43:38,832 --> 00:43:40,540
maybe you could hope
for grid unfoldings.

849
00:43:40,540 --> 00:43:42,081
Edge unfoldings
obviously don't work.

850
00:43:42,081 --> 00:43:45,600
We had lots of examples
where those fail,

851
00:43:45,600 --> 00:43:49,990
like the cube with little
bites taken out of the edges.

852
00:43:49,990 --> 00:43:52,220
But grid unfolding, you
get lots of subdivision.

853
00:43:52,220 --> 00:43:53,590
It might be easy.

854
00:43:53,590 --> 00:43:55,660
Well, it's not easy.

855
00:43:55,660 --> 00:43:58,760
I would guess, actually,
it's not possible.

856
00:43:58,760 --> 00:44:01,810
The next best thing you
could hope for is to refine.

857
00:44:08,490 --> 00:44:14,880
So you take each of
the grid rectangles

858
00:44:14,880 --> 00:44:20,350
and divide it into
k by k, so subgrid.

859
00:44:26,460 --> 00:44:29,780
So ideally, k is one, and
you're not subdividing at all.

860
00:44:29,780 --> 00:44:33,420
But maybe, you take every
rectangle, divide it in half.

861
00:44:33,420 --> 00:44:36,640
Maybe that's enough to
then be edge unfoldable.

862
00:44:36,640 --> 00:44:42,220
That would be sort of a refined
level two grid-like unfolding.

863
00:44:42,220 --> 00:44:45,590
There are a ton of
results about this.

864
00:44:45,590 --> 00:44:47,670
They're all partial.

865
00:44:47,670 --> 00:44:49,610
Obviously, in the
general situation--

866
00:44:49,610 --> 00:44:51,480
I mean this algorithm
we just covered--

867
00:44:51,480 --> 00:44:53,720
you can achieve a
refinement of only 2

868
00:44:53,720 --> 00:44:57,170
to the theta n exponential.

869
00:44:57,170 --> 00:44:59,030
When can you do better?

870
00:44:59,030 --> 00:45:03,040
Ideally, you get 1 by 1, but
maybe, you can do something.

871
00:45:10,875 --> 00:45:11,375
What?

872
00:45:14,870 --> 00:45:15,630
OK.

873
00:45:15,630 --> 00:45:16,130
Interesting.

874
00:45:22,420 --> 00:45:28,620
One thing you could do, with
merely 5 by 4 refinement,

875
00:45:28,620 --> 00:45:30,690
is something called
Manhattan Towers.

876
00:45:30,690 --> 00:45:34,329
Let me show you a picture
of Manhattan Tower.

877
00:45:34,329 --> 00:45:36,120
No that's not a picture
of Manhattan Tower.

878
00:45:36,120 --> 00:45:39,410
This is more crazy examples
of what it's like to visit.

879
00:45:39,410 --> 00:45:42,260
This is not-- this
probably is complete,

880
00:45:42,260 --> 00:45:45,250
but here, there isn't too much
doubling because there's only

881
00:45:45,250 --> 00:45:47,920
a single child, more
or less, everywhere.

882
00:45:47,920 --> 00:45:51,350
But the unfolding looks
something like that.

883
00:45:51,350 --> 00:45:53,160
Here is Manhattan Tower.

884
00:45:53,160 --> 00:45:59,070
So it has a connected
base on the x-y plane.

885
00:45:59,070 --> 00:46:04,780
And I want to consider z-slices
as I go up in z-coordinate,

886
00:46:04,780 --> 00:46:07,810
and I want those z-slices to
get smaller and smaller, always

887
00:46:07,810 --> 00:46:09,930
contained in what I had before.

888
00:46:09,930 --> 00:46:11,260
So I never have overhang.

889
00:46:11,260 --> 00:46:13,550
That's a Manhattan Tower.

890
00:46:13,550 --> 00:46:16,240
And in that case,
5 by 4 refinement

891
00:46:16,240 --> 00:46:19,080
is enough to unfold
these things.

892
00:46:19,080 --> 00:46:22,020
So that's pretty good, still
not perfect, but pretty good.

893
00:46:22,020 --> 00:46:24,620
And this is by the same
authors, like a year

894
00:46:24,620 --> 00:46:27,974
before the general result.

895
00:46:27,974 --> 00:46:29,650
Let's see.

896
00:46:29,650 --> 00:46:31,350
I think maybe I have a movie.

897
00:46:31,350 --> 00:46:31,995
Yeah.

898
00:46:31,995 --> 00:46:33,850
So this algorithm
has been implemented,

899
00:46:33,850 --> 00:46:36,740
at least in some
simple examples.

900
00:46:36,740 --> 00:46:37,990
And it kind of nicely unrolls.

901
00:46:37,990 --> 00:46:41,320
You can see a 5 by 4 refinement
in that little staircase.

902
00:46:41,320 --> 00:46:44,750
It's, again, to make everything
keep going to the right,

903
00:46:44,750 --> 00:46:47,630
but here, they find a clever
way to visit all the faces

904
00:46:47,630 --> 00:46:51,415
without having to revisit,
basically, at all,

905
00:46:51,415 --> 00:46:53,540
just visiting each face a
constant number of times.

906
00:46:59,990 --> 00:47:03,530
And then, we can zoom out,
and you get the unfolding.

907
00:47:03,530 --> 00:47:05,354
So it looks very
similar in spirit.

908
00:47:05,354 --> 00:47:07,770
Of course, the details is how
do you do all that visiting.

909
00:47:07,770 --> 00:47:10,970
I'm not going to
cover that here,

910
00:47:10,970 --> 00:47:13,050
but you get substantially
less refinement

911
00:47:13,050 --> 00:47:14,100
for that special case.

912
00:47:18,150 --> 00:47:18,810
Yeah.

913
00:47:18,810 --> 00:47:20,710
Another case looks like this.

914
00:47:20,710 --> 00:47:21,380
Boom!

915
00:47:21,380 --> 00:47:22,555
AUDIENCE: Woah.

916
00:47:22,555 --> 00:47:23,680
PROFESSOR: Isn't that cool?

917
00:47:23,680 --> 00:47:24,780
I'll play it again.

918
00:47:24,780 --> 00:47:27,190
This is just slightly
more special.

919
00:47:27,190 --> 00:47:30,907
So again-- I have three of them.

920
00:47:30,907 --> 00:47:31,740
They're so much fun.

921
00:47:31,740 --> 00:47:33,591
It's like exploding a city.

922
00:47:33,591 --> 00:47:34,090
Boom!

923
00:47:37,220 --> 00:47:40,300
So here, the floor
is a rectangle.

924
00:47:40,300 --> 00:47:42,590
That's the only
additional requirement,

925
00:47:42,590 --> 00:47:47,270
and again, as you slice upwards,
things only get smaller.

926
00:47:47,270 --> 00:47:48,140
Here's a bigger one.

927
00:47:50,760 --> 00:47:52,850
Boom!

928
00:47:52,850 --> 00:47:54,680
So exciting.

929
00:47:54,680 --> 00:47:55,700
I could do this all day.

930
00:47:58,860 --> 00:48:02,970
So I'm not going to
describe how this works,

931
00:48:02,970 --> 00:48:06,040
but you could almost reconstruct
it from these diagrams.

932
00:48:06,040 --> 00:48:10,060
This is what we call
an orthogonal terrain.

933
00:48:10,060 --> 00:48:13,840
This result by Joe O'Rourke.

934
00:48:13,840 --> 00:48:17,190
Here, you don't need any
refinement, grid unfolding, one

935
00:48:17,190 --> 00:48:18,097
by one.

936
00:48:18,097 --> 00:48:18,930
That's pretty sweet.

937
00:48:21,671 --> 00:48:22,170
All right.

938
00:48:22,170 --> 00:48:26,400
Next one is what we
call orthostacks.

939
00:48:26,400 --> 00:48:30,460
These are like a stack of a
bunch of orthogonal polygons,

940
00:48:30,460 --> 00:48:31,670
a bunch of bands, basically.

941
00:48:31,670 --> 00:48:33,420
This is where the idea
of bands came from.

942
00:48:33,420 --> 00:48:37,546
This is from an old paper
in 1998, from the beginning.

943
00:48:37,546 --> 00:48:39,170
So it's just I have
a band, and then, I

944
00:48:39,170 --> 00:48:40,350
stack another band on top.

945
00:48:40,350 --> 00:48:44,372
So each z cross-section
is connected.

946
00:48:44,372 --> 00:48:45,580
So that's a little different.

947
00:48:45,580 --> 00:48:48,090
With towers, I could have
multiple towers here.

948
00:48:48,090 --> 00:48:51,404
I really only want one
tower built slab by slab.

949
00:48:51,404 --> 00:48:53,320
These things we don't
know how to grid unfold.

950
00:48:53,320 --> 00:48:56,330
That's an open problem,
but if you refine just

951
00:48:56,330 --> 00:49:01,360
in z by a factor of 2,
that's enough to unfold.

952
00:49:01,360 --> 00:49:03,690
So 1 by 2 refinement is
enough for orthostacks.

953
00:49:12,840 --> 00:49:15,420
Now, you could go
a little crazier

954
00:49:15,420 --> 00:49:20,030
and allow vertex unfolding
of orthostacks with some grid

955
00:49:20,030 --> 00:49:24,420
refinement, and in that case,
you don't need any refinement.

956
00:49:24,420 --> 00:49:27,190
So grid vertex
unfolding orthostacks

957
00:49:27,190 --> 00:49:30,510
was the title of paper.

958
00:49:30,510 --> 00:49:33,570
These guys, John Iacono
and Stefan Langerman.

959
00:49:33,570 --> 00:49:36,050
And it looks
something like this.

960
00:49:36,050 --> 00:49:38,920
He uses vertex unfolding
to fix the direction.

961
00:49:38,920 --> 00:49:41,730
Here you were going up, but
you really wanted to go right.

962
00:49:41,730 --> 00:49:43,180
So it makes things easier.

963
00:49:43,180 --> 00:49:46,830
And in fact, the other
guys, Damian, Flatland,

964
00:49:46,830 --> 00:49:50,130
and O'Rourke-- It's got to
be awesome doing geometry

965
00:49:50,130 --> 00:49:55,030
and your last name is
Flatland, unless she

966
00:49:55,030 --> 00:49:57,490
does a lot of three
dimensional stuff.

967
00:49:57,490 --> 00:49:59,110
Irony.

968
00:49:59,110 --> 00:50:04,234
You can do vertex grid unfolding
of any orthogonal polyhedron

969
00:50:04,234 --> 00:50:05,400
is what I have written here.

970
00:50:05,400 --> 00:50:06,608
I haven't actually read that.

971
00:50:06,608 --> 00:50:08,400
I should read that paper.

972
00:50:08,400 --> 00:50:09,650
What else do I have?

973
00:50:09,650 --> 00:50:11,770
Orthotubes.

974
00:50:11,770 --> 00:50:15,420
Orthotubes, this is
again in the old paper.

975
00:50:15,420 --> 00:50:20,590
Orthotube is just sort of
thickness one orthogonal tube.

976
00:50:20,590 --> 00:50:23,095
It could even be closed,
I think, in a loop,

977
00:50:23,095 --> 00:50:25,320
but here I've shown it open.

978
00:50:25,320 --> 00:50:27,200
And here, grid
unfolding is enough.

979
00:50:27,200 --> 00:50:29,090
You just do all the
grid refinement.

980
00:50:29,090 --> 00:50:32,030
You could even
just do it locally.

981
00:50:32,030 --> 00:50:35,412
Technically,
there's a slice here

982
00:50:35,412 --> 00:50:36,870
that might slice
over here, but you

983
00:50:36,870 --> 00:50:38,161
don't have to worry about that.

984
00:50:38,161 --> 00:50:40,330
You just subdivide
into a bunch of boxes,

985
00:50:40,330 --> 00:50:45,250
and this can be unfolded in
a sort of zigzag fashion.

986
00:50:45,250 --> 00:50:46,850
So 1 by 1.

987
00:50:46,850 --> 00:50:48,620
You can generalize
this to trees also,

988
00:50:48,620 --> 00:50:50,110
though, it's not totally known.

989
00:50:50,110 --> 00:50:52,750
As long as you have
a tree of cubes

990
00:50:52,750 --> 00:50:56,870
with fairly long connectors
in between the branch points--

991
00:50:56,870 --> 00:50:59,990
those are called
well-separated orthotrees--

992
00:50:59,990 --> 00:51:02,090
that works grid unfolding.

993
00:51:02,090 --> 00:51:03,540
If you don't have
that condition,

994
00:51:03,540 --> 00:51:05,814
so just have cubes connect
in some treelike fashion,

995
00:51:05,814 --> 00:51:07,730
it's open whether you
can do a grid unfolding.

996
00:51:07,730 --> 00:51:09,104
We've worked on
that in the past.

997
00:51:09,104 --> 00:51:15,925
My conjecture's, in general,
you need to omega n refinement.

998
00:51:24,830 --> 00:51:28,460
My belief is that this
kind of exponential blow

999
00:51:28,460 --> 00:51:32,720
up is necessary, but that you
can do it only a balance tree.

1000
00:51:32,720 --> 00:51:35,089
So this would give
like linear refinement.

1001
00:51:35,089 --> 00:51:36,880
I think that's necessary,
but we can't even

1002
00:51:36,880 --> 00:51:40,670
prove that you need 2 buy 1
refinement in any example.

1003
00:51:40,670 --> 00:51:44,210
We don't have anything where
grid unfolding is definitely

1004
00:51:44,210 --> 00:51:46,190
impossible.

1005
00:51:46,190 --> 00:51:49,630
It's quite sad state, I guess.

1006
00:51:52,940 --> 00:51:55,440
It's very hard to prove that
there aren't unfoldings, accept

1007
00:51:55,440 --> 00:51:59,372
by exhaustive
enumeration, and that's

1008
00:51:59,372 --> 00:52:00,580
hard to do because it's slow.

1009
00:52:09,604 --> 00:52:11,520
We've come up with lots
of candidate examples,

1010
00:52:11,520 --> 00:52:13,370
but eventually, we
unfold them all.

1011
00:52:16,690 --> 00:52:21,430
I have a bunch of
other open problems.

1012
00:52:21,430 --> 00:52:23,080
This was genus 0.

1013
00:52:23,080 --> 00:52:27,620
Interesting question is can
you do genus higher than 0?

1014
00:52:27,620 --> 00:52:30,670
Orthogonal polyhedra.

1015
00:52:30,670 --> 00:52:32,530
I would guess so,
but I'm not sure.

1016
00:52:32,530 --> 00:52:34,196
I think the biggest
question is, can you

1017
00:52:34,196 --> 00:52:35,440
make this non-orthogonal?

1018
00:52:35,440 --> 00:52:37,850
But then, the bands get messy.

1019
00:52:37,850 --> 00:52:39,661
Haven't been able to do that.

1020
00:52:39,661 --> 00:52:40,160
All right.

1021
00:52:40,160 --> 00:52:42,167
I think those were
the main problems.

1022
00:52:42,167 --> 00:52:42,750
Any questions?

1023
00:52:45,960 --> 00:52:48,250
I'm going to take a
break from unfolding now

1024
00:52:48,250 --> 00:52:50,340
and switch the other
direction of folding.

1025
00:52:53,100 --> 00:53:09,695
So with folding, we're imagining
we're given some polygon,

1026
00:53:09,695 --> 00:53:11,570
and we'd like to make
a polyhedron out of it.

1027
00:53:11,570 --> 00:53:14,840
It's exactly the reverse of
what we've been thinking about.

1028
00:53:17,730 --> 00:53:19,202
When is this possible?

1029
00:53:19,202 --> 00:53:21,660
Now, the rules of the game here
are different from origami.

1030
00:53:21,660 --> 00:53:24,250
With origami, we
showed from any shape,

1031
00:53:24,250 --> 00:53:27,350
you can make anything
if you scale it down.

1032
00:53:27,350 --> 00:53:29,840
But here, I really
want exact coverage.

1033
00:53:29,840 --> 00:53:34,360
Every point here, or let's
say every little patch here,

1034
00:53:34,360 --> 00:53:37,800
should be covered by
exactly one layer over here,

1035
00:53:37,800 --> 00:53:39,330
not allowing multiple layers.

1036
00:53:39,330 --> 00:53:41,590
So this means not
everything is possible.

1037
00:53:41,590 --> 00:53:43,950
I also care about
scale factor, but--

1038
00:53:43,950 --> 00:53:46,190
This one, of course, you
can crease like this,

1039
00:53:46,190 --> 00:53:49,600
and more importantly, you
glue these edges together.

1040
00:53:49,600 --> 00:53:51,240
You glue these edges together.

1041
00:53:51,240 --> 00:53:54,564
The opposite of
cutting is gluing.

1042
00:53:54,564 --> 00:53:56,730
We'll be more formal about
defining gluing, I think,

1043
00:53:56,730 --> 00:53:58,520
next lecture.

1044
00:53:58,520 --> 00:54:00,810
This is just a sort of prelude.

1045
00:54:00,810 --> 00:54:03,440
But you end up gluing--
I want to make something,

1046
00:54:03,440 --> 00:54:05,280
let's say-- in
fact, we're always

1047
00:54:05,280 --> 00:54:08,820
going to talk about
folding convex polyhedra.

1048
00:54:08,820 --> 00:54:14,459
There's very little work
on the non-convex case,

1049
00:54:14,459 --> 00:54:16,250
though, there was
actually a recent result.

1050
00:54:16,250 --> 00:54:19,419
I'll mention that next lecture.

1051
00:54:19,419 --> 00:54:20,960
If you want to make
something convex,

1052
00:54:20,960 --> 00:54:22,510
and therefore,
sphere-like, you have

1053
00:54:22,510 --> 00:54:25,400
to get rid of all the
boundary, so you've got a glue

1054
00:54:25,400 --> 00:54:25,927
every edge.

1055
00:54:25,927 --> 00:54:28,010
You don't have to glue
whole edges to whole edges.

1056
00:54:28,010 --> 00:54:29,180
Maybe you just glue
part of an edge

1057
00:54:29,180 --> 00:54:31,390
to another part of an
edge, but somehow, boundary

1058
00:54:31,390 --> 00:54:32,650
has to get glued up.

1059
00:54:32,650 --> 00:54:34,140
And if you want something
sphere-like, in fact,

1060
00:54:34,140 --> 00:54:35,723
those gluings have
to be non-crossing.

1061
00:54:35,723 --> 00:54:38,130
I have to be able to
draw a picture like this.

1062
00:54:38,130 --> 00:54:41,840
Question is when do these
gluings make a polyhedron?

1063
00:54:41,840 --> 00:54:44,870
That is the question we will
be answering next class.

1064
00:54:44,870 --> 00:54:47,930
But first question is suppose I
gave you one of these pictures.

1065
00:54:47,930 --> 00:54:50,200
I give you a polygon,
and I give you a gluing.

1066
00:54:50,200 --> 00:54:54,030
What this tells you is sort
of how to locally walk around.

1067
00:54:54,030 --> 00:54:57,060
So if I'm here, I
could walk over here.

1068
00:54:57,060 --> 00:55:01,080
I could walk over here, teleport
over there, walk over here,

1069
00:55:01,080 --> 00:55:02,520
teleport over here.

1070
00:55:02,520 --> 00:55:03,410
Whatever.

1071
00:55:03,410 --> 00:55:03,910
OK?

1072
00:55:03,910 --> 00:55:06,720
The gluing tells you locally
what the surface looks like,

1073
00:55:06,720 --> 00:55:09,400
even though you don't know
what it looks like yet in 3D.

1074
00:55:09,400 --> 00:55:13,320
In particular, you can
compute shortest paths here.

1075
00:55:13,320 --> 00:55:15,920
I could compute the shortest
path from this point

1076
00:55:15,920 --> 00:55:18,790
to this point you might
think is a straight line.

1077
00:55:18,790 --> 00:55:20,950
But no, it's not.

1078
00:55:24,375 --> 00:55:25,160
I don't think.

1079
00:55:25,160 --> 00:55:26,410
Or maybe it is.

1080
00:55:26,410 --> 00:55:27,812
Let's see.

1081
00:55:27,812 --> 00:55:31,310
A little tricky.

1082
00:55:31,310 --> 00:55:34,141
AUDIENCE: Should be diagonal
with the bottom square.

1083
00:55:34,141 --> 00:55:35,890
PROFESSOR: Diagonal
with the bottom square

1084
00:55:35,890 --> 00:55:37,640
because these guys are
both in the bottom.

1085
00:55:37,640 --> 00:55:38,230
Very good.

1086
00:55:38,230 --> 00:55:43,820
So this point is the same as
this point, or no, this point.

1087
00:55:43,820 --> 00:55:46,190
Because these get
zipped together.

1088
00:55:46,190 --> 00:55:48,840
This point is the
same as this point,

1089
00:55:48,840 --> 00:55:52,490
so in fact, that diagonal is the
shortest path between those two

1090
00:55:52,490 --> 00:55:53,310
points.

1091
00:55:53,310 --> 00:55:55,060
So you have to think
about it for a while,

1092
00:55:55,060 --> 00:55:57,310
but it turns out, in polynomial
time, you can do that.

1093
00:55:59,890 --> 00:56:02,070
That's cool.

1094
00:56:02,070 --> 00:56:08,670
What I want to show
now is that suppose

1095
00:56:08,670 --> 00:56:10,860
you could make a convex
polyhedron in this way.

1096
00:56:10,860 --> 00:56:14,370
I claim you can only make
one, never more than one

1097
00:56:14,370 --> 00:56:16,440
from the same gluing.

1098
00:56:16,440 --> 00:56:18,376
So I've defined locally
what this thing is.

1099
00:56:18,376 --> 00:56:19,500
It's like a piece of paper.

1100
00:56:19,500 --> 00:56:20,710
I can mangle it around.

1101
00:56:20,710 --> 00:56:22,760
If I want to make
something convex,

1102
00:56:22,760 --> 00:56:24,820
there's only one thing
it could possibly make.

1103
00:56:31,290 --> 00:56:34,390
Finding out what that one
thing is quite a challenge,

1104
00:56:34,390 --> 00:56:37,800
but at least, we can
prove that it's unique.

1105
00:56:37,800 --> 00:56:38,925
Sorry about the screeching.

1106
00:56:41,830 --> 00:56:45,190
This is Cauchy's
rigidity theorem.

1107
00:56:45,190 --> 00:56:49,880
I think we mentioned it
in a previous lecture when

1108
00:56:49,880 --> 00:56:52,266
we were talking about
rigidity and three dimensions

1109
00:56:52,266 --> 00:56:56,169
and like why domes stand up.

1110
00:56:56,169 --> 00:56:57,460
But now, we're going to use it.

1111
00:56:57,460 --> 00:56:58,750
We're actually going
to prove this theorem,

1112
00:56:58,750 --> 00:57:01,140
and we're going to use it to
study these kinds of gluings

1113
00:57:01,140 --> 00:57:04,467
and say there's, at
most, one way to do this.

1114
00:57:04,467 --> 00:57:06,300
There's a lot of ways
to state this theorem,

1115
00:57:06,300 --> 00:57:11,400
but one way is to say, suppose
you have two convex polyhedra,

1116
00:57:11,400 --> 00:57:13,810
and suppose they came
from the same sort

1117
00:57:13,810 --> 00:57:17,580
of intrinsic geometry.

1118
00:57:17,580 --> 00:57:20,450
So there's the
geometry of the faces,

1119
00:57:20,450 --> 00:57:23,060
and there's how they're
connected together.

1120
00:57:23,060 --> 00:57:24,920
So here, initially,
I drew a tree

1121
00:57:24,920 --> 00:57:26,980
of how the faces were
connected together,

1122
00:57:26,980 --> 00:57:30,820
and then, I drew some other
connections like this.

1123
00:57:30,820 --> 00:57:33,630
So the dual here is, of
course, an octahedron.

1124
00:57:33,630 --> 00:57:36,960
But if you have two
convex polyhedra

1125
00:57:36,960 --> 00:57:39,360
and they are combinatorally
equivalent, they

1126
00:57:39,360 --> 00:57:41,570
have the same way that
things are connected

1127
00:57:41,570 --> 00:57:51,710
together, the same graph, and
they have congruent faces,

1128
00:57:51,710 --> 00:57:57,060
so the geometries
match also, then, they

1129
00:57:57,060 --> 00:58:06,954
are actually the same
thing, the same polyhedron.

1130
00:58:06,954 --> 00:58:11,070
AUDIENCE: Is that
something that--

1131
00:58:11,070 --> 00:58:14,137
PROFESSOR: Up to
rotation and translation.

1132
00:58:14,137 --> 00:58:17,217
AUDIENCE: Is it the same
set of congruent faces?

1133
00:58:17,217 --> 00:58:18,050
PROFESSOR: Oh, yeah.

1134
00:58:18,050 --> 00:58:19,591
So when I say
congruent faces, I mean

1135
00:58:19,591 --> 00:58:21,150
according to this equivalence.

1136
00:58:21,150 --> 00:58:22,600
If you take a face
on one that has

1137
00:58:22,600 --> 00:58:24,250
a corresponding
face on the other,

1138
00:58:24,250 --> 00:58:27,100
those two faces should be
congruent, not just any pair,

1139
00:58:27,100 --> 00:58:28,860
and they're not all the same.

1140
00:58:28,860 --> 00:58:30,380
Different faces
can be different,

1141
00:58:30,380 --> 00:58:33,450
but they're identical in pairs.

1142
00:58:33,450 --> 00:58:36,930
So I'm just saying basically,
if you have this picture,

1143
00:58:36,930 --> 00:58:38,330
there's only one realization.

1144
00:58:38,330 --> 00:58:41,450
So this picture defines what
are the geometry of the faces,

1145
00:58:41,450 --> 00:58:43,810
how are they connected together.

1146
00:58:43,810 --> 00:58:45,800
And so if you had
two convex polyhedra

1147
00:58:45,800 --> 00:58:47,880
with that same
underlying diagram,

1148
00:58:47,880 --> 00:58:49,890
they have to be the
same polyhedron.

1149
00:58:49,890 --> 00:58:53,115
That's what we're claiming.

1150
00:58:53,115 --> 00:58:54,490
That's what we're
going to prove.

1151
00:59:04,591 --> 00:59:05,698
Is this one any better?

1152
00:59:05,698 --> 00:59:06,448
Yeah, it's better.

1153
00:59:16,420 --> 00:59:18,370
This is an old theorem.

1154
00:59:18,370 --> 00:59:21,820
You may have heard of Cauchy,
famous French mathematician.

1155
00:59:21,820 --> 00:59:24,262
Cauchy-Schwarz inequality,
all those good things.

1156
00:59:24,262 --> 00:59:25,470
You don't need to know those.

1157
00:59:25,470 --> 00:59:27,240
He proved a lot of things.

1158
00:59:27,240 --> 00:59:29,480
This theorem he
didn't actually prove.

1159
00:59:29,480 --> 00:59:31,880
He wrote a paper about it
or, I think, partly a letter,

1160
00:59:31,880 --> 00:59:34,890
partly a paper in 1813.

1161
00:59:34,890 --> 00:59:38,730
Proof was wrong, and
it was fixed in 1934,

1162
00:59:38,730 --> 00:59:41,010
over 100 years
later, by Steinitz.

1163
00:59:41,010 --> 00:59:44,880
So sometimes it's called
Cauchy-Steinitz rigidity

1164
00:59:44,880 --> 00:59:47,250
theorem, although,
usually, Cauchy.

1165
00:59:47,250 --> 00:59:49,116
There's a lemma in
here that's often

1166
00:59:49,116 --> 00:59:50,240
attributed to both of them.

1167
00:59:55,190 --> 01:00:00,550
So it's sort of a
proof by contradiction.

1168
01:00:00,550 --> 01:00:02,490
We want to prove uniqueness.

1169
01:00:02,490 --> 01:00:04,420
So we're supposing,
well, maybe, there's

1170
01:00:04,420 --> 01:00:09,510
two polyhedra, p and
p prime, and they're

1171
01:00:09,510 --> 01:00:12,030
combinatory equivalent and
have matching faces congruent.

1172
01:00:15,370 --> 01:00:24,890
What I want to do is look
at corresponding vertices,

1173
01:00:24,890 --> 01:00:30,760
let's say a vertex v and p and
a vertex v prime and p prime.

1174
01:00:30,760 --> 01:00:35,170
So there corresponding in the
combinatorial equivalence.

1175
01:00:35,170 --> 01:00:41,075
And then, I want to
slice the polyhedra, p

1176
01:00:41,075 --> 01:00:49,820
and p prime, with
an epsilon sphere,

1177
01:00:49,820 --> 01:00:55,530
epsilon radius sphere
centered at v and v prime.

1178
01:01:02,460 --> 01:01:04,750
Quick mention, this
is not true if you

1179
01:01:04,750 --> 01:01:06,500
allow non-convex realizations.

1180
01:01:06,500 --> 01:01:08,770
You may have seen
that example before.

1181
01:01:08,770 --> 01:01:11,910
These have exactly the same
combinatorial structure,

1182
01:01:11,910 --> 01:01:14,824
same geometry on each
face, but one's non-convex,

1183
01:01:14,824 --> 01:01:15,740
and they're different.

1184
01:01:15,740 --> 01:01:18,450
But as long as they're convex,
they're going to be the same.

1185
01:01:18,450 --> 01:01:21,170
In a convex situation, here's
what the slice looks like.

1186
01:01:21,170 --> 01:01:27,610
So here's little
vertex, degree like 5,

1187
01:01:27,610 --> 01:01:30,360
and I slice the polyhedron
with this tiny sphere centered

1188
01:01:30,360 --> 01:01:35,630
at v, not the interior, just
the boundary of the sphere.

1189
01:01:35,630 --> 01:01:37,100
What I get are these arcs.

1190
01:01:37,100 --> 01:01:40,130
They'll be great circular arcs.

1191
01:01:40,130 --> 01:01:42,250
Here we can see
all five of them.

1192
01:01:42,250 --> 01:01:43,610
They're great circular arcs.

1193
01:01:43,610 --> 01:01:44,990
There on the sphere.

1194
01:01:44,990 --> 01:01:46,680
I get a polygon.

1195
01:01:46,680 --> 01:01:48,980
I get a convex
polygon on the sphere.

1196
01:01:55,030 --> 01:01:58,466
Convex spherical
polygons, convex

1197
01:01:58,466 --> 01:01:59,840
because the
polyhedra are convex.

1198
01:01:59,840 --> 01:02:01,130
I actually get two of them.

1199
01:02:01,130 --> 01:02:01,630
Right?

1200
01:02:01,630 --> 01:02:04,695
One for p, one for p prime.

1201
01:02:04,695 --> 01:02:05,195
Polygons.

1202
01:02:07,960 --> 01:02:09,500
What do I know about
those polygons?

1203
01:02:09,500 --> 01:02:13,050
I know they're edge lengths
because the length of an edge

1204
01:02:13,050 --> 01:02:16,300
here is equal to-- if
this is a unit sphere,

1205
01:02:16,300 --> 01:02:18,800
if I rescale the
epsilon to be 1,

1206
01:02:18,800 --> 01:02:21,660
that edge length is
actually this angle.

1207
01:02:21,660 --> 01:02:25,440
That angle is an angle of
the face at the vertex.

1208
01:02:25,440 --> 01:02:26,686
So I know all the angles.

1209
01:02:26,686 --> 01:02:28,310
I know all the
geometries of the faces.

1210
01:02:28,310 --> 01:02:29,820
I know they're congruent.

1211
01:02:29,820 --> 01:02:34,130
So I know what these edge
lengths are in the sphere.

1212
01:02:34,130 --> 01:02:37,080
What I don't know are these
angles of the polygon.

1213
01:02:37,080 --> 01:02:38,850
I know the lengths
of the polygon,

1214
01:02:38,850 --> 01:02:40,600
but I don't know the
angles of the polygon

1215
01:02:40,600 --> 01:02:46,010
because that's essentially the
dihedral angle of that edge.

1216
01:02:46,010 --> 01:02:48,610
And the worry is, well, maybe
all the edge lengths match.

1217
01:02:48,610 --> 01:02:50,110
We could do this
at every vertex,

1218
01:02:50,110 --> 01:02:52,070
but maybe the dihedral
angles are different.

1219
01:02:52,070 --> 01:02:54,000
Maybe the convex
polyhedron could flex.

1220
01:02:54,000 --> 01:02:55,720
That's why this
is about rigidity.

1221
01:02:55,720 --> 01:02:57,750
Maybe it's flexible,
and then, maybe there

1222
01:02:57,750 --> 01:02:59,916
are two different states.

1223
01:02:59,916 --> 01:03:02,290
All the faces are the same,
and therefore, all those edge

1224
01:03:02,290 --> 01:03:05,644
lengths are the same, but
the angles might differ.

1225
01:03:05,644 --> 01:03:07,560
If p and p prime are
supposed to be different,

1226
01:03:07,560 --> 01:03:10,440
then there must be two
angles that differ.

1227
01:03:10,440 --> 01:03:13,330
So I want to look
at those angles,

1228
01:03:13,330 --> 01:03:31,270
and I want to label vertex
of a spherical polygon.

1229
01:03:31,270 --> 01:03:36,630
So I have n of these spherical
polygons for p n of them

1230
01:03:36,630 --> 01:03:37,900
and p prime.

1231
01:03:37,900 --> 01:03:39,890
Let's look at them in p.

1232
01:03:39,890 --> 01:03:47,780
I'm going to label it plus if
the angle there in p is bigger

1233
01:03:47,780 --> 01:03:50,886
than the angle in p prime.

1234
01:03:50,886 --> 01:03:53,010
Remember I have a correspondence
between everything

1235
01:03:53,010 --> 01:03:54,770
in p and everything and p prime.

1236
01:03:54,770 --> 01:04:00,964
Minus, if the angle is less,
and 0, if the angles are equal.

1237
01:04:00,964 --> 01:04:02,380
So what I want to
show is actually

1238
01:04:02,380 --> 01:04:03,421
all the angles are equal.

1239
01:04:03,421 --> 01:04:05,810
Therefore, the polyhedra
will be identical.

1240
01:04:05,810 --> 01:04:08,937
If all the faces are the same
and all the angles at which you

1241
01:04:08,937 --> 01:04:10,770
join them are the same
and they're connected

1242
01:04:10,770 --> 01:04:12,686
in the same way, they
are the same polyhedron.

1243
01:04:12,686 --> 01:04:15,710
There's no flexibility
there, but it

1244
01:04:15,710 --> 01:04:17,399
could be there are
pluses and minuses.

1245
01:04:17,399 --> 01:04:19,690
But if there's going to be
a problem with this theorem,

1246
01:04:19,690 --> 01:04:21,189
there have to be
pluses and minuses.

1247
01:04:21,189 --> 01:04:23,170
That's the proof
by contradiction.

1248
01:04:23,170 --> 01:04:24,799
So let's look at one of them.

1249
01:04:24,799 --> 01:04:27,090
Let's look at a vertex that
has some pluses or minuses.

1250
01:04:44,470 --> 01:04:47,260
So we have-- it's a
spherical polygon.

1251
01:04:47,260 --> 01:04:53,520
I'm going to draw it more like
a polygon, maybe some pluses,

1252
01:04:53,520 --> 01:04:57,160
some zeroes, some
minuses, whatever.

1253
01:04:57,160 --> 01:05:02,327
First question is could it
be all pluses and zeroes?

1254
01:05:02,327 --> 01:05:03,035
Is that possible?

1255
01:05:09,630 --> 01:05:12,345
No.

1256
01:05:12,345 --> 01:05:13,220
It doesn't look good.

1257
01:05:13,220 --> 01:05:14,530
What does that mean?

1258
01:05:14,530 --> 01:05:16,270
It would mean this
is like a linkage.

1259
01:05:16,270 --> 01:05:17,710
We're used to linkages.

1260
01:05:17,710 --> 01:05:19,190
It just happens
to be on a sphere.

1261
01:05:19,190 --> 01:05:20,260
Forget this on a sphere.

1262
01:05:20,260 --> 01:05:23,074
Think of it is as almost flat.

1263
01:05:23,074 --> 01:05:24,990
What that would mean is
there's some other way

1264
01:05:24,990 --> 01:05:26,146
to draw this thing.

1265
01:05:26,146 --> 01:05:28,020
Basically, there's a
way to flex this linkage

1266
01:05:28,020 --> 01:05:30,320
so that all of these angles
increase and this one

1267
01:05:30,320 --> 01:05:32,070
stays the same.

1268
01:05:32,070 --> 01:05:35,380
How could I get a polygon
where all the angles increase

1269
01:05:35,380 --> 01:05:37,610
and still be convex?

1270
01:05:37,610 --> 01:05:38,860
Ain't possible.

1271
01:05:38,860 --> 01:05:40,360
Why is it not possible?

1272
01:05:40,360 --> 01:05:45,650
I think we've used this
fact a couple lectures ago.

1273
01:05:45,650 --> 01:05:48,161
It's not possible by something
called the Cauchy Arm Lemma.

1274
01:05:51,107 --> 01:05:52,800
This is the part that
Cauchy got wrong,

1275
01:05:52,800 --> 01:05:55,880
so it's also called the
Cauchy-Steinitz Arm Lemma.

1276
01:05:55,880 --> 01:05:57,870
And here's the thing.

1277
01:05:57,870 --> 01:06:04,530
If you have a convex
chain but open chain here.

1278
01:06:04,530 --> 01:06:06,580
There's a missing bar.

1279
01:06:06,580 --> 01:06:08,370
So suppose that even
when you add the bar,

1280
01:06:08,370 --> 01:06:09,470
it's a convex polygon.

1281
01:06:09,470 --> 01:06:11,740
But then, I take a convex
polygon, remove an edge.

1282
01:06:11,740 --> 01:06:14,940
This is what we
call convex chain.

1283
01:06:14,940 --> 01:06:21,280
And if you open all the angles,
increasing all the angles,

1284
01:06:21,280 --> 01:06:34,125
in a convex chain, then
this distance increases.

1285
01:06:42,044 --> 01:06:42,960
It's pretty intuitive.

1286
01:06:42,960 --> 01:06:45,660
I open all these angles.

1287
01:06:45,660 --> 01:06:46,670
So I put plus.

1288
01:06:46,670 --> 01:06:48,940
Some of them could
stay the same,

1289
01:06:48,940 --> 01:06:51,465
but then, this
distance will increase,

1290
01:06:51,465 --> 01:06:52,590
as long as you stay convex.

1291
01:06:55,270 --> 01:06:56,590
I should mention that.

1292
01:06:56,590 --> 01:06:59,650
But in this situation, we know
that both the initial position

1293
01:06:59,650 --> 01:07:02,125
in p and its target position
in p prime are both convex.

1294
01:07:08,420 --> 01:07:09,010
OK.

1295
01:07:09,010 --> 01:07:11,500
Let's just take
that lemma as given.

1296
01:07:11,500 --> 01:07:12,840
This is the one I don't like.

1297
01:07:25,250 --> 01:07:27,660
Then, I know, in particular,
I can't have all pluses

1298
01:07:27,660 --> 01:07:32,530
because then, if I just pretend
one of these edges wasn't here,

1299
01:07:32,530 --> 01:07:36,614
I know that that
distance must increase.

1300
01:07:36,614 --> 01:07:37,530
But it can't increase.

1301
01:07:37,530 --> 01:07:38,821
It's supposed to stay the same.

1302
01:07:38,821 --> 01:07:40,370
The edge lengths are fixed.

1303
01:07:40,370 --> 01:07:42,800
So if they're all pluses
and zeroes, or all minuses

1304
01:07:42,800 --> 01:07:46,335
and zeroes, the same is true
just viewing p prime as p and p

1305
01:07:46,335 --> 01:07:49,009
as p prime.

1306
01:07:49,009 --> 01:07:50,050
They can't be all pluses.

1307
01:07:50,050 --> 01:07:52,040
They can't be all minuses.

1308
01:07:52,040 --> 01:07:54,180
So if there's anything in
there other than zeroes,

1309
01:07:54,180 --> 01:07:57,540
there has to be at least one
plus, at least one minus.

1310
01:07:57,540 --> 01:08:01,050
In particular, there have to
be at least two alternations.

1311
01:08:03,620 --> 01:08:07,390
Alternation is either going
from plus to minus or from minus

1312
01:08:07,390 --> 01:08:08,800
to plus.

1313
01:08:08,800 --> 01:08:10,353
So it could be
something like plus,

1314
01:08:10,353 --> 01:08:14,950
plus, plus, minus, minus,
minus, minus, plus, plus.

1315
01:08:14,950 --> 01:08:16,050
Whatever.

1316
01:08:16,050 --> 01:08:16,550
OK?

1317
01:08:16,550 --> 01:08:19,399
Maybe that's your polygon,
and that's your labeling.

1318
01:08:26,410 --> 01:08:27,190
Is that possible?

1319
01:08:31,089 --> 01:08:33,130
So once you have at least
one plus and one minus,

1320
01:08:33,130 --> 01:08:38,380
you have to have these
two switches at least.

1321
01:08:38,380 --> 01:08:39,693
Could this happen?

1322
01:08:39,693 --> 01:08:40,660
AUDIENCE: No.

1323
01:08:40,660 --> 01:08:42,370
PROFESSOR: No.

1324
01:08:42,370 --> 01:08:43,950
Right.

1325
01:08:43,950 --> 01:08:48,590
Because you pick some chord.

1326
01:08:48,590 --> 01:08:52,002
What do I do here?

1327
01:08:52,002 --> 01:08:53,000
Just subdivide.

1328
01:08:53,000 --> 01:08:53,500
All right.

1329
01:08:53,500 --> 01:08:54,890
Whatever.

1330
01:08:54,890 --> 01:08:58,505
Pick some chord like this one.

1331
01:08:58,505 --> 01:08:59,505
I'm not sure it matters.

1332
01:08:59,505 --> 01:09:00,609
Maybe here?

1333
01:09:00,609 --> 01:09:03,020
Whatever.

1334
01:09:03,020 --> 01:09:05,550
These angles down
here are decreasing.

1335
01:09:05,550 --> 01:09:08,229
Therefore, this
distance decreases.

1336
01:09:08,229 --> 01:09:10,870
The angles up here
are all increasing.

1337
01:09:10,870 --> 01:09:13,200
Therefore, this
distance increases.

1338
01:09:13,200 --> 01:09:15,310
Can't have both.

1339
01:09:15,310 --> 01:09:18,109
So this is also not possible.

1340
01:09:18,109 --> 01:09:20,767
So in fact, you have to have
at least four alternations.

1341
01:09:20,767 --> 01:09:21,475
It's always even.

1342
01:09:27,020 --> 01:09:30,760
And so it has to be at least a
bunch of pluses, then a bunch

1343
01:09:30,760 --> 01:09:32,910
of minuses, then a
bunch of pluses, then

1344
01:09:32,910 --> 01:09:34,560
a bunch of minuses.

1345
01:09:34,560 --> 01:09:37,910
And so we're counting these
transitions from plus to minus.

1346
01:09:37,910 --> 01:09:41,380
This is a lemma that we used
when we were locking trees,

1347
01:09:41,380 --> 01:09:42,000
I think.

1348
01:09:42,000 --> 01:09:43,000
We had a convex polygon.

1349
01:09:43,000 --> 01:09:45,654
I said at least two of the
angles have to decrease,

1350
01:09:45,654 --> 01:09:48,279
or at least two of them have to
increase, one way or the other.

1351
01:09:48,279 --> 01:09:51,460
That's because there have to be
at least two groups of pluses,

1352
01:09:51,460 --> 01:09:55,382
not only at least two pluses
and at least two minuses.

1353
01:09:55,382 --> 01:09:56,840
You might think,
well, what happens

1354
01:09:56,840 --> 01:09:59,465
if there's only three vertices.

1355
01:09:59,465 --> 01:10:01,340
Then, you can't have
these four alternations.

1356
01:10:01,340 --> 01:10:03,450
Well, yeah, you can't have
those four alternation

1357
01:10:03,450 --> 01:10:05,830
because if you have a
triangle, even on the sphere,

1358
01:10:05,830 --> 01:10:07,240
triangles are rigid.

1359
01:10:07,240 --> 01:10:11,917
So you would know if I had
a degree 3 vertex, locally

1360
01:10:11,917 --> 01:10:12,750
that thing is rigid.

1361
01:10:12,750 --> 01:10:13,830
It can't flex at all.

1362
01:10:13,830 --> 01:10:17,090
We're only interested in cases
where it might flex locally

1363
01:10:17,090 --> 01:10:20,230
at a vertex like the pentagon,
like a quadrilateral.

1364
01:10:22,760 --> 01:10:23,550
All right.

1365
01:10:23,550 --> 01:10:25,480
So what?

1366
01:10:25,480 --> 01:10:30,610
This was true at every vertex
that was not entirely zero.

1367
01:10:30,610 --> 01:10:35,807
So if we look at-- let
me write this down.

1368
01:10:35,807 --> 01:10:37,640
I really just care about
the non-zero edges,

1369
01:10:37,640 --> 01:10:38,973
the ones that are plus or minus.

1370
01:10:42,510 --> 01:10:51,520
So I'll call the subgraph
of plus and minus edges.

1371
01:10:54,480 --> 01:11:02,482
The number of alternations
is at least 4 times

1372
01:11:02,482 --> 01:11:03,440
the number of vertices.

1373
01:11:03,440 --> 01:11:10,083
I'm going to denote the number
of vertices by a capital V. OK?

1374
01:11:10,083 --> 01:11:11,030
That's page one.

1375
01:11:13,670 --> 01:11:16,620
We're going to use a trick.

1376
01:11:16,620 --> 01:11:18,120
It has many names, I suppose.

1377
01:11:18,120 --> 01:11:21,000
I usually call it double
counting in combinatorics,

1378
01:11:21,000 --> 01:11:22,500
where you have one
quantity, namely,

1379
01:11:22,500 --> 01:11:23,583
the number of alterations.

1380
01:11:23,583 --> 01:11:25,830
We're going to count it
in two different ways.

1381
01:11:25,830 --> 01:11:27,455
We'll get two different
answers, but we

1382
01:11:27,455 --> 01:11:29,150
know they must end
up being the same.

1383
01:11:29,150 --> 01:11:30,650
And then, we'll get
a contradiction.

1384
01:11:33,410 --> 01:11:35,760
So what's the second
way of counting.

1385
01:11:35,760 --> 01:11:39,390
Well, we counted
local to a vertex.

1386
01:11:39,390 --> 01:11:42,420
The other natural
way to count angles

1387
01:11:42,420 --> 01:11:43,936
is by looking at the faces.

1388
01:11:43,936 --> 01:11:44,810
There are also faces.

1389
01:11:44,810 --> 01:11:46,226
It's sort of the
dual perspective.

1390
01:11:46,226 --> 01:11:48,610
Every phase has a bunch of
angles that have some degree

1391
01:11:48,610 --> 01:11:50,767
or whatever.

1392
01:11:50,767 --> 01:11:52,350
They're really kind
of the same thing.

1393
01:11:52,350 --> 01:11:53,250
Oh, here was Cauchy's Arm Lemma.

1394
01:11:53,250 --> 01:11:53,750
Beautiful.

1395
01:11:56,230 --> 01:11:57,830
If you look at the
alternations as you

1396
01:11:57,830 --> 01:12:02,420
walk around a vertex versus
as you walk around a face,

1397
01:12:02,420 --> 01:12:04,810
you'll end up counting
them the same number.

1398
01:12:04,810 --> 01:12:05,310
Right?

1399
01:12:05,310 --> 01:12:09,290
Here was an alternation
from plus to minus.

1400
01:12:09,290 --> 01:12:10,857
What's interesting
here-- before we

1401
01:12:10,857 --> 01:12:12,440
were thinking of
labeling the vertices

1402
01:12:12,440 --> 01:12:15,380
of the spherical polygon, but in
fact, whatever this edge does,

1403
01:12:15,380 --> 01:12:18,300
it does the same thing local to
that vertex as the thing local

1404
01:12:18,300 --> 01:12:19,020
to that vertex.

1405
01:12:19,020 --> 01:12:21,640
So really the labels are
on the edges of the graph.

1406
01:12:21,640 --> 01:12:24,030
They could be zero,
plus, or minus.

1407
01:12:24,030 --> 01:12:26,660
And if I have an alternation
from plus to minus,

1408
01:12:26,660 --> 01:12:28,710
view from the vertex,
it's also an alternation

1409
01:12:28,710 --> 01:12:30,650
as I walk around the face.

1410
01:12:30,650 --> 01:12:32,800
So instead of counting
by walking around all

1411
01:12:32,800 --> 01:12:35,216
the vertices-- which are just
did, and I got at least four

1412
01:12:35,216 --> 01:12:37,579
at every vertex-- let's
do it from the perspective

1413
01:12:37,579 --> 01:12:38,120
of the faces.

1414
01:12:41,215 --> 01:12:43,590
And we're in this weird subgraph
of plus and minus edges,

1415
01:12:43,590 --> 01:12:44,870
so assume there are no zeroes.

1416
01:12:49,410 --> 01:12:49,910
All right.

1417
01:12:49,910 --> 01:13:02,110
If I have a face of 2
k or 2 k plus 1 edges,

1418
01:13:02,110 --> 01:13:08,500
then it will have, at
most, 2 k alternations.

1419
01:13:13,710 --> 01:13:16,600
So I already have a lower
bound number of alternations.

1420
01:13:16,600 --> 01:13:18,360
I'm going to try and
prove an upper bound,

1421
01:13:18,360 --> 01:13:20,901
sandwich it between, and show
that, actually, the upper bound

1422
01:13:20,901 --> 01:13:23,850
is smaller than the lower bound,
and that's a contradiction.

1423
01:13:23,850 --> 01:13:25,760
So this is kind of obvious.

1424
01:13:25,760 --> 01:13:26,260
Right?

1425
01:13:26,260 --> 01:13:29,350
If you have 2 k vertices, no
more than 2 k alternations,

1426
01:13:29,350 --> 01:13:31,770
slight, the place where we're
making a little improvement

1427
01:13:31,770 --> 01:13:32,975
is for the odd case.

1428
01:13:32,975 --> 01:13:34,600
We know because a
number of alterations

1429
01:13:34,600 --> 01:13:38,990
has to be even you can't get
up to 2 k plus 1 alternations.

1430
01:13:38,990 --> 01:13:41,360
Has to be even because
for every plus to minus,

1431
01:13:41,360 --> 01:13:44,220
there has to be a matching minus
to plus because it's cyclic.

1432
01:13:44,220 --> 01:13:46,460
So even here, you can
only get 2 k alternations.

1433
01:13:46,460 --> 01:13:50,670
That helps us a little
bit because now, we

1434
01:13:50,670 --> 01:13:59,430
can talk about
number of alternation

1435
01:13:59,430 --> 01:14:03,730
is at most two times
the number of triangles,

1436
01:14:03,730 --> 01:14:08,320
f sub 3 is going to be the
number of faces of degree 3,

1437
01:14:08,320 --> 01:14:14,010
plus 4 times the number
of quadrilaterals

1438
01:14:14,010 --> 01:14:17,490
plus 4 times the
number of pentagons

1439
01:14:17,490 --> 01:14:21,937
plus 6 times the number
of hexagons plus--

1440
01:14:21,937 --> 01:14:22,770
why'd I write seven?

1441
01:14:22,770 --> 01:14:23,850
I wrote seven.

1442
01:14:23,850 --> 01:14:26,150
I'm being sloppy.

1443
01:14:26,150 --> 01:14:27,860
At that point, I don't care.

1444
01:14:27,860 --> 01:14:31,050
7 f 7, 8 f 8, 9 f
9-- I'm not going

1445
01:14:31,050 --> 01:14:35,580
to try to be clever
from 6 on, but I'm

1446
01:14:35,580 --> 01:14:38,270
going to be clever at 5 and 3.

1447
01:14:38,270 --> 01:14:39,750
Why 5 and 3?

1448
01:14:39,750 --> 01:14:41,330
As you may remember
from way back

1449
01:14:41,330 --> 01:14:44,160
when, you have a planar
graph-- because polyhedra

1450
01:14:44,160 --> 01:14:45,190
are planar graphs.

1451
01:14:45,190 --> 01:14:47,020
They're convex.

1452
01:14:47,020 --> 01:14:53,700
The average degree is 5?

1453
01:14:53,700 --> 01:14:57,680
Slightly under 6, 4, 3, 2, 1?

1454
01:14:57,680 --> 01:14:59,940
One of those numbers.

1455
01:14:59,940 --> 01:15:01,080
Let's see.

1456
01:15:01,080 --> 01:15:06,120
Should be like 3 n minus 6
edges, so that should be 3.

1457
01:15:09,025 --> 01:15:10,600
Yeah, three.

1458
01:15:10,600 --> 01:15:13,830
So most of the faces are
going to have low degree.

1459
01:15:13,830 --> 01:15:14,580
That's the points.

1460
01:15:14,580 --> 01:15:17,062
So 3 and 5 really
matter, but out here, it

1461
01:15:17,062 --> 01:15:18,020
doesn't matter so much.

1462
01:15:36,750 --> 01:15:39,160
This is kind of a magical proof.

1463
01:15:39,160 --> 01:15:43,130
It shouldn't be intuitive
where it came from,

1464
01:15:43,130 --> 01:15:44,440
but it's really beautiful.

1465
01:15:44,440 --> 01:15:47,390
You'll see as it
all comes together.

1466
01:15:47,390 --> 01:15:48,670
Fun.

1467
01:15:48,670 --> 01:15:51,680
What do I do next?

1468
01:15:51,680 --> 01:15:52,400
Right.

1469
01:15:52,400 --> 01:15:56,000
I want to relate-- I
have a vertex count here.

1470
01:15:56,000 --> 01:16:00,330
I have a face count here,
and I know Euler's formula.

1471
01:16:00,330 --> 01:16:01,902
Hopefully, we know it.

1472
01:16:01,902 --> 01:16:02,735
It's a cool formula.

1473
01:16:06,340 --> 01:16:07,350
V minus E plus F is 2.

1474
01:16:07,350 --> 01:16:09,350
This is the number of
vertices, number of edges,

1475
01:16:09,350 --> 01:16:11,630
number faces is two.

1476
01:16:11,630 --> 01:16:16,494
[? For ?] connected,
planar graphs.

1477
01:16:16,494 --> 01:16:18,910
There are other versions when
you have multiple components

1478
01:16:18,910 --> 01:16:22,420
or when you have
tori, genus, whatever,

1479
01:16:22,420 --> 01:16:24,930
but for convex
polyhedra, this is true.

1480
01:16:24,930 --> 01:16:28,430
So this conveniently
relates vertices to faces,

1481
01:16:28,430 --> 01:16:30,790
but it involves edges.

1482
01:16:30,790 --> 01:16:33,616
So somehow, I have to
bring edges into the mix.

1483
01:16:33,616 --> 01:16:34,370
All right.

1484
01:16:34,370 --> 01:16:38,930
Well, edges.

1485
01:16:38,930 --> 01:16:41,190
I want to count
the number of edges

1486
01:16:41,190 --> 01:16:43,667
in terms of the number of faces.

1487
01:16:43,667 --> 01:16:45,500
I could do it in terms
of vertices or faces.

1488
01:16:45,500 --> 01:16:47,710
The number of edges
is half the sum

1489
01:16:47,710 --> 01:16:49,310
of the degrees of the vertices.

1490
01:16:49,310 --> 01:16:52,240
Remember, that's handshaking
lemma from way back when.

1491
01:16:52,240 --> 01:16:57,200
It's also half the sum of
the degrees of the faces.

1492
01:16:57,200 --> 01:17:00,170
If I look at every face and
I count the number of edges,

1493
01:17:00,170 --> 01:17:02,560
I will end up counting
every edge twice, once

1494
01:17:02,560 --> 01:17:07,515
from each side, so this is
half the number of faces.

1495
01:17:07,515 --> 01:17:10,240
Well, number faces I want
to write in this form.

1496
01:17:10,240 --> 01:17:15,445
So this is half 2-- No, sorry.

1497
01:17:15,445 --> 01:17:16,320
Not the number faces.

1498
01:17:16,320 --> 01:17:18,460
What am I doing?

1499
01:17:18,460 --> 01:17:20,680
Half the sum of the
degrees of the faces.

1500
01:17:23,810 --> 01:17:28,690
So this is half-- what is
the degree of degree 3 faces?

1501
01:17:28,690 --> 01:17:30,340
3.

1502
01:17:30,340 --> 01:17:32,110
What is the degree
of degree 4 faces?

1503
01:17:32,110 --> 01:17:32,881
4.

1504
01:17:32,881 --> 01:17:33,380
And so on.

1505
01:17:41,795 --> 01:17:42,295
Exactly.

1506
01:17:45,567 --> 01:17:47,400
So now, things are
starting to look similar,

1507
01:17:47,400 --> 01:17:51,300
and I want to get some
cancellation going on.

1508
01:17:51,300 --> 01:17:53,530
Use my cheat sheet here.

1509
01:17:53,530 --> 01:17:56,120
I'm going to
rewrite this formula

1510
01:17:56,120 --> 01:18:02,830
as V equals 2 plus E minus F.
Hopefully, that's the same.

1511
01:18:02,830 --> 01:18:04,270
I put E over here.

1512
01:18:04,270 --> 01:18:05,495
I put F over there.

1513
01:18:05,495 --> 01:18:07,650
We'll get this.

1514
01:18:07,650 --> 01:18:08,260
OK.

1515
01:18:08,260 --> 01:18:12,790
So now, I have E
minus F. E is this.

1516
01:18:12,790 --> 01:18:29,390
F-- I have F. Am I
missing something here?

1517
01:18:29,390 --> 01:18:31,430
I'll write the
next one, and I'll

1518
01:18:31,430 --> 01:18:32,680
figure out where it came from.

1519
01:18:52,800 --> 01:18:55,690
Oh, duh, I just skipped a step.

1520
01:18:55,690 --> 01:18:59,620
F is the sum of f3 plus
f4 plus f5 plus f6.

1521
01:18:59,620 --> 01:19:00,215
OK?

1522
01:19:00,215 --> 01:19:03,020
All I did here was
decrease by-- well,

1523
01:19:03,020 --> 01:19:05,540
because there's a half out here,
I decrease each coefficient

1524
01:19:05,540 --> 01:19:09,204
by 2, nothing surprising.

1525
01:19:09,204 --> 01:19:11,470
Whew!

1526
01:19:11,470 --> 01:19:16,610
So I took E, I subtracted F,
just took away one of each.

1527
01:19:16,610 --> 01:19:17,670
Now, I have this formula.

1528
01:19:17,670 --> 01:19:18,690
That's V.

1529
01:19:18,690 --> 01:19:23,850
Now, I also know
that 4V is at most,

1530
01:19:23,850 --> 01:19:26,538
the number of alternations.

1531
01:19:26,538 --> 01:19:27,430
Hm.

1532
01:19:27,430 --> 01:19:29,291
So I could get a
formula for 4V here.

1533
01:19:29,291 --> 01:19:29,790
Right?

1534
01:19:29,790 --> 01:19:36,230
4V is going to be 8 plus
this is going to be like 2.

1535
01:19:36,230 --> 01:19:43,670
So I get 2 f3 plus-- oh
boy, 1 times-- 4 times f4

1536
01:19:43,670 --> 01:19:47,890
plus-- just double
these numbers-- 6 times

1537
01:19:47,890 --> 01:19:50,990
f5 plus-- what's the next one?

1538
01:19:50,990 --> 01:19:54,810
4-- 8 times f6, and so on.

1539
01:20:04,470 --> 01:20:06,240
Yes.

1540
01:20:06,240 --> 01:20:06,790
OK.

1541
01:20:06,790 --> 01:20:15,620
Now, these guys look very
similar to these guys.

1542
01:20:18,560 --> 01:20:20,101
Now, here it gets bigger.

1543
01:20:20,101 --> 01:20:20,600
8.

1544
01:20:20,600 --> 01:20:23,670
It's going to go 10,
instead of 6 and 7.

1545
01:20:23,670 --> 01:20:25,080
We also have a plus 8.

1546
01:20:25,080 --> 01:20:27,660
We don't know whether there
any faces of degree 6 or more,

1547
01:20:27,660 --> 01:20:29,790
so we can't rely on
that, but we have plus 8.

1548
01:20:29,790 --> 01:20:33,402
So somehow, 4V,
which is, at most,

1549
01:20:33,402 --> 01:20:34,860
the number of
alternations-- that's

1550
01:20:34,860 --> 01:20:37,070
the reverse of what
we said before-- 4V

1551
01:20:37,070 --> 01:20:38,970
is, at most, this
number, and yet, it's

1552
01:20:38,970 --> 01:20:40,670
also equal to this number.

1553
01:20:40,670 --> 01:20:41,620
It can't be both.

1554
01:20:41,620 --> 01:20:43,680
This number's at least 8
larger than this number.

1555
01:20:43,680 --> 01:20:46,380
It could be even more
larger, but at least

1556
01:20:46,380 --> 01:20:49,320
that contradiction done.

1557
01:20:49,320 --> 01:20:51,750
This works as long as
there's at least one face,

1558
01:20:51,750 --> 01:20:54,310
meaning there's at
least 1 plus or minus

1559
01:20:54,310 --> 01:20:56,700
because we're only looking at
the subgraph plus and minus

1560
01:20:56,700 --> 01:20:57,530
edges.

1561
01:20:57,530 --> 01:20:59,320
And that is Cauchy's
rigidity theorem.

1562
01:20:59,320 --> 01:21:04,944
Let me quickly tell
you in our situation,

1563
01:21:04,944 --> 01:21:06,610
here, we don't actually
necessarily know

1564
01:21:06,610 --> 01:21:07,526
where the creases are.

1565
01:21:07,526 --> 01:21:10,302
We just know how things
are glued together.

1566
01:21:10,302 --> 01:21:12,010
Even in that situation,
you could sort of

1567
01:21:12,010 --> 01:21:14,890
figure out where the creases
must be because as I said,

1568
01:21:14,890 --> 01:21:17,360
you can compute shortest paths
once you have the gluing.

1569
01:21:17,360 --> 01:21:18,440
So you compute
the shortest paths

1570
01:21:18,440 --> 01:21:20,898
between all pairs of vertices,
something like this picture,

1571
01:21:20,898 --> 01:21:23,040
except you don't know
what it looks like in 3D.

1572
01:21:23,040 --> 01:21:24,748
You can still compute
the shortest paths.

1573
01:21:24,748 --> 01:21:27,030
You know every edge
must be a shortest path.

1574
01:21:27,030 --> 01:21:29,660
So the edges are some
subset of these guys.

1575
01:21:29,660 --> 01:21:33,500
And so you've got lots of
little convex polygons here.

1576
01:21:33,500 --> 01:21:35,360
We know it must make
a convex polyhedron.

1577
01:21:35,360 --> 01:21:38,710
If it made two, Cauchy's
rigidity theorem

1578
01:21:38,710 --> 01:21:40,330
would tell you that
they're the same.

1579
01:21:40,330 --> 01:21:42,090
So even once you
fix the gluing, you

1580
01:21:42,090 --> 01:21:46,590
know that there's a
unique convex realization,

1581
01:21:46,590 --> 01:21:49,710
and there will be a unique
set of edges from the shortest

1582
01:21:49,710 --> 01:21:51,340
paths that actually realize it.

1583
01:21:51,340 --> 01:21:53,184
The next class will
be all about how

1584
01:21:53,184 --> 01:21:55,600
to actually find those gluings
and know that they actually

1585
01:21:55,600 --> 01:21:58,290
will fold into some convex shape
and how to find that convex

1586
01:21:58,290 --> 01:22:01,880
shape, but that's it for today.