1
00:00:36,370 --> 00:00:39,200
PROFESSOR: All right,
let's get started.

2
00:00:39,200 --> 00:00:43,120
So today we're continuing
the theme of locked linkages.

3
00:00:43,120 --> 00:00:45,970
Last time we talked about the
Carpenter's Rule Theorem, which

4
00:00:45,970 --> 00:00:48,420
brought together all of the
rigidity theory and tensegrity

5
00:00:48,420 --> 00:00:50,620
theory we had
built, essentially,

6
00:00:50,620 --> 00:00:55,090
and showed that there
were no locked 2D

7
00:00:55,090 --> 00:00:59,040
chains, paths or
cycles, as graphs.

8
00:00:59,040 --> 00:01:07,130
And in general you can think
in 2D, you can think in 3D,

9
00:01:07,130 --> 00:01:10,210
and you can think
in 4D and higher.

10
00:01:10,210 --> 00:01:14,560
I drew this table way at
the beginning of class.

11
00:01:14,560 --> 00:01:18,560
And you can think
about chains, which

12
00:01:18,560 --> 00:01:24,470
are-- these are so-called open
chains, like robotic arms.

13
00:01:24,470 --> 00:01:26,690
Paths is a graph.

14
00:01:26,690 --> 00:01:32,630
And polygons are closed chains.

15
00:01:32,630 --> 00:01:34,960
And then another case it's
interesting to think about,

16
00:01:34,960 --> 00:01:37,820
and there's been a lot of
work on, are trees where

17
00:01:37,820 --> 00:01:40,190
you don't have any
cycles in your graph,

18
00:01:40,190 --> 00:01:44,890
but you could have branching
points with degree more than 2.

19
00:01:44,890 --> 00:01:47,670
You may recall Carpenter's
Rule Theorem worked whenever

20
00:01:47,670 --> 00:01:51,050
you had a graph of
maximum degree 2.

21
00:01:51,050 --> 00:01:53,940
So it allowed even
multiple chains

22
00:01:53,940 --> 00:01:56,980
in one graph that
could be disconnected.

23
00:01:56,980 --> 00:02:02,020
But it forbade any kind
of tree branching stuff.

24
00:02:02,020 --> 00:02:03,920
So Carpenter's Rule
Theorem is that there's

25
00:02:03,920 --> 00:02:07,990
no locked chains in 2D.

26
00:02:07,990 --> 00:02:11,640
So we did that.

27
00:02:11,640 --> 00:02:16,020
But trees, there are
locked trees in 2D.

28
00:02:16,020 --> 00:02:20,140
3D there are locked chains.

29
00:02:20,140 --> 00:02:21,720
And these are two
results that we

30
00:02:21,720 --> 00:02:26,380
will talk about in
today's lecture.

31
00:02:26,380 --> 00:02:31,190
And that implies that there are
locked trees, sort of obvious.

32
00:02:31,190 --> 00:02:34,620
Trees are more general
than open chains.

33
00:02:34,620 --> 00:02:36,700
And because there's a
locked open chain in 3D,

34
00:02:36,700 --> 00:02:39,260
there's also a locked tree.

35
00:02:39,260 --> 00:02:42,450
In 4D, though, it switches.

36
00:02:42,450 --> 00:02:44,700
So none of these
things are locked.

37
00:02:44,700 --> 00:02:48,000
Of course, there are
locked graphs in general.

38
00:02:48,000 --> 00:02:51,070
But if you look at trees, or
paths, or cycles, none of them

39
00:02:51,070 --> 00:02:51,760
can lock.

40
00:02:51,760 --> 00:02:54,057
We won't have time to talk
about that today, maybe

41
00:02:54,057 --> 00:02:54,640
another class.

42
00:02:57,166 --> 00:02:58,540
Essentially we're
always thinking

43
00:02:58,540 --> 00:03:00,890
about an intrinsically
one dimensional structure.

44
00:03:00,890 --> 00:03:04,260
These edges, bars, are
always one dimensional.

45
00:03:04,260 --> 00:03:07,980
In 2D, where the
space are living in,

46
00:03:07,980 --> 00:03:10,720
the ambient dimension is just
one more than the dimension

47
00:03:10,720 --> 00:03:12,700
of your edges, it's
really interesting.

48
00:03:12,700 --> 00:03:15,070
And that's what
we saw for chains

49
00:03:15,070 --> 00:03:17,370
and what we'll see for trees.

50
00:03:17,370 --> 00:03:19,740
In 3D it gets tricky.

51
00:03:19,740 --> 00:03:22,480
This mismatch of
two in dimension

52
00:03:22,480 --> 00:03:25,340
allows you to lock things,
kind of like knots.

53
00:03:25,340 --> 00:03:28,480
And in 4D there's such--
there's so much freedom

54
00:03:28,480 --> 00:03:30,540
in the way you move,
somehow, nothing

55
00:03:30,540 --> 00:03:33,125
can lock, in the same way
that there are no knots in 4D.

56
00:03:35,840 --> 00:03:37,340
Cool.

57
00:03:37,340 --> 00:03:40,310
So that's the reminder
summary about locked linkages,

58
00:03:40,310 --> 00:03:43,920
what they are, and
what's known about them.

59
00:03:43,920 --> 00:03:46,690
Now we've seen that
chains aren't locked.

60
00:03:46,690 --> 00:03:49,360
But before I get trees I
want to talk about algorithms

61
00:03:49,360 --> 00:03:50,415
for actually doing this.

62
00:03:50,415 --> 00:03:53,430
This is, after all,
geometric folding algorithms.

63
00:03:53,430 --> 00:03:55,866
We've seen a proof
that they exist,

64
00:03:55,866 --> 00:03:57,490
that in fact if we
had a chain, there's

65
00:03:57,490 --> 00:04:00,490
an expansive motion
increases all the distances

66
00:04:00,490 --> 00:04:04,300
and therefore doesn't
collide with itself,

67
00:04:04,300 --> 00:04:08,550
and eventually straightens
all the outermost open chains

68
00:04:08,550 --> 00:04:12,260
and convexifies all the closed
chains that are outermost,

69
00:04:12,260 --> 00:04:15,780
not containing anything.

70
00:04:15,780 --> 00:04:19,570
But how do we actually
find such a motion?

71
00:04:19,570 --> 00:04:21,430
And this has been the
subject of some study.

72
00:04:21,430 --> 00:04:23,096
There are three
algorithms for doing it.

73
00:04:28,590 --> 00:04:35,940
So we'll call this
unfolding 2D chains,

74
00:04:35,940 --> 00:04:42,040
unfolding being the goal of
straightening or convexifying.

75
00:04:42,040 --> 00:04:45,830
And the first algorithm is just
to mimic the proof that we saw.

76
00:04:59,700 --> 00:05:03,060
I call that proof CDR, because
it's Connelly-Demain-Rote.

77
00:05:03,060 --> 00:05:07,100
And if you think
about what we proved,

78
00:05:07,100 --> 00:05:09,650
we just looked at the sort
of infinitesimal case.

79
00:05:09,650 --> 00:05:12,890
We said, if we're at some
position then at least,

80
00:05:12,890 --> 00:05:14,914
for an infinitesimal
amount of time,

81
00:05:14,914 --> 00:05:16,330
we can increase
all the distances.

82
00:05:16,330 --> 00:05:18,320
All the struts can
increase in length,

83
00:05:18,320 --> 00:05:21,000
and the bars stay
the same length.

84
00:05:21,000 --> 00:05:23,550
That's really only specifying
the first derivative

85
00:05:23,550 --> 00:05:24,410
of a motion.

86
00:05:24,410 --> 00:05:27,170
So if you think of motion
as being, I don't know,

87
00:05:27,170 --> 00:05:30,430
some configuration C
that's a function of time,

88
00:05:30,430 --> 00:05:34,010
and you take the derivative
with respect to t,

89
00:05:34,010 --> 00:05:36,107
then we computed what
that derivative should be.

90
00:05:36,107 --> 00:05:37,440
That was the first order motion.

91
00:05:37,440 --> 00:05:42,270
I think we called it d,
which is a little confusing.

92
00:05:42,270 --> 00:05:45,380
This is the d that we computed.

93
00:05:45,380 --> 00:05:49,710
And we said, well that's
an infinitesimal motion

94
00:05:49,710 --> 00:05:51,155
of some tensegrity.

95
00:05:51,155 --> 00:05:52,780
We can find it with
linear programming.

96
00:05:52,780 --> 00:05:55,050
So at any moment in
time we can figure out

97
00:05:55,050 --> 00:05:58,210
what our derivative should be.

98
00:05:58,210 --> 00:06:00,720
This is what's called an
ordinary differential equation,

99
00:06:00,720 --> 00:06:03,650
for those who have
taking 1803 or whatever.

100
00:06:03,650 --> 00:06:05,474
You just solve it.

101
00:06:05,474 --> 00:06:06,890
And a very easy
way to solve it is

102
00:06:06,890 --> 00:06:10,610
if you imagine configuration
space, you start somewhere,

103
00:06:10,610 --> 00:06:12,349
you compute which
way to go, you move

104
00:06:12,349 --> 00:06:13,640
a little bit in that direction.

105
00:06:13,640 --> 00:06:15,931
You're only supposed to go
for an infinitesimal amount.

106
00:06:15,931 --> 00:06:18,070
But it goes some
positive epsilon.

107
00:06:18,070 --> 00:06:20,020
Then recompute
your new direction,

108
00:06:20,020 --> 00:06:22,170
follow that, recompute
your new direction,

109
00:06:22,170 --> 00:06:24,500
follow that for a little
bit, and keep going.

110
00:06:24,500 --> 00:06:26,960
And you're almost tracking
the intended curve.

111
00:06:26,960 --> 00:06:28,510
It's not quite perfect.

112
00:06:28,510 --> 00:06:31,440
But as your step
size, this epsilon,

113
00:06:31,440 --> 00:06:35,970
goes to 0 you approximate
the intended path.

114
00:06:35,970 --> 00:06:38,514
And that's, I mean how
close you approximate

115
00:06:38,514 --> 00:06:39,680
is computed in the textbook.

116
00:06:39,680 --> 00:06:41,380
You can see that for details.

117
00:06:41,380 --> 00:06:43,350
This is a method
called forward Euler.

118
00:06:43,350 --> 00:06:46,810
It's like the simplest way to
solve an ordinary differential

119
00:06:46,810 --> 00:06:47,310
equation.

120
00:06:47,310 --> 00:06:48,850
There are much better ways.

121
00:06:48,850 --> 00:06:51,810
But it's easiest to think
about, and reason about.

122
00:06:51,810 --> 00:06:54,835
And so it gives you some
approximate solution.

123
00:06:57,670 --> 00:06:58,760
Let me show it to you.

124
00:07:03,130 --> 00:07:04,750
I'm going to show
you three methods.

125
00:07:04,750 --> 00:07:06,800
The top one is the
one I'm talking about.

126
00:07:06,800 --> 00:07:08,425
So we're starting
with the same polygon

127
00:07:08,425 --> 00:07:11,637
on the left, simple
little v shape.

128
00:07:11,637 --> 00:07:13,595
A second method is called
pseudo-triangulation.

129
00:07:13,595 --> 00:07:14,840
We'll talk about that next.

130
00:07:14,840 --> 00:07:16,310
The third method's
called energy.

131
00:07:16,310 --> 00:07:17,970
That one's one you've seen.

132
00:07:17,970 --> 00:07:19,810
Let's start with
the first method,

133
00:07:19,810 --> 00:07:25,080
where we take some example
like this teeth example,

134
00:07:25,080 --> 00:07:28,850
and we, at each
step, compute what

135
00:07:28,850 --> 00:07:31,960
is the, in some
sense, the biggest

136
00:07:31,960 --> 00:07:33,359
expanse of motion you could do.

137
00:07:33,359 --> 00:07:34,900
We go a little bit
in that direction.

138
00:07:34,900 --> 00:07:36,560
So in these animations,
the edge links

139
00:07:36,560 --> 00:07:37,957
are changing a little bit.

140
00:07:37,957 --> 00:07:40,290
And I've chosen the step size
so the edge lengths change

141
00:07:40,290 --> 00:07:42,840
by, at most, 10%,
something like that.

142
00:07:42,840 --> 00:07:44,390
And here, this one,
left side we're

143
00:07:44,390 --> 00:07:47,120
zooming, right side
we're not zooming.

144
00:07:47,120 --> 00:07:48,620
These are somewhat
older animations.

145
00:07:48,620 --> 00:07:51,580
So apologies for the frame rate.

146
00:07:51,580 --> 00:07:53,010
What else do I want to say?

147
00:07:53,010 --> 00:07:54,930
This is an example--
well, notice

148
00:07:54,930 --> 00:07:57,650
it's pulling the teeth
off sort of from the end.

149
00:07:57,650 --> 00:08:00,260
If you want to be expensive you
cannot pull the teeth apart.

150
00:08:00,260 --> 00:08:02,702
We're going to see other ways
to do the teeth in a moment.

151
00:08:02,702 --> 00:08:03,660
Here's another example.

152
00:08:03,660 --> 00:08:05,730
Both of these
examples at some point

153
00:08:05,730 --> 00:08:08,430
we're conjectured to be locked.

154
00:08:08,430 --> 00:08:10,420
Before the Carpenter's
Rule Theorem was proved,

155
00:08:10,420 --> 00:08:13,862
a lot of people were trying
to find counter examples.

156
00:08:13,862 --> 00:08:14,570
There aren't any.

157
00:08:18,080 --> 00:08:18,775
So this is nice.

158
00:08:18,775 --> 00:08:22,560
It preserve five-fold
rotational symmetry.

159
00:08:22,560 --> 00:08:26,570
And you can do it for
multiple shapes as well.

160
00:08:26,570 --> 00:08:29,890
So here we have one polygon
in the center, four arcs,

161
00:08:29,890 --> 00:08:33,940
four paths on the outside.

162
00:08:33,940 --> 00:08:36,380
In this case, the simulation
stopped when one of them

163
00:08:36,380 --> 00:08:41,064
convexified because I didn't
implement all of the algorithm.

164
00:08:41,064 --> 00:08:43,230
At that point you just have
to rigidify that polygon

165
00:08:43,230 --> 00:08:46,600
and expand the rest.

166
00:08:46,600 --> 00:08:49,140
So that gives you an
idea sort of visually

167
00:08:49,140 --> 00:08:51,200
what this looks like.

168
00:08:51,200 --> 00:08:53,780
Qualitatively, we can say
a bunch of cool things

169
00:08:53,780 --> 00:08:55,070
about this method.

170
00:08:55,070 --> 00:08:58,680
One is that's it's the
only method that's strictly

171
00:08:58,680 --> 00:09:06,100
expansive, strictly meaning not
only do the struts not decrease

172
00:09:06,100 --> 00:09:07,990
in length, but they
also don't stay the same

173
00:09:07,990 --> 00:09:09,530
whenever that's possible.

174
00:09:09,530 --> 00:09:11,150
They strictly increase.

175
00:09:11,150 --> 00:09:12,510
Now, there are some exceptions.

176
00:09:12,510 --> 00:09:16,750
If I have a bar, obviously,
if I add a strut here

177
00:09:16,750 --> 00:09:18,680
it's not going to strictly
increase in length.

178
00:09:18,680 --> 00:09:22,020
Also, if I have a bunch
of bars that are collinear

179
00:09:22,020 --> 00:09:24,156
and I add a strut
from here to here,

180
00:09:24,156 --> 00:09:25,780
that's not going to
increase in length.

181
00:09:25,780 --> 00:09:27,090
It's not possible.

182
00:09:27,090 --> 00:09:30,317
But all other struts will
strictly increase in length.

183
00:09:30,317 --> 00:09:30,900
You can check.

184
00:09:30,900 --> 00:09:35,150
That's what we proved here.

185
00:09:35,150 --> 00:09:38,190
So that's nice.

186
00:09:38,190 --> 00:09:42,860
Another fun fact, I
didn't write it here,

187
00:09:42,860 --> 00:09:47,740
but it preserves symmetry
if you do it right.

188
00:09:51,350 --> 00:09:53,485
You may notice in
this example actually,

189
00:09:53,485 --> 00:09:55,610
it was originally four-fold
rotationally symmetric.

190
00:09:55,610 --> 00:09:58,360
It does rotate a little bit.

191
00:09:58,360 --> 00:10:00,220
That's from computational error.

192
00:10:00,220 --> 00:10:01,660
And you could correct for that.

193
00:10:01,660 --> 00:10:05,050
I didn't correct for it.

194
00:10:05,050 --> 00:10:11,700
It should, in theory, preserve
the four-fold symmetry there.

195
00:10:11,700 --> 00:10:12,660
Good.

196
00:10:12,660 --> 00:10:14,640
We can compute one step.

197
00:10:14,640 --> 00:10:18,310
So how fast can you
compute this thing?

198
00:10:18,310 --> 00:10:20,250
We can compute one step
in polynomial time.

199
00:10:20,250 --> 00:10:23,262
That's finding an infinitesimal
motion of a tensegrity.

200
00:10:23,262 --> 00:10:24,345
That's linear programming.

201
00:10:27,970 --> 00:10:31,820
But how many steps does it make?

202
00:10:31,820 --> 00:10:32,850
We don't really know.

203
00:10:32,850 --> 00:10:35,180
There is a bound
in the textbook.

204
00:10:35,180 --> 00:10:38,390
It's not very small.

205
00:10:38,390 --> 00:10:44,960
It's also-- with forward Euler
you lose a bit of accuracy.

206
00:10:44,960 --> 00:10:47,100
Now you could
correct for accuracy,

207
00:10:47,100 --> 00:10:48,850
force all the edge
lengths to be the same,

208
00:10:48,850 --> 00:10:50,080
then you won't
preserve the symmetry.

209
00:10:50,080 --> 00:10:51,720
There's a bit of trade off here.

210
00:10:51,720 --> 00:10:53,469
I don't want to spend
too much time on it.

211
00:10:53,469 --> 00:10:55,080
This is the first algorithm.

212
00:10:55,080 --> 00:10:58,280
Although it's the only one
that's strictly expensive,

213
00:10:58,280 --> 00:10:59,550
to me it's not the nicest.

214
00:10:59,550 --> 00:11:06,976
So I'm going to move on
to the next method which

215
00:11:06,976 --> 00:11:08,350
is pointed
pseudo-triangulations.

216
00:11:13,970 --> 00:11:16,360
This is a method
by Ileana Streinu,

217
00:11:16,360 --> 00:11:20,350
basically at the same
time as the original

218
00:11:20,350 --> 00:11:22,830
Carpenter's Rule Theorem.

219
00:11:22,830 --> 00:11:24,280
And these are
examples of what are

220
00:11:24,280 --> 00:11:26,970
called pointed
pseudo-triangulations

221
00:11:26,970 --> 00:11:28,830
for a given set of points.

222
00:11:28,830 --> 00:11:34,040
So in general, a
pseudo-triangle is a polygon

223
00:11:34,040 --> 00:11:36,980
that has three convex vertices.

224
00:11:36,980 --> 00:11:40,020
So a triangle has three
convex vertices and that's it.

225
00:11:40,020 --> 00:11:44,670
A pseudo-triangle, I allow
a bunch of reflex vertices

226
00:11:44,670 --> 00:11:47,260
in between those guys.

227
00:11:47,260 --> 00:11:48,550
So this should be a polygon.

228
00:11:48,550 --> 00:11:51,200
I've drawn it curved.

229
00:11:51,200 --> 00:11:55,430
And if you check in this
figure, all of the faces

230
00:11:55,430 --> 00:11:56,460
are pseudo-triangles.

231
00:11:56,460 --> 00:11:59,180
Most of them here are
actually quadrilaterals.

232
00:11:59,180 --> 00:12:01,090
But in particular,
that's a pseudo-triangle.

233
00:12:01,090 --> 00:12:04,610
Sometimes there's
actual triangles.

234
00:12:04,610 --> 00:12:07,130
Here they're all
quads or triangles.

235
00:12:07,130 --> 00:12:10,040
And a pointed
pseudo-triangulation

236
00:12:10,040 --> 00:12:14,630
has the property
that at every vertex

237
00:12:14,630 --> 00:12:19,410
all of the edges incident
to it lie in a half plane.

238
00:12:19,410 --> 00:12:21,340
They all lie on that side.

239
00:12:21,340 --> 00:12:26,315
In other words, there's an angle
that's bigger than 180 degrees.

240
00:12:26,315 --> 00:12:32,150
So this was a pseudo-triangle,
and this is pointed.

241
00:12:34,870 --> 00:12:38,330
And if you check this
example, every vertex

242
00:12:38,330 --> 00:12:40,275
has an angle that's
bigger than 180 degrees.

243
00:12:44,510 --> 00:12:46,850
Obviously the ones on
the outside have to.

244
00:12:46,850 --> 00:12:49,290
They're always convex.

245
00:12:49,290 --> 00:12:51,240
And pointed
pseudo-triangulations

246
00:12:51,240 --> 00:12:55,850
are-- they're not only supposed
to have these two properties,

247
00:12:55,850 --> 00:12:58,346
but you're also supposed to
have as many edges as possible,

248
00:12:58,346 --> 00:12:59,970
subject to having
these two properties.

249
00:12:59,970 --> 00:13:03,900
If you tried to add any edge
to either of these examples,

250
00:13:03,900 --> 00:13:06,980
you would violate
the pointed property.

251
00:13:06,980 --> 00:13:11,630
If I added this edge, then I'd
destroy this 180 degree angle.

252
00:13:11,630 --> 00:13:13,080
Any edge you
consider, either you

253
00:13:13,080 --> 00:13:14,990
get a crossing--
which is not allowed--

254
00:13:14,990 --> 00:13:21,250
or one of the vertices
loses a 180 degree angle.

255
00:13:21,250 --> 00:13:23,480
So this is actually a
really powerful concept.

256
00:13:23,480 --> 00:13:27,470
And it was introduced by this
paper by Streinu in 2000.

257
00:13:27,470 --> 00:13:30,164
It has lots of theory
built around it by now.

258
00:13:30,164 --> 00:13:31,830
I'm not going to talk
about that theory.

259
00:13:31,830 --> 00:13:34,466
I want to talk about how it
applies to Carpenter's Rule

260
00:13:34,466 --> 00:13:34,965
problem.

261
00:13:39,546 --> 00:13:40,920
I guess the first
thing to say is

262
00:13:40,920 --> 00:13:48,590
that if you have,
say, a polygon-- maybe

263
00:13:48,590 --> 00:13:51,290
I should do a real example,
so I'll make it not so giant.

264
00:13:56,230 --> 00:13:58,330
OK, you take a polygon.

265
00:13:58,330 --> 00:13:59,950
So far it's pointed, right?

266
00:13:59,950 --> 00:14:01,980
Every vertex-- one
of the two sides

267
00:14:01,980 --> 00:14:05,090
has an angle bigger than 180.

268
00:14:05,090 --> 00:14:10,485
So just add as many edges as you
can while still being pointed.

269
00:14:10,485 --> 00:14:12,200
Do, do, do.

270
00:14:12,200 --> 00:14:15,570
This one, that looks OK.

271
00:14:19,480 --> 00:14:20,590
I think that's it.

272
00:14:20,590 --> 00:14:22,534
I think this is a
pseudo-triangulation.

273
00:14:22,534 --> 00:14:24,950
I've added as many edges as I
can while still being point.

274
00:14:24,950 --> 00:14:27,060
It turns out if you start
with something that's pointed,

275
00:14:27,060 --> 00:14:29,270
you'll be able to keep adding
edges, keep it pointed.

276
00:14:29,270 --> 00:14:32,930
And in the end all of the
faces will be pseudo-triangles.

277
00:14:32,930 --> 00:14:34,347
So mission accomplished.

278
00:14:34,347 --> 00:14:35,930
This is a pointed
pseudo-triangulation

279
00:14:35,930 --> 00:14:40,160
that contains, as a subgraph,
my original polygon.

280
00:14:43,190 --> 00:14:46,120
So these are the edge
lengths I want to preserve.

281
00:14:46,120 --> 00:14:47,720
Those should stay rigid.

282
00:14:47,720 --> 00:14:50,340
I'm going to be kind of
crazy and also preserve

283
00:14:50,340 --> 00:14:53,070
the lengths of all
of these edges.

284
00:14:53,070 --> 00:14:55,910
Now if you did that, this
thing would be rigid,

285
00:14:55,910 --> 00:14:57,290
which is not so exciting.

286
00:14:57,290 --> 00:15:01,300
All you need to do to fix that
is remove one of these added

287
00:15:01,300 --> 00:15:05,530
edges that's on the convex hull
on the outer boundary here.

288
00:15:05,530 --> 00:15:08,069
Then it will have one
degree of freedom.

289
00:15:08,069 --> 00:15:09,610
So there will be
two ways to move it.

290
00:15:09,610 --> 00:15:12,690
One is to expand this distance,
the other is to contract it.

291
00:15:12,690 --> 00:15:15,250
And the one that
expands that distance

292
00:15:15,250 --> 00:15:17,770
will expand all distances.

293
00:15:17,770 --> 00:15:22,730
So if you look at some
other pair, like this guy,

294
00:15:22,730 --> 00:15:27,790
every other pair of distances
will expand or stay the same.

295
00:15:27,790 --> 00:15:30,690
So this thing is expansive.

296
00:15:33,930 --> 00:15:35,670
But not strictly
expensive, because we're

297
00:15:35,670 --> 00:15:37,250
preserving these lengths.

298
00:15:37,250 --> 00:15:39,720
These ones will stay the same.

299
00:15:39,720 --> 00:15:43,990
So let me show you some of
the things that can happen.

300
00:15:43,990 --> 00:15:46,742
So this is sort of at the
heart of the algorithm.

301
00:15:46,742 --> 00:15:48,450
I'm not going to go
into all the details.

302
00:15:48,450 --> 00:15:52,230
This can happen like n
cubed times, where you say,

303
00:15:52,230 --> 00:15:53,862
OK I've added in
all these edges.

304
00:15:53,862 --> 00:15:55,320
Here's a nice
pseudo-triangle here.

305
00:15:55,320 --> 00:15:57,470
It happens to be a triangle with
some stuff on the bottom, stuff

306
00:15:57,470 --> 00:15:58,230
on the top.

307
00:15:58,230 --> 00:16:00,325
As I flex this to
expand the distances

308
00:16:00,325 --> 00:16:01,866
this is going to move like that.

309
00:16:01,866 --> 00:16:03,240
At some point,
those three points

310
00:16:03,240 --> 00:16:06,720
will become collinear,
got to deal with that.

311
00:16:06,720 --> 00:16:10,230
And the way you deal with it
is as you continue moving,

312
00:16:10,230 --> 00:16:13,480
it's no longer the case
that those two blue edges

313
00:16:13,480 --> 00:16:15,530
are the edges in the
pseudo-triangulation.

314
00:16:15,530 --> 00:16:17,574
Now it's this one red edge.

315
00:16:17,574 --> 00:16:18,490
This is called a flip.

316
00:16:18,490 --> 00:16:20,323
And you see, we used
to have a quadrilateral

317
00:16:20,323 --> 00:16:22,180
up here an a triangle down here.

318
00:16:22,180 --> 00:16:26,030
Now we have a triangle up here
and a quadrilateral over here.

319
00:16:26,030 --> 00:16:27,655
That's pretty intuitive.

320
00:16:27,655 --> 00:16:29,530
You preserve that it's
a pseudo-triangulation

321
00:16:29,530 --> 00:16:32,610
as you move this.

322
00:16:32,610 --> 00:16:35,660
If you tried to go
too far here you

323
00:16:35,660 --> 00:16:38,130
would actually get a
convex quadrilateral.

324
00:16:38,130 --> 00:16:40,897
If this kept going
then this would

325
00:16:40,897 --> 00:16:41,980
be a convex quadrilateral.

326
00:16:41,980 --> 00:16:43,396
That's not a
pseudo-triangulation,

327
00:16:43,396 --> 00:16:44,265
that's bad.

328
00:16:44,265 --> 00:16:45,640
But you can always
do these flips

329
00:16:45,640 --> 00:16:46,724
to fix it as you go along.

330
00:16:46,724 --> 00:16:48,306
And it turns out
this will only happen

331
00:16:48,306 --> 00:16:49,520
a polynomial number of times.

332
00:16:49,520 --> 00:16:51,830
AUDIENCE: [INAUDIBLE].

333
00:16:51,830 --> 00:16:54,051
PROFESSOR: Like shown,
is how I flipped it.

334
00:16:54,051 --> 00:16:56,410
AUDIENCE: Is it
[INAUDIBLE] to a red?

335
00:16:56,410 --> 00:17:02,310
PROFESSOR: Yeah, the red edge
now goes from here to there.

336
00:17:02,310 --> 00:17:04,280
Whereas before there
were two segments that

337
00:17:04,280 --> 00:17:05,863
went from here to
there, now I'm going

338
00:17:05,863 --> 00:17:08,060
to go straight from
there to there.

339
00:17:08,060 --> 00:17:11,230
That's the difference.

340
00:17:11,230 --> 00:17:14,480
This is the general picture,
you have to believe me,

341
00:17:14,480 --> 00:17:17,310
because I want to talk
about other things.

342
00:17:17,310 --> 00:17:18,619
And, cool.

343
00:17:18,619 --> 00:17:21,890
So here's an example
of it actually running.

344
00:17:21,890 --> 00:17:25,510
So we have some polygon
in the top left.

345
00:17:25,510 --> 00:17:28,750
We've deleted the
edge that's cyan.

346
00:17:28,750 --> 00:17:31,930
So that one's going to expand.

347
00:17:31,930 --> 00:17:34,940
And we move a little bit.

348
00:17:34,940 --> 00:17:36,810
This is moving a little bit.

349
00:17:36,810 --> 00:17:38,770
This is moving a
little bit more.

350
00:17:38,770 --> 00:17:40,770
At this point, these three
points are collinear.

351
00:17:40,770 --> 00:17:43,061
Actually, the way it's drawn
it looks like a little bit

352
00:17:43,061 --> 00:17:44,010
too far.

353
00:17:44,010 --> 00:17:45,326
But they should be collinear.

354
00:17:45,326 --> 00:17:46,950
That would be a
violation, because this

355
00:17:46,950 --> 00:17:48,510
would be-- if we go
any farther-- that

356
00:17:48,510 --> 00:17:50,100
would be a violation because
this would be a convex

357
00:17:50,100 --> 00:17:51,180
quadrilateral.

358
00:17:51,180 --> 00:17:55,357
So we do a flip here, and we
end up with that edge instead.

359
00:17:55,357 --> 00:17:56,940
So now everything's
a pseudo-triangle,

360
00:17:56,940 --> 00:17:59,700
everything's pointed,
everything's happy.

361
00:17:59,700 --> 00:18:02,970
You keep going, then these
three points become collinear.

362
00:18:02,970 --> 00:18:04,960
So you end up getting that edge.

363
00:18:04,960 --> 00:18:06,800
And you keep going.

364
00:18:06,800 --> 00:18:11,040
And all throughout this
motion all pairwise distances

365
00:18:11,040 --> 00:18:12,250
are not decreasing.

366
00:18:12,250 --> 00:18:14,800
Everything's expanding
or staying the same.

367
00:18:14,800 --> 00:18:17,530
And then the wrap
up looks like this.

368
00:18:17,530 --> 00:18:19,714
In the end, you get
a convex polygon.

369
00:18:19,714 --> 00:18:21,130
Then you're stuck,
because there's

370
00:18:21,130 --> 00:18:23,817
no edge to remove
on the outside.

371
00:18:23,817 --> 00:18:25,400
You've got to keep
all of those edges.

372
00:18:27,970 --> 00:18:30,750
So that is the
pseudo-triangulation method

373
00:18:30,750 --> 00:18:31,670
in a nutshell.

374
00:18:31,670 --> 00:18:34,400
And it's expansive.

375
00:18:34,400 --> 00:18:41,150
It has a polynomial
in n number of moves.

376
00:18:48,640 --> 00:18:52,920
It's not clear how quickly
you can compute the moves.

377
00:18:57,750 --> 00:18:59,420
I know an exponential
time algorithm.

378
00:18:59,420 --> 00:19:01,070
There might be faster one.

379
00:19:01,070 --> 00:19:02,620
I don't think this
has been resolved.

380
00:19:02,620 --> 00:19:05,930
Let me just check my notes here.

381
00:19:05,930 --> 00:19:08,290
It says best algorithm
is exponential to compute

382
00:19:08,290 --> 00:19:08,790
one move.

383
00:19:08,790 --> 00:19:10,340
In these examples,
where they're all

384
00:19:10,340 --> 00:19:12,590
quadrilaterals or
triangles, it's

385
00:19:12,590 --> 00:19:14,190
really easy to compute them.

386
00:19:14,190 --> 00:19:16,950
In general, if you get
complicated things,

387
00:19:16,950 --> 00:19:17,494
it's unclear.

388
00:19:17,494 --> 00:19:19,910
Pseudo-triangulations though
have a lot of nice structure.

389
00:19:19,910 --> 00:19:24,660
So maybe it's easier than
general linkage folding.

390
00:19:24,660 --> 00:19:27,050
This, I have here,
implementing this algorithm

391
00:19:27,050 --> 00:19:30,430
would be a cool project.

392
00:19:30,430 --> 00:19:33,210
Oh, another open problem
is how many steps

393
00:19:33,210 --> 00:19:34,990
does this algorithm really need?

394
00:19:34,990 --> 00:19:37,045
Can you prove a
pseudo-polynomial bound?

395
00:19:37,045 --> 00:19:38,880
I'm not sure.

396
00:19:38,880 --> 00:19:43,150
But let me tell you about my
personal favorite algorithm,

397
00:19:43,150 --> 00:19:46,340
the energy method.

398
00:19:46,340 --> 00:19:52,980
This is also the
most recent, in 2004.

399
00:19:52,980 --> 00:19:54,480
The weird thing
about this algorithm

400
00:19:54,480 --> 00:19:55,745
is that it is not expansive.

401
00:20:01,586 --> 00:20:03,835
It's really easy to compute
one step of the algorithm.

402
00:20:06,420 --> 00:20:10,700
I can do it in quadratic
time, even exactly.

403
00:20:14,880 --> 00:20:17,690
This one you have to specify
some error tolerance epsilon.

404
00:20:17,690 --> 00:20:20,160
This one you have to specify
some error tolerance epsilon.

405
00:20:20,160 --> 00:20:24,480
Here you can do it perfectly
if you have exact square roots.

406
00:20:24,480 --> 00:20:27,800
And the number of
steps is also small.

407
00:20:38,570 --> 00:20:41,250
It has a pseudo-polynomial
number of steps.

408
00:20:41,250 --> 00:20:44,070
Have I talked about
pseudo-polynomial yet?

409
00:20:44,070 --> 00:20:45,510
Maybe, briefly at some point.

410
00:20:45,510 --> 00:20:49,890
Let me tell you what I mean.

411
00:20:49,890 --> 00:20:52,290
I want to be polynomial
in the number vertices

412
00:20:52,290 --> 00:20:55,672
n and another parameter r.

413
00:20:55,672 --> 00:20:59,370
r is going to be, basically,
the maximum distance

414
00:20:59,370 --> 00:21:01,780
in your initial
configuration divided

415
00:21:01,780 --> 00:21:04,050
by the minimum distance.

416
00:21:04,050 --> 00:21:07,040
Exactly how you define
distance, don't worry about it.

417
00:21:07,040 --> 00:21:08,600
Not a big deal.

418
00:21:08,600 --> 00:21:10,810
So this is a geometric
feature of the input.

419
00:21:10,810 --> 00:21:12,116
It has nothing to do with n.

420
00:21:12,116 --> 00:21:13,740
n could stay fixed
and you could change

421
00:21:13,740 --> 00:21:15,780
this to whatever
you want by making

422
00:21:15,780 --> 00:21:17,620
nastier and nastier examples.

423
00:21:17,620 --> 00:21:21,900
Nastiness is measured here by
how big your linkages versus

424
00:21:21,900 --> 00:21:24,130
how tight things get.

425
00:21:24,130 --> 00:21:25,890
So as long as that's
reasonable, this

426
00:21:25,890 --> 00:21:28,090
is going to be a
polynomial number of moves.

427
00:21:28,090 --> 00:21:29,110
So this is a good bound.

428
00:21:29,110 --> 00:21:32,940
It's better than all the others
that's been proved anyway.

429
00:21:32,940 --> 00:21:35,980
It might hold for some
of the other methods.

430
00:21:35,980 --> 00:21:38,370
Here, of course, we have a
better bound on the number

431
00:21:38,370 --> 00:21:38,870
moves.

432
00:21:38,870 --> 00:21:40,470
But each move is
kind of complicated.

433
00:21:40,470 --> 00:21:43,090
It's pseudo-triangulation.

434
00:21:43,090 --> 00:21:44,840
So there is a trade
off between complexity

435
00:21:44,840 --> 00:21:46,737
of move and number of moves.

436
00:21:46,737 --> 00:21:48,570
But this gets a decent
bound for everything.

437
00:21:52,201 --> 00:21:53,200
You've seen this method.

438
00:21:53,200 --> 00:21:55,690
I showed it in lecture one.

439
00:21:55,690 --> 00:22:01,450
But it's interesting to contrast
it with the CDR method, which

440
00:22:01,450 --> 00:22:04,130
sort of unrolled the
teeth from the end.

441
00:22:04,130 --> 00:22:08,490
Here, this method basically
pulls the teeth right apart.

442
00:22:08,490 --> 00:22:12,940
And that's possible because it's
not required to be expansive.

443
00:22:12,940 --> 00:22:16,850
And we have the double tree.

444
00:22:19,529 --> 00:22:21,070
This one looks more
or less the same.

445
00:22:21,070 --> 00:22:23,572
It's a little smoother, I would
say, than the other method.

446
00:22:23,572 --> 00:22:24,780
I don't have it side by side.

447
00:22:24,780 --> 00:22:27,674
So it's a little hard to guess.

448
00:22:27,674 --> 00:22:29,840
And what's really exciting
about this method, to me,

449
00:22:29,840 --> 00:22:33,230
is that you can run it
for really large examples

450
00:22:33,230 --> 00:22:34,910
in like a minute or so.

451
00:22:34,910 --> 00:22:37,620
500 vertices, no big deal.

452
00:22:37,620 --> 00:22:39,820
Whereas, because each
step is pretty easy

453
00:22:39,820 --> 00:22:42,420
to compute in quadratic
time, with this method

454
00:22:42,420 --> 00:22:46,470
you need it to solve
a convex program.

455
00:22:46,470 --> 00:22:49,655
And that's a little
costly for large examples.

456
00:22:53,200 --> 00:22:54,370
Cool.

457
00:22:54,370 --> 00:22:55,930
So that's the energy method.

458
00:22:55,930 --> 00:22:57,930
Now I'm going to actually
tell you how it works.

459
00:23:01,934 --> 00:23:03,475
Big difference is
it's not expansive.

460
00:23:09,510 --> 00:23:12,240
And the idea is,
well, being expansive

461
00:23:12,240 --> 00:23:14,710
on every edge,
that's kind of hard.

462
00:23:14,710 --> 00:23:15,780
How do I figure that out?

463
00:23:15,780 --> 00:23:17,100
Oh, it makes my brain hurt.

464
00:23:17,100 --> 00:23:18,930
I've got to solve
this tensegrity.

465
00:23:18,930 --> 00:23:22,800
But if I was just
expansive on average,

466
00:23:22,800 --> 00:23:24,570
maybe that would be good enough.

467
00:23:24,570 --> 00:23:27,150
That's the crazy idea.

468
00:23:27,150 --> 00:23:38,610
So we define this energy
function that does just that.

469
00:23:38,610 --> 00:23:40,156
It's a function
on configurations.

470
00:23:47,290 --> 00:23:54,230
So I'm going to write E of
C. No M in this equation.

471
00:23:54,230 --> 00:24:00,390
We're going to sum over all
edges Ew, and then sum over all

472
00:24:00,390 --> 00:24:06,300
vertices U different
from V and W.

473
00:24:06,300 --> 00:24:11,600
And we're going to take 1
over the distance from U

474
00:24:11,600 --> 00:24:13,990
to the edge.

475
00:24:13,990 --> 00:24:17,610
This is just saying sum over
all non-incident vertices

476
00:24:17,610 --> 00:24:20,784
and edges, take their
distance-- there's

477
00:24:20,784 --> 00:24:23,200
many ways you could define
that, like the minimum distance

478
00:24:23,200 --> 00:24:25,970
between them would be
fine-- take the reciprocal,

479
00:24:25,970 --> 00:24:27,450
add them all up.

480
00:24:27,450 --> 00:24:29,900
This is my energy function.

481
00:24:29,900 --> 00:24:30,920
Kind of weird.

482
00:24:30,920 --> 00:24:34,560
But it's averaging
not of distances,

483
00:24:34,560 --> 00:24:36,640
but 1 over distances.

484
00:24:36,640 --> 00:24:38,150
So why is that so interesting?

485
00:24:38,150 --> 00:24:41,110
Because distances
are always positive.

486
00:24:41,110 --> 00:24:42,790
So I only have to go over here.

487
00:24:42,790 --> 00:24:45,590
If I write distance
in the x-axis,

488
00:24:45,590 --> 00:24:49,940
and 1 over the
distance in the y-axis,

489
00:24:49,940 --> 00:24:52,550
the plot of that
function looks like this.

490
00:24:52,550 --> 00:24:56,190
It has this vertical
asymptote at 0.

491
00:24:56,190 --> 00:24:59,740
So if the distance were to
go to 0, which is bad for me

492
00:24:59,740 --> 00:25:02,620
because that's one thing start
crossing-- they will start

493
00:25:02,620 --> 00:25:05,760
touching at d equals 0, and
after that they might cross--

494
00:25:05,760 --> 00:25:09,360
the energy shoots to
positive infinity.

495
00:25:09,360 --> 00:25:12,630
So if I take the sum over all
these, if any of the distances

496
00:25:12,630 --> 00:25:17,890
decrease to 0 then the energy
will shoot to infinity.

497
00:25:17,890 --> 00:25:20,960
What if my distances increase?

498
00:25:20,960 --> 00:25:24,717
Because I know
expansive motions exist.

499
00:25:24,717 --> 00:25:26,050
That's what we proved last time.

500
00:25:28,560 --> 00:25:30,390
I'm not going to compute them.

501
00:25:30,390 --> 00:25:33,240
But I know they're
out there somewhere.

502
00:25:33,240 --> 00:25:36,080
If I had an expansive motion,
all of these distances

503
00:25:36,080 --> 00:25:38,210
increase.

504
00:25:38,210 --> 00:25:41,810
So that means this
reciprocal decreases.

505
00:25:41,810 --> 00:25:43,320
So I have-- it's
hard to imagine--

506
00:25:43,320 --> 00:25:46,150
I have this giant dimensional
configuration space.

507
00:25:46,150 --> 00:25:48,295
It has d times n
dimensions, whatever.

508
00:25:48,295 --> 00:25:49,140
A lot of stuff.

509
00:25:51,660 --> 00:25:53,910
I'm somewhere.

510
00:25:53,910 --> 00:25:57,280
But if I now plotted--
imagine that's

511
00:25:57,280 --> 00:26:00,190
in two dimensions,
in the plane here,

512
00:26:00,190 --> 00:26:02,580
so that's there-- and then
in the third dimension I

513
00:26:02,580 --> 00:26:04,430
plot what is this
energy landscape.

514
00:26:04,430 --> 00:26:06,840
For each of these configurations
I compute some height.

515
00:26:06,840 --> 00:26:08,080
What is my energy?

516
00:26:08,080 --> 00:26:10,690
I get some 3D surface here.

517
00:26:10,690 --> 00:26:15,050
In general, it's going to be
d times n plus 1 dimensions.

518
00:26:15,050 --> 00:26:16,350
Here's the configuration space.

519
00:26:16,350 --> 00:26:18,540
We plot over that.

520
00:26:18,540 --> 00:26:21,880
What I'm saying is, because
expansive motions exist,

521
00:26:21,880 --> 00:26:24,030
there has to be a motion
for every point that

522
00:26:24,030 --> 00:26:25,966
decreases energy.

523
00:26:25,966 --> 00:26:27,590
Because if you decrease
every distance,

524
00:26:27,590 --> 00:26:31,310
you also decrease them
on average in this sense.

525
00:26:31,310 --> 00:26:34,740
If every term decreases,
then of course, the sum will.

526
00:26:34,740 --> 00:26:45,230
So this means energy
decreasing motions exist.

527
00:26:49,616 --> 00:26:51,950
Now energy decreasing
feels like a good thing.

528
00:26:51,950 --> 00:26:57,480
Because I start somewhere, it
has some energy, not infinity.

529
00:26:57,480 --> 00:26:59,990
If I decrease the energy,
it's really hard for my energy

530
00:26:59,990 --> 00:27:01,912
to go to plus infinity.

531
00:27:01,912 --> 00:27:03,880
It can't happen.

532
00:27:03,880 --> 00:27:12,890
So all I need to do is follow
any energy decreasing motion.

533
00:27:18,830 --> 00:27:21,790
It might be an expansive
one, but probably not.

534
00:27:21,790 --> 00:27:23,790
We know that there are
energy decreasing motions

535
00:27:23,790 --> 00:27:25,732
because there are
expansive motions.

536
00:27:25,732 --> 00:27:27,440
But let's just take
any energy decreasing

537
00:27:27,440 --> 00:27:30,960
motion, sort of
expansive on average.

538
00:27:30,960 --> 00:27:38,262
Then it won't self-intersect
because if energy decreases

539
00:27:38,262 --> 00:27:39,720
it will never get
to plus infinity.

540
00:27:39,720 --> 00:27:41,761
And therefore none of the
distances will go to 0.

541
00:27:44,500 --> 00:27:48,820
So what this algorithm does
is follow the gradient.

542
00:27:57,500 --> 00:28:00,500
So I should say something
about what this notation is.

543
00:28:00,500 --> 00:28:02,860
This is whatever,
higher order calculus.

544
00:28:06,340 --> 00:28:09,120
You live in some space.

545
00:28:09,120 --> 00:28:10,940
You're on like some
hill, or whatever.

546
00:28:10,940 --> 00:28:13,380
You're on some surface.

547
00:28:13,380 --> 00:28:16,322
And you say well I'd really
like to go downhill from here.

548
00:28:16,322 --> 00:28:18,030
And there might be
many downhill options.

549
00:28:18,030 --> 00:28:19,488
There might be many
uphill options.

550
00:28:19,488 --> 00:28:21,960
This is some crazy high
dimensional choice.

551
00:28:21,960 --> 00:28:24,790
Just take the option
that decreases energy

552
00:28:24,790 --> 00:28:27,210
the fastest, the most downhill.

553
00:28:27,210 --> 00:28:32,260
That is negative gradient of
E. Believe me that it exists.

554
00:28:32,260 --> 00:28:34,990
It exists because this
energy function is smooth,

555
00:28:34,990 --> 00:28:35,790
in some sense.

556
00:28:38,180 --> 00:28:39,930
You're basically taking
first derivatives.

557
00:28:39,930 --> 00:28:41,910
That gives you the
highest chain--

558
00:28:41,910 --> 00:28:44,390
or you take the place that has
the highest change and boom.

559
00:28:44,390 --> 00:28:45,264
That gives it to you.

560
00:28:45,264 --> 00:28:47,510
It's very easy to compute
because this function has

561
00:28:47,510 --> 00:28:48,690
quadratically many terms.

562
00:28:48,690 --> 00:28:51,930
It takes about n
squared time to find it.

563
00:28:51,930 --> 00:28:56,550
And that gives you some
energy decrease in motion.

564
00:28:56,550 --> 00:28:57,990
It's just an easy one to find.

565
00:28:57,990 --> 00:29:00,480
And that's what we're
animating all the time.

566
00:29:00,480 --> 00:29:02,570
So we just find energy
decreasing motion,

567
00:29:02,570 --> 00:29:04,230
move a little bit
in that direction.

568
00:29:04,230 --> 00:29:06,040
Not too much,
because if you go--

569
00:29:06,040 --> 00:29:07,960
it's again a first derivative.

570
00:29:07,960 --> 00:29:09,710
This is only an
infinitesimal motion.

571
00:29:09,710 --> 00:29:12,330
But we're actually going
to move in that direction

572
00:29:12,330 --> 00:29:13,740
for a positive amount of time.

573
00:29:13,740 --> 00:29:16,010
As long as we don't go too
far, and we can find how

574
00:29:16,010 --> 00:29:19,370
far by binary search,
we won't self-intersect.

575
00:29:19,370 --> 00:29:21,350
Because we know locally
it's decreasing energy.

576
00:29:21,350 --> 00:29:25,140
If we go small enough step it
really will decrease energy,

577
00:29:25,140 --> 00:29:26,660
and then life is good.

578
00:29:26,660 --> 00:29:30,600
So it's really easy to do this.

579
00:29:30,600 --> 00:29:33,170
Super simple algorithm.

580
00:29:33,170 --> 00:29:35,150
Good.

581
00:29:35,150 --> 00:29:37,460
It's non-expansive
because we're only

582
00:29:37,460 --> 00:29:41,932
decreasing the average,
not each of the terms.

583
00:29:41,932 --> 00:29:44,730
Do, do, do, do, do.

584
00:29:44,730 --> 00:29:47,470
You can prove the
number of steps--

585
00:29:47,470 --> 00:29:53,130
this is the part I'm most proud
of here-- is pseudo-polynomial,

586
00:29:53,130 --> 00:29:58,220
polynomial in n and r.

587
00:29:58,220 --> 00:30:00,420
What is the polynomial?

588
00:30:00,420 --> 00:30:06,010
n to the 123 times r to the 81.

589
00:30:06,010 --> 00:30:07,880
This is the largest
polynomial bound

590
00:30:07,880 --> 00:30:11,960
I know of that's not
dependent on dimension.

591
00:30:11,960 --> 00:30:12,710
So I'm very proud.

592
00:30:12,710 --> 00:30:15,270
In practice, the number of
steps is much, much smaller

593
00:30:15,270 --> 00:30:15,770
than this.

594
00:30:15,770 --> 00:30:17,870
This is what we could
prove in an easy way.

595
00:30:17,870 --> 00:30:20,000
If you want a project,
it's a little tedious,

596
00:30:20,000 --> 00:30:21,700
but it would be easy,
I'm pretty sure.

597
00:30:21,700 --> 00:30:24,070
You could decrease
this to, I don't know,

598
00:30:24,070 --> 00:30:26,670
at least n to the
20 or something

599
00:30:26,670 --> 00:30:28,720
by being a little more
careful on the analysis.

600
00:30:28,720 --> 00:30:30,886
But once we got something
that was pseudo-polynomial

601
00:30:30,886 --> 00:30:32,010
we were happy.

602
00:30:32,010 --> 00:30:34,822
So we could leave others to
figure out the right bound.

603
00:30:34,822 --> 00:30:37,280
It's a little tricky with these
gradient descent algorithms

604
00:30:37,280 --> 00:30:38,734
to get good bounds.

605
00:30:38,734 --> 00:30:40,150
In practice they
work really well,

606
00:30:40,150 --> 00:30:41,830
because the gradient
is not as nasty

607
00:30:41,830 --> 00:30:44,640
as you might imagine it
to be in the worst case.

608
00:30:44,640 --> 00:30:47,670
But the worst case bound,
hey, it's pseudo-polynomial.

609
00:30:47,670 --> 00:30:49,290
It's a nice
theoretical guarantee.

610
00:30:49,290 --> 00:30:51,350
And in practice it
also happens to work.

611
00:30:51,350 --> 00:30:53,230
Like these examples
only take, I don't know,

612
00:30:53,230 --> 00:30:55,950
a few hundred steps,
1,000 steps, whatever.

613
00:30:55,950 --> 00:30:58,510
AUDIENCE: Where do
those numbers come from?

614
00:30:58,510 --> 00:31:00,370
PROFESSOR: Where
does 123 come from?

615
00:31:00,370 --> 00:31:02,140
We really wanted
it to be 1, 2, 3.

616
00:31:02,140 --> 00:31:03,796
81 is my birth year.

617
00:31:03,796 --> 00:31:05,404
[LAUGHTER]

618
00:31:05,404 --> 00:31:07,445
PROFESSOR: That's why I
like this number so much.

619
00:31:07,445 --> 00:31:09,760
Oh, actually here
I have 41 written.

620
00:31:09,760 --> 00:31:11,080
I'm pretty sure it's 81 though.

621
00:31:11,080 --> 00:31:15,362
I should correct the notes,
double check, something.

622
00:31:15,362 --> 00:31:18,730
It's really you're just adding
up twos and threes a lot,

623
00:31:18,730 --> 00:31:21,310
and fours, and things like
that a whole bunch of times.

624
00:31:21,310 --> 00:31:24,090
And then just luckily it
came out to a nice number.

625
00:31:24,090 --> 00:31:26,894
As far as I know,
not intentional.

626
00:31:26,894 --> 00:31:28,810
There are few authors
though, so I don't know.

627
00:31:28,810 --> 00:31:31,870
Maybe one of them increased
a bound to make it a little

628
00:31:31,870 --> 00:31:35,480
cooler number in the end.

629
00:31:35,480 --> 00:31:37,767
Good.

630
00:31:37,767 --> 00:31:39,600
So that's pseudo-polynomial
number of steps.

631
00:31:39,600 --> 00:31:41,000
It's interesting,
each of the steps

632
00:31:41,000 --> 00:31:42,291
is actually a very nice motion.

633
00:31:42,291 --> 00:31:44,080
It just moves along
a straight line

634
00:31:44,080 --> 00:31:45,580
in the configuration space.

635
00:31:45,580 --> 00:31:47,580
I would be nice to know
whether you can actually

636
00:31:47,580 --> 00:31:50,800
achieve a polynomial number
of steps independent of r.

637
00:31:50,800 --> 00:31:52,220
I think the answer
is no, it's not

638
00:31:52,220 --> 00:31:55,629
possible for some linkages
that are really tight.

639
00:31:55,629 --> 00:31:57,920
I think you need a dependence,
a polynomial dependence,

640
00:31:57,920 --> 00:32:00,075
on r-- at least a
linear dependence on r.

641
00:32:00,075 --> 00:32:01,550
But I don't know
how to prove that.

642
00:32:01,550 --> 00:32:03,830
It's a nice open problem.

643
00:32:03,830 --> 00:32:07,210
Another fun problem, these
examples with polygons

644
00:32:07,210 --> 00:32:12,210
end up with a particular
convex shape in the end.

645
00:32:12,210 --> 00:32:13,650
Is that shape unique?

646
00:32:13,650 --> 00:32:15,650
If I gave you the sequence
of edge lengths here,

647
00:32:15,650 --> 00:32:18,190
they're all 1, or
they're almost 1.

648
00:32:18,190 --> 00:32:19,196
Not quite the same.

649
00:32:19,196 --> 00:32:21,320
No, sorry, they get tinier
as you go to the center.

650
00:32:21,320 --> 00:32:22,903
But for that sequence
of edge lengths,

651
00:32:22,903 --> 00:32:25,390
is there a unique minimum
energy configuration?

652
00:32:25,390 --> 00:32:26,430
I think so.

653
00:32:26,430 --> 00:32:29,390
But I don't know.

654
00:32:29,390 --> 00:32:31,640
We know that this method
will get to convex,

655
00:32:31,640 --> 00:32:33,930
because only at the
convex configuration

656
00:32:33,930 --> 00:32:35,640
do you no longer have
an expansive motion

657
00:32:35,640 --> 00:32:40,320
and no longer have a decreasing
energy motion, maybe.

658
00:32:40,320 --> 00:32:40,820
Cool.

659
00:32:40,820 --> 00:32:42,150
Those are the three algorithms.

660
00:32:42,150 --> 00:32:43,300
Any questions about them?

661
00:32:49,230 --> 00:32:52,715
Before we go to trees, again,
tell you a cool application

662
00:32:52,715 --> 00:33:02,905
to origami, which
is to rigid origami.

663
00:33:07,040 --> 00:33:10,900
Remember, rigid origami, we're
not allowed to add any creases.

664
00:33:10,900 --> 00:33:15,370
And all of the faces between
creases have to stay flat.

665
00:33:15,370 --> 00:33:17,130
They're like made
of-- imagine you

666
00:33:17,130 --> 00:33:19,920
have pieces of metal
representing the faces.

667
00:33:19,920 --> 00:33:22,730
You have piano hinges
making the edges.

668
00:33:22,730 --> 00:33:24,010
That's rigid origami.

669
00:33:24,010 --> 00:33:25,270
That's kind of like a linkage.

670
00:33:25,270 --> 00:33:27,600
In particular, when you
have a single vertex

671
00:33:27,600 --> 00:33:35,060
origami-- say that's five
foldable, I don't know.

672
00:33:35,060 --> 00:33:37,020
And if you say each
of these wedges

673
00:33:37,020 --> 00:33:41,560
is a rigid piece of
metal, well we already

674
00:33:41,560 --> 00:33:46,380
know this kind of set up can
be modeled by its boundary.

675
00:33:46,380 --> 00:33:47,980
Ignore the interior.

676
00:33:47,980 --> 00:33:51,260
Just think of there
being hinges of this one

677
00:33:51,260 --> 00:33:56,040
dimensional structure like that.

678
00:33:56,040 --> 00:33:59,110
So that looks a
lot like a linkage.

679
00:33:59,110 --> 00:34:01,170
These have to stay rigid.

680
00:34:01,170 --> 00:34:02,400
They're not straight lines.

681
00:34:02,400 --> 00:34:03,780
It's a little different.

682
00:34:03,780 --> 00:34:05,400
It's almost the same.

683
00:34:05,400 --> 00:34:08,227
If you think about how you're
allowed to fold these things--

684
00:34:08,227 --> 00:34:09,810
really you should
think about it here,

685
00:34:09,810 --> 00:34:14,310
I guess-- what happens, by
a continuous motion what

686
00:34:14,310 --> 00:34:17,600
happens is that you're
living on a sphere.

687
00:34:17,600 --> 00:34:22,290
So you start out on the
equator of the sphere.

688
00:34:22,290 --> 00:34:24,040
This thing, just plop
it down the equator.

689
00:34:24,040 --> 00:34:26,139
The boundary lies on the
boundary of the sphere.

690
00:34:26,139 --> 00:34:28,659
The interior of the paper
is inside the sphere,

691
00:34:28,659 --> 00:34:31,659
right along the flat part there.

692
00:34:31,659 --> 00:34:35,909
As you fold, these points
will stay on the sphere.

693
00:34:35,909 --> 00:34:38,590
And these edge links
will be preserved,

694
00:34:38,590 --> 00:34:39,940
their arcs on the spheres.

695
00:34:39,940 --> 00:34:42,580
There will be great
circular arcs at all times.

696
00:34:42,580 --> 00:34:44,710
So folding this
thing in three space

697
00:34:44,710 --> 00:34:50,150
is really equivalent to folding
this linkage on the sphere.

698
00:34:50,150 --> 00:34:52,650
And that looks an awful lot
like a polygon on a sphere.

699
00:34:52,650 --> 00:34:55,659
What we really want is a
spherical Carpenter's Rule

700
00:34:55,659 --> 00:34:56,159
Theorem.

701
00:35:02,980 --> 00:35:05,430
And there is.

702
00:35:05,430 --> 00:35:07,190
This is by Streinu and Whiteley.

703
00:35:10,940 --> 00:35:22,370
So if you have a closed chain,
a polygon, of total length

704
00:35:22,370 --> 00:35:32,290
at most 2 pi on a
unit sphere then you

705
00:35:32,290 --> 00:35:35,195
have a connected
configuration space.

706
00:35:41,320 --> 00:35:44,421
So in the plane,
closed chain always

707
00:35:44,421 --> 00:35:45,920
had a connected
configuration space.

708
00:35:45,920 --> 00:35:50,002
On the sphere you need that
the chain is not too long,

709
00:35:50,002 --> 00:35:52,210
because there's only sort
of a bounded amount of room

710
00:35:52,210 --> 00:35:53,310
on the sphere.

711
00:35:53,310 --> 00:36:03,720
The equator has total length 2
pi, perimeter of the equator.

712
00:36:03,720 --> 00:36:06,230
For unit sphere it's 2 pi.

713
00:36:06,230 --> 00:36:08,540
So we're just canonically
making it a unit sphere.

714
00:36:08,540 --> 00:36:10,940
And 2 pi really corresponds
to this situation,

715
00:36:10,940 --> 00:36:12,520
which is 360 degrees.

716
00:36:12,520 --> 00:36:14,190
That's how much
total length we have.

717
00:36:14,190 --> 00:36:16,629
So it just fits in the
equator, life is good.

718
00:36:16,629 --> 00:36:18,170
The reason you want
it to have length

719
00:36:18,170 --> 00:36:20,700
at most 2 pi is because
then you actually do

720
00:36:20,700 --> 00:36:22,880
have a convex configuration.

721
00:36:22,880 --> 00:36:25,040
In the case of 2 pi, it
lies along the equator.

722
00:36:25,040 --> 00:36:28,980
If it's smaller than 2 pi it'll
be like some smaller portion.

723
00:36:28,980 --> 00:36:32,100
This would be less than
2 pi on the sphere.

724
00:36:35,600 --> 00:36:38,917
If it's greater than 2 pi,
you can't draw it convexly

725
00:36:38,917 --> 00:36:39,500
on the sphere.

726
00:36:39,500 --> 00:36:40,480
You can draw something.

727
00:36:40,480 --> 00:36:42,410
Like I could draw
something like this,

728
00:36:42,410 --> 00:36:46,090
that'll have really long
length on the sphere.

729
00:36:46,090 --> 00:36:48,160
But there will be no
way to convexify it.

730
00:36:48,160 --> 00:36:50,480
You get stuck at some point.

731
00:36:50,480 --> 00:36:52,670
And as long as you don't
have that situation,

732
00:36:52,670 --> 00:36:55,210
you can take this--
what this actually

733
00:36:55,210 --> 00:36:57,300
implies is that your
polygon at all times

734
00:36:57,300 --> 00:36:58,940
will lie in a hemisphere.

735
00:36:58,940 --> 00:37:01,400
You take that hemisphere
and you sort of unroll it,

736
00:37:01,400 --> 00:37:04,010
you splay it out, you
project it to the plane.

737
00:37:04,010 --> 00:37:06,630
You apply the planar
Carpenter's Rule Theorem.

738
00:37:06,630 --> 00:37:09,770
You apply the fact that
unrolling operation,

739
00:37:09,770 --> 00:37:11,970
or that projection
if you will-- I

740
00:37:11,970 --> 00:37:17,200
think it's like
projection from some point

741
00:37:17,200 --> 00:37:21,730
here out to the plane--
that projection preserves

742
00:37:21,730 --> 00:37:26,520
infinitesimal flexibility
of tensegrities.

743
00:37:26,520 --> 00:37:29,630
So it'll still have the
expensive motion in the plane.

744
00:37:29,630 --> 00:37:33,590
And then you could turn in to an
expansive motion on the sphere.

745
00:37:33,590 --> 00:37:36,620
And eventually you'll
get a convex polygon.

746
00:37:36,620 --> 00:37:38,850
So that's sort of
how this is proved.

747
00:37:38,850 --> 00:37:41,020
Then you can apply
it to rigid origami,

748
00:37:41,020 --> 00:37:42,870
and say for single
vertex origami,

749
00:37:42,870 --> 00:37:44,920
it's always rigidly foldable.

750
00:37:44,920 --> 00:37:46,950
Any state you want
to reach can be

751
00:37:46,950 --> 00:37:50,332
reached by continuous
motion, without bending

752
00:37:50,332 --> 00:37:51,040
any of the faces.

753
00:37:53,890 --> 00:37:55,190
So a fun little application.

754
00:38:01,320 --> 00:38:04,250
And one of the few things
we know about rigid origami,

755
00:38:04,250 --> 00:38:08,170
for multiple vertices
it gets a lot harder.

756
00:38:08,170 --> 00:38:11,527
Not always possible,
and we don't

757
00:38:11,527 --> 00:38:12,735
have a nice characterization.

758
00:38:24,480 --> 00:38:29,900
So finally, let's move onto
locked trees in the plane.

759
00:38:34,910 --> 00:38:38,100
So these are sort of
the classic examples.

760
00:38:38,100 --> 00:38:41,520
The top two were
in a 1998 paper.

761
00:38:41,520 --> 00:38:46,010
This was when I started
working on folding stuff, very

762
00:38:46,010 --> 00:38:47,366
beginning.

763
00:38:47,366 --> 00:38:52,120
A whole bunch of people
from a big workshop.

764
00:38:52,120 --> 00:38:53,970
First example is this one.

765
00:38:53,970 --> 00:39:00,080
And it's a bunch of sort of
arms tucked into their armpits,

766
00:39:00,080 --> 00:39:02,550
and in a cyclic way.

767
00:39:02,550 --> 00:39:06,680
And the dotted circles
mean objects in this mirror

768
00:39:06,680 --> 00:39:08,530
are closer than they appear.

769
00:39:08,530 --> 00:39:13,270
So imagine that
all of these points

770
00:39:13,270 --> 00:39:17,050
are actually really, really
close and tight in here.

771
00:39:17,050 --> 00:39:19,140
And this guy's actually
really close and tight

772
00:39:19,140 --> 00:39:21,212
against this edge.

773
00:39:21,212 --> 00:39:22,670
So I've drawn it
with lots of slack

774
00:39:22,670 --> 00:39:24,461
so you can see the
combinatorial structure.

775
00:39:24,461 --> 00:39:27,070
But geometrically
it's much tighter.

776
00:39:27,070 --> 00:39:29,730
Then the intuition is that you
can't get any of these arms

777
00:39:29,730 --> 00:39:35,430
open unless you could somehow
expand one of these wedges.

778
00:39:35,430 --> 00:39:38,160
It's like, if you could
expand this angle then

779
00:39:38,160 --> 00:39:40,159
this guy would have
room to come out.

780
00:39:40,159 --> 00:39:41,450
But how do I expand that angle?

781
00:39:41,450 --> 00:39:43,533
Well I'd have to compress
the other angles because

782
00:39:43,533 --> 00:39:44,470
of the cyclic picture.

783
00:39:44,470 --> 00:39:47,450
And I can't, if I look at some
other angle like this one,

784
00:39:47,450 --> 00:39:50,830
I can't compress it because
the arm is in the way.

785
00:39:50,830 --> 00:39:54,962
So I can't open an arm until
I've closed some other arm.

786
00:39:54,962 --> 00:39:56,920
And I can't close an arm
before I've opened it.

787
00:39:56,920 --> 00:39:58,090
And so nothing can happen.

788
00:39:58,090 --> 00:40:00,390
That's the intuition
behind the proof.

789
00:40:00,390 --> 00:40:02,050
The details are very
messy, because you

790
00:40:02,050 --> 00:40:05,210
have to define open
and closed, and what

791
00:40:05,210 --> 00:40:08,140
things must happen
before what things.

792
00:40:08,140 --> 00:40:09,550
This example should
look familiar

793
00:40:09,550 --> 00:40:12,410
because if you double it, if
you replace every edge with two

794
00:40:12,410 --> 00:40:14,300
edges, we get one
of the examples that

795
00:40:14,300 --> 00:40:19,260
was expanding with the
five-fold symmetry.

796
00:40:19,260 --> 00:40:22,770
So people have sort
of known for a while,

797
00:40:22,770 --> 00:40:26,550
and then we finally proved,
that this is locked.

798
00:40:26,550 --> 00:40:29,250
If you double it, people thought
well maybe it's still locked.

799
00:40:29,250 --> 00:40:34,730
Turns out no, because the center
vertex can expand in a polygon,

800
00:40:34,730 --> 00:40:36,600
but with a tree it can't.

801
00:40:36,600 --> 00:40:38,900
OK, big deal.

802
00:40:38,900 --> 00:40:42,910
These examples, this is just
yet another way to do that.

803
00:40:42,910 --> 00:40:47,060
But what's interesting is if
you double some of the edges

804
00:40:47,060 --> 00:40:48,700
you get this tree.

805
00:40:48,700 --> 00:40:51,610
So you have 1 degree 3
vertex in the center.

806
00:40:51,610 --> 00:40:54,160
And you go around,
you visit this arm.

807
00:40:54,160 --> 00:40:56,040
You go back, you visit this arm.

808
00:40:56,040 --> 00:40:58,860
Turns out this really
does simulate this tree,

809
00:40:58,860 --> 00:41:01,720
because we haven't touched
the central degree 3 vertex.

810
00:41:01,720 --> 00:41:05,300
And this is nice because
it has one degree 3 vertex.

811
00:41:05,300 --> 00:41:07,890
Everything else
is degree 2 or 1.

812
00:41:07,890 --> 00:41:09,420
So in the Carpenter's
Rule theorem

813
00:41:09,420 --> 00:41:12,330
where we said maximum
degree 2, it's really tight.

814
00:41:12,330 --> 00:41:14,810
And as soon as you add
one vertex of degree 3

815
00:41:14,810 --> 00:41:16,800
you can be locked.

816
00:41:16,800 --> 00:41:20,000
That was the point
of this example.

817
00:41:20,000 --> 00:41:22,010
What other fun things
can you do with trees?

818
00:41:22,010 --> 00:41:24,610
Well, three years
ago in this class

819
00:41:24,610 --> 00:41:27,360
we started thinking
about other locked trees.

820
00:41:27,360 --> 00:41:29,510
How low could you go?

821
00:41:29,510 --> 00:41:31,980
How many edges do
need to get locked?

822
00:41:31,980 --> 00:41:36,579
This example-- no I guess
this example would be minimum.

823
00:41:36,579 --> 00:41:38,120
Here I've drawn it
with eight petals.

824
00:41:38,120 --> 00:41:40,770
You can get away
with five petals,

825
00:41:40,770 --> 00:41:43,200
or are five arms,
each of length 3.

826
00:41:43,200 --> 00:41:44,500
So that's 15 edges.

827
00:41:44,500 --> 00:41:46,170
That was the state of the art.

828
00:41:46,170 --> 00:41:50,650
And then we had this crazy
idea in a problem session

829
00:41:50,650 --> 00:41:55,010
of this locked
chain, locked tree.

830
00:41:55,010 --> 00:41:57,530
It has two degree 3 vertices.

831
00:41:57,530 --> 00:41:59,360
And it kind of just
winds in there.

832
00:41:59,360 --> 00:42:01,360
We're going to see why
these things are locked.

833
00:42:01,360 --> 00:42:02,170
But OK, that's kind of neat.

834
00:42:02,170 --> 00:42:03,740
11, that's much better than 15.

835
00:42:03,740 --> 00:42:04,820
Can we do better?

836
00:42:04,820 --> 00:42:08,890
And every week we improved
it, for a few weeks.

837
00:42:08,890 --> 00:42:13,140
This one looks messier,
but it has one fewer edge.

838
00:42:13,140 --> 00:42:16,930
And then this one came
along, and we're like whoa.

839
00:42:16,930 --> 00:42:20,860
So symmetric, so beautiful.

840
00:42:20,860 --> 00:42:25,580
What's interesting is it doesn't
have the cyclic structure.

841
00:42:25,580 --> 00:42:27,422
It's almost flat, in fact.

842
00:42:27,422 --> 00:42:29,630
You could think of all these
guys being in one point.

843
00:42:29,630 --> 00:42:31,030
All these guys
being in one point.

844
00:42:31,030 --> 00:42:32,446
All those guys
being in one point.

845
00:42:32,446 --> 00:42:34,460
It's like they're,
all three, collinear.

846
00:42:34,460 --> 00:42:36,360
So it's a very different
kind of example.

847
00:42:36,360 --> 00:42:38,670
Before we thought
locked trees required

848
00:42:38,670 --> 00:42:40,425
this kind of cyclic condition.

849
00:42:43,170 --> 00:42:46,950
And so, for example,
we conjectured--

850
00:42:46,950 --> 00:42:48,810
or I guess Poon,
one of the authors

851
00:42:48,810 --> 00:42:51,516
here-- conjectured that
there would be no way

852
00:42:51,516 --> 00:42:53,140
to lock something if
all the edges were

853
00:42:53,140 --> 00:42:54,750
horizontal and vertical.

854
00:42:54,750 --> 00:42:57,770
Because if you have
that you couldn't

855
00:42:57,770 --> 00:42:59,770
have five things in a circle.

856
00:42:59,770 --> 00:43:02,030
Now suddenly we think,
oh, this is interesting.

857
00:43:02,030 --> 00:43:05,970
Because it has two
degree 3 vertices.

858
00:43:05,970 --> 00:43:07,140
It's so symmetric.

859
00:43:07,140 --> 00:43:09,220
Surely this is the
fewest possible edges

860
00:43:09,220 --> 00:43:11,840
we could get away with.

861
00:43:11,840 --> 00:43:16,550
But no, then we found
the eight edge example.

862
00:43:16,550 --> 00:43:18,580
And this is kind of funny.

863
00:43:18,580 --> 00:43:22,940
It's like instead of
being nice and symmetric,

864
00:43:22,940 --> 00:43:25,110
essentially, what we're
doing is removing this edge.

865
00:43:25,110 --> 00:43:27,229
And we want to instead
attach it in here.

866
00:43:27,229 --> 00:43:29,270
But then we have to futz
around with the vertices

867
00:43:29,270 --> 00:43:31,070
to make it work
out and be a tree.

868
00:43:31,070 --> 00:43:35,020
And now has only
one degree 3 vertex.

869
00:43:35,020 --> 00:43:36,751
So it also has
that nice property.

870
00:43:36,751 --> 00:43:37,750
It only has eight edges.

871
00:43:37,750 --> 00:43:39,124
And this, we
believe, is optimal.

872
00:43:43,140 --> 00:43:47,120
And we can actually prove
something along those lines.

873
00:43:47,120 --> 00:43:59,870
So a linear tree is one where
all the vertices lie nearly

874
00:43:59,870 --> 00:44:00,370
on the line.

875
00:44:04,930 --> 00:44:07,360
So this is a linear locked tree.

876
00:44:07,360 --> 00:44:09,640
And we can prove that among
all linear locked trees,

877
00:44:09,640 --> 00:44:11,820
they must have at
least eight edges.

878
00:44:15,952 --> 00:44:24,130
Locked linear tree has
at least eight edges.

879
00:44:24,130 --> 00:44:28,390
So at least among linear locked
trees that example is optimal.

880
00:44:28,390 --> 00:44:32,490
But maybe you could use
the second dimension

881
00:44:32,490 --> 00:44:35,400
to do something better
than eight edges.

882
00:44:35,400 --> 00:44:36,840
But I don't think so.

883
00:44:36,840 --> 00:44:39,230
That's an open problem.

884
00:44:39,230 --> 00:44:41,110
What other good
things can we do?

885
00:44:41,110 --> 00:44:43,110
You can make it orthogonal.

886
00:44:43,110 --> 00:44:45,570
You can mimic exactly
this structure

887
00:44:45,570 --> 00:44:47,440
with an orthogonal
structure, all the edges

888
00:44:47,440 --> 00:44:48,840
horizontal and vertical.

889
00:44:48,840 --> 00:44:52,170
Just expand each of these
vertices into very tiny,

890
00:44:52,170 --> 00:44:54,120
and if this is drawn
really squished,

891
00:44:54,120 --> 00:44:55,450
these are super short edges.

892
00:44:55,450 --> 00:44:58,550
So they really don't change
the motion space hardly at all.

893
00:44:58,550 --> 00:44:59,910
You can actually prove that.

894
00:44:59,910 --> 00:45:01,222
And this is also locked.

895
00:45:01,222 --> 00:45:03,430
That's one of the nice things
you do once you get out

896
00:45:03,430 --> 00:45:04,810
of the cyclic kind of structure.

897
00:45:07,540 --> 00:45:10,110
I think I have
something else, no.

898
00:45:10,110 --> 00:45:11,300
OK, stay there.

899
00:45:15,180 --> 00:45:17,130
Yeah, actually maybe
I do want to go there.

900
00:45:21,870 --> 00:45:24,980
This is an example, it has
a cyclic structure again.

901
00:45:24,980 --> 00:45:30,020
Same paper, last year.

902
00:45:30,020 --> 00:45:31,414
This is a newer example.

903
00:45:31,414 --> 00:45:33,580
And it has the property
that all of the edge lengths

904
00:45:33,580 --> 00:45:38,960
are the same, and
nothing touches.

905
00:45:38,960 --> 00:45:43,160
Now you'll appreciate this in
a little while, how crazy this

906
00:45:43,160 --> 00:45:43,730
is.

907
00:45:43,730 --> 00:45:47,190
Because all of the
previous examples

908
00:45:47,190 --> 00:45:49,560
required things to be
very close and tight.

909
00:45:49,560 --> 00:45:54,230
And we crucially use
very tight proximity

910
00:45:54,230 --> 00:45:57,150
in order to prove
things are locked.

911
00:45:57,150 --> 00:46:05,370
And if you take the example
that looks like this, with six

912
00:46:05,370 --> 00:46:14,250
arms-- I didn't
draw it very well.

913
00:46:14,250 --> 00:46:15,700
But if you draw
it with six arms,

914
00:46:15,700 --> 00:46:17,790
these are like
equilateral triangles,

915
00:46:17,790 --> 00:46:19,902
the edge lengths will
be almost all the same.

916
00:46:19,902 --> 00:46:21,360
In fact, if you
allow them to touch

917
00:46:21,360 --> 00:46:23,247
they will be exactly the same.

918
00:46:23,247 --> 00:46:24,830
So there was this
open question, well,

919
00:46:24,830 --> 00:46:27,740
if I want them to be exactly
the same but not touch,

920
00:46:27,740 --> 00:46:29,180
can you still lock?

921
00:46:29,180 --> 00:46:31,940
We thought no, but in
fact you can do it.

922
00:46:31,940 --> 00:46:33,710
And this is very
tricky to prove locked

923
00:46:33,710 --> 00:46:36,645
because we can't make
these arbitrarily tight.

924
00:46:36,645 --> 00:46:39,020
It really has to look like
this, because the edge lengths

925
00:46:39,020 --> 00:46:40,350
are all the same.

926
00:46:40,350 --> 00:46:43,275
Now this has
seven-fold symmetry.

927
00:46:46,030 --> 00:46:48,181
Because sixfold they
would all touch.

928
00:46:48,181 --> 00:46:48,680
It's tricky.

929
00:46:52,170 --> 00:46:52,670
Cool.

930
00:46:55,870 --> 00:46:57,370
A big open question
here, of course,

931
00:46:57,370 --> 00:46:59,300
is to characterize locked trees.

932
00:46:59,300 --> 00:47:02,149
I think that's quite hard,
because in particular,

933
00:47:02,149 --> 00:47:03,940
if you're given a tree
and you want to know

934
00:47:03,940 --> 00:47:05,440
does that go from
this configuration

935
00:47:05,440 --> 00:47:07,770
to this configuration,
that's known to be p space

936
00:47:07,770 --> 00:47:10,400
complete, which is
really, really hard.

937
00:47:10,400 --> 00:47:12,180
But all of those
examples are locked.

938
00:47:12,180 --> 00:47:15,520
So maybe unlocked trees
have some special structure

939
00:47:15,520 --> 00:47:17,020
that's easy to find.

940
00:47:17,020 --> 00:47:18,090
I guess not.

941
00:47:18,090 --> 00:47:20,200
But who knows?

942
00:47:20,200 --> 00:47:21,700
What I do think has
a nice structure

943
00:47:21,700 --> 00:47:23,884
are these linear locked trees.

944
00:47:23,884 --> 00:47:24,800
They're pretty simple.

945
00:47:24,800 --> 00:47:26,230
They're basically
one dimensional.

946
00:47:26,230 --> 00:47:28,790
And I think we could
characterize linear lock trees

947
00:47:28,790 --> 00:47:30,170
in polynomial time.

948
00:47:30,170 --> 00:47:33,430
Maybe we'll work on
that this afternoon.

949
00:47:33,430 --> 00:47:36,590
But that is open.

950
00:47:36,590 --> 00:47:38,460
All right, next thing
I want to talk about

951
00:47:38,460 --> 00:47:41,364
is how the heck do you prove
that these things are locked?

952
00:47:41,364 --> 00:47:43,780
Now historically there have
been lots of different proofs.

953
00:47:43,780 --> 00:47:46,860
And this tree still does
not have a nice proof

954
00:47:46,860 --> 00:47:48,000
that it's locked.

955
00:47:48,000 --> 00:47:50,420
But all the other trees I
can give you very succinct

956
00:47:50,420 --> 00:47:52,182
proofs that they're locked.

957
00:47:52,182 --> 00:47:53,890
And so I want to tell
you how we do that.

958
00:48:03,230 --> 00:48:06,980
Because it uses tensegrity
theory, our good friend.

959
00:48:14,120 --> 00:48:18,365
This is the idea of
infinitesimally locked

960
00:48:18,365 --> 00:48:18,865
linkages.

961
00:48:38,199 --> 00:48:39,990
So ignore this part of
the picture for now.

962
00:48:39,990 --> 00:48:43,240
If we take some tree,
most of the examples

963
00:48:43,240 --> 00:48:45,370
I drew these little
scion circles to say,

964
00:48:45,370 --> 00:48:47,700
well, these guys
are really tight.

965
00:48:47,700 --> 00:48:49,350
And the intuition
is the tighter you

966
00:48:49,350 --> 00:48:52,506
make it, the less that
configuration can move.

967
00:48:52,506 --> 00:48:54,380
If you look, we're
claiming the configuration

968
00:48:54,380 --> 00:48:55,460
space is disconnected.

969
00:48:55,460 --> 00:48:57,335
But not only that, we
have this configuration

970
00:48:57,335 --> 00:49:00,265
and we say there's a
small ball of motions

971
00:49:00,265 --> 00:49:01,390
that you could possibly do.

972
00:49:01,390 --> 00:49:03,667
It does wiggle a little bit.

973
00:49:03,667 --> 00:49:05,250
And then there's
other stuff over here

974
00:49:05,250 --> 00:49:07,790
that you can never reach because
this can't move very much.

975
00:49:12,270 --> 00:49:18,060
Well in fact, if you make these
circles tighter and tighter,

976
00:49:18,060 --> 00:49:21,830
the claim is that this,
in the space of motions,

977
00:49:21,830 --> 00:49:24,750
you get less and less freedom.

978
00:49:24,750 --> 00:49:26,950
How could we formalize that?

979
00:49:26,950 --> 00:49:30,040
As you draw the thing tighter
you get less and less motion.

980
00:49:32,730 --> 00:49:35,540
Well, let's go to the limit.

981
00:49:35,540 --> 00:49:37,540
Suppose we went all
the way to the point

982
00:49:37,540 --> 00:49:40,890
that these things are touching.

983
00:49:40,890 --> 00:49:43,670
Now this is going to be a
little tricky mathematically,

984
00:49:43,670 --> 00:49:46,770
because we have to remember that
this vertex is in this wedge,

985
00:49:46,770 --> 00:49:49,330
even though it's actually
right on top of this point.

986
00:49:49,330 --> 00:49:53,190
So we have to remember, sort
of, how things look locally.

987
00:49:53,190 --> 00:50:04,390
But geometrically the picture
is going to look like that.

988
00:50:04,390 --> 00:50:06,740
If you look from
afar, you won't be

989
00:50:06,740 --> 00:50:09,290
able to tell that there's
three edges along here,

990
00:50:09,290 --> 00:50:11,630
because they're right
on top of each other.

991
00:50:11,630 --> 00:50:14,860
But if you, sort of, imagine
zooming in infinitesimally

992
00:50:14,860 --> 00:50:16,420
in each of these
vertices, it's going

993
00:50:16,420 --> 00:50:18,720
to be whatever the heck it is.

994
00:50:24,504 --> 00:50:26,420
So you remember the fact
that right here there

995
00:50:26,420 --> 00:50:27,440
are four vertices.

996
00:50:27,440 --> 00:50:29,606
And this is how they're
connected to incident edges.

997
00:50:29,606 --> 00:50:32,750
There's actually three edges
here, and two edges here,

998
00:50:32,750 --> 00:50:33,250
and so on.

999
00:50:36,080 --> 00:50:38,140
So that's how you
describe one of these,

1000
00:50:38,140 --> 00:50:44,890
we call them,
self-touching linkages.

1001
00:50:44,890 --> 00:50:47,822
Because the edges are
touching each other.

1002
00:50:47,822 --> 00:50:51,170
They're right on
top of each other.

1003
00:50:51,170 --> 00:50:56,330
Now normally, if you wanted to
capture this notion of wiggling

1004
00:50:56,330 --> 00:50:59,100
a little bit, we
could define the idea

1005
00:50:59,100 --> 00:51:01,640
of being locked within epsilon.

1006
00:51:05,880 --> 00:51:10,520
So a configuration is
locked within epsilon

1007
00:51:10,520 --> 00:51:23,440
if it's impossible
to get farther

1008
00:51:23,440 --> 00:51:27,610
than epsilon in
configuration space.

1009
00:51:33,290 --> 00:51:36,910
So that's this little ball,
thinking about radius epsilon

1010
00:51:36,910 --> 00:51:38,076
here.

1011
00:51:38,076 --> 00:51:39,700
And if you can't get
outside that ball,

1012
00:51:39,700 --> 00:51:41,440
you have some weird
space you can get to.

1013
00:51:41,440 --> 00:51:44,480
But if it's bounded by
a ball of radius epsilon

1014
00:51:44,480 --> 00:51:47,432
then I say you're
locked within epsilon.

1015
00:51:47,432 --> 00:51:48,890
And if epsilon is
small that really

1016
00:51:48,890 --> 00:51:51,650
means there are other
things you can't get to.

1017
00:51:51,650 --> 00:51:53,710
Well as soon as I
get to self-touching

1018
00:51:53,710 --> 00:51:57,090
I can actually think about
being locked within epsilon

1019
00:51:57,090 --> 00:52:05,250
for epsilon equals 0,
also known as being rigid.

1020
00:52:07,839 --> 00:52:09,880
We've already talked about
being locked within 0.

1021
00:52:09,880 --> 00:52:12,560
If you can't move it
all, that's rigid.

1022
00:52:12,560 --> 00:52:14,630
So this thing could
actually be rigid.

1023
00:52:14,630 --> 00:52:16,314
Whereas all the
locked trees, they

1024
00:52:16,314 --> 00:52:17,730
can never be rigid
because I don't

1025
00:52:17,730 --> 00:52:18,910
want them to really touch.

1026
00:52:18,910 --> 00:52:19,970
This is an analysis tool.

1027
00:52:19,970 --> 00:52:21,720
I don't like trees
that are self-touching.

1028
00:52:21,720 --> 00:52:23,760
It's kind of ugly and cheating.

1029
00:52:26,750 --> 00:52:29,370
And so real trees,
non-self-touching touching

1030
00:52:29,370 --> 00:52:32,400
trees could only be locked
within some positive epsilon.

1031
00:52:32,400 --> 00:52:34,360
But if we consider
for the moment

1032
00:52:34,360 --> 00:52:36,850
the extreme when
they're touching then

1033
00:52:36,850 --> 00:52:38,110
we could hope for rigidity.

1034
00:52:38,110 --> 00:52:40,734
Rigidity is good because we know
how to prove things are rigid.

1035
00:52:40,734 --> 00:52:43,070
We know how to test things
are rigid in two dimensions.

1036
00:52:43,070 --> 00:52:44,590
It's pretty easy.

1037
00:52:44,590 --> 00:52:46,620
We could test is it
infinitesimally rigid?

1038
00:52:46,620 --> 00:52:50,770
If it's infinitesimally
rigid we know it's rigid.

1039
00:52:50,770 --> 00:52:55,280
It's a little bit
trickier because we

1040
00:52:55,280 --> 00:52:58,100
have to represent the
non-crossing constraints.

1041
00:52:58,100 --> 00:53:00,650
And that's what these
purple edges do.

1042
00:53:00,650 --> 00:53:04,830
So the idea is,
well I definitely

1043
00:53:04,830 --> 00:53:06,780
want to preserve the
lengths of these edges.

1044
00:53:06,780 --> 00:53:08,490
I'm not interested
in expansive motions,

1045
00:53:08,490 --> 00:53:10,770
because that's a subset
of possible motions.

1046
00:53:10,770 --> 00:53:14,860
But I do know that this
vertex should move away

1047
00:53:14,860 --> 00:53:16,530
from this edge, or
stay on the edge.

1048
00:53:16,530 --> 00:53:19,610
It's not allowed to go to
the other side of the edge.

1049
00:53:19,610 --> 00:53:21,850
So I imagine there's a
little tiny strut here.

1050
00:53:21,850 --> 00:53:23,840
It's of infinitesimal
length right now.

1051
00:53:23,840 --> 00:53:26,890
It can get longer, but it has
to get longer in that direction.

1052
00:53:26,890 --> 00:53:28,590
You have to move
away-- you have to stay

1053
00:53:28,590 --> 00:53:30,425
on the right side of this edge.

1054
00:53:30,425 --> 00:53:32,550
And you have to stay on
the left side of this edge.

1055
00:53:32,550 --> 00:53:36,170
It turns out you really can
represent that by a strut.

1056
00:53:36,170 --> 00:53:38,150
It's now a strut between
a vertex and an edge,

1057
00:53:38,150 --> 00:53:41,730
because this guy can slide along
the edge, or move away from it.

1058
00:53:41,730 --> 00:53:43,680
But it's a strut.

1059
00:53:43,680 --> 00:53:45,840
And you can think of
this as a tensegrity,

1060
00:53:45,840 --> 00:53:48,170
all the usual tensegrity
theory applies.

1061
00:53:48,170 --> 00:53:52,690
You can define equilibrium
stresses, polyhedral liftings,

1062
00:53:52,690 --> 00:53:54,590
infinitesimal rigidity.

1063
00:53:54,590 --> 00:53:58,926
And so what's
actually drawn here--

1064
00:53:58,926 --> 00:54:00,800
you should look at the
book for more details.

1065
00:54:00,800 --> 00:54:02,466
I don't want to go
into details on this.

1066
00:54:02,466 --> 00:54:05,470
But originally the state of
the art for proving something

1067
00:54:05,470 --> 00:54:08,090
like this is locked
is you basically

1068
00:54:08,090 --> 00:54:11,130
give-- you show an
equilibrium stress that

1069
00:54:11,130 --> 00:54:13,714
is positive on all the struts.

1070
00:54:13,714 --> 00:54:15,380
We know that if you
find a stress that's

1071
00:54:15,380 --> 00:54:17,640
positive on all struts
than all the struts in fact

1072
00:54:17,640 --> 00:54:19,670
must act as bars.

1073
00:54:19,670 --> 00:54:24,350
What that means is that this
length, which is currently 0,

1074
00:54:24,350 --> 00:54:26,370
must stay 0.

1075
00:54:26,370 --> 00:54:27,890
Therefore this
vertex is actually

1076
00:54:27,890 --> 00:54:31,060
pinned because it has
to both be on this edge

1077
00:54:31,060 --> 00:54:32,020
and be on this edge.

1078
00:54:32,020 --> 00:54:34,959
That's what a 0 length bar
here and here would mean.

1079
00:54:34,959 --> 00:54:36,250
It really has to be right here.

1080
00:54:36,250 --> 00:54:37,580
It can't move it all.

1081
00:54:37,580 --> 00:54:40,790
And therefore this whole thing
is rigid, and nothing moves.

1082
00:54:47,480 --> 00:54:48,730
Clear?

1083
00:54:48,730 --> 00:54:50,810
So we can use all
the tensegrity stuff

1084
00:54:50,810 --> 00:54:53,420
to prove that when this thing
is actually self-touching,

1085
00:54:53,420 --> 00:54:57,020
and all these distances
are 0, it is rigid.

1086
00:54:57,020 --> 00:54:58,510
But so what?

1087
00:54:58,510 --> 00:55:00,560
Who cares about the
self-touching thing

1088
00:55:00,560 --> 00:55:01,890
being rigid?

1089
00:55:01,890 --> 00:55:03,790
It's nice, but what
I really care about

1090
00:55:03,790 --> 00:55:06,150
is the non-self-touching
configurations

1091
00:55:06,150 --> 00:55:09,540
are locked within epsilon,
for some tiny epsilon.

1092
00:55:09,540 --> 00:55:10,285
Well good news.

1093
00:55:15,800 --> 00:55:21,120
Rigidity of the
self-touching configuration

1094
00:55:21,120 --> 00:55:28,620
implies what's called
strongly locked,

1095
00:55:28,620 --> 00:55:32,690
yet another term which
I need to define.

1096
00:55:32,690 --> 00:55:45,240
Strongly locked means
that sufficiently small,

1097
00:55:45,240 --> 00:56:01,755
sufficiently small
perturbations of the linkage,

1098
00:56:01,755 --> 00:56:15,140
or I should say of the
linkage configuration,

1099
00:56:15,140 --> 00:56:21,980
are locked within
epsilon for any epsilon.

1100
00:56:26,170 --> 00:56:30,420
This is exactly the property
I wanted to formalize,

1101
00:56:30,420 --> 00:56:34,710
saying that if you draw
the example tighter

1102
00:56:34,710 --> 00:56:36,660
it can move less.

1103
00:56:36,660 --> 00:56:39,390
For any epsilon, we
want to say you can only

1104
00:56:39,390 --> 00:56:43,150
move within epsilon,
there's some notion

1105
00:56:43,150 --> 00:56:47,640
of sufficiently small such
that if I take this example

1106
00:56:47,640 --> 00:56:49,570
and imagine currently
all the vertices are

1107
00:56:49,570 --> 00:56:50,854
on top of each other.

1108
00:56:50,854 --> 00:56:52,270
But now I perturb
it a little bit,

1109
00:56:52,270 --> 00:56:54,720
which involves changing not
only the vertex coordinates,

1110
00:56:54,720 --> 00:56:58,730
but also the edge lengths--
but a tiny amount, some delta

1111
00:56:58,730 --> 00:57:01,740
that's a function of epsilon.

1112
00:57:01,740 --> 00:57:03,360
Whatever epsilon
you choose, there's

1113
00:57:03,360 --> 00:57:05,870
a very small disk I
can draw like this,

1114
00:57:05,870 --> 00:57:09,600
such that as long as all the
vertices stay within that disk

1115
00:57:09,600 --> 00:57:13,150
your example will be
locked within epsilon.

1116
00:57:13,150 --> 00:57:15,870
This is great because it lets
us analyze rigidity, which

1117
00:57:15,870 --> 00:57:18,940
is easy for self-touching
configurations, which are not

1118
00:57:18,940 --> 00:57:20,940
interesting in some sense.

1119
00:57:20,940 --> 00:57:24,470
But we get to conclude something
about the perturbations which

1120
00:57:24,470 --> 00:57:26,607
are not self-touching
and therefore nice.

1121
00:57:26,607 --> 00:57:28,440
And we get the property
we want, that you're

1122
00:57:28,440 --> 00:57:30,720
locked within epsilon
for any epsilon you want.

1123
00:57:30,720 --> 00:57:32,747
You just draw it
tighter you'll be

1124
00:57:32,747 --> 00:57:34,080
locked within a smaller epsilon.

1125
00:57:37,540 --> 00:57:39,400
So this is pretty cool.

1126
00:57:39,400 --> 00:57:41,900
And I didn't defined
perturbation.

1127
00:57:41,900 --> 00:57:54,691
But I just mean every vertex
stays within a radius delta

1128
00:57:54,691 --> 00:57:55,190
disk.

1129
00:57:59,520 --> 00:58:03,369
And sufficiently
small here is delta,

1130
00:58:03,369 --> 00:58:04,660
which is a function of epsilon.

1131
00:58:09,890 --> 00:58:12,130
So this is pretty cool.

1132
00:58:12,130 --> 00:58:14,970
And it turns out also--
this is proved much later--

1133
00:58:14,970 --> 00:58:16,570
these results are like 2002.

1134
00:58:16,570 --> 00:58:21,320
And then 2006 proved
that if you take

1135
00:58:21,320 --> 00:58:23,900
any self-touching configuration,
which is like this,

1136
00:58:23,900 --> 00:58:26,410
you specify the geometry
where things are all

1137
00:58:26,410 --> 00:58:27,580
on top of each other.

1138
00:58:27,580 --> 00:58:31,130
And then you'd say what you
want every vertex to look like.

1139
00:58:31,130 --> 00:58:35,980
It turns out there really
is a valid perturbation that

1140
00:58:35,980 --> 00:58:38,040
preserves that
combinatorial structure

1141
00:58:38,040 --> 00:58:39,821
and is arbitrarily small.

1142
00:58:39,821 --> 00:58:41,820
So here, of course, I've
drawn it so it's clear.

1143
00:58:41,820 --> 00:58:43,457
You can perturb things.

1144
00:58:43,457 --> 00:58:45,290
And I changed the edge
lengths a little bit.

1145
00:58:45,290 --> 00:58:49,570
But I can actually realize
this combinatorial structure.

1146
00:58:49,570 --> 00:58:52,360
It turns out that's
always possible.

1147
00:58:52,360 --> 00:58:54,110
So you can take any
self-touching linkage,

1148
00:58:54,110 --> 00:58:56,040
you can perturb it so
it's not self-touching,

1149
00:58:56,040 --> 00:58:58,950
and it is arbitrarily
locked within epsilon.

1150
00:59:02,680 --> 00:59:06,300
And the way you
prove it, or one way

1151
00:59:06,300 --> 00:59:08,240
to prove that
something is rigid,

1152
00:59:08,240 --> 00:59:10,320
is to say well this
proof is infinitesimally

1153
00:59:10,320 --> 00:59:13,120
rigid by constructing an
equilibrium stress that's

1154
00:59:13,120 --> 00:59:15,345
positive on all these
0 length struts.

1155
00:59:15,345 --> 00:59:17,270
As long as it's positive
all those 0 length

1156
00:59:17,270 --> 00:59:21,860
struts you know that
they're effectively bars.

1157
00:59:21,860 --> 00:59:24,310
Usually once they're
bars it's really obvious

1158
00:59:24,310 --> 00:59:28,140
that the thing is rigid because
it pins vertices into corners.

1159
00:59:28,140 --> 00:59:30,640
Then you know that the whole
thing is infinitesimally rigid,

1160
00:59:30,640 --> 00:59:33,850
therefore it's rigid,
therefore it's strongly locked.

1161
00:59:33,850 --> 00:59:36,339
And you could see some examples
of doing that in the book.

1162
00:59:36,339 --> 00:59:38,130
But this is no longer
the state of the art.

1163
00:59:38,130 --> 00:59:39,546
There are now
easier ways to prove

1164
00:59:39,546 --> 00:59:41,005
that trees are
locked-- some trees.

1165
00:59:41,005 --> 00:59:41,962
It doesn't always work.

1166
00:59:41,962 --> 00:59:44,060
Of course, it might be
infinitesimally flexible.

1167
00:59:44,060 --> 00:59:46,360
Lots of things could fail.

1168
00:59:46,360 --> 00:59:48,964
But if it succeeds in
proving something is rigid,

1169
00:59:48,964 --> 00:59:51,130
then you know it's strongly
locked and you're happy.

1170
00:59:54,350 --> 00:59:56,300
So it's a conservative
test, you might say.

1171
00:59:56,300 --> 00:59:57,710
It would be a cool
thing to implement.

1172
00:59:57,710 --> 00:59:59,668
This is not hard, it's
just linear programming.

1173
01:00:02,150 --> 01:00:07,110
But there's a cooler way
that's even more-- like one

1174
01:00:07,110 --> 01:00:08,980
where I can really
draw the pictures here,

1175
01:00:08,980 --> 01:00:11,100
you know we had to
draw all these diagrams

1176
01:00:11,100 --> 01:00:13,530
and figure out that stress
positive numbers worked on all

1177
01:00:13,530 --> 01:00:17,035
these edges, and
eh, it's tedious.

1178
01:00:21,080 --> 01:00:22,230
There's a much slicker way.

1179
01:00:33,270 --> 01:00:38,780
And for whatever reason, they
have become known as the rules.

1180
01:00:38,780 --> 01:00:40,160
There are two of them.

1181
01:00:40,160 --> 01:00:44,480
Although we've tried to come
up with various rule threes,

1182
01:00:44,480 --> 01:00:46,870
the ones that have been
tried and tested and used

1183
01:00:46,870 --> 01:00:49,360
all over the place are
rule one and rule two.

1184
01:00:53,420 --> 01:00:56,712
So rule one.

1185
01:00:56,712 --> 01:00:58,670
You'll see why this is
interesting in a moment.

1186
01:01:01,420 --> 01:01:04,810
I have some linkage,
self-touching linkage.

1187
01:01:04,810 --> 01:01:06,500
We're in the same framework.

1188
01:01:06,500 --> 01:01:10,690
Suppose these two edges
have the same length.

1189
01:01:10,690 --> 01:01:12,800
I'm drawing this
one slightly smaller

1190
01:01:12,800 --> 01:01:16,207
just so I can show you,
which is on which side.

1191
01:01:16,207 --> 01:01:17,790
But suppose they
have the same length.

1192
01:01:17,790 --> 01:01:21,570
And suppose that
both of these angles

1193
01:01:21,570 --> 01:01:25,490
are acute, strictly,
less than 90.

1194
01:01:28,180 --> 01:01:29,930
What do you think
happens in this picture?

1195
01:01:29,930 --> 01:01:31,450
I have this bar floating around.

1196
01:01:31,450 --> 01:01:33,700
And I have these three bars.

1197
01:01:33,700 --> 01:01:36,404
What happens to this bar?

1198
01:01:36,404 --> 01:01:38,570
AUDIENCE: It's confined to
be against the other bar.

1199
01:01:38,570 --> 01:01:40,861
PROFESSOR: It's confined to
be right against this edge.

1200
01:01:40,861 --> 01:01:46,220
It can't move at all until
this angle-- both of these,

1201
01:01:46,220 --> 01:01:48,730
and I guess at least one of
them would have to get to 90,

1202
01:01:48,730 --> 01:01:50,820
then you could try
to slide it out.

1203
01:01:50,820 --> 01:01:54,720
Like if this one goes beyond
90 then you can slide out.

1204
01:01:54,720 --> 01:02:01,670
As long as they're both less
than 90 it's pinned there.

1205
01:02:01,670 --> 01:02:08,460
So what I'm going to do
is redraw this diagram as

1206
01:02:08,460 --> 01:02:13,292
with two edges there, which
I mean, you can just ignore.

1207
01:02:13,292 --> 01:02:15,250
The point is if there's
something attached here

1208
01:02:15,250 --> 01:02:19,390
and here-- maybe many things,
it's a tree, who knows?--

1209
01:02:19,390 --> 01:02:22,770
you can just attach
them right there.

1210
01:02:22,770 --> 01:02:24,440
This is a simplification
to the linkage.

1211
01:02:24,440 --> 01:02:26,190
It does not behave the same.

1212
01:02:26,190 --> 01:02:29,260
But it has the same rigidity.

1213
01:02:29,260 --> 01:02:31,640
Because if all I care
about is rigidity,

1214
01:02:31,640 --> 01:02:34,290
I care about can I move at all?

1215
01:02:34,290 --> 01:02:36,780
So in order for this
guy to move and all,

1216
01:02:36,780 --> 01:02:38,870
these guys would
first have to move

1217
01:02:38,870 --> 01:02:41,710
for quite a while, a
positive amount of time.

1218
01:02:41,710 --> 01:02:44,640
I just want to know can
I move 0 or more than 0?

1219
01:02:44,640 --> 01:02:47,830
If I can move more than 0 I
could move this guy more than 0

1220
01:02:47,830 --> 01:02:49,760
without this guy moving at all.

1221
01:02:49,760 --> 01:02:51,760
So really I don't care
about how this guy moves.

1222
01:02:51,760 --> 01:02:55,240
He's effectively pinned for at
least a small amount of time.

1223
01:02:55,240 --> 01:02:57,480
I really care about can
the rest move it all?

1224
01:02:57,480 --> 01:02:59,760
So you can simplify
your linkage like this,

1225
01:02:59,760 --> 01:03:02,180
and the rigidity
will be preserved.

1226
01:03:02,180 --> 01:03:05,784
This is awesome because it's
easy to see when this applies.

1227
01:03:05,784 --> 01:03:07,200
And you just
simplify your linkage

1228
01:03:07,200 --> 01:03:10,520
until you can just tell
whether it's rigid.

1229
01:03:10,520 --> 01:03:14,310
All right rule two is
sort of a special case.

1230
01:03:14,310 --> 01:03:16,760
It looks like this.

1231
01:03:16,760 --> 01:03:21,280
So here, this bar and this bar
actually share an endpoint.

1232
01:03:21,280 --> 01:03:23,980
And again, I need that
this angle is acute.

1233
01:03:23,980 --> 01:03:26,990
And I need that these
two have the same length.

1234
01:03:31,060 --> 01:03:38,040
And I can simplify
to that, where

1235
01:03:38,040 --> 01:03:41,030
if anything was attached here
it just gets attached there.

1236
01:03:43,590 --> 01:03:44,390
Let's try it out.

1237
01:03:50,145 --> 01:03:51,770
Actually, I could
use some more boards.

1238
01:04:12,950 --> 01:04:19,380
So I'm just going to copy the
example we had from before.

1239
01:04:19,380 --> 01:04:21,380
And this will work on
basically all the examples

1240
01:04:21,380 --> 01:04:23,130
I've shown you except
the equilateral one.

1241
01:04:39,000 --> 01:04:43,496
This is the one, two, three,
four, five, six, seven,

1242
01:04:43,496 --> 01:04:44,700
eight bar example.

1243
01:04:44,700 --> 01:04:48,022
This is conjectured minimum.

1244
01:04:48,022 --> 01:04:49,230
Let's prove that it's locked.

1245
01:04:52,160 --> 01:04:53,640
See any rule one's
we could apply?

1246
01:04:57,300 --> 01:05:00,040
Do you see any edges
that are effectively

1247
01:05:00,040 --> 01:05:01,810
wedged against another edge?

1248
01:05:01,810 --> 01:05:05,250
Remember, all of these
vertices are actually

1249
01:05:05,250 --> 01:05:08,147
on top of each
other, we're thinking

1250
01:05:08,147 --> 01:05:09,480
about the self-touching version.

1251
01:05:09,480 --> 01:05:11,280
These guys are
actually touching,

1252
01:05:11,280 --> 01:05:13,562
and these guys are
actually touching.

1253
01:05:13,562 --> 01:05:16,510
AUDIENCE: This rule
one [INAUDIBLE].

1254
01:05:16,510 --> 01:05:19,477
PROFESSOR: There's a rule one--

1255
01:05:19,477 --> 01:05:20,560
AUDIENCE: That one, there.

1256
01:05:20,560 --> 01:05:21,450
PROFESSOR: --here?

1257
01:05:21,450 --> 01:05:22,600
Yeah, good.

1258
01:05:22,600 --> 01:05:25,960
This edge is pinned
against this edge.

1259
01:05:25,960 --> 01:05:30,100
Because look, we have acute
angles here, namely 0.

1260
01:05:30,100 --> 01:05:32,660
There's an acute
angle here, that's 0.

1261
01:05:32,660 --> 01:05:34,910
And this edge, if these guys
are on top of each other,

1262
01:05:34,910 --> 01:05:35,910
and these guys aer
on top of each other,

1263
01:05:35,910 --> 01:05:37,720
these two edges have
the same length.

1264
01:05:37,720 --> 01:05:41,760
Therefore these are
pinned together.

1265
01:05:41,760 --> 01:05:45,940
So let me redraw it when
they're pinned together.

1266
01:05:58,472 --> 01:06:00,930
I'm drawing things with curves
just so it's easier to draw,

1267
01:06:00,930 --> 01:06:04,620
but you understand this
is also basically flat.

1268
01:06:04,620 --> 01:06:08,380
Again these guys are all
on top of each other,

1269
01:06:08,380 --> 01:06:12,730
these are on top of each other,
these are on top of each other.

1270
01:06:12,730 --> 01:06:14,200
Did I do that right?

1271
01:06:14,200 --> 01:06:15,680
I think so, yep.

1272
01:06:15,680 --> 01:06:16,994
Anymore?

1273
01:06:16,994 --> 01:06:17,910
AUDIENCE: [INAUDIBLE].

1274
01:06:17,910 --> 01:06:19,700
PROFESSOR: Yes,
on the right side.

1275
01:06:19,700 --> 01:06:21,890
It's another rule one.

1276
01:06:21,890 --> 01:06:25,290
Here it's actually
pretty symmetric.

1277
01:06:25,290 --> 01:06:26,060
Is this locked?

1278
01:06:26,060 --> 01:06:26,770
Is this rigid?

1279
01:06:26,770 --> 01:06:27,811
It's kind of hard to say.

1280
01:06:27,811 --> 01:06:30,410
But now it's going
to be pretty obvious.

1281
01:06:30,410 --> 01:06:32,440
Because when these
guys join together

1282
01:06:32,440 --> 01:06:33,940
that means these
two vertices really

1283
01:06:33,940 --> 01:06:35,731
are on top of each
other for positive time.

1284
01:06:35,731 --> 01:06:37,000
Same for these.

1285
01:06:37,000 --> 01:06:41,230
So a new example looks
like two triangles

1286
01:06:41,230 --> 01:06:43,827
with an edge floating there.

1287
01:06:43,827 --> 01:06:45,410
Now it's pretty
obvious this is rigid.

1288
01:06:45,410 --> 01:06:47,580
But if you really want
to make it obvious

1289
01:06:47,580 --> 01:06:53,080
you can apply rule
two to this guy.

1290
01:06:53,080 --> 01:06:55,770
And then these guys
are pinned together.

1291
01:06:55,770 --> 01:07:02,560
And then you have two triangles.

1292
01:07:02,560 --> 01:07:05,640
Two triangles are rigid.

1293
01:07:05,640 --> 01:07:06,954
I think that's pretty obvious.

1294
01:07:06,954 --> 01:07:09,370
You could check it, whether
they're infinitesimally rigid,

1295
01:07:09,370 --> 01:07:10,580
whatever you feel like.

1296
01:07:10,580 --> 01:07:13,845
But because that's rigid,
this is rigid, this is rigid,

1297
01:07:13,845 --> 01:07:14,470
this is ridged.

1298
01:07:14,470 --> 01:07:16,120
Because these operations
preserve rigidity.

1299
01:07:16,120 --> 01:07:17,744
They don't preserve
locked or whatever,

1300
01:07:17,744 --> 01:07:19,101
but they preserve rigidity.

1301
01:07:19,101 --> 01:07:20,600
Once you know that
this is rigid you

1302
01:07:20,600 --> 01:07:21,860
know that it's strongly locked.

1303
01:07:21,860 --> 01:07:23,693
So when you perturb it
so these guys are not

1304
01:07:23,693 --> 01:07:26,285
on top of each other, but
they're slightly spread out,

1305
01:07:26,285 --> 01:07:27,660
it will be locked
within epsilon,

1306
01:07:27,660 --> 01:07:29,960
for any epsilon you want.

1307
01:07:29,960 --> 01:07:31,216
Now this is super easy.

1308
01:07:31,216 --> 01:07:32,840
And this is how we
were able to iterate

1309
01:07:32,840 --> 01:07:35,650
through all those locked
trees, and say, oh yeah,

1310
01:07:35,650 --> 01:07:38,950
this is still rigid, so
it's still strongly locked.

1311
01:07:38,950 --> 01:07:41,220
Now this doesn't always work.

1312
01:07:41,220 --> 01:07:42,604
But it seems pretty good.

1313
01:07:42,604 --> 01:07:44,270
And one of the
conjectures on the tables

1314
01:07:44,270 --> 01:07:47,870
is that for linear
trees rules one and two

1315
01:07:47,870 --> 01:07:49,530
are kind of almost enough.

1316
01:07:49,530 --> 01:07:51,020
It's not literally true.

1317
01:07:51,020 --> 01:07:52,822
But it's hopefully mostly true.

1318
01:07:52,822 --> 01:07:55,190
AUDIENCE: It seems that
that argument would

1319
01:07:55,190 --> 01:08:00,720
mean you wouldn't need
one of the n's on--

1320
01:08:00,720 --> 01:08:03,260
PROFESSOR: Right, so we could
think about this example.

1321
01:08:03,260 --> 01:08:06,790
And you're asking do you need--
does this need to be that long

1322
01:08:06,790 --> 01:08:10,010
or could we throw
away the last bar?

1323
01:08:10,010 --> 01:08:11,670
Right, it looks
like that's good.

1324
01:08:11,670 --> 01:08:14,020
If you throw away this
bar it's still the case

1325
01:08:14,020 --> 01:08:18,720
that this edge you
can apply rule one,

1326
01:08:18,720 --> 01:08:21,580
and say that it's pinned
against this edge.

1327
01:08:21,580 --> 01:08:24,060
And once those are there,
this thing basically

1328
01:08:24,060 --> 01:08:26,979
acts as a single triangle,
and life is good.

1329
01:08:26,979 --> 01:08:29,580
Yeah, so you can
remove this one edge.

1330
01:08:29,580 --> 01:08:34,090
You could not remove both of
the edges, at least for rule one

1331
01:08:34,090 --> 01:08:36,819
to apply, because then this
guy's not wedged into anything.

1332
01:08:36,819 --> 01:08:40,020
He's wedged on this side but
he's not wedged on that side.

1333
01:08:40,020 --> 01:08:41,610
But you can remove this edge.

1334
01:08:41,610 --> 01:08:43,110
And that's a super
easy way to prove

1335
01:08:43,110 --> 01:08:43,960
that this thing is locked.

1336
01:08:43,960 --> 01:08:45,918
We didn't know this when
we wrote the textbook,

1337
01:08:45,918 --> 01:08:47,740
otherwise we would
have given that proof.

1338
01:08:47,740 --> 01:08:50,740
There's one based on
stresses in the textbook.

1339
01:08:50,740 --> 01:08:53,700
But here, yeah, you can make
very easy judgments like that.

1340
01:08:53,700 --> 01:08:55,830
Now it doesn't
mean that it's not

1341
01:08:55,830 --> 01:08:57,529
locked when you remove
two of the edges.

1342
01:09:00,609 --> 01:09:03,090
I'm not sure, no I
think it's not locked.

1343
01:09:03,090 --> 01:09:04,960
But just because the
rules don't apply

1344
01:09:04,960 --> 01:09:06,729
doesn't tell you
that's not locked.

1345
01:09:06,729 --> 01:09:09,184
But it at least makes
it hard to prove.

1346
01:09:09,184 --> 01:09:10,600
And it's a good
sort of guideline.

1347
01:09:14,970 --> 01:09:16,234
Questions?

1348
01:09:16,234 --> 01:09:17,202
Yeah.

1349
01:09:17,202 --> 01:09:20,533
AUDIENCE: Did the perturbations
that are positive and finite

1350
01:09:20,533 --> 01:09:21,074
have numbers?

1351
01:09:21,074 --> 01:09:24,125
Like, what's-- it's sort of
disturbing that they get very,

1352
01:09:24,125 --> 01:09:26,819
very small-- what are they?

1353
01:09:26,819 --> 01:09:30,160
PROFESSOR: So you want
to know how big is delta?

1354
01:09:30,160 --> 01:09:31,149
How big is epsilon?

1355
01:09:31,149 --> 01:09:37,010
Well it depends, of course, how
much motion you want to allow,

1356
01:09:37,010 --> 01:09:39,310
how small the
perturbations have to be.

1357
01:09:39,310 --> 01:09:40,760
And I don't have a great answer.

1358
01:09:40,760 --> 01:09:44,710
I do recall that we computed
a bound, probably in terms

1359
01:09:44,710 --> 01:09:48,640
of something like r, the
maximum distance divided

1360
01:09:48,640 --> 01:09:51,600
by the smallest non 0 distance.

1361
01:09:51,600 --> 01:09:55,959
So it depends how close to
tight you are in other places.

1362
01:09:55,959 --> 01:09:57,250
And it depends on your epsilon.

1363
01:09:57,250 --> 01:09:58,917
There is an explicit bound.

1364
01:09:58,917 --> 01:10:00,750
I think it's polynomial
on those two things.

1365
01:10:00,750 --> 01:10:02,970
But I don't quite remember.

1366
01:10:02,970 --> 01:10:05,110
So you can actually compute
how much perturbation

1367
01:10:05,110 --> 01:10:08,060
will give you locked
within epsilon.

1368
01:10:08,060 --> 01:10:10,870
But it's certainly not clean.

1369
01:10:10,870 --> 01:10:13,850
It's actually not too hard
to prove this statement

1370
01:10:13,850 --> 01:10:14,700
just using topology.

1371
01:10:21,140 --> 01:10:22,900
Should I try to remember
how to prove it?

1372
01:10:27,730 --> 01:10:34,610
Basically, yeah, so
there's this fun fact.

1373
01:10:34,610 --> 01:10:37,280
Suppose you have
some tensegrity.

1374
01:10:37,280 --> 01:10:39,370
So tensegrities
are kind of hard.

1375
01:10:39,370 --> 01:10:44,057
They say, look, a bar has
this length, ain't changing.

1376
01:10:44,057 --> 01:10:45,390
Let's be a little more flexible.

1377
01:10:45,390 --> 01:10:48,444
What if you said, oh, this
bar can change within epsilon?

1378
01:10:48,444 --> 01:10:50,610
Because in reality you could
probably pull the metal

1379
01:10:50,610 --> 01:10:53,970
a little bit, just
not very much.

1380
01:10:53,970 --> 01:10:56,330
Struts, it's not
supposed to get smaller.

1381
01:10:56,330 --> 01:11:00,050
Let's say it can get epsilon
smaller than it's supposed to.

1382
01:11:00,050 --> 01:11:02,900
It turns out if you take
some tensegrity that's rigid,

1383
01:11:02,900 --> 01:11:04,920
and then you add this
little bit of flexibility

1384
01:11:04,920 --> 01:11:08,210
so the edges can change
in length a tiny amount,

1385
01:11:08,210 --> 01:11:13,387
then before you made this change
your configuration was a point,

1386
01:11:13,387 --> 01:11:14,970
and there might have
been other stuff.

1387
01:11:14,970 --> 01:11:17,070
But locally you
couldn't move at all.

1388
01:11:17,070 --> 01:11:21,090
If you add this
flexibility, the new picture

1389
01:11:21,090 --> 01:11:24,500
is a point with a
tiny ball around it.

1390
01:11:24,500 --> 01:11:28,067
And whatever, I mean this
might change a little bit also.

1391
01:11:28,067 --> 01:11:30,400
But the point is this point
doesn't get much bigger when

1392
01:11:30,400 --> 01:11:32,170
you add just a little
bit of flexibility.

1393
01:11:32,170 --> 01:11:34,780
This is a fact that was
known in rigidity theory.

1394
01:11:34,780 --> 01:11:36,920
It's called sloppy rigidity.

1395
01:11:36,920 --> 01:11:39,170
And it's essentially what's
going on here-- that we're

1396
01:11:39,170 --> 01:11:40,670
adding a little bit
of perturbation.

1397
01:11:40,670 --> 01:11:42,030
Before you couldn't move at all.

1398
01:11:42,030 --> 01:11:42,960
Now you can move a little bit.

1399
01:11:42,960 --> 01:11:45,001
We just had to generalize
from regular tensegrity

1400
01:11:45,001 --> 01:11:49,620
so these weird tensegrities
with sliding struts.

1401
01:11:49,620 --> 01:11:52,160
And this is kind of intuitive.

1402
01:11:52,160 --> 01:11:54,520
To really check it
you just need to check

1403
01:11:54,520 --> 01:11:57,650
that the constraints on edges
and struts are closed sets,

1404
01:11:57,650 --> 01:11:59,840
and then the have
to behave this way.

1405
01:11:59,840 --> 01:12:03,000
But you can actually compute
how quickly they change.

1406
01:12:03,000 --> 01:12:05,470
It's just messy.

1407
01:12:05,470 --> 01:12:07,090
So I'll leave it at that.

1408
01:12:07,090 --> 01:12:09,795
Hey, there's a little
proof addition,

1409
01:12:09,795 --> 01:12:12,084
glad I still remember it.

1410
01:12:12,084 --> 01:12:12,750
Other questions?

1411
01:12:12,750 --> 01:12:14,970
This is the end of 2D trees.

1412
01:12:14,970 --> 01:12:19,920
And now I want to talk
briefly about 3D chains.

1413
01:12:19,920 --> 01:12:23,438
So we did this one, now
I want to do this one.

1414
01:12:40,230 --> 01:12:43,860
Actually this is one of the
oldest results, from 1998.

1415
01:12:43,860 --> 01:12:46,110
So right around the same
time as the locked trees.

1416
01:13:04,190 --> 01:13:07,040
I may have shown this example
last time, or in lecture one.

1417
01:13:07,040 --> 01:13:09,040
But here it is again.

1418
01:13:09,040 --> 01:13:11,770
Three bars in the
center, and two

1419
01:13:11,770 --> 01:13:13,210
really long bars in the ends.

1420
01:13:13,210 --> 01:13:26,430
We call this knitting needles,
because it's like two knitting

1421
01:13:26,430 --> 01:13:28,620
needles with a short
string connecting them,

1422
01:13:28,620 --> 01:13:30,400
tied in a knot, sort of.

1423
01:13:30,400 --> 01:13:31,980
Topologically this
is not knotted.

1424
01:13:31,980 --> 01:13:34,410
If you could add
extra creases here

1425
01:13:34,410 --> 01:13:36,640
you could pull that
through, no problem.

1426
01:13:36,640 --> 01:13:40,090
But if this is rigid,
like a linkage,

1427
01:13:40,090 --> 01:13:51,060
then this thing
is locked provided

1428
01:13:51,060 --> 01:14:02,550
each end bar has
length strictly greater

1429
01:14:02,550 --> 01:14:04,820
than the sum of the middle bars.

1430
01:14:11,730 --> 01:14:15,230
So there's three middle bars
here, add up their lengths,

1431
01:14:15,230 --> 01:14:16,680
it should be shorter
than this one

1432
01:14:16,680 --> 01:14:19,050
and it should be
shorter than this one.

1433
01:14:19,050 --> 01:14:20,570
That's the cutoff.

1434
01:14:20,570 --> 01:14:23,430
And I believe once it's
the other way around,

1435
01:14:23,430 --> 01:14:24,290
this is not locked.

1436
01:14:36,370 --> 01:14:39,900
Sadly, rules one and do not
prove this thing is locked.

1437
01:14:39,900 --> 01:14:41,920
They only work in
two dimensions.

1438
01:14:41,920 --> 01:14:45,340
But for this one example--
and to tell you the truth,

1439
01:14:45,340 --> 01:14:48,540
this is pretty much the only
example of a locked open chain

1440
01:14:48,540 --> 01:14:54,180
that we have-- there's a
really simple, nice proof.

1441
01:14:54,180 --> 01:14:55,996
Let me draw the picture again.

1442
01:14:55,996 --> 01:14:59,191
AUDIENCE: Do you
need the bottom one?

1443
01:14:59,191 --> 01:15:01,690
PROFESSOR: You're asking do I
need three bars in the center,

1444
01:15:01,690 --> 01:15:04,010
or could I get away with two?

1445
01:15:04,010 --> 01:15:07,780
In fact, to draw this in
3D you need three bars.

1446
01:15:07,780 --> 01:15:08,940
That's maybe not obvious.

1447
01:15:13,370 --> 01:15:16,990
But if you tried to draw
it would just four bars,

1448
01:15:16,990 --> 01:15:22,780
like this, it's
not really possible

1449
01:15:22,780 --> 01:15:25,290
because these three
points are coplanar,

1450
01:15:25,290 --> 01:15:29,672
as all three points
are, and then yeah.

1451
01:15:29,672 --> 01:15:31,199
You can't get this
weaving pattern.

1452
01:15:31,199 --> 01:15:33,490
This guy's going to be either
above or below the plane.

1453
01:15:33,490 --> 01:15:35,656
And this guy's going to be
above or below the plane.

1454
01:15:35,656 --> 01:15:39,030
And in all cases, you
don't get this weaving.

1455
01:15:39,030 --> 01:15:41,407
So you really need the five.

1456
01:15:41,407 --> 01:15:42,240
This is the minimum.

1457
01:15:42,240 --> 01:15:46,370
You can prove all four
bar 3D chains to not lock.

1458
01:15:46,370 --> 01:15:47,760
But with five you can do it.

1459
01:15:50,320 --> 01:15:52,264
Good question.

1460
01:15:52,264 --> 01:15:53,680
So here's how we're
going to prove

1461
01:15:53,680 --> 01:15:54,763
that this thing is locked.

1462
01:15:58,634 --> 01:16:01,050
So there are these three edges,
they have various lengths,

1463
01:16:01,050 --> 01:16:01,980
who knows?

1464
01:16:01,980 --> 01:16:06,130
But add up the lengths, divide
by 2, and measure out that far.

1465
01:16:06,130 --> 01:16:08,515
So I'm going to call that
the midpoint of those three

1466
01:16:08,515 --> 01:16:09,015
segments.

1467
01:16:12,390 --> 01:16:15,750
I want to center a ball here.

1468
01:16:15,750 --> 01:16:22,780
It's going to look
something like that.

1469
01:16:22,780 --> 01:16:25,900
It's a 3D ball centered there.

1470
01:16:25,900 --> 01:16:38,010
So it's a ball, diameter equal
to the sum of the middle bars.

1471
01:16:42,190 --> 01:16:43,640
So radius is half that.

1472
01:16:43,640 --> 01:16:55,215
And the center is the midpoint
of the sum of the middle bars,

1473
01:16:55,215 --> 01:16:55,715
whatever.

1474
01:16:59,970 --> 01:17:02,420
So the radius is half the
sum of the middle bars.

1475
01:17:02,420 --> 01:17:05,050
And this is at half the
sum of the middle bars.

1476
01:17:05,050 --> 01:17:11,290
Therefore these middle
bars stay inside the ball.

1477
01:17:11,290 --> 01:17:13,280
Maybe they touch the boundary.

1478
01:17:13,280 --> 01:17:16,570
But they're inside or on
the boundary of the ball.

1479
01:17:16,570 --> 01:17:22,555
So that means the middle
bars are in the ball.

1480
01:17:26,640 --> 01:17:30,340
Maybe just barely, but no
matter how you move this thing--

1481
01:17:30,340 --> 01:17:33,870
I mean if you move this point
then the ball moves with it.

1482
01:17:33,870 --> 01:17:35,640
So no matter how
this thing folds,

1483
01:17:35,640 --> 01:17:37,390
those three bar stay
inside the ball.

1484
01:17:37,390 --> 01:17:38,470
What about these bars?

1485
01:17:38,470 --> 01:17:40,937
Or what about the endpoints?

1486
01:17:40,937 --> 01:17:42,770
Well if this point is
inside the ball, which

1487
01:17:42,770 --> 01:17:45,500
it is, and this point is
inside the ball, which it is,

1488
01:17:45,500 --> 01:17:50,530
then this thing is longer
than the diameter of the ball.

1489
01:17:50,530 --> 01:17:52,880
This thing is greater than
the sum of the middle bars.

1490
01:17:52,880 --> 01:17:55,171
The diameter of the ball is
the sum of the middle bars.

1491
01:17:55,171 --> 01:17:56,820
So if I take any
point inside the ball

1492
01:17:56,820 --> 01:18:02,170
and move straight from there
by the radius of the-- more

1493
01:18:02,170 --> 01:18:06,500
than the diameter of the ball,
I must go outside the ball.

1494
01:18:06,500 --> 01:18:13,115
So the endpoints are
outside the ball.

1495
01:18:16,494 --> 01:18:18,410
How the heck are you
going untie that not when

1496
01:18:18,410 --> 01:18:21,120
all of the interior vertices
stay inside of the ball,

1497
01:18:21,120 --> 01:18:22,920
and these guys stay
outside the ball?

1498
01:18:22,920 --> 01:18:28,325
To formalize how the
heck, you can say well

1499
01:18:28,325 --> 01:18:30,870
if you ignore what's inside the
ball-- something's happening

1500
01:18:30,870 --> 01:18:31,960
there, who knows?

1501
01:18:31,960 --> 01:18:33,600
But outside it's
like there's a ball

1502
01:18:33,600 --> 01:18:36,250
and there's two sticks
coming out of it.

1503
01:18:36,250 --> 01:18:39,310
So you can just
imagine, for example,

1504
01:18:39,310 --> 01:18:43,640
tying a string
between the two ends.

1505
01:18:43,640 --> 01:18:46,160
And something happens
in the inside.

1506
01:18:46,160 --> 01:18:48,600
But if you think of just
from the outside perspective,

1507
01:18:48,600 --> 01:18:49,850
these sticks just move around.

1508
01:18:49,850 --> 01:18:53,580
It's really easy to
not tangle this cord

1509
01:18:53,580 --> 01:18:56,120
when these sticks move around.

1510
01:18:56,120 --> 01:18:59,260
So now, in order for
this thing to become

1511
01:18:59,260 --> 01:19:02,690
unlocked, to straighten
out for example,

1512
01:19:02,690 --> 01:19:06,050
somehow inside you
have to do some magic

1513
01:19:06,050 --> 01:19:08,120
to get rid of this topology.

1514
01:19:08,120 --> 01:19:09,190
Well what do I mean?

1515
01:19:09,190 --> 01:19:14,120
Well if you could do it inside,
you could do that motion even

1516
01:19:14,120 --> 01:19:16,210
when there's a
string tied out here.

1517
01:19:16,210 --> 01:19:18,330
But when I tie the string
out here, it is a knot.

1518
01:19:18,330 --> 01:19:19,830
It's a trefoil know.

1519
01:19:19,830 --> 01:19:23,070
There's no way to untie a
trefoil know without crossing.

1520
01:19:23,070 --> 01:19:24,932
So either there's
crossing in here--

1521
01:19:24,932 --> 01:19:26,890
which better be-- or
there's crossing out here.

1522
01:19:26,890 --> 01:19:28,265
There can't be
crossing out here.

1523
01:19:28,265 --> 01:19:30,052
It's just two
sticks and a string.

1524
01:19:30,052 --> 01:19:32,010
You can easily arrange
the motion of the string

1525
01:19:32,010 --> 01:19:35,130
to not cross the two sticks.

1526
01:19:35,130 --> 01:19:38,820
Therefore this thing
in fact cannot untie.

1527
01:19:38,820 --> 01:19:43,880
Therefore this thing is
locked, cannot straighten out.

1528
01:19:43,880 --> 01:19:47,060
So that's locked 3D chains.

1529
01:19:47,060 --> 01:19:49,490
Let me tell you a bunch
of cool open problems.

1530
01:19:49,490 --> 01:19:52,120
This is really the only good
example of a locked 3D chain

1531
01:19:52,120 --> 01:19:53,150
we have.

1532
01:19:53,150 --> 01:19:56,040
And it has length ratios,
the best you could do

1533
01:19:56,040 --> 01:19:59,040
is like 1 to 3 plus epsilon.

1534
01:19:59,040 --> 01:20:00,800
If each of these
is length 1, these

1535
01:20:00,800 --> 01:20:02,580
have to be 3 plus epsilon.

1536
01:20:02,580 --> 01:20:04,330
Is that the best?

1537
01:20:04,330 --> 01:20:07,400
Or could it be, for
example, that all the edges

1538
01:20:07,400 --> 01:20:10,590
or between length 1 and 2,
and the chain is locked?

1539
01:20:10,590 --> 01:20:11,920
We don't know.

1540
01:20:11,920 --> 01:20:14,830
In the extreme case, what if all
the edge lengths of the same,

1541
01:20:14,830 --> 01:20:16,550
all length 1?

1542
01:20:16,550 --> 01:20:18,120
Is there a locked 3D chain?

1543
01:20:18,120 --> 01:20:21,260
We now know there's
a locked 2D tree,

1544
01:20:21,260 --> 01:20:23,600
but for chains it's tricky.

1545
01:20:23,600 --> 01:20:26,170
It's even open if
you add thickness.

1546
01:20:26,170 --> 01:20:30,970
You say, hey, let's
think about a 3D chain,

1547
01:20:30,970 --> 01:20:32,630
all the edge lengths
are the same,

1548
01:20:32,630 --> 01:20:36,187
and you get to specify
some radius of the bars.

1549
01:20:36,187 --> 01:20:38,020
For a while I thought
maybe this was locked.

1550
01:20:38,020 --> 01:20:40,990
I don't think it is.

1551
01:20:40,990 --> 01:20:42,920
We can unfold.

1552
01:20:42,920 --> 01:20:46,130
All of these questions are
open, and pretty fascinating.

1553
01:20:46,130 --> 01:20:50,300
Especially because proteins
are a lot like equilateral--

1554
01:20:50,300 --> 01:20:53,840
like all the edge lengths
the same-- 3D chains.

1555
01:20:53,840 --> 01:20:57,580
It's even open for
equilateral 3D trees.

1556
01:20:57,580 --> 01:20:59,590
So we know 2D equilateral
trees can lock.

1557
01:20:59,590 --> 01:21:03,140
But 3D, it's open.

1558
01:21:03,140 --> 01:21:04,640
I think that's
enough open problems.

1559
01:21:04,640 --> 01:21:06,456
Lots of cool questions here.

1560
01:21:06,456 --> 01:21:07,580
All right, I have one more.

1561
01:21:07,580 --> 01:21:08,800
It's just fun.

1562
01:21:08,800 --> 01:21:15,890
Even if you have a 3D chain
where all the edges are

1563
01:21:15,890 --> 01:21:17,790
on top of each other
on a line segment--

1564
01:21:17,790 --> 01:21:19,370
it's like a linear
tree but now it's

1565
01:21:19,370 --> 01:21:24,880
a linear 3D chain-- it's
like a bundle of segments.

1566
01:21:24,880 --> 01:21:26,780
Is that locked?

1567
01:21:26,780 --> 01:21:27,580
I don't think so.

1568
01:21:27,580 --> 01:21:30,080
But even that is tricky
to come up with algorithm.

1569
01:21:30,080 --> 01:21:32,940
Maybe we'll work on some of
these in the problem session.

1570
01:21:32,940 --> 01:21:35,000
That's it.