1
00:00:03,310 --> 00:00:07,320
PROFESSOR: All right, lecture
19 is about mostly refolding,

2
00:00:07,320 --> 00:00:11,330
common unfoldings of polyhedra,
convex polyhedra, and also

3
00:00:11,330 --> 00:00:12,840
about Mozartkugel.

4
00:00:12,840 --> 00:00:17,390
But most questions were
about the common unfolding

5
00:00:17,390 --> 00:00:20,280
stuff, so let me
start with that.

6
00:00:20,280 --> 00:00:23,330
First question-- is
this fractal unfolding

7
00:00:23,330 --> 00:00:27,150
of regular tetrahedron
and the cube,

8
00:00:27,150 --> 00:00:33,350
which looks like
this-- is it resolved?

9
00:00:33,350 --> 00:00:36,230
The answer is no, but it
is at least published now.

10
00:00:36,230 --> 00:00:38,500
This is at EuroCG.

11
00:00:38,500 --> 00:00:40,970
One of the authors is
here, Ryuhei Uehara.

12
00:00:40,970 --> 00:00:45,740
And it's still
conjecture, although there

13
00:00:45,740 --> 00:00:47,465
is a way to explicitly
construct this.

14
00:00:47,465 --> 00:00:49,820
This does actually
fold into two things.

15
00:00:49,820 --> 00:00:53,270
It folds into a cube and a
tetramonohedron, a tetrahedron

16
00:00:53,270 --> 00:00:55,110
with equal sides.

17
00:00:55,110 --> 00:00:58,270
It's not a regular
tetrahedron, though,

18
00:00:58,270 --> 00:01:02,130
but this is actually
an unfolding.

19
00:01:02,130 --> 00:01:06,180
And they have another one, which
they haven't drawn because it

20
00:01:06,180 --> 00:01:12,050
has 10 to the 180 edges or so,
but it is within 3 times 10

21
00:01:12,050 --> 00:01:16,690
to the minus 1796 of regular.

22
00:01:16,690 --> 00:01:21,116
So that's pretty good evidence
that this is gonna work out.

23
00:01:21,116 --> 00:01:22,240
Let me write that down.

24
00:01:27,140 --> 00:01:29,820
So they have a particular
iteration procedure.

25
00:01:29,820 --> 00:01:33,230
It's not always guaranteed to
make a connected unfolding, is

26
00:01:33,230 --> 00:01:34,770
the troubling part.

27
00:01:34,770 --> 00:01:36,960
I think it will converge,
but the challenge

28
00:01:36,960 --> 00:01:40,030
is to find an infinite
sequence of examples

29
00:01:40,030 --> 00:01:43,950
like this one, that form
a connected polygon, that

30
00:01:43,950 --> 00:01:46,520
would limit to the fractal.

31
00:01:46,520 --> 00:01:47,870
They have that instance.

32
00:01:47,870 --> 00:01:51,600
They have another instance
which is very, very good.

33
00:01:51,600 --> 00:01:53,897
This is the error.

34
00:01:53,897 --> 00:01:55,980
But we don't yet know
whether there are infinitely

35
00:01:55,980 --> 00:02:00,720
many examples the converge all
the way to some fractal set.

36
00:02:00,720 --> 00:02:03,590
So that is a state
of the fractal.

37
00:02:03,590 --> 00:02:07,220
Next we have a
question about boxes.

38
00:02:07,220 --> 00:02:09,490
When that lecture was
given, it was open

39
00:02:09,490 --> 00:02:11,700
whether there was a common
unfolding of three boxes.

40
00:02:11,700 --> 00:02:14,130
That problem is now solved.

41
00:02:14,130 --> 00:02:15,790
There's two papers
about this topic.

42
00:02:15,790 --> 00:02:19,070
The first one we wrote together.

43
00:02:19,070 --> 00:02:21,180
And this is just
a fun example, not

44
00:02:21,180 --> 00:02:22,790
yet related to the
three box problem,

45
00:02:22,790 --> 00:02:25,820
but you may recall there
were some unfoldings of two

46
00:02:25,820 --> 00:02:29,530
boxes where one was 45
degrees to the other.

47
00:02:29,530 --> 00:02:31,010
This was the
challenge of making it

48
00:02:31,010 --> 00:02:33,300
not 45 degrees, something else.

49
00:02:33,300 --> 00:02:39,430
And here it's a 3 by 1
triangle that is the axis.

50
00:02:39,430 --> 00:02:44,670
So you end up with root 10, root
of 3 squared plus 1 squared.

51
00:02:44,670 --> 00:02:48,080
So that's still two boxes,
but at least at a funny angle.

52
00:02:48,080 --> 00:02:50,590
And this was actually
found computationally,

53
00:02:50,590 --> 00:02:53,720
with an integer linear
program for this.

54
00:02:53,720 --> 00:02:56,360
Integer linear
programming-- set up

55
00:02:56,360 --> 00:02:58,860
the problem of which pixels
are in and which pixels are out

56
00:02:58,860 --> 00:03:00,510
in this representation,
and could you

57
00:03:00,510 --> 00:03:03,719
fit-- we started with
these parameters.

58
00:03:03,719 --> 00:03:05,260
You have to be very
careful about how

59
00:03:05,260 --> 00:03:07,960
you choose the numbers to
make the surface areas match.

60
00:03:07,960 --> 00:03:10,780
So we just chose that
by hand, and then tried

61
00:03:10,780 --> 00:03:13,031
to find whether there
was such a thing.

62
00:03:13,031 --> 00:03:15,280
You could formulate that as
an integer linear program.

63
00:03:15,280 --> 00:03:17,330
It's not guaranteed
to be solvable

64
00:03:17,330 --> 00:03:19,750
especially efficiently, but
this one we found a solution.

65
00:03:19,750 --> 00:03:22,514
For fun we also turned it into
a puzzle, where we were just

66
00:03:22,514 --> 00:03:24,430
modified the boundaries
in corresponding ways,

67
00:03:24,430 --> 00:03:28,214
so now finding either folding
is actually pretty challenging.

68
00:03:28,214 --> 00:03:29,630
Well, one of them's
probably easy,

69
00:03:29,630 --> 00:03:34,910
but the off center
one is challenging.

70
00:03:34,910 --> 00:03:38,250
But this paper also had
a first kind of solution

71
00:03:38,250 --> 00:03:39,560
to the three box problem.

72
00:03:39,560 --> 00:03:44,360
So this is one net-- it's
a grid unfolding, even--

73
00:03:44,360 --> 00:03:49,740
and it folds into 1 by 1 by 5,
1 by 2 by 3, and 0 by 1 by 11.

74
00:03:49,740 --> 00:03:52,290
So this is a flat box, a
doubly covered rectangle.

75
00:03:52,290 --> 00:03:54,516
It's also not technically
a grid unfolding-- we're

76
00:03:54,516 --> 00:03:56,140
cutting in the middle
of a pixel, here.

77
00:03:56,140 --> 00:04:01,310
But if you just refine by 2
and call this 0 by 2 by 22,

78
00:04:01,310 --> 00:04:04,530
2 by 4 by 6 and then
2 by 2 by 10, then

79
00:04:04,530 --> 00:04:06,939
this is a grid unfolding.

80
00:04:06,939 --> 00:04:09,230
I don't know if you care
whether it's a grid unfolding,

81
00:04:09,230 --> 00:04:12,090
but you can make
it one if you want.

82
00:04:12,090 --> 00:04:15,260
So this is the first
solution, and this is actually

83
00:04:15,260 --> 00:04:16,435
found by computer search.

84
00:04:19,810 --> 00:04:22,070
This came from a long
series of research

85
00:04:22,070 --> 00:04:23,320
in the two box problem.

86
00:04:23,320 --> 00:04:25,820
We started by looking at
all common unfoldings of 1

87
00:04:25,820 --> 00:04:29,240
by 1 by 5, 1 by 2 by 3, and
then saw whether any of them

88
00:04:29,240 --> 00:04:31,930
folded in to 0 by one by 11.

89
00:04:31,930 --> 00:04:34,445
So the just for fun,
the way that search

90
00:04:34,445 --> 00:04:37,350
was done-- which is
in the same paper--

91
00:04:37,350 --> 00:04:41,020
is essentially to try to
construct incrementally

92
00:04:41,020 --> 00:04:43,270
a common unfolding
of those two polygons

93
00:04:43,270 --> 00:04:46,155
so they both have
surface area 22.

94
00:04:46,155 --> 00:04:47,905
The idea is, you start
with a single pixel

95
00:04:47,905 --> 00:04:51,100
and you say, oh, a single
pixel can fit on both surfaces.

96
00:04:51,100 --> 00:04:53,722
Then you try all possible
ways to add a second pixel,

97
00:04:53,722 --> 00:04:55,430
and make sure that at
all times the thing

98
00:04:55,430 --> 00:04:59,306
you have is a partial
unfolding of both shapes.

99
00:04:59,306 --> 00:05:00,680
So that way you
get to throw away

100
00:05:00,680 --> 00:05:02,440
anything that's
guaranteed to be bad,

101
00:05:02,440 --> 00:05:05,060
that won't fit on either shape.

102
00:05:05,060 --> 00:05:12,690
And so as you grow your polygon,
you get more and more options.

103
00:05:12,690 --> 00:05:16,600
So the max we get to
here is about 5 million

104
00:05:16,600 --> 00:05:19,850
common partial
unfoldings of size 17.

105
00:05:19,850 --> 00:05:21,850
And for comparison,
the total number

106
00:05:21,850 --> 00:05:26,700
of ways to build
polyominals of size 17--

107
00:05:26,700 --> 00:05:28,640
this is only looking
at grid unfoldings--

108
00:05:28,640 --> 00:05:33,450
so all possible polyominals
of size 17 is 10 times that.

109
00:05:33,450 --> 00:05:35,620
So we're getting a
lot of savings in time

110
00:05:35,620 --> 00:05:39,580
by throwing away things that are
not common partial unfoldings.

111
00:05:39,580 --> 00:05:42,230
And then, conveniently,
things start getting smaller.

112
00:05:42,230 --> 00:05:44,430
You get to throw away
more and more stuff,

113
00:05:44,430 --> 00:05:51,640
and you end up with only 2,263
common unfoldings of these two

114
00:05:51,640 --> 00:05:53,640
shapes, top.

115
00:05:53,640 --> 00:05:57,110
And surprisingly, exactly
one of them folds into a 0

116
00:05:57,110 --> 00:05:58,870
by 1 by 11 box.

117
00:05:58,870 --> 00:06:02,680
So this is unique for
these three parameters.

118
00:06:02,680 --> 00:06:04,570
And unfortunately
that's kind of the limit

119
00:06:04,570 --> 00:06:07,110
of how far we could easily
go with a computer search.

120
00:06:07,110 --> 00:06:09,000
And we could maybe get
a little bit bigger,

121
00:06:09,000 --> 00:06:10,500
but it seemed like
we were not going

122
00:06:10,500 --> 00:06:13,430
to be able to find three
boxes just by computer search

123
00:06:13,430 --> 00:06:16,757
if we wanted a non-flat example,
because numbers are just

124
00:06:16,757 --> 00:06:18,340
going to have to get
bigger, and we're

125
00:06:18,340 --> 00:06:21,570
not to be able to do this
enumeration procedure.

126
00:06:21,570 --> 00:06:26,680
So we need to think,
instead of computer search.

127
00:06:26,680 --> 00:06:29,590
Before we get to that,
one other fun result

128
00:06:29,590 --> 00:06:34,130
we got from thinking is, if you
allow these flat doubly covered

129
00:06:34,130 --> 00:06:39,220
rectangles, there's one
polygon, namely a long strip,

130
00:06:39,220 --> 00:06:44,610
that will fold into arbitrarily
many distinct rectangles.

131
00:06:44,610 --> 00:06:48,410
So if you take a long strip,
you can do this zigzag pattern.

132
00:06:48,410 --> 00:06:52,610
In this case it's height 2, and
you alternate covered on top,

133
00:06:52,610 --> 00:06:55,210
covered on the bottom, covered
on top, covered on the bottom,

134
00:06:55,210 --> 00:06:58,020
and you turn around
and do the same.

135
00:06:58,020 --> 00:07:01,200
And if provided your dimensions
are properly divisible,

136
00:07:01,200 --> 00:07:04,200
this will make a doubly
covered rectangle of this size.

137
00:07:04,200 --> 00:07:06,650
And you could also
do it with-- this

138
00:07:06,650 --> 00:07:09,920
is height 1, height 2,
height 3, and so on.

139
00:07:09,920 --> 00:07:14,970
In general , if you have and l
by one strip you can make about

140
00:07:14,970 --> 00:07:18,140
l different rectangles,
depending on how you choose

141
00:07:18,140 --> 00:07:22,040
your angle to be one
of some integer thing.

142
00:07:22,040 --> 00:07:22,832
So that's cool.

143
00:07:22,832 --> 00:07:24,290
If you allow doubly
covered things,

144
00:07:24,290 --> 00:07:26,680
then you can actually
make many, many boxes.

145
00:07:26,680 --> 00:07:29,180
If you really want
3D boxes, that's

146
00:07:29,180 --> 00:07:31,090
where the next paper comes in.

147
00:07:31,090 --> 00:07:35,460
So next paper is about three
different orthogonal boxes,

148
00:07:35,460 --> 00:07:37,115
all not flat.

149
00:07:37,115 --> 00:07:41,510
It's by Shirakawa
and Uehara again.

150
00:07:41,510 --> 00:07:42,860
And here's one of the solutions.

151
00:07:42,860 --> 00:07:47,110
So this one polygon is a grid
unfolding-- folds into a 7

152
00:07:47,110 --> 00:07:51,910
by 8 by 14, a 2 by 4 by
43, and a 2 by 13 by 16.

153
00:07:51,910 --> 00:07:54,500
And this is almost certainly
beyond what we could ever

154
00:07:54,500 --> 00:07:58,430
do by computer search,
by just exhaustion.

155
00:07:58,430 --> 00:08:02,420
This is found using human
intelligence, and this grid

156
00:08:02,420 --> 00:08:03,670
pattern.

157
00:08:03,670 --> 00:08:04,930
And they have another example.

158
00:08:04,930 --> 00:08:09,720
This one falls into 2 by 13 by
58, 7 by 14 by 38, and 7 by 56.

159
00:08:09,720 --> 00:08:12,900
This is actually-- this a
little more complicated,

160
00:08:12,900 --> 00:08:15,510
or a little bigger,
I should say.

161
00:08:15,510 --> 00:08:18,780
They're both based on a
common principle, which

162
00:08:18,780 --> 00:08:22,260
is to start from a solution
to the two box problem

163
00:08:22,260 --> 00:08:25,770
and then modify it
to make three boxes.

164
00:08:25,770 --> 00:08:30,120
So this is a common unfolding
event a by b by 8a box,

165
00:08:30,120 --> 00:08:32,799
and an a by two a
by 2a plus 3b box

166
00:08:32,799 --> 00:08:36,419
for, I think, for any
values of a and b.

167
00:08:36,419 --> 00:08:40,929
And essentially the way you
see that is, you either wrap it

168
00:08:40,929 --> 00:08:43,409
around this way, or you
wrap it around this way.

169
00:08:43,409 --> 00:08:46,350
I think that's how
the two foldings work.

170
00:08:46,350 --> 00:08:49,150
You have to figure out which
of these lines to keep,

171
00:08:49,150 --> 00:08:51,470
and which ones to throw.

172
00:08:51,470 --> 00:08:53,020
I haven't color coded them here.

173
00:08:53,020 --> 00:08:57,800
But that's a general
thing that's known before.

174
00:08:57,800 --> 00:08:59,610
I think maybe we
even saw in lecture

175
00:08:59,610 --> 00:09:02,740
this-- there was an example
where you could just repeat,

176
00:09:02,740 --> 00:09:06,010
and get infinitely many
examples of two different boxes.

177
00:09:06,010 --> 00:09:09,730
Now we want to modify it to
make three different boxes,

178
00:09:09,730 --> 00:09:12,340
or to add a third box.

179
00:09:12,340 --> 00:09:16,230
It's not known how to do a
fourth box of nonzero volume.

180
00:09:16,230 --> 00:09:19,620
So the basic idea,
which doesn't work,

181
00:09:19,620 --> 00:09:22,840
is following-- if
I built some box,

182
00:09:22,840 --> 00:09:26,270
and I slit it in this
sort of h pattern,

183
00:09:26,270 --> 00:09:31,010
and then I refold
it to be like this,

184
00:09:31,010 --> 00:09:32,880
I should get a different box.

185
00:09:32,880 --> 00:09:34,530
That would be really cool.

186
00:09:34,530 --> 00:09:36,290
The problem is, if
you look at what

187
00:09:36,290 --> 00:09:37,750
lengths have to
be equal for this.

188
00:09:37,750 --> 00:09:42,220
So these two lengths end
up being these two lengths,

189
00:09:42,220 --> 00:09:44,360
so those sums must be equal.

190
00:09:44,360 --> 00:09:47,800
And over here that
same length must

191
00:09:47,800 --> 00:09:51,810
equal the two
heights of the boxes.

192
00:09:51,810 --> 00:09:54,530
That implies that actually
this length equals this length,

193
00:09:54,530 --> 00:09:57,040
and so this is
actually a square.

194
00:09:57,040 --> 00:09:59,625
It's drawn here magically
to not be squares,

195
00:09:59,625 --> 00:10:01,000
but in fact they
must be squares,

196
00:10:01,000 --> 00:10:03,946
so it's good example
of misleading figure.

197
00:10:03,946 --> 00:10:06,320
Think about it as like, oh,
man, they have to be squares.

198
00:10:06,320 --> 00:10:08,910
That means you're converting
a 2 by 1 rectangle to a 1

199
00:10:08,910 --> 00:10:10,112
by 2 rectangle.

200
00:10:10,112 --> 00:10:12,070
And that's really not
going to change anything.

201
00:10:12,070 --> 00:10:15,310
You just rotated your
box, which is frustrating.

202
00:10:15,310 --> 00:10:18,260
Fortunately, there's a
tweak to make it work.

203
00:10:18,260 --> 00:10:24,680
The tweak is, you add wiggly
tabs and you refold things.

204
00:10:24,680 --> 00:10:30,610
So this is a-- in terms of
the grid, here, we've got an 8

205
00:10:30,610 --> 00:10:35,820
by 7 rectangle
originally on this face.

206
00:10:35,820 --> 00:10:39,280
We do all these cuts
that are in bold,

207
00:10:39,280 --> 00:10:42,510
and then when we refold we
also add in these dash creases.

208
00:10:42,510 --> 00:10:44,670
That's kind of
part of the magic.

209
00:10:44,670 --> 00:10:47,030
And we end up with
a 2 by 13 rectangle.

210
00:10:47,030 --> 00:10:49,350
These don't even have
the same surface area.

211
00:10:49,350 --> 00:10:53,662
Some of the material got
moved into the adjacent sides.

212
00:10:53,662 --> 00:10:55,120
But if you stare
at it long enough,

213
00:10:55,120 --> 00:10:59,070
this is a valid transformation
of an existing box.

214
00:10:59,070 --> 00:11:01,700
Now, of course, the challenge
is to get this trick

215
00:11:01,700 --> 00:11:07,029
to work with this
unfolding of two boxes.

216
00:11:07,029 --> 00:11:08,570
And it'll be really
nice if you could

217
00:11:08,570 --> 00:11:10,111
make it work for
both the boxes, then

218
00:11:10,111 --> 00:11:11,450
you'd get four boxes, total.

219
00:11:11,450 --> 00:11:12,900
We don't know how to do that.

220
00:11:12,900 --> 00:11:15,370
But with this particular
augmentation--

221
00:11:15,370 --> 00:11:19,030
so, same picture, particular
values of a and b chosen here,

222
00:11:19,030 --> 00:11:23,740
but these extra tabs and
pockets are put in there.

223
00:11:23,740 --> 00:11:28,330
Then we get a common
unfolding of three boxes.

224
00:11:28,330 --> 00:11:30,210
And you can see from
the numbers here you've

225
00:11:30,210 --> 00:11:34,100
got the 2 by 13 and
the 7 by 8 appearing,

226
00:11:34,100 --> 00:11:38,380
And this comes from
the other folding.

227
00:11:38,380 --> 00:11:40,640
So, pretty cool, right?

228
00:11:40,640 --> 00:11:43,466
And the same gadget
is being used

229
00:11:43,466 --> 00:11:45,090
in this the first
example I showed you,

230
00:11:45,090 --> 00:11:47,200
which is a somewhat smaller one.

231
00:11:47,200 --> 00:11:50,335
Again you get 7 by
8 here, and 2 by 13.

232
00:11:53,360 --> 00:11:58,504
So, this is how we
know how to do boxes.

233
00:11:58,504 --> 00:11:59,920
Now, in fact, there
are infinitely

234
00:11:59,920 --> 00:12:02,590
many examples along these lines.

235
00:12:02,590 --> 00:12:04,360
You can change that grid.

236
00:12:04,360 --> 00:12:08,620
Parities have to be right and
maintain some proportions,

237
00:12:08,620 --> 00:12:11,660
but there are lots of
examples of taking that grid,

238
00:12:11,660 --> 00:12:16,670
applying it to this net, and
then you get these three boxes.

239
00:12:16,670 --> 00:12:20,950
For any integer, k greater
or equal to zero will work.

240
00:12:20,950 --> 00:12:27,080
And so you get a nice infinite
series of examples, I think.

241
00:12:27,080 --> 00:12:31,212
The rough ratios are almost
the same, but-- is that right?

242
00:12:31,212 --> 00:12:33,295
No, no, actually they're
changing the proportions.

243
00:12:33,295 --> 00:12:40,805
You've got k, 4k, 16k, versus
4k, 4k, 7k, versus 4k, k, 4k.

244
00:12:40,805 --> 00:12:44,224
So-- sorry?

245
00:12:44,224 --> 00:12:49,841
Oh, 4k, 2k, and 24k,
sorry-- and this is 32.

246
00:12:49,841 --> 00:12:50,340
Yeah, sorry.

247
00:12:50,340 --> 00:12:53,250
I need to expand on that that.

248
00:12:53,250 --> 00:12:55,240
So, pretty cool, and
this is the state

249
00:12:55,240 --> 00:12:57,490
of the art of common unfoldings.

250
00:12:57,490 --> 00:13:01,110
Next open problem is for
non-zero volume boxes.

251
00:13:01,110 --> 00:13:04,930
Can you prove that 100 nonzero
volume boxes are impossible?

252
00:13:04,930 --> 00:13:07,100
Who knows.

253
00:13:07,100 --> 00:13:10,300
Maybe even four or
five is impossible.

254
00:13:10,300 --> 00:13:12,540
It seems pretty tricky.

255
00:13:12,540 --> 00:13:16,180
OK, I have one more
question, which

256
00:13:16,180 --> 00:13:18,160
is about the smooth
unfolding stuff.

257
00:13:21,850 --> 00:13:25,490
I call this a limit, and there's
this is issue about the area.

258
00:13:25,490 --> 00:13:26,640
So I want to go into this.

259
00:13:26,640 --> 00:13:29,620
Remember this picture-- we
take a smooth prismatoid, which

260
00:13:29,620 --> 00:13:34,090
is two convex-- not
polygons, but two convex

261
00:13:34,090 --> 00:13:35,330
bodies, parallel.

262
00:13:35,330 --> 00:13:36,550
Take the convex hull.

263
00:13:36,550 --> 00:13:38,830
You get the smooth surface.

264
00:13:38,830 --> 00:13:41,820
And then we unfold
it in the volcano way

265
00:13:41,820 --> 00:13:44,860
of taking every rib
here, every rule line,

266
00:13:44,860 --> 00:13:48,480
and developing it out flat.

267
00:13:48,480 --> 00:13:50,540
And when you do that,
you take the union

268
00:13:50,540 --> 00:13:53,040
of the ribs and the
unfolding, does not

269
00:13:53,040 --> 00:13:55,410
have the same area as
the original surface

270
00:13:55,410 --> 00:13:57,960
area of the polyhedron,
which seems a little weird.

271
00:14:01,255 --> 00:14:03,630
Sometimes we might call I
might have called this a limit.

272
00:14:03,630 --> 00:14:06,599
It's not really a limit
of the discrete case.

273
00:14:06,599 --> 00:14:09,140
And this is why we don't know
how to solve the discrete case,

274
00:14:09,140 --> 00:14:13,710
and yet, we can solve
the smooth case.

275
00:14:13,710 --> 00:14:17,330
The issue is, if you wanted
to form this structure

276
00:14:17,330 --> 00:14:20,960
as a limit of
discrete prismatoids,

277
00:14:20,960 --> 00:14:23,150
you want to subdivide this
into a very fine polygon,

278
00:14:23,150 --> 00:14:25,810
subdivide this into
a very fine polygon,

279
00:14:25,810 --> 00:14:27,710
and then try to unfold
those discreetly,

280
00:14:27,710 --> 00:14:30,210
and then take the limit as
you take finer and finer,

281
00:14:30,210 --> 00:14:32,660
closer and closer approximations
of the smooth things

282
00:14:32,660 --> 00:14:35,080
as polygons.

283
00:14:35,080 --> 00:14:39,250
It really depends how you
discretize-- maybe one way

284
00:14:39,250 --> 00:14:42,360
to think about that
is, if I start here

285
00:14:42,360 --> 00:14:45,460
and I do uniform spacing
from this point--

286
00:14:45,460 --> 00:14:48,390
or I do it from this, point or
from this point-- each of those

287
00:14:48,390 --> 00:14:50,810
is going to have a different--
it's going to capture

288
00:14:50,810 --> 00:14:53,612
a somewhat different
discrete structure.

289
00:14:53,612 --> 00:14:55,070
I think the way to
think about that

290
00:14:55,070 --> 00:14:57,690
is, the gaps will be spread
out in different ways.

291
00:14:57,690 --> 00:14:59,650
If I always cut here,
there will always

292
00:14:59,650 --> 00:15:02,500
be a gap incident to that point,
whereas if I always cut here,

293
00:15:02,500 --> 00:15:04,583
there will always be a gap
incident to this point,

294
00:15:04,583 --> 00:15:07,830
and not necessarily
at this point.

295
00:15:07,830 --> 00:15:10,810
Depending on your
discrete approximation,

296
00:15:10,810 --> 00:15:13,020
you'll get a very
different unfolded diagram.

297
00:15:13,020 --> 00:15:14,020
You're going to
get holes out here,

298
00:15:14,020 --> 00:15:15,645
because when you do
the discrete thing,

299
00:15:15,645 --> 00:15:17,740
the discrete thing
does preserve area.

300
00:15:17,740 --> 00:15:20,150
And yet, you imagine, as
you take the limit somehow

301
00:15:20,150 --> 00:15:21,440
the area jumps at the end.

302
00:15:21,440 --> 00:15:24,480
That shouldn't happen for a
limit, so this is not a limit.

303
00:15:24,480 --> 00:15:26,820
What happens is, you have
many different limit points

304
00:15:26,820 --> 00:15:29,110
depending on your
discrete representation.

305
00:15:29,110 --> 00:15:30,630
And this green
thing is essentially

306
00:15:30,630 --> 00:15:35,760
the union of all
those limit points.

307
00:15:35,760 --> 00:15:37,820
When you take the
union, the area goes up.

308
00:15:37,820 --> 00:15:40,290
That's essentially
what's going on here.

309
00:15:40,290 --> 00:15:41,540
That's my best interpretation.

310
00:15:45,750 --> 00:15:48,554
But yeah, shouldn't
use the word limit.

311
00:15:48,554 --> 00:15:50,220
OK that's it for
questions, unless there

312
00:15:50,220 --> 00:15:54,460
are other questions
about this or anything

313
00:15:54,460 --> 00:15:57,250
we talked about in lecture.

314
00:15:57,250 --> 00:16:01,780
So the next topic I want to
talk about-- this is a bonus,

315
00:16:01,780 --> 00:16:04,190
extra topic hasn't been
covered in this class before,

316
00:16:04,190 --> 00:16:05,630
but I've always wanted to.

317
00:16:05,630 --> 00:16:07,975
I think there was a final
project about it once.

318
00:16:07,975 --> 00:16:11,980
It's this guy,
Theo Jansen, and he

319
00:16:11,980 --> 00:16:13,695
builds these crazy
linkage structures.

320
00:16:13,695 --> 00:16:16,070
And the next two classes are
about protein folding, which

321
00:16:16,070 --> 00:16:21,080
is more on the linkage
folding aspect, and so

322
00:16:21,080 --> 00:16:24,280
thought it'd be fun
to transition over.

323
00:16:24,280 --> 00:16:26,000
Maybe first I'll
just show you some.

324
00:16:26,000 --> 00:16:32,620
These are two kits you can
buy of his constructions.

325
00:16:32,620 --> 00:16:36,476
This guy-- I will blow.

326
00:16:36,476 --> 00:16:40,380
[BLOWS]

327
00:16:49,180 --> 00:16:50,450
This guy's on here.

328
00:16:59,190 --> 00:17:03,500
If I turn this guy,
he walks-- and walks

329
00:17:03,500 --> 00:17:06,230
pretty elegantly if I
turn this very uniformly.

330
00:17:06,230 --> 00:17:06,983
It's pretty cool.

331
00:17:06,983 --> 00:17:08,149
It can walk both directions.

332
00:17:13,220 --> 00:17:14,329
It's also unpowered.

333
00:17:14,329 --> 00:17:15,589
You blow it, and it works.

334
00:17:15,589 --> 00:17:16,970
Here's another example.

335
00:17:16,970 --> 00:17:18,261
You see just the legs.

336
00:17:18,261 --> 00:17:20,510
It's more clearly a linkage,
although this is actually

337
00:17:20,510 --> 00:17:22,599
just implementing
the same linkage.

338
00:17:22,599 --> 00:17:25,800
It's just got some thicker--
filled in some rigid triangles

339
00:17:25,800 --> 00:17:26,934
here.

340
00:17:26,934 --> 00:17:30,140
[BLOWS]

341
00:17:30,140 --> 00:17:33,380
This is fun, because the
wind's going in one direction,

342
00:17:33,380 --> 00:17:35,710
and the guy's going in
the opposite direction.

343
00:17:35,710 --> 00:17:37,650
Here it's going in the
direction of the wind.

344
00:17:37,650 --> 00:17:39,884
Here it's going
perpendicular to the wind.

345
00:17:39,884 --> 00:17:41,700
[BLOWS]

346
00:17:41,700 --> 00:17:43,523
A little bit slower, though.

347
00:17:48,150 --> 00:17:53,280
I'll just manually
turn this guy.

348
00:17:53,280 --> 00:17:54,785
Is he going a little faster?

349
00:17:54,785 --> 00:17:58,000
So it's got really
cool leg mechanism.

350
00:18:02,330 --> 00:18:04,060
Looks almost like an animal.

351
00:18:04,060 --> 00:18:08,290
Theo calls these his
children, his animals.

352
00:18:08,290 --> 00:18:10,570
His goal is for them
to be self sufficient

353
00:18:10,570 --> 00:18:13,440
and to live on their
own in the beach.

354
00:18:13,440 --> 00:18:15,270
How many people have
seen his TED Talk?

355
00:18:15,270 --> 00:18:16,057
A few people.

356
00:18:16,057 --> 00:18:16,890
You should watch it.

357
00:18:16,890 --> 00:18:19,181
I'm not going to show it
here, because some people have

358
00:18:19,181 --> 00:18:20,820
seen it.

359
00:18:20,820 --> 00:18:23,300
He has this vision.

360
00:18:23,300 --> 00:18:24,970
He makes them all out of PVC.

361
00:18:24,970 --> 00:18:31,259
These are just injection molded
plastic kits which you can buy.

362
00:18:31,259 --> 00:18:32,050
They're super cool.

363
00:18:32,050 --> 00:18:33,740
Let me show you the bigger ones.

364
00:18:33,740 --> 00:18:36,511
This is living on the beach.

365
00:18:36,511 --> 00:18:38,260
At this point they're
not self sufficient,

366
00:18:38,260 --> 00:18:41,220
he has to correct
them occasionally. .

367
00:18:41,220 --> 00:18:45,320
But this guy's just walking
in the direction of the wind.

368
00:18:45,320 --> 00:18:50,420
I think here the wind slows down
a little bit, so lose momentum,

369
00:18:50,420 --> 00:18:53,170
then picks up.

370
00:18:53,170 --> 00:18:54,600
He does this in the Netherlands.

371
00:18:54,600 --> 00:18:56,470
He's got this crazy beach.

372
00:18:56,470 --> 00:18:59,620
Here's a much bigger one.

373
00:18:59,620 --> 00:19:03,160
It's got a lot more wind
collectors at the top.

374
00:19:03,160 --> 00:19:05,060
So these are all self powered.

375
00:19:05,060 --> 00:19:10,330
Originally he pushed them
along, but now they've

376
00:19:10,330 --> 00:19:12,540
learned to move by themselves.

377
00:19:12,540 --> 00:19:16,540
And very cool legs-- I want
to talk more about the leg

378
00:19:16,540 --> 00:19:21,050
linkage, because that's
the more mathematical part.

379
00:19:21,050 --> 00:19:23,900
So this is his original
drawing of the leg.

380
00:19:23,900 --> 00:19:25,580
It's a pretty simple idea.

381
00:19:25,580 --> 00:19:27,350
It's a pretty simple linkage.

382
00:19:27,350 --> 00:19:34,740
You've got a quadrilateral
here, 4 bar linkage, C-K-J-B,

383
00:19:34,740 --> 00:19:37,090
and you've got a rigid
triangle attached here.

384
00:19:37,090 --> 00:19:40,670
Then another quadrilateral
here, and then a rigid triangle

385
00:19:40,670 --> 00:19:41,750
at the bottom.

386
00:19:41,750 --> 00:19:44,120
So it's really just
two quadrilaterals

387
00:19:44,120 --> 00:19:45,980
connected by two
rigid triangles.

388
00:19:45,980 --> 00:19:48,230
And so it's a one degree
of freedom mechanism.

389
00:19:48,230 --> 00:19:50,370
As you flex this
quad, this one has

390
00:19:50,370 --> 00:19:53,960
to adapt because that
quad controls all three

391
00:19:53,960 --> 00:19:56,000
of these points because
of the rigid triangles,

392
00:19:56,000 --> 00:19:58,400
and so the fourth
one is determined.

393
00:19:58,400 --> 00:19:59,980
And then the crank is here.

394
00:20:02,880 --> 00:20:07,850
So this says vaste printer,
which means fixed points.

395
00:20:07,850 --> 00:20:09,940
So you've got these
two fixed points--

396
00:20:09,940 --> 00:20:11,810
this is actually also fixed.

397
00:20:11,810 --> 00:20:14,360
This is just an x to y ratio.

398
00:20:14,360 --> 00:20:17,240
So these two guys are
pinned, and this guy

399
00:20:17,240 --> 00:20:24,150
is rotating around this fixed
point-- that's the actuation,

400
00:20:24,150 --> 00:20:28,130
and that causes this
quadrilateral to fold.

401
00:20:28,130 --> 00:20:31,070
We are basically setting the
center of rotation of this guy

402
00:20:31,070 --> 00:20:32,760
relative to this
guy by measuring

403
00:20:32,760 --> 00:20:34,270
these lengths, a and l.

404
00:20:36,930 --> 00:20:40,170
He has these numbers, which
he calls the 11 holy numbers.

405
00:20:40,170 --> 00:20:42,820
It's a little cause because
there's actually 13 of them.

406
00:20:42,820 --> 00:20:45,305
There are 11 edge
lengths, and then there's

407
00:20:45,305 --> 00:20:47,180
these two, which are
not really edge lengths.

408
00:20:47,180 --> 00:20:49,810
It's just a measurement of how
to put the center of rotation

409
00:20:49,810 --> 00:20:53,330
relative to this guy.

410
00:20:53,330 --> 00:20:55,160
And the reason why
x and y matter here

411
00:20:55,160 --> 00:20:57,410
is because there's actually
a floor at the bottom,

412
00:20:57,410 --> 00:21:01,650
and you have to orient yourself
correctly relative to the floor

413
00:21:01,650 --> 00:21:04,300
so you can actually walk.

414
00:21:04,300 --> 00:21:09,310
So he found these numbers
by genetic algorithm.

415
00:21:09,310 --> 00:21:12,170
In the early days there
weren't-- I guess,

416
00:21:12,170 --> 00:21:14,590
before there were a lot of
genetic algorithms around.

417
00:21:14,590 --> 00:21:16,349
So he tried lots of
different values,

418
00:21:16,349 --> 00:21:17,890
simulated them on
the computer, found

419
00:21:17,890 --> 00:21:21,420
which ones walked
best, basically.

420
00:21:21,420 --> 00:21:23,830
Ones that were walking well
in various kinds of terrain

421
00:21:23,830 --> 00:21:25,750
would survive to
the next generation,

422
00:21:25,750 --> 00:21:28,260
try various random
mutations, perturbing

423
00:21:28,260 --> 00:21:32,350
all of these numbers, and
eventually finding this set

424
00:21:32,350 --> 00:21:35,690
to be especially
good in simulation,

425
00:21:35,690 --> 00:21:38,930
and that's what all of the
built linkages are based on.

426
00:21:42,160 --> 00:21:44,730
This is his computer
drawing, plus some markings

427
00:21:44,730 --> 00:21:46,390
on top of it.

428
00:21:46,390 --> 00:21:48,800
If you take this linkage
and go through the one

429
00:21:48,800 --> 00:21:51,260
degree of freedom,
this point, of course,

430
00:21:51,260 --> 00:21:52,730
moves along a circle.

431
00:21:52,730 --> 00:21:55,860
Then we're just drawing lots of
instances on top of each other.

432
00:21:55,860 --> 00:21:58,750
The foot-- this
guy down here moves

433
00:21:58,750 --> 00:22:01,790
in this pattern,
which is pretty cool.

434
00:22:01,790 --> 00:22:05,240
This is basically lifting
your foot up off the ground,

435
00:22:05,240 --> 00:22:08,199
pushing it down, and then
dragging along the floor.

436
00:22:08,199 --> 00:22:09,990
That's how you walk,
if you think about it.

437
00:22:09,990 --> 00:22:14,540
You push and propel
yourself forward.

438
00:22:14,540 --> 00:22:16,710
So this scraping
along the floor,

439
00:22:16,710 --> 00:22:18,330
if you've got a
high friction foot,

440
00:22:18,330 --> 00:22:21,100
will actually move you forward.

441
00:22:21,100 --> 00:22:24,162
In some sense, all of this
design-- and the one way

442
00:22:24,162 --> 00:22:25,620
you would choose
these numbers-- is

443
00:22:25,620 --> 00:22:27,590
to guarantee that
A- you get lift,

444
00:22:27,590 --> 00:22:30,840
because if you have non-uniform
terrain, you want to walk.

445
00:22:30,840 --> 00:22:33,730
He likes to call this mechanism
better than the wheel.

446
00:22:33,730 --> 00:22:37,000
It's like every invention
of something like the wheel,

447
00:22:37,000 --> 00:22:39,100
but even better because
wheels don't really

448
00:22:39,100 --> 00:22:40,990
work if you have
steps on the ground.

449
00:22:40,990 --> 00:22:42,740
Walking is superior,
cause you could

450
00:22:42,740 --> 00:22:46,050
walk over bumps and steps
and things like that.

451
00:22:46,050 --> 00:22:47,810
So you want a good
amount of lift,

452
00:22:47,810 --> 00:22:51,320
and then you want a good
amount, and ideally-- you

453
00:22:51,320 --> 00:22:52,790
can kind of see this here.

454
00:22:52,790 --> 00:22:55,600
These are uniformly spaced
as you rotate around,

455
00:22:55,600 --> 00:22:57,590
and these are all--
they look uniformly

456
00:22:57,590 --> 00:23:00,560
spaced on the bottom,
which means you're applying

457
00:23:00,560 --> 00:23:04,420
uniform force all along
the floor contact.

458
00:23:04,420 --> 00:23:10,120
It's not uniform here, but
it's uniform down here, mostly.

459
00:23:10,120 --> 00:23:12,370
It's not actually straight
in the true mechanism.

460
00:23:12,370 --> 00:23:15,630
I think this is another
drawing, two other people's

461
00:23:15,630 --> 00:23:17,467
drawings of the same mechanism.

462
00:23:17,467 --> 00:23:19,050
Here the triangles
have been filled in

463
00:23:19,050 --> 00:23:23,900
to be-- I think this is a
design for a metal version.

464
00:23:23,900 --> 00:23:25,860
And then this is a
mathematical simulation.

465
00:23:25,860 --> 00:23:30,480
Again, uniformly spaced
rotary cranking here.

466
00:23:30,480 --> 00:23:32,410
You get to see the spacing
of the points here.

467
00:23:32,410 --> 00:23:35,970
So it's moving a little
bit faster over the lift,

468
00:23:35,970 --> 00:23:38,350
and then pushing down and
spending a fair amount of time

469
00:23:38,350 --> 00:23:42,940
here, and pretty uniformly
applying . force.

470
00:23:42,940 --> 00:23:45,430
Also not completely flat.

471
00:23:45,430 --> 00:23:47,490
It looks like it's
convex-- in fact

472
00:23:47,490 --> 00:23:50,180
there's a little
upward dip here.

473
00:23:50,180 --> 00:23:53,850
But it's pretty good, pretty
straight, and a really good

474
00:23:53,850 --> 00:23:57,160
walking mechanism, possibly
the best within this family.

475
00:23:57,160 --> 00:24:00,770
Depends how you define
best, of course.

476
00:24:00,770 --> 00:24:05,370
And here's a fun
animation of how

477
00:24:05,370 --> 00:24:07,860
the parts of the linkage and
adding multiple legs, which

478
00:24:07,860 --> 00:24:09,910
is I wanted to show it.

479
00:24:09,910 --> 00:24:12,110
So you start with
one 4 bar mechanism.

480
00:24:12,110 --> 00:24:15,570
These are probably not quite
the right lengths, but close.

481
00:24:15,570 --> 00:24:19,090
You add two rigid
triangles it's a little

482
00:24:19,090 --> 00:24:22,190
add another quadrilateral on
there you see the spitting

483
00:24:22,190 --> 00:24:25,940
part offset relative to
that and then here you

484
00:24:25,940 --> 00:24:28,250
get the trajectory of the foot.

485
00:24:28,250 --> 00:24:31,660
And you see on a terrain how
it would move you forward,

486
00:24:31,660 --> 00:24:34,170
especially if you have
multiple legs, then one of them

487
00:24:34,170 --> 00:24:37,650
is always in contact so
you're moving forward.

488
00:24:37,650 --> 00:24:40,570
And then if you want
to be even cooler,

489
00:24:40,570 --> 00:24:45,120
you go to three dimensions,
use even more legs.

490
00:24:45,120 --> 00:24:48,290
And you have a little
crank in the center that

491
00:24:48,290 --> 00:24:53,229
offsets the timing of the feet
so they're all sporadically

492
00:24:53,229 --> 00:24:54,770
moving, and that's
essentially what's

493
00:24:54,770 --> 00:24:59,100
going on in this structure
where they're nicely,

494
00:24:59,100 --> 00:25:01,800
evenly distributed.

495
00:25:01,800 --> 00:25:03,810
There's a central
crank here that

496
00:25:03,810 --> 00:25:08,900
just offsets the timing
of all these feet,

497
00:25:08,900 --> 00:25:11,174
and it's super cool.

498
00:25:11,174 --> 00:25:12,590
I mean, some ways
the coolest part

499
00:25:12,590 --> 00:25:16,360
is how it looks like a
real animal walking--

500
00:25:16,360 --> 00:25:18,764
the funny timing, and the
way you lift and then spend

501
00:25:18,764 --> 00:25:20,930
most your time with a foot
on the ground, and so on.

502
00:25:20,930 --> 00:25:25,280
So, fun stuff.

503
00:25:25,280 --> 00:25:26,004
Let's see.

504
00:25:26,004 --> 00:25:27,420
Of course, other
people have tried

505
00:25:27,420 --> 00:25:28,880
to improve it in various ways.

506
00:25:28,880 --> 00:25:32,410
There's a couple of papers
proposing alternate walking

507
00:25:32,410 --> 00:25:34,030
linkage mechanisms.

508
00:25:34,030 --> 00:25:37,490
This is the Theo
Jansen version, this

509
00:25:37,490 --> 00:25:40,750
is another version by Ghassaei.

510
00:25:40,750 --> 00:25:44,060
And it's actually a
simpler linkage, just

511
00:25:44,060 --> 00:25:48,100
a 4 bar linkage over here too
rigid triangles and another 4

512
00:25:48,100 --> 00:25:49,360
bar linkage.

513
00:25:49,360 --> 00:25:51,160
I guess roughly the
same number of parts

514
00:25:51,160 --> 00:25:54,352
but, I think, one fewer
edge if I counted correctly,

515
00:25:54,352 --> 00:25:56,060
just because of the
way they're combined.

516
00:25:59,160 --> 00:26:02,960
And some fun things as you get--
this is the foot pattern down

517
00:26:02,960 --> 00:26:05,890
here, still has a nice straight
part, looks pretty uniform,

518
00:26:05,890 --> 00:26:07,240
speeds up for the lift.

519
00:26:07,240 --> 00:26:12,506
Does not lift as high-- this one
has a 30 percent higher step,

520
00:26:12,506 --> 00:26:13,630
which is considered better.

521
00:26:13,630 --> 00:26:15,900
You could move over
rougher terrain.

522
00:26:15,900 --> 00:26:18,750
This one has other Jansen's
has other advantages,

523
00:26:18,750 --> 00:26:20,807
like if you have two of
them in the same plane

524
00:26:20,807 --> 00:26:22,390
they will not collide
with each other.

525
00:26:22,390 --> 00:26:24,682
Two legs reflected
around each other.

526
00:26:24,682 --> 00:26:25,890
This one, they would collide.

527
00:26:25,890 --> 00:26:28,982
You have to move them
to separate planes.

528
00:26:28,982 --> 00:26:30,440
But some fun things
here as you get

529
00:26:30,440 --> 00:26:34,170
that this foot pattern is
symmetric, down the y-axis,

530
00:26:34,170 --> 00:26:38,030
and you get much less
variation in the height,

531
00:26:38,030 --> 00:26:42,680
sort of more uniform,
and a little less

532
00:26:42,680 --> 00:26:46,956
velocity variation on the
stride part, the walking.

533
00:26:46,956 --> 00:26:48,330
And then the other
thing, which I

534
00:26:48,330 --> 00:26:50,250
think was the main
purpose of the design,

535
00:26:50,250 --> 00:26:52,690
was that the center of
mass of the whole structure

536
00:26:52,690 --> 00:26:54,450
doesn't move very much.

537
00:26:54,450 --> 00:26:56,090
The blue dots here
are the center

538
00:26:56,090 --> 00:26:59,960
of mass over the motion
of the Jansen mechanism.

539
00:26:59,960 --> 00:27:05,471
And this one, the blue dots are
much smaller areas, 85% less

540
00:27:05,471 --> 00:27:05,970
movement.

541
00:27:05,970 --> 00:27:09,920
I'm not sure of the exact
definition of 85% there.

542
00:27:12,850 --> 00:27:15,400
This should be, if you were
like riding on the thing,

543
00:27:15,400 --> 00:27:18,740
you should feel less
wobble, in principle.

544
00:27:18,740 --> 00:27:22,110
They did build one of these
out of PVC tubing as well,

545
00:27:22,110 --> 00:27:25,260
but I haven't seen any video
so I don't know how it looks.

546
00:27:25,260 --> 00:27:27,830
Be kind of neat to see

547
00:27:27,830 --> 00:27:34,000
So this is a two fat
example, apparently.

548
00:27:34,000 --> 00:27:39,175
Said this one was over eating,
and so it tends to fall over.

549
00:27:39,175 --> 00:27:41,550
So it's a challenge to get
these things to work and you--

550
00:27:41,550 --> 00:27:44,250
I can't imagine how many hours
he spends building these,

551
00:27:44,250 --> 00:27:47,680
and then for one of
them to not live.

552
00:27:47,680 --> 00:27:52,759
He does things like, he'll
run these experiments.

553
00:27:52,759 --> 00:27:54,300
Like genetic algorithms
is one thing,

554
00:27:54,300 --> 00:27:56,070
but he'll do the
real genetics thing

555
00:27:56,070 --> 00:27:58,820
of building lots of
different variations,

556
00:27:58,820 --> 00:28:01,920
put them on the sand, see
which one goes the fastest.

557
00:28:01,920 --> 00:28:03,450
The other ones he kills.

558
00:28:03,450 --> 00:28:05,750
The one who went was
the fastest survives

559
00:28:05,750 --> 00:28:06,740
to the next generation.

560
00:28:06,740 --> 00:28:09,610
But each experiment must
take hours and hours--

561
00:28:09,610 --> 00:28:10,360
really impressive.

562
00:28:13,020 --> 00:28:16,140
So there are issues in
getting these things to work,

563
00:28:16,140 --> 00:28:19,590
and just having the leg
mechanism is one thing.

564
00:28:19,590 --> 00:28:21,800
These guys have a lot of
very interesting mechanisms

565
00:28:21,800 --> 00:28:25,150
beyond the legs, in order to--
they're sensors, essentially.

566
00:28:25,150 --> 00:28:28,830
They're purely mechanical--
there's no electronics here

567
00:28:28,830 --> 00:28:30,900
that are physical, I should say.

568
00:28:30,900 --> 00:28:33,550
There are sensors to detect
things like the wind direction.

569
00:28:33,550 --> 00:28:37,310
Recently we had the honor of
co-exhibiting with Theo Jansen.

570
00:28:37,310 --> 00:28:40,850
So this is some of our curve
crease paper folding in this,

571
00:28:40,850 --> 00:28:42,570
or some of his structures.

572
00:28:42,570 --> 00:28:45,510
Some of these small kits and
photographs of his larger ones

573
00:28:45,510 --> 00:28:50,640
were at this show in Stonybrook,
New York, earlier this year.

574
00:28:50,640 --> 00:28:52,610
It's kind of fun.

575
00:28:52,610 --> 00:28:56,460
And while we're on the
topic of kinetic sculpture,

576
00:28:56,460 --> 00:28:58,685
I have to show Arthur Ganson.

577
00:28:58,685 --> 00:29:02,680
It's funny, Ganson and
Jensen are so close.

578
00:29:02,680 --> 00:29:06,480
This is our local kinetic
expert-- kinetic sculpture

579
00:29:06,480 --> 00:29:06,980
expert.

580
00:29:06,980 --> 00:29:10,950
These are four examples
I took from the web,

581
00:29:10,950 --> 00:29:13,020
but all of these examples
and many, many more

582
00:29:13,020 --> 00:29:13,900
at the MIT Museum.

583
00:29:13,900 --> 00:29:16,820
If you haven't seen
his exhibit, you must.

584
00:29:16,820 --> 00:29:21,960
He hand builds lots
of gears and cranks,

585
00:29:21,960 --> 00:29:24,330
and uses them to
build crazy things.

586
00:29:24,330 --> 00:29:28,120
This is called the
self-oiling machine.

587
00:29:28,120 --> 00:29:30,430
it just grabbed a
bunch of oil and it'

588
00:29:30,430 --> 00:29:33,460
about to pour all
that oil onto itself.

589
00:29:33,460 --> 00:29:35,935
It's really gross.

590
00:29:35,935 --> 00:29:40,190
There it goes, pour-- and
should never get rusty.

591
00:29:40,190 --> 00:29:45,700
This is a very simple mechanism
that's just the random and cool

592
00:29:45,700 --> 00:29:48,057
how the changes kind of
move, wiggles around.

593
00:29:48,057 --> 00:29:49,640
It's a force to be
planar, essentially

594
00:29:49,640 --> 00:29:51,360
because of that chain.

595
00:29:51,360 --> 00:29:54,350
This one-- you've got various
handmade gears and so on,

596
00:29:54,350 --> 00:29:59,510
and hand cut pieces of rough
paper, to make them fly.

597
00:29:59,510 --> 00:30:02,460
And this is another beautiful,
relatively simple one.

598
00:30:02,460 --> 00:30:06,690
You've got a big counterweight
to make this chair almost zero

599
00:30:06,690 --> 00:30:09,871
weight, and then there's
a little cat here

600
00:30:09,871 --> 00:30:11,620
that's just rigid and
going back and forth

601
00:30:11,620 --> 00:30:15,350
and it causes that chair to
dance around in very cool ways.

602
00:30:19,367 --> 00:30:20,200
It's pretty amazing.

603
00:30:20,200 --> 00:30:24,310
He lives locally, and has made
a huge array of sculptures,

604
00:30:24,310 --> 00:30:27,570
one which I don't have a video
of, which is a lot of fun,

605
00:30:27,570 --> 00:30:30,700
is just-- there's a
big crank, and then

606
00:30:30,700 --> 00:30:33,510
a series of like 20
to 1 gear reductions,

607
00:30:33,510 --> 00:30:35,440
and at the other end
the turning element

608
00:30:35,440 --> 00:30:37,095
is in a solid block of concrete.

609
00:30:37,095 --> 00:30:38,970
And there's some
computation, like it'll

610
00:30:38,970 --> 00:30:42,950
take 100,000 years before that
before it turns that concrete

611
00:30:42,950 --> 00:30:45,760
around, which it can't do.

612
00:30:45,760 --> 00:30:47,585
Initially the gears
are going pretty fast,

613
00:30:47,585 --> 00:30:48,970
and then, slower
and slower, and then

614
00:30:48,970 --> 00:30:50,850
you just can't see them
for like five iterations

615
00:30:50,850 --> 00:30:52,270
and then it's
wedged into concrete

616
00:30:52,270 --> 00:31:00,050
It's a fun, practical example--
practical Anyway, lots

617
00:31:00,050 --> 00:31:03,840
of very cool kinetic sculpture
and all somehow based

618
00:31:03,840 --> 00:31:08,150
around linkages although not
necessarily strictly following

619
00:31:08,150 --> 00:31:08,890
our definition.

620
00:31:08,890 --> 00:31:12,680
And next week we'll be
talking about protein folding.