1
00:00:16,920 --> 00:00:23,070
I'd like to welcome you all to the Mathematical Institute for another Oxford Mathematics Public Lecture.

2
00:00:23,580 --> 00:00:27,930
My name is I don't go really. And I'm in charge of the lecture for the institute.

3
00:00:28,650 --> 00:00:33,510
It's great to see so many of you. Just a few point of order.

4
00:00:34,260 --> 00:00:40,320
I just want to point to the exits. These are what we call full British exits.

5
00:00:40,800 --> 00:00:43,940
You'll have to be careful if you take this one.

6
00:00:43,950 --> 00:00:49,470
I think I've seen Boris Johnson's trying to hide from voting in the Commons today.

7
00:00:49,860 --> 00:00:58,460
So you'll have to step over him. If you go that way today, we'll talk about origami.

8
00:00:58,470 --> 00:01:01,260
It's a topic that I've always been fascinated by.

9
00:01:01,260 --> 00:01:08,190
I'm completely rubbish at it, but I always love it and I've always wanted to have a public lecture on it.

10
00:01:08,190 --> 00:01:16,050
And I'm extremely grateful for Professor James today to give a lecture on this topic.

11
00:01:16,500 --> 00:01:18,030
If you're interested in this topic,

12
00:01:18,030 --> 00:01:27,749
and I'm sure you'll be after it tells you all that he has to tell you today there is another event that is organised by both the university,

13
00:01:27,750 --> 00:01:31,649
I think the engineering department and the British Origami Society.

14
00:01:31,650 --> 00:01:37,920
There is a British origami society with 700 members, so there'll be an event in September, early September.

15
00:01:37,920 --> 00:01:43,020
You can find that on the website of the bourse, as it's called, British Origami Society,

16
00:01:43,620 --> 00:01:48,179
where there'll be talk both about mathematics but also about more the playful aspect of origami.

17
00:01:48,180 --> 00:01:51,479
So if you're interested in things you should definitely go with.

18
00:01:51,480 --> 00:01:55,290
I just wanted to advertise and be short.

19
00:01:55,290 --> 00:02:05,099
I just want to point out that our Oxford Mathematics Public Lecture, funded in part by X markets and we are very grateful for their help.

20
00:02:05,100 --> 00:02:10,470
It's a financial company with offices in New York, London and Singapore.

21
00:02:12,710 --> 00:02:20,460
Today, I will not give you a full introduction. I've admired the words of Professor James for many years.

22
00:02:20,480 --> 00:02:29,420
I've heard it can give many talks, and I've always been fascinated not only by his research, but also what I would call this clarity of thought.

23
00:02:29,750 --> 00:02:36,650
There are few people, when you hear them, you realise both that they understand the topic extremely deeply,

24
00:02:36,860 --> 00:02:42,260
but also they can phrase it in a way that makes you feel you also understand deeply.

25
00:02:42,650 --> 00:02:46,550
And so I always wanted him to give a public lecture here, and particularly.

26
00:02:47,270 --> 00:02:56,510
But since he has been a very long time friend and collaborator to Professor John Ball, I thought it would be best for him to introduce him.

27
00:02:56,690 --> 00:03:00,080
So please help me introduce John Bode. We will do a proper introduction.

28
00:03:08,630 --> 00:03:14,480
So it's my great honour and pleasure to introduce my scientific colleague and good friend,

29
00:03:15,110 --> 00:03:22,820
Professor Richard James, who is a distinguished McKnight University professor at the University of Minnesota.

30
00:03:24,020 --> 00:03:35,059
So I think James is a truly remarkable scientist, an undoubted world leader in both theoretical and experimental mechanics.

31
00:03:35,060 --> 00:03:38,540
And that combination itself makes him unusual.

32
00:03:39,140 --> 00:03:46,280
And he has a whole string of extraordinarily original contributions over a range of problems and mechanics,

33
00:03:46,280 --> 00:03:56,599
mostly centred on on on the behaviour of alloys that undergo solid phase transformations, which is a very important practical issue,

34
00:03:56,600 --> 00:04:02,959
but stretching into other areas such as the structure of viruses and parts of statistical

35
00:04:02,960 --> 00:04:09,620
physics now I think belongs to is does not belong to a mathematics department,

36
00:04:09,620 --> 00:04:13,399
but the Department of Aerospace, Engineering and Mechanics.

37
00:04:13,400 --> 00:04:16,610
And though he will deny it, he's also a very fine mathematician.

38
00:04:17,840 --> 00:04:30,440
And I think that one thing that I admire of him is that perhaps more than most mathematicians who work and in mathematics departments,

39
00:04:30,440 --> 00:04:34,070
he believes in the power of mathematics for describing nature.

40
00:04:35,230 --> 00:04:42,559
So silly special cases that mathematicians would ignore and his hands turn into discoveries of new materials.

41
00:04:42,560 --> 00:04:47,620
So I think that's a remarkable ability to to to do that.

42
00:04:47,620 --> 00:05:07,970
And so it's a great pleasure to invite him to give his lecture. They're just number two.

43
00:05:07,990 --> 00:05:11,620
Okay. Thank you very much for this very kind introduction.

44
00:05:11,620 --> 00:05:20,620
And everyone can hear me. Is that clear? And it's a great pleasure to deliver this lecture and a great honour.

45
00:05:21,700 --> 00:05:30,160
I'm going to speak about in particular. It's a it's a it's a great pleasure to speak in this room, this beautiful room with this beautiful ceiling,

46
00:05:30,730 --> 00:05:38,379
and also to speak in this wonderful math institute and with its many references to mathematics.

47
00:05:38,380 --> 00:05:47,640
In fact, this glass ceiling that you see there has a nice reference to to the eigenvalues of the plus in their eigenvectors,

48
00:05:48,850 --> 00:05:52,120
but it also has a connection, a nice connection to origami.

49
00:05:52,450 --> 00:05:55,390
So, in fact, I'm going to start with a homework problem.

50
00:05:55,840 --> 00:06:04,030
Is this this you can think of these glass panels as rigid, but I'm going to allow the joints to be flexible.

51
00:06:04,390 --> 00:06:07,540
And so my question is, can this be folded? Okay.

52
00:06:07,810 --> 00:06:12,660
So we'll we'll I'll give you enough information that you'll be able to decide whether this can be folded or not.

53
00:06:12,670 --> 00:06:16,840
And at the end, I'll tell you I'll tell you all about it. Okay.

54
00:06:16,840 --> 00:06:22,479
So first a little bit about myself. I come from a kind of place maybe I've not heard of.

55
00:06:22,480 --> 00:06:29,050
It's it's Minnesota. It's in the United States. It's this little red state up there in the very far north.

56
00:06:29,830 --> 00:06:33,430
And it's known for these things.

57
00:06:33,430 --> 00:06:39,999
You know, people like to take vacations in Minnesota, and they they they like to go canoeing because it's the land of 10,000 lakes.

58
00:06:40,000 --> 00:06:47,559
In fact, there are 15,000 lakes. And Minnesota not not not including small ponds, you know, so we really have a lot of lakes.

59
00:06:47,560 --> 00:06:52,590
So almost everybody has a canoe and they go around in these lakes.

60
00:06:52,600 --> 00:07:03,370
So just to convince you that origami and folding has penetrated all areas of technology, I want to show you a little video here.

61
00:07:03,640 --> 00:07:09,370
That was the the Ori Canoe Company was extremely pleased that I was going to show this video.

62
00:07:09,370 --> 00:07:12,729
But you can even fold the canoe up.

63
00:07:12,730 --> 00:07:17,140
So if you have a studio apartment, you can own a canoe and then you can go canoeing.

64
00:07:18,340 --> 00:07:23,080
So. Secondly, I'd like to make an acknowledgement.

65
00:07:23,500 --> 00:07:30,660
I have actually a project on origami structures, the design of origami structures, and it's one of the participants.

66
00:07:30,670 --> 00:07:35,200
It's about five or six people. One of the participants is Robert J.

67
00:07:35,200 --> 00:07:43,180
Lang. You may know him from his many books on origami, and he's an interesting collaborator,

68
00:07:43,920 --> 00:07:51,070
a very interesting person, and he does something that we are so far away from doing mathematically.

69
00:07:52,480 --> 00:08:00,450
He does this, he makes a rhinoceros. So you tell him to make a rhinoceros and he will make you a beautiful rhinoceros.

70
00:08:00,460 --> 00:08:07,480
So he will decide what the fold lines are on a piece of paper, such that when you fold it up, you get a rhinoceros.

71
00:08:07,570 --> 00:08:13,059
That's what in mathematics we would call the inverse problem. We have no way to solve the inverse problem.

72
00:08:13,060 --> 00:08:17,049
And in particular, you know,

73
00:08:17,050 --> 00:08:28,680
he solves the inverse problem in such a way that with a great dose of asceticism and beauty and simplicity that leads to these origami structures.

74
00:08:28,690 --> 00:08:35,380
So in fact, this collaboration with him is I feel like it's working with a real with a true genius.

75
00:08:36,190 --> 00:08:41,240
One can say that Robert de Lange is a true, true genius. Okay.

76
00:08:41,260 --> 00:08:48,540
So this this talk is about origami, but it's inspired by Adam Mystics,

77
00:08:48,550 --> 00:08:52,629
which may seem like an unusual place to get some inspiration for folding things.

78
00:08:52,630 --> 00:08:54,310
But that's that's exactly what I'm going to do.

79
00:08:55,370 --> 00:09:03,380
So I'm going to start with the optimistic and I start there with with one of the most important atomistic structures, a carbon nanotube.

80
00:09:03,540 --> 00:09:10,100
It's it's one of the three forms of carbon that did not get the Nobel Prize.

81
00:09:10,130 --> 00:09:14,030
That's of lower dimension. In fact, I think it's the most interesting one.

82
00:09:14,660 --> 00:09:21,890
Or nanotubes in general are are very interesting. And what I would like to do is point out a feature of its structure.

83
00:09:22,310 --> 00:09:27,110
So I want you to imagine. So I put a little couple dots there.

84
00:09:27,140 --> 00:09:33,800
I want you to imagine that you're they actually sitting on one of those atoms and you're looking out at the structure.

85
00:09:34,950 --> 00:09:41,790
And of course. So what do you say? You see the nearest neighbours and then you see the atoms bit further away and a bit further away and so forth.

86
00:09:43,550 --> 00:09:47,520
Now you go to another point on the structure. So you're going to get you started here.

87
00:09:47,520 --> 00:09:56,240
And we did this. We go to this other point and we sit on that point and we reorient ourselves in exactly the right way.

88
00:09:56,600 --> 00:09:59,960
You see exactly the same picture as you do on the first one.

89
00:10:00,620 --> 00:10:02,570
And that's true of every atom of the structure.

90
00:10:03,050 --> 00:10:10,250
If you if you orient yourself in the right way, you take a picture, you see exactly the same environment out to infinity.

91
00:10:11,190 --> 00:10:19,470
Okay. That's interesting. I mean, the reason I chose the two red dots is because it looks like this can't possibly be true,

92
00:10:19,950 --> 00:10:23,100
because how can this be the same environment as this? This is.

93
00:10:23,550 --> 00:10:29,580
So, in fact, what you have to do is on the top one, you have to look down on the bottom and you have to look up.

94
00:10:30,150 --> 00:10:34,430
And if you do, then you'll see that that's the orientation at which you'll see exactly the same thing.

95
00:10:34,440 --> 00:10:40,860
And if you turn sideways, you'll see exactly the same thing. It's a property of that particular structure.

96
00:10:42,390 --> 00:10:49,670
But it's also a property of many, many structures, and particularly these kind of nanostructures that people are discovering these days.

97
00:10:49,680 --> 00:10:54,570
So here's. So first, I'm going to put it in mathematical terms that I'll show you a bunch of pictures of.

98
00:10:54,960 --> 00:11:00,510
I call these objective structures. And so let's let's try to put that in mathematical terms.

99
00:11:00,870 --> 00:11:04,919
So here it is. Here's a so I'm going to think of positions in space.

100
00:11:04,920 --> 00:11:08,790
You can think of three coordinates of the position or in two dimensions like this.

101
00:11:09,120 --> 00:11:17,759
It's the idea is in either case and suppose X one is a point of the structure and there's other points,

102
00:11:17,760 --> 00:11:25,410
many other points, and I'm not drawing them all. I'll just draw some of them. And now I draw with vectors to to every other point.

103
00:11:26,100 --> 00:11:31,409
So this and so. Okay, fine. Now, now I go to another point of the structure.

104
00:11:31,410 --> 00:11:37,290
Any other point say XY and I draw vectors to all the points of the structure.

105
00:11:37,860 --> 00:11:44,579
Okay. So I get this spray of vectors coming from X one and of course you know this there would be a vector from here to here.

106
00:11:44,580 --> 00:11:52,280
I just have a draw on it, so forth. An objective structure has the property that I can take that spray of vectors coming from x one,

107
00:11:52,280 --> 00:11:57,469
I can rigidly rotate it and I can rotate it into into the spray of vectors coming

108
00:11:57,470 --> 00:12:02,030
from X to that's an objective structure and you can write that down mathematically.

109
00:12:03,740 --> 00:12:07,129
The notation, I mean, I think some of you will understand the notation.

110
00:12:07,130 --> 00:12:08,180
I'm not sure everybody will,

111
00:12:08,180 --> 00:12:16,150
but it's you can you can read this on two different levels of the picture level and the linear algebra level, but it's like that.

112
00:12:16,160 --> 00:12:19,280
It goes like this. So this is the structure.

113
00:12:19,280 --> 00:12:29,000
That's the full set of points. And they have the property that for any one of the points, for example EXI you can take all the points,

114
00:12:30,020 --> 00:12:34,340
you can take the, the, the arrows going from X1 to all the points.

115
00:12:34,910 --> 00:12:42,270
You can reorient them depending on the choice of AI and you can add them back to EXI and you get the same structure back again.

116
00:12:43,130 --> 00:12:46,970
Okay, so that's exactly what I said in words is that's the mathematical definition.

117
00:12:47,470 --> 00:12:49,100
Of course you have a mathematical definition.

118
00:12:49,100 --> 00:12:55,520
You can study it and you can do quantum mechanics with this definition, and you can do many things, many interesting things.

119
00:12:55,520 --> 00:13:05,059
But what I will show you is a picture. So all these structures are objective structures, and I'll tell you about some of them.

120
00:13:05,060 --> 00:13:10,000
Of course, these are Nobel Prize winning structures here, graphene and buckyball.

121
00:13:11,330 --> 00:13:15,620
I've already discussed carbon nanotubes. These are the sort of classic carbon nanotubes.

122
00:13:15,620 --> 00:13:23,900
This is carbon nanotube with chirality. That's they're all objective structures and here are some helical structures and some viruses and so forth.

123
00:13:24,140 --> 00:13:29,240
The only thing I should mention is, is like these viruses, that's the bird flu virus, by the way.

124
00:13:29,240 --> 00:13:33,290
This is this is phosphor green, which is also a very, very interesting structure.

125
00:13:33,680 --> 00:13:39,799
And again, you might not think that every atom sees the same environment, whether it's on the lower level or the upper level.

126
00:13:39,800 --> 00:13:42,920
But that's true. Okay.

127
00:13:44,780 --> 00:13:49,610
And this this amyloid protein is also extremely important protein.

128
00:13:50,000 --> 00:13:53,629
But this is one of the bird flu viruses.

129
00:13:53,630 --> 00:14:01,450
And you can see this is made of molecules. So that the definition I gave you was was for an atomic structure and a molecular structure.

130
00:14:01,460 --> 00:14:05,240
The definition goes like this. You have identical molecules.

131
00:14:06,530 --> 00:14:10,370
You number the atoms in each molecule, say 1 to 100.

132
00:14:10,910 --> 00:14:19,700
And then you you might assume the numbering is done in a very good way so that I go to the 27th atom of of this molecule.

133
00:14:20,540 --> 00:14:31,030
Now I go to the 27th atom of a different molecule, any other molecule in the structure, and I and I look in an appropriate direction.

134
00:14:31,040 --> 00:14:34,340
Then the 27th atom of every molecule sees the same environment.

135
00:14:34,970 --> 00:14:41,660
Okay, so that's the definite 26th atom sees a different environment from the 27th, but all the 26th atom sees the same.

136
00:14:42,740 --> 00:14:48,170
And then there is a huge number of, of molecular structures which which satisfy that definition.

137
00:14:48,170 --> 00:14:52,100
Most of all do the atomic case because it's a bit easier to to explain things.

138
00:14:52,100 --> 00:14:55,130
But those are those are objective structures.

139
00:14:55,370 --> 00:14:58,970
So now we're going to play a game with the periodic table.

140
00:15:01,190 --> 00:15:08,760
So there's a periodic table. And my little game would fail miserably if I included these radioactive elements down here.

141
00:15:08,790 --> 00:15:18,469
That's why. So I raised them. Okay. And I also asked the team because ascertain there's only one gram in the entire Earth's

142
00:15:18,470 --> 00:15:22,130
crust that any crust at any one moment and no one knows the crystal structure.

143
00:15:22,310 --> 00:15:25,970
Okay. So this is all about the structure of the elements. Okay.

144
00:15:26,690 --> 00:15:31,820
How do most people think of the periodic table? You can open a book on atomic structure.

145
00:15:31,830 --> 00:15:36,020
They think this way. They think in terms of Broadway lattices.

146
00:15:36,590 --> 00:15:40,100
They they build up the structures from from previous lattices.

147
00:15:40,910 --> 00:15:41,840
So mathematically,

148
00:15:41,840 --> 00:15:52,640
you take three factors that do not lie on a plane and you take integers coefficients on those factors and you add them up or and more pictorial terms.

149
00:15:52,700 --> 00:15:57,620
You start with this atom. Those are the three factors in this particular case.

150
00:15:58,310 --> 00:16:04,430
And, you know, and so this this atom can be shifted to this position by by adding E three.

151
00:16:04,670 --> 00:16:13,969
So this that means this middle atom is also part of the structure. It and this one is because it's 23 and and so you take integers all possible

152
00:16:13,970 --> 00:16:17,420
combinations of integers on these three vectors and you get the structure,

153
00:16:17,420 --> 00:16:20,930
let's call the phase centred cubic structure FCC.

154
00:16:21,410 --> 00:16:24,830
You might not think if you, if you, if you're not really quick, like,

155
00:16:24,830 --> 00:16:31,219
like I'm not you might not think you could get this atom, but this atom is obtained by e1e2 minus c three.

156
00:16:31,220 --> 00:16:36,230
So it's integers on those three vectors. And this is the phase centred cubic structure.

157
00:16:37,550 --> 00:16:42,860
So as I say, most people think of the periodic table in terms of rough lattices.

158
00:16:43,370 --> 00:16:52,700
So I want to, I want to take the periodic table again and I'm going to blacking out all the elements that are not of lattices.

159
00:16:53,300 --> 00:17:00,379
Okay. So there you go. So of course lots of elements are not don't crystallise.

160
00:17:00,380 --> 00:17:03,350
And I did this kind of rationally somehow I,

161
00:17:04,130 --> 00:17:09,120
I took the most common structure at room temperature and if the material is not solid at room temperature,

162
00:17:09,130 --> 00:17:13,280
I took the structure accepted structure at zero temperature. So I tried not to fiddle anything.

163
00:17:13,280 --> 00:17:19,010
And there's, there's my blackened out. So many elements do not crystallise as a lattices.

164
00:17:20,770 --> 00:17:25,090
Now, many of those are actually the same crystal structure.

165
00:17:26,110 --> 00:17:38,680
About a third of the periodic table. So roughly speaking, half of those black, black and out elements prefer the hexagonal close pack structure.

166
00:17:39,010 --> 00:17:42,310
Okay, so let me explain what hexagonal close packed is.

167
00:17:43,510 --> 00:17:48,549
We take we take a bunch of balls and we put them down in a close, packed fashion,

168
00:17:48,550 --> 00:17:54,790
fit together with the closest the smallest volume, smallest mass, and a large volume say.

169
00:17:55,810 --> 00:17:59,230
In in this array. And now that's that's layer one.

170
00:17:59,270 --> 00:18:01,690
And now I'm going to put layer to layer two.

171
00:18:02,140 --> 00:18:08,740
I'm going to put balls in the holes in the depressions of the first layer, and they'll fit together perfectly, as you see there.

172
00:18:09,310 --> 00:18:13,240
And the same. Same on the right. In fact, I could it doesn't matter what I wear.

173
00:18:13,240 --> 00:18:17,140
I put the first one. If I put the first one here, I would still get exactly the same picture.

174
00:18:17,410 --> 00:18:22,420
Okay. So it does doesn't that mean the fact that they look the same is no loss of generality?

175
00:18:24,490 --> 00:18:29,830
Okay. Now I'll put layer three on, but layer three I have a decision.

176
00:18:30,700 --> 00:18:34,870
And layer three, I could do that. In other words, I could.

177
00:18:34,870 --> 00:18:40,930
I could put a ball right in the depression of the previous layer, but not over one of the first layers.

178
00:18:42,090 --> 00:18:47,980
Or. I could do that. I could put the the atoms directly over the first layer.

179
00:18:50,090 --> 00:18:54,110
And the left hand structure is is actually phase centre cubic.

180
00:18:54,380 --> 00:18:59,180
So it's a little hard to see. But if you go back to the picture before.

181
00:19:00,290 --> 00:19:02,210
Um. Where is it? Right there.

182
00:19:03,140 --> 00:19:10,310
If you look down the body diagonal of that of that structure, you'll see exactly the picture that I show on this, this other slide.

183
00:19:10,310 --> 00:19:15,379
So that's the this picture on the left and on the right, of course, it's something different.

184
00:19:15,380 --> 00:19:22,670
And on the right is is hexagonal, close, packed. And what I mean by that is, if you continue now, you've got the first layer.

185
00:19:22,730 --> 00:19:26,600
So you only alternate between two different structures.

186
00:19:26,930 --> 00:19:31,460
You might say a baobab and you might call this stacking ABC, ABC, ABC.

187
00:19:32,930 --> 00:19:42,650
The structure on the right is not above a lattice. So that's why that's that explains a number of the positions on the periodic table.

188
00:19:43,910 --> 00:19:47,810
So why is that? What's the proof? It's not a Bravo lattice. It's very easy.

189
00:19:49,520 --> 00:19:56,809
If you have a very lattice and you have a vector that goes from one atom to another, you have an arrow and you double that arrow.

190
00:19:56,810 --> 00:20:00,080
You always get to an atom that follows directly from the definition.

191
00:20:01,170 --> 00:20:11,210
So I'll have a little arrow and that's going from the centre of the bottom atom to the centre of the layer to atom, the green one.

192
00:20:11,220 --> 00:20:18,780
So it's going from the centre of the black one underneath there to the centre of the, the, the, the red one.

193
00:20:19,710 --> 00:20:22,770
If I double that vector, you see there's no atom there.

194
00:20:23,400 --> 00:20:26,640
So hexagonal, close packed is not a break lattice.

195
00:20:27,330 --> 00:20:30,540
So but hexagonal close packed is an objective structure.

196
00:20:30,990 --> 00:20:34,770
So every atom and hexagonal close back sees exactly the same environment.

197
00:20:36,230 --> 00:20:41,960
Okay. All right, good. So there's the definition I already gave of objective structure.

198
00:20:41,970 --> 00:20:46,100
And I want to know, obviously, what I'm going to do is I'm going to go back to the periodic table and I'm going

199
00:20:46,100 --> 00:20:50,570
to blacken out all the elements that do not crystallise as objective structures.

200
00:20:52,950 --> 00:20:57,719
So that's there's an objective structure, a ring of atoms. This is one example, simple example.

201
00:20:57,720 --> 00:21:03,180
So now I go back to the periodic table, I block it out. All the elements which are not objective structures.

202
00:21:04,850 --> 00:21:10,700
Okay. There's one. Manganese. Manganese is not an objective structure.

203
00:21:10,700 --> 00:21:17,059
It's accepted. Crystal structure at room temperature is not an objective structure.

204
00:21:17,060 --> 00:21:20,570
So that's quite interesting. But there's also many, many cases here.

205
00:21:20,810 --> 00:21:31,490
And and in addition to these standard crystal structures, boron likes these icosahedron carbon is the diamond lattice is an objective structure.

206
00:21:32,630 --> 00:21:39,280
The buckyballs are objective structures. Of course I already discussed this carbon nanotubes first.

207
00:21:39,320 --> 00:21:41,360
Fostering is an object of structure.

208
00:21:41,360 --> 00:21:48,650
Sulphur likes this kind of double ring, which is and the halogen compounds like these layered structures which which also have this property.

209
00:21:50,600 --> 00:21:57,409
So you would think that some really smart mathematician would have figured out why this is.

210
00:21:57,410 --> 00:22:02,260
But this is one of the this is one of the outstanding open problems in mathematics.

211
00:22:02,270 --> 00:22:07,129
It's one of those problems in mathematics that every couple of years people simplify the problem

212
00:22:07,130 --> 00:22:12,930
more so they can try to get more information and they get just painfully little information,

213
00:22:12,960 --> 00:22:24,050
then they simplify it more. And so now it's the problem of showing that that with Leonard with some some you know bad variant of of Leonard

214
00:22:24,050 --> 00:22:31,010
Jones potential that that that that you get the FCC structure as the ground state the lowest energy structure.

215
00:22:32,240 --> 00:22:35,720
And so and that's this problem is called the crystallisation problem.

216
00:22:35,840 --> 00:22:38,270
So it's it's really a fundamental open problem.

217
00:22:38,810 --> 00:22:46,060
Why should all these elements which have this, you know, they have very directional bonds to to other to other atoms.

218
00:22:46,070 --> 00:22:50,570
Why should they? Why should they prefer the same environment?

219
00:22:50,690 --> 00:22:56,940
It's completely unknown. Okay. But it's a it's a very good natured love, subjective structure,

220
00:22:57,440 --> 00:23:03,500
but it's a very good way to begin to to to be domestically inspired to make our economy.

221
00:23:03,680 --> 00:23:07,630
So we'll do that. So I have to talk about I symmetries.

222
00:23:07,640 --> 00:23:10,970
I symmetries are fundamental to the construction of origami.

223
00:23:12,650 --> 00:23:17,660
I symmetry is people would use this notation and again, you can think of this on several different levels.

224
00:23:17,660 --> 00:23:25,310
If you want to think about it on the picture level, I'll show you some pictures. And you can also think of R as a rotation matrix and C is a vector.

225
00:23:25,580 --> 00:23:29,480
And I see matrix consists of a rotation and a translation.

226
00:23:31,140 --> 00:23:35,250
So in two dimensions, the rotation might look like that, you know.

227
00:23:35,910 --> 00:23:39,540
So just something and the translation may look like that.

228
00:23:40,830 --> 00:23:44,190
So that's that's a nice symmetry and it's so simplistic.

229
00:23:44,550 --> 00:23:48,390
One of the simple there are more general notions of asymmetries, but the simplest example.

230
00:23:49,600 --> 00:23:54,580
If you have a nice symmetry, you can act on positions in three or two dimensions depending.

231
00:23:55,060 --> 00:23:58,840
And that's the that's the, the orbit of the point.

232
00:23:58,840 --> 00:24:03,730
Our that's the position of the point X acted on by that asymmetry.

233
00:24:05,450 --> 00:24:11,140
So now let's suppose we have two cemeteries and we're going to have our 1c1 on our two C2.

234
00:24:12,410 --> 00:24:20,240
I want to multiply them. So one of the great things about mathematics is you can you can invent your own multiplication rules.

235
00:24:20,530 --> 00:24:25,040
Okay, so here's the multiplication rule. I'm going to choose for G one and G2.

236
00:24:26,020 --> 00:24:29,499
Looks. Looks a bit weird, right? Well, this this kind of makes sense.

237
00:24:29,500 --> 00:24:33,190
I should multiply the rotations, rotate ones, maybe rotate again.

238
00:24:33,820 --> 00:24:37,270
But what's this all about? I mean, there's oh, there should be a left.

239
00:24:37,270 --> 00:24:41,170
A C2 should be a C2 right there. So. Sorry about that.

240
00:24:42,820 --> 00:24:50,710
Anyway, makes it more bizarre anyway. So what is that multiplication rule?

241
00:24:50,740 --> 00:24:57,670
It's a it's actually the most natural multiplication rule. It's it's the the fact is, the multiplication of G1.

242
00:24:57,670 --> 00:25:04,149
G2 is first you do g of. In fact, I switched them the first.

243
00:25:04,150 --> 00:25:09,610
I should first do G2 and then G1. I'm sorry. So I'm actually showing G2.

244
00:25:09,610 --> 00:25:16,000
G1. But anyway, you first rotate with with R one and then you translate C one.

245
00:25:17,930 --> 00:25:24,620
And then you take the result where you are and you rotate again by R two and you translate by see two.

246
00:25:25,460 --> 00:25:29,660
Okay. So, so, so the that multiplication rule means exactly this.

247
00:25:29,960 --> 00:25:34,910
You rotate and translate and then you take the result and you rotate and translate.

248
00:25:35,150 --> 00:25:39,410
Okay. So it's in mathematical terms, it's called composition of mappings.

249
00:25:40,790 --> 00:25:46,520
So it's exactly this this rule. So it's a very natural rule for for multiplying asymmetries.

250
00:25:50,300 --> 00:25:53,510
Now there's a there's a very fundamental notion in mathematics.

251
00:25:53,510 --> 00:25:57,050
The notion of a group is a fantastic idea.

252
00:25:57,680 --> 00:26:01,250
And always it's it's usually comes up in considerations of symmetry.

253
00:26:01,670 --> 00:26:05,150
And usually people would show the platonic solids when they want to describe groups.

254
00:26:05,510 --> 00:26:11,260
I happen to be a big aficionado of groups that describe things that look completely symmetric.

255
00:26:11,270 --> 00:26:16,610
But anyway, we have two asymmetries we just described.

256
00:26:16,940 --> 00:26:21,260
In fact, I will not quite described how to multiply them.

257
00:26:22,950 --> 00:26:26,400
And you can have a list of asymmetries.

258
00:26:26,880 --> 00:26:33,270
And the definition of a group tells you if you multiply any two by this rule, then you'll get back in the group.

259
00:26:33,790 --> 00:26:43,440
Okay, so that's just a closure kind of relation. And and groups have to have the I submit the elements of the group have to have inverses.

260
00:26:43,450 --> 00:26:50,950
It turns out if if this is the isometric rotation our translations see that the inverse of the symmetry is rotation are inverse.

261
00:26:50,950 --> 00:26:55,419
That just means instead of doing this you go back and, and it turns out minus R,

262
00:26:55,420 --> 00:27:00,550
minus one C and that's with that product is the inverse and that's the identity.

263
00:27:01,000 --> 00:27:05,739
And there's a repository for all the groups, all the discrete groups.

264
00:27:05,740 --> 00:27:15,790
I won't tell you what discrete is. It's a bit hard to explain, but all the sort of reasonable groups that that can be formed with asymmetries,

265
00:27:15,790 --> 00:27:18,340
it's the international tables of crystallography.

266
00:27:18,610 --> 00:27:26,709
And so in the library it's this big, it's online and you can look at those groups organised in a bizarre way.

267
00:27:26,710 --> 00:27:31,150
But anyway, that's where they are, so they're nice to use.

268
00:27:31,150 --> 00:27:34,299
Okay, so I mentioned two things.

269
00:27:34,300 --> 00:27:40,930
I mentioned objective structures, that's each atom sees the same environment and I mentioned something completely different.

270
00:27:41,230 --> 00:27:47,980
I saw symmetries and I saw symmetries can form groups. So the point is that there's a there's a very strong connection.

271
00:27:48,860 --> 00:27:51,040
Every object of structure you could.

272
00:27:51,230 --> 00:27:56,480
In other words, you could try to make them by looking at doing these different environments, but that would be very hard.

273
00:27:56,990 --> 00:28:00,260
But this is a very easy way to construct all objective structures.

274
00:28:01,540 --> 00:28:06,250
You take a single point, you take a nice symmetry group with the product I just mentioned,

275
00:28:06,610 --> 00:28:15,310
and you take the you take the each group element and you apply it to X with this rule of g on x axis,

276
00:28:15,460 --> 00:28:22,390
our x plus cft is our and C, so x is there g one of x might be their g, two of x might be their two.

277
00:28:22,390 --> 00:28:28,060
Three of x might be it's called the orbit of the point x, which is a very natural one.

278
00:28:28,060 --> 00:28:35,590
You see this picture. In fact, I made a kind of helical structure, which is an objective structure, and you can get all objective structures that way.

279
00:28:35,590 --> 00:28:42,370
So that's the very simple way is first you figure out the isometric groups and then

280
00:28:42,370 --> 00:28:46,240
you you apply them to points and you get many different objective structures.

281
00:28:46,840 --> 00:29:00,060
Okay. Now objective structures are very important for for I mean I symmetry groups are very important for origami and it's and it's for this reason.

282
00:29:01,610 --> 00:29:09,340
And there's also an eye I symmetry that I haven't told you about, a very important eye symmetry, which in which it's not exactly a rotation.

283
00:29:09,350 --> 00:29:13,610
I'll tell you about it now. So so this is the this is the rotation.

284
00:29:13,610 --> 00:29:18,950
And I let's make the translation equal to zero. So that's a there's another thing you can do with a star.

285
00:29:19,960 --> 00:29:28,960
You can you can divide it in half by a by a line, this dashed line, and you can flip it over.

286
00:29:29,930 --> 00:29:34,100
You can you can reflect across that line and you get the structure.

287
00:29:34,940 --> 00:29:41,330
Now that cannot be obtained by rotation because of the fact you can see this, this, this, this diamond, this here.

288
00:29:41,750 --> 00:29:47,000
And if you rotated to this position, the diamond would not be pointing down.

289
00:29:47,000 --> 00:29:51,620
So there's no way you can do this by rotation. But it's incredibly important for origami.

290
00:29:51,980 --> 00:29:57,799
Why? Because this this kind of asymmetry where this kind of, you know,

291
00:29:57,800 --> 00:30:04,400
this is what you we could call it a reflection is exactly what happens when you take a piece of paper and you flip it over.

292
00:30:05,030 --> 00:30:10,339
Okay. That's that's if I if I drew triangles and so forth on this piece of piece of paper.

293
00:30:10,340 --> 00:30:16,250
That's exactly. Of course, if I can flip it over and also move it, I can allow a translation as well.

294
00:30:17,540 --> 00:30:24,610
So and those are those are all the things you get to do in at least piecewise rigid origami.

295
00:30:24,980 --> 00:30:30,050
Okay. So in fact, that's represented by this matrix. The rotation is represented by this matrix.

296
00:30:31,420 --> 00:30:34,990
So you can make a little diagram. I find this kind of diagram very useful.

297
00:30:35,170 --> 00:30:39,370
I take and imagine the square is all two by two matrices.

298
00:30:39,760 --> 00:30:47,169
And again, you can think of this also in intuitive terms, but and this this circle, I mean, there's just a one parameter.

299
00:30:47,170 --> 00:30:48,549
It's the angle of rotation.

300
00:30:48,550 --> 00:30:57,820
So as you change the angle of rotation, you can imagine there's there's a curve in the four dimensional space of two by two matrices that that,

301
00:30:57,820 --> 00:31:03,250
that closes on itself goes from the identity. Each point on that curve is a rotation matrix like that.

302
00:31:03,400 --> 00:31:04,660
It's just a representation,

303
00:31:05,020 --> 00:31:14,079
but I find a useful representation and then you can do this flipping and that's this particular flipping is represented by this matrix.

304
00:31:14,080 --> 00:31:20,580
But once you flip, of course, just as in origami, once you flip, you can also rotate as well.

305
00:31:21,010 --> 00:31:24,579
Okay. And if you can, you can say, well, maybe it's more complicated.

306
00:31:24,580 --> 00:31:30,490
Maybe if instead of flipping this way, I flip this way and rotate it, I could get something new.

307
00:31:30,670 --> 00:31:34,510
No, that's. You don't get anything. So you just get this second. Second circle.

308
00:31:34,540 --> 00:31:39,460
One parameter family. Okay, of.

309
00:31:40,750 --> 00:31:44,430
One parameter family of of flips. Okay.

310
00:31:44,560 --> 00:31:48,730
So those are the three, three matrices. Okay. So that's that's fine.

311
00:31:48,730 --> 00:31:51,730
So now, of course, I didn't say anything about the C,

312
00:31:51,730 --> 00:31:56,890
I'm just talking about the R and I'm going to allow the R when I do origami to be either through this one or this one.

313
00:31:58,300 --> 00:32:03,010
And so at the beginning, I'm just going to talk about folding from two dimensions back to two dimensions.

314
00:32:03,070 --> 00:32:07,330
In other words, I take the piece of paper, I fold it up, and I put it back into two dimensions.

315
00:32:08,890 --> 00:32:14,290
Okay. Now, origami requires more than I saw.

316
00:32:14,290 --> 00:32:17,710
Symmetries the same. I saw symmetry somehow have to fit together.

317
00:32:18,580 --> 00:32:23,700
So we have to think about what that means. So let's suppose we have the nice symmetry.

318
00:32:23,740 --> 00:32:29,050
Let's have the trivial asymmetry, which is the identity, which is identity matrix and zero.

319
00:32:29,200 --> 00:32:33,940
That means that means it doesn't rotate and it doesn't translate.

320
00:32:34,120 --> 00:32:37,920
So that's the most trivial asymmetry.

321
00:32:37,930 --> 00:32:41,860
And now let's that's that that's this one. And now let's let's choose this one.

322
00:32:41,860 --> 00:32:50,420
And I want to choose it to fit. So I draw a line on this piece of paper, and I flip this over the line that's folding and.

323
00:32:51,520 --> 00:32:59,470
Now that's governed by by an I cemetery and you can write that I cemetery down but it's not any old I cemetery.

324
00:32:59,620 --> 00:33:03,699
The fact that those two asymmetries agree on this line is a very strong restriction.

325
00:33:03,700 --> 00:33:07,159
That's called compatibility. Now I'll write that.

326
00:33:07,160 --> 00:33:14,090
I'm going to write it as a normal. If people are know matrices, it means that first of all, the translations don't matter at all.

327
00:33:14,090 --> 00:33:16,630
You can always adjust them that are compatible.

328
00:33:16,640 --> 00:33:26,450
The real condition is that the two matrices, the two rotation and reflection matrices have to differ by a rank one matrix.

329
00:33:26,450 --> 00:33:34,099
So this would be if you wrote this in its components by J minus AJ is high and dry or if you want to think

330
00:33:34,100 --> 00:33:41,030
of it and completely intuitively then then exactly it's this it's the meaning of that is this picture.

331
00:33:41,030 --> 00:33:44,780
But the point is we can put it into into analytical form.

332
00:33:46,390 --> 00:33:51,340
Okay. Here's the theorem. Which is which is which we should prove.

333
00:33:51,340 --> 00:33:54,760
It says that every pair. So you take these two orbits.

334
00:33:54,760 --> 00:33:58,900
This was the rotations, this was the reflections plus rotations.

335
00:33:59,410 --> 00:34:02,470
And I choose the matrix over here. I choose the matrix over here.

336
00:34:03,010 --> 00:34:09,040
And as I say, you can always adjust the the little and the little b the translations to make an origami.

337
00:34:09,040 --> 00:34:13,090
But but it turns out that B is always compatible with a.

338
00:34:13,990 --> 00:34:19,310
Okay. Proof. We can do this. Okay, so I.

339
00:34:20,330 --> 00:34:27,310
I take it again, I can without loss. And generally I can take be to be the identity, or I could take it to be a rotation and a translation.

340
00:34:27,730 --> 00:34:35,890
And now I flip this over and before I push down, I can rotate it any way I want and then I can push down.

341
00:34:37,720 --> 00:34:48,600
And that that construction, which is simple construction is shows that shows that given any matrix over there,

342
00:34:48,610 --> 00:34:55,990
given any reflection plus possible rotation and any rotation, any pure rotation, they're always compatible.

343
00:34:59,730 --> 00:35:06,090
Can I invent a little notation for that? I'm going to I'm going to draw a blue line between these two matrices, if they're compatible,

344
00:35:06,090 --> 00:35:12,180
if they satisfy this condition, or more intuitively, if this if this, if the two symmetries can agree on a line.

345
00:35:14,210 --> 00:35:24,230
Okay. So now we do the simplest economy. And I think what I'm trying to do is okay, so first of all, let me let me let me do it on analytically.

346
00:35:24,230 --> 00:35:32,990
So there's our one. I pick a rotation that's doing this and and I pick a cue two, I pick a reflection.

347
00:35:33,320 --> 00:35:38,600
So and as I say, I can always adjust the translations to work.

348
00:35:39,290 --> 00:35:45,500
So I pick, pick a rotation reflection. They're always compatible because I said any two are compatible so I can draw a blue line.

349
00:35:46,100 --> 00:35:48,959
Now I pick a rotation over here and it's compatible with.

350
00:35:48,960 --> 00:35:56,570
Q to pick a I pick a reflection over there, it's compatible with our three and sure enough, Q four is compatible with our one.

351
00:35:57,170 --> 00:35:59,600
Okay, I satisfied all compatibility.

352
00:35:59,600 --> 00:36:06,530
Just I can choose anything I want, you know, and that means analytically that those four conditions are satisfied.

353
00:36:07,100 --> 00:36:13,909
And even if you really don't quite understand those four conditions, if I add them up, the two cancels, the one cancels.

354
00:36:13,910 --> 00:36:16,070
There, the four cancels and three cancels.

355
00:36:17,540 --> 00:36:25,040
And that means that the sum of the right hand side is zero and there's some kind of restriction on the normals to the interfaces.

356
00:36:25,850 --> 00:36:30,830
Okay, so, so now I'm going to make I saw Matrix. Now tell you what the translations are, I'm going to make them all zero.

357
00:36:31,190 --> 00:36:34,930
And I can do that because they meet at this point. Okay.

358
00:36:34,940 --> 00:36:38,190
Now I'm going to do a very, very famous construction in origami.

359
00:36:38,210 --> 00:36:44,920
I'm going to take a piece of paper and and I'm going to just do like this.

360
00:36:44,940 --> 00:36:50,700
I'm going to push it because I'm describing origami mapping from R2 to R2.

361
00:36:50,720 --> 00:36:54,200
Right. I'm in two dimensions, so I just do this. Okay.

362
00:36:54,320 --> 00:36:59,300
So you can do this at home. And what happens is this is a parchment paper.

363
00:36:59,330 --> 00:37:04,400
Works pretty good for this. And you can see what are our fault lines.

364
00:37:05,750 --> 00:37:08,239
So you have a lot of interesting fold lines.

365
00:37:08,240 --> 00:37:16,310
We have places that are kind of destroyed, but we also have some quite simple places where four folds come together,

366
00:37:16,790 --> 00:37:21,470
like down here, and there's one right there and this one right there.

367
00:37:22,040 --> 00:37:24,530
And there's there's one right there as well.

368
00:37:25,040 --> 00:37:32,600
And if you if you study them for a while, you'd realise that they there's some particular geometric relation for those guys.

369
00:37:34,220 --> 00:37:37,280
And this is a famous theorem in origami. It says that.

370
00:37:38,360 --> 00:37:45,290
You can fold this fully folded back into to to start in two dimensions and fully fold it back into two dimensions.

371
00:37:45,290 --> 00:37:51,919
If, if, and only if. Alpha plus beta equals pi. So the sum of opposite angles is pi.

372
00:37:51,920 --> 00:37:55,790
And of course, the sum of those R is pi two necessarily.

373
00:37:56,890 --> 00:38:00,170
And there's two solutions and those are that. Those are four for two.

374
00:38:00,820 --> 00:38:04,840
Typical case, those are the two solutions. So that's a something you see all the time in origami.

375
00:38:04,840 --> 00:38:10,180
It's a very, very typical construction if you want to fold from two dimensions back into two dimensions.

376
00:38:11,180 --> 00:38:19,430
Okay. And the interesting thing is there's more than that is that if you satisfy alpha plus beta equals pi.

377
00:38:20,530 --> 00:38:24,849
Then not only is there some kind of mapping from two dimensions to two dimensions,

378
00:38:24,850 --> 00:38:32,590
but there's a mapping from two dimensions up into three dimensions and I symmetry all the way and back to back into two dimensions.

379
00:38:32,620 --> 00:38:38,200
In fact, there are two exactly two such contiguous mappings.

380
00:38:39,130 --> 00:38:42,820
There's also two trivial solutions where you fold it and you fold it.

381
00:38:42,820 --> 00:38:48,760
But okay, that's the you can write. Exactly. And you could write formulas for all the information that that you have.

382
00:38:49,670 --> 00:38:52,610
Okay. So that's a well-known thing in our gallery, but it's nice.

383
00:38:52,850 --> 00:38:58,160
I hope it's nice to see you in a now maybe more mathematical way than you would see in origami books.

384
00:38:58,970 --> 00:39:03,320
So here's something that you won't see in origami books. So this is.

385
00:39:03,740 --> 00:39:08,450
So. So now we say some of op at opposite angles is PI.

386
00:39:08,510 --> 00:39:13,940
Right. And so I'm going to make a lot of four fold things and I'm going to arrange that sum of opposite angles as pi.

387
00:39:14,270 --> 00:39:18,830
Can I fold it? So if I cut out one of those little regions, I can fold it.

388
00:39:19,370 --> 00:39:22,579
But turns out I can't fold that and I can fold that.

389
00:39:22,580 --> 00:39:25,640
But in general, I won't be able to do it on a piece of paper.

390
00:39:25,670 --> 00:39:30,160
You won't be able to fold it. So what is the what is the restriction?

391
00:39:30,310 --> 00:39:33,400
It's it's very nice, actually. It's this.

392
00:39:34,470 --> 00:39:39,090
So I look on the edge and I look at these guys, those four fold guys.

393
00:39:39,100 --> 00:39:46,200
They're. And I arrange, arrange, arrange them to have any, any angles.

394
00:39:46,560 --> 00:39:51,450
But samba of opposite angles is pie for this point, for this point, for all the points on that edge.

395
00:39:52,840 --> 00:40:01,780
And now I arrange. Now I pick this edge and I arrange the sum of opposite angles as pie for for all the points along this edge.

396
00:40:02,950 --> 00:40:11,140
And the theorem says. And okay, one more thing you get to choose for each point on this edge, you get to choose a plus or minus one as well.

397
00:40:12,580 --> 00:40:14,020
And with that amount of freedom,

398
00:40:14,410 --> 00:40:24,610
then the entire fold crease pattern is determined that these are all ways to fold it and you can write formulas for everything.

399
00:40:24,610 --> 00:40:34,659
This is a particularly simple one, but you may not be able to decide how it gets folded, but that's the situation.

400
00:40:34,660 --> 00:40:40,299
So it's very restricted. You can't just go ahead and put these together in some arbitrary way.

401
00:40:40,300 --> 00:40:45,820
With Alpha plus equals pi, you can only choose them here and here, and then everything's determined.

402
00:40:46,780 --> 00:40:54,129
It's a little bit like differential equations. You kind of assign boundary conditions and then you you see that, you see the possible,

403
00:40:54,130 --> 00:40:58,990
you get one solution, unique solution, if you would, those plus or minus, as I've mentioned.

404
00:41:00,830 --> 00:41:06,320
Okay. So that's that's what that folds now that that's that structure falls in this way.

405
00:41:08,880 --> 00:41:13,500
In fact, if it falls flat, I just it just doesn't the movie doesn't go all the way.

406
00:41:14,970 --> 00:41:20,520
And you can do more complicated things like this. So you can turn this case along this edge.

407
00:41:20,520 --> 00:41:27,390
So see those guys right there? They were assigned such an alpha plus spider equals pi and some kind of quasi random way.

408
00:41:27,660 --> 00:41:33,150
And these were assigned along this edge. And then all these all these fold lines were determined.

409
00:41:34,070 --> 00:41:37,070
And then it can be folded. That's.

410
00:41:38,090 --> 00:41:41,570
Okay. So that's gives you an idea about that.

411
00:41:41,610 --> 00:41:45,690
You can see it again, that fold that folds flat, by the way. Okay.

412
00:41:46,050 --> 00:41:50,550
So nature has another has there's all kinds of folding in nature.

413
00:41:50,590 --> 00:42:02,990
And this was mentioned by John Paul, but it turns out that that phase transformations and crystals are very closely related at some level to origami.

414
00:42:03,000 --> 00:42:07,920
So what is a phase transformation in crystals and how does it differ from origami and how is it the same?

415
00:42:08,880 --> 00:42:17,760
So many transformations in crystals are transformations where where the crystal structure changes at some particular temperature.

416
00:42:18,290 --> 00:42:24,780
Okay. So maybe very often this cubic structure like this body centred cubic structure, which, by the way, this very lattice.

417
00:42:25,760 --> 00:42:28,610
Is the stable structure at high temperature.

418
00:42:29,390 --> 00:42:37,280
And what happens when you reach a you cool this material, you reach a certain temperature and the and the material distorts.

419
00:42:37,280 --> 00:42:45,860
And that just shows one example in this particular material, which is here it starts by shortening on this vertical axis.

420
00:42:46,130 --> 00:42:49,730
And the top face here, which is a square here, becomes a rhombus.

421
00:42:51,380 --> 00:43:00,200
And now you can think by symmetry that there should be actually six ways that can distort in this with this kind of face transformation.

422
00:43:00,200 --> 00:43:03,590
And those are the six matrices that describe those distortion.

423
00:43:03,800 --> 00:43:09,020
So what are the six ways? So instead of shortening along this axis, I could shorten along this axis sucker.

424
00:43:09,020 --> 00:43:13,250
I shorten along this axis. And for each one of those, I can make two rhombus.

425
00:43:13,550 --> 00:43:17,990
I can pull it out like this or I can pull it out like this. And those are the six distortions.

426
00:43:19,290 --> 00:43:25,860
So now these are described not exactly like like like I saw cemeteries, but quite similar in some way.

427
00:43:26,250 --> 00:43:31,970
You know, you first have to you first have to distort and that's the end.

428
00:43:31,990 --> 00:43:39,330
You do a linear transformation described by one of those matrices, and then you're allowed to rotate because, of course, it's a crystal.

429
00:43:39,630 --> 00:43:46,470
You can you can rotate it however you want and you're not going to it's going to be equally possible before and after rotation.

430
00:43:47,840 --> 00:43:53,140
So. But then you can also have translations, of course. So it's a little bit more complicated than origami.

431
00:43:53,150 --> 00:43:58,760
We have not only a rotation here or a part or a reflection in a a translation.

432
00:43:58,760 --> 00:44:02,110
We have also a distortion, but the distortions are discrete.

433
00:44:02,120 --> 00:44:09,710
There's only six of them. Okay. And exactly the same mathematics work.

434
00:44:09,740 --> 00:44:15,620
So instead of two of those orbits, we have three by three matrices, we have six orbits.

435
00:44:15,620 --> 00:44:25,130
We have this free rotation. This is kind of just a representation of of all the distortion plus rotations the crystal can undergo.

436
00:44:25,820 --> 00:44:35,719
And there's a particular microstructure of this crystal. And if you if you go in here and you you notice this, in fact, it's a very unusual situation.

437
00:44:35,720 --> 00:44:39,680
You find out that this this matrix is compatible with this matrix.

438
00:44:39,680 --> 00:44:45,990
And this is compatible with this one. This is compatible with this one. The normal cases and this would not be compatible back to air.

439
00:44:46,040 --> 00:44:49,069
So in that sense, it's very different from origami.

440
00:44:49,070 --> 00:44:55,890
But for this material, it's this fact be it goes right back to a and that means all those four interfaces are compatible.

441
00:44:56,330 --> 00:44:58,610
And once you realise that you can make the whole structure.

442
00:44:58,610 --> 00:45:05,899
So and as John was saying, I mean, these, these, these phase transformations are incredibly interesting.

443
00:45:05,900 --> 00:45:13,370
And I'm going to show you I'm going to show you one in which you can imagine that, you know, just like origami,

444
00:45:13,370 --> 00:45:21,200
if you get to play with those six matrices by changing these so-called lattice parameters, alpha, beta, gamma, you get to change the distortions.

445
00:45:21,770 --> 00:45:27,559
Then maybe you could have very special distortions in which the two phases fit together in many, many ways.

446
00:45:27,560 --> 00:45:31,670
You got many, many origami. So, in fact, that is the truth.

447
00:45:32,120 --> 00:45:40,669
That is exactly the truth, is if you tune Alphabet and cannot have some special values, then you'll get many.

448
00:45:40,670 --> 00:45:43,870
Origami is possible with these crystal structures. Okay.

449
00:45:44,890 --> 00:45:48,580
So how do you tune your tune? By changing the composition of the material.

450
00:45:48,910 --> 00:45:53,310
So if the material is is some alloy, then you change the composition.

451
00:45:53,320 --> 00:46:01,180
You very carefully measure these things, this, and you go towards your mathematically determined lattice parameters.

452
00:46:02,140 --> 00:46:05,860
So. Very good. So. That's that's how you do it.

453
00:46:05,860 --> 00:46:10,200
And I want to show you material that's been tuned. This is one of the most or.

454
00:46:11,290 --> 00:46:13,449
There are two sort of spectacular materials.

455
00:46:13,450 --> 00:46:20,979
This is this is one of them that had been obtained by this kind of tuning of lattice parameters so that there were many,

456
00:46:20,980 --> 00:46:24,460
many ways to fit together completely non generic conditions.

457
00:46:24,790 --> 00:46:27,850
And it's fun to watch a movie of it transforming back and forth.

458
00:46:27,880 --> 00:46:31,960
By the way, I had six distortions in the case I mentioned before.

459
00:46:32,080 --> 00:46:38,260
This material is one of the distortions, but it's very similar, very similar otherwise to to what I was saying before.

460
00:46:39,210 --> 00:46:46,000
So this is a movie where you were I'm just heating and cooling the specimens.

461
00:46:46,000 --> 00:46:50,400
So there's a there's a there's a little thin film heater underneath this this piece of material.

462
00:46:50,730 --> 00:46:54,390
It's heating it uniformly as possible, and it's cooling.

463
00:46:54,870 --> 00:46:59,140
And one of the wonderful things about this movie is that the every picture is the different.

464
00:46:59,340 --> 00:47:05,000
Different. So in normal phase transforming materials, every picture is the same.

465
00:47:05,240 --> 00:47:11,330
But this particular material somehow has so many different origami to make that it can do a different each time.

466
00:47:11,330 --> 00:47:17,809
And this this particular material is one of the most reversible materials that that's known.

467
00:47:17,810 --> 00:47:24,830
As I say, it's one of two materials that okay, to explain it a bit more in that phase.

468
00:47:24,830 --> 00:47:27,650
That's the cubic phase, the boring grey one.

469
00:47:29,040 --> 00:47:34,920
You can raise the temperature to some higher temperature and you're always in the grey phase, that cubic phase.

470
00:47:35,280 --> 00:47:43,230
But you can cause it to transform by putting stress. And that's the most demanding test of a phase transformation is do it under stress.

471
00:47:43,740 --> 00:47:49,200
This material will, even though it's zinc, gold, copper, which is never expected to do this,

472
00:47:49,830 --> 00:47:57,030
this material can go 100,000 cycles with a stress which would make the steel in this building fail.

473
00:47:57,450 --> 00:48:02,730
Okay. It's way about twice the failure stress of the steel in this building.

474
00:48:04,560 --> 00:48:13,890
So, okay, so okay. Now let's go back to two to origami and let's try to be again, Adam mystically inspired.

475
00:48:14,820 --> 00:48:20,400
And so I arrange I arrange that, that the sum of those two angles is PI.

476
00:48:21,060 --> 00:48:22,920
And I'm also going to do it on a parallelogram.

477
00:48:22,920 --> 00:48:29,130
So I'm going to arrange that, that, that this length equals this length and this length equals this length.

478
00:48:31,380 --> 00:48:37,410
And you can follow that because it's a it's got four regions and some have opposite angles as pie.

479
00:48:37,440 --> 00:48:41,520
There's the two solutions I was describing earlier. This is partly folded.

480
00:48:42,480 --> 00:48:46,740
You can you can fully fold it. And as I say, those are the two ways.

481
00:48:47,340 --> 00:48:51,149
So if you have that, you could do so you can think of doing something interesting.

482
00:48:51,150 --> 00:48:57,480
So let's let's partly fold it. So we take this guy, we take maybe this guy here, maybe that's what I did.

483
00:48:57,750 --> 00:48:59,700
Now I guess it's the top one. But anyway, it doesn't matter.

484
00:49:00,180 --> 00:49:09,050
And remember, after it's been folded, this length equals equals this length here, and this length here equals this length.

485
00:49:09,550 --> 00:49:14,750
And in fact, it's down here. The way I hope it's clear, the way the folding goes, the perspective.

486
00:49:14,760 --> 00:49:23,620
But anyway, this this length equals this one, and this one equals this one. You know, I saw cemeteries preserve length.

487
00:49:24,580 --> 00:49:32,230
So you can imagine you may be able to find a nice symmetry that maps this line into this line and this line into this line.

488
00:49:32,830 --> 00:49:37,820
Turns out you can do that. That's not that hard. I'll tell you what they are in a minute.

489
00:49:38,620 --> 00:49:42,680
And not only that, you can find these two asymmetries and they commute.

490
00:49:42,700 --> 00:49:46,580
Actually. So G1. G2 is G2. G1 we're using the same product for.

491
00:49:48,090 --> 00:49:55,530
If two asymmetries compute and you can just take powers of them and that automatically forms a group.

492
00:49:55,530 --> 00:50:03,340
So this sapele in case the the community elements is just wonderful because if you take two things of the form G one to the page,

493
00:50:03,360 --> 00:50:08,790
two to the Q and you take another one from here like ag1 to the one to the as you write it out,

494
00:50:09,360 --> 00:50:13,590
since they come here, you can switch these two guys and then of course you realise you can switch,

495
00:50:13,860 --> 00:50:17,970
you can push all the g ones to the left and put push all the details to the right.

496
00:50:18,300 --> 00:50:25,710
You get something like this, it's back in this set. Okay. So a very easy way to form a group if you have if you have the elements compute.

497
00:50:26,370 --> 00:50:32,460
So in fact, I can make asymmetries which both which both do that oops, there it is.

498
00:50:33,000 --> 00:50:36,330
Which both map the side and to the side and which form a group.

499
00:50:37,380 --> 00:50:44,100
And they look something like that. But this is the wonderful thing about groups is is.

500
00:50:45,200 --> 00:50:49,279
This this isometric G1, for example, maps this side into this side,

501
00:50:49,280 --> 00:50:54,830
but I can apply it to the whole partly folded origami and it fits on there perfectly.

502
00:50:54,840 --> 00:51:01,310
It satisfies compatibility. And then I can take G2 and G2 fits on there perfectly.

503
00:51:01,700 --> 00:51:05,930
And the wonder of a group is that G1 times G2.

504
00:51:06,990 --> 00:51:13,260
Applied to this original or partly folded origami perfectly fits with all its neighbours.

505
00:51:13,590 --> 00:51:19,260
And I can keep going and going and going and make big origami so we can do that.

506
00:51:19,440 --> 00:51:26,830
And let me tell you about the asymmetries that that so we have to be sure that they compute and they they.

507
00:51:26,850 --> 00:51:31,080
So the symmetries that do that are our members of the helical groups.

508
00:51:31,590 --> 00:51:34,229
And there's a very interesting theorem about the helical groups.

509
00:51:34,230 --> 00:51:41,340
I won't go into detail about it, but those are the four helical groups and they're the details of the asymmetries are there.

510
00:51:41,340 --> 00:51:47,250
But you don't want to read that. I mean, it's just it's just it's just explaining to you that you can explicitly write them down.

511
00:51:49,370 --> 00:51:57,049
This group, for example. So I explain it by I take this blue ball and I take it to orbit under this group and the age is there.

512
00:51:57,050 --> 00:51:59,750
So age to the piece. So I get I get that structure.

513
00:52:00,350 --> 00:52:08,389
And in this case, I took I took the group B, I took the orbit of this under this group and and I coloured,

514
00:52:08,390 --> 00:52:12,680
I actually coloured the atoms according to the power. M Which could be one or two.

515
00:52:12,680 --> 00:52:20,479
That's why you got the colours. And C Is, is, is the orbit of, of one of these points and it's coloured according to the power.

516
00:52:20,480 --> 00:52:27,440
Q And so forth and so on. It turns out that we're going to the groups that satisfy the conditions on the previous slide.

517
00:52:27,440 --> 00:52:32,510
They, they, they compute and they, they can map edges into edges.

518
00:52:33,820 --> 00:52:40,560
Our members of this group see. So it shows that I can write them down explicitly.

519
00:52:41,280 --> 00:52:45,510
So then immediately you can just form lots and lots of our gummies and that's them.

520
00:52:47,100 --> 00:52:53,219
So and this is this is one, for example, where the where the thing is folded and then it folded a little bit more.

521
00:52:53,220 --> 00:53:00,170
And you get the structure here. But you can imagine, as I said, you apply these group elements.

522
00:53:00,170 --> 00:53:03,560
You're building up the structure when you get around the other side. Of course, they may not meet.

523
00:53:04,470 --> 00:53:08,670
And so there's additional conditions and actually they're quite restrictive.

524
00:53:09,000 --> 00:53:15,630
So I won't go into detail. Now, this is a this is a bi stable case like I was just describing.

525
00:53:15,640 --> 00:53:20,820
You have folding and you get this structure, you fold it a little bit more and you can get this structure.

526
00:53:20,910 --> 00:53:25,770
So it's kind of bi stable and in fact, you can write the solutions.

527
00:53:25,770 --> 00:53:31,020
You can write solutions in terms of two parameters. And the solutions are represented by these dots.

528
00:53:31,740 --> 00:53:40,350
These dots are isolated. And that means, in fact, there's a little picture there so you can get so that's jumping between different solutions.

529
00:53:40,380 --> 00:53:49,980
You can do it again here and you can see you can't continuously, you know, morph this helical structure by, by, by any method.

530
00:53:51,160 --> 00:53:57,130
So of course. So that's the situation I just was.

531
00:53:57,850 --> 00:54:01,420
So. But it would be fun to morph it. So. So let's try.

532
00:54:01,660 --> 00:54:08,530
Let's try another method of morphing. So and again, we'll be we'll be out of mystically inspired.

533
00:54:08,680 --> 00:54:12,640
So this is a this is a virus actually studied this virus at one point.

534
00:54:13,150 --> 00:54:17,490
This is bacteriophage T4. It has a capsid.

535
00:54:17,880 --> 00:54:26,100
It has its very complicated virus and attacks. Bacteria like E Coli is a favourite target and it has.

536
00:54:26,380 --> 00:54:29,670
There is the real thing. It attacks them in the following way.

537
00:54:30,120 --> 00:54:33,390
It lands on the surface of the bacterium.

538
00:54:33,750 --> 00:54:41,250
And these these these tail fibres are sticky to the surface of the of the bacterium.

539
00:54:41,460 --> 00:54:44,790
And these long tail fibres are are also sticky.

540
00:54:44,790 --> 00:54:53,280
So they come down. And what they do is they and as soon as those come down, a nucleus, a phase transformation in the tail of this virus.

541
00:54:53,730 --> 00:55:01,530
And so the phase transformation and there's a picture of theoretic from a from a theory, but quite, quite,

542
00:55:01,680 --> 00:55:07,110
quite accurate representation of the phase transformation which occurs in the tail sheath of bacteriophage.

543
00:55:07,110 --> 00:55:12,720
T4 Inside this tail sheath is a very stiff tail tube.

544
00:55:13,050 --> 00:55:17,220
So this is joined very rigidly onto the surface of the bacterium.

545
00:55:17,250 --> 00:55:23,160
So when it shortens, it drives the tail tube through the cell wall.

546
00:55:23,550 --> 00:55:28,590
In fact, it not only drives it through, but it twists it. It twists the knife as it's going through.

547
00:55:28,980 --> 00:55:35,820
And then the DNA of the of the bacteriophage T4 enters the cell and then it does its usual thing of replication.

548
00:55:37,210 --> 00:55:42,310
So we could be we could be optimistically inspired and we could say, okay,

549
00:55:42,310 --> 00:55:47,350
there could be phase transformations and helical structures, kind of like the phase transformations I showed before.

550
00:55:47,740 --> 00:55:57,549
And it's not hard to work out all possible compatible structures which undergo phase transformations and you're seeing representatives of Bose,

551
00:55:57,550 --> 00:56:02,920
they're very similar to the idea of compatibility is exactly the same.

552
00:56:02,920 --> 00:56:07,480
In fact, idea of compatibility we're using for origami, but it's for two helical structures.

553
00:56:09,010 --> 00:56:17,600
Okay. So that provides a nice way of of morphing and a face morphing origami structures.

554
00:56:17,620 --> 00:56:25,299
We can have we can arrange a compatible interface and then we can by moving that interface, we can do morphing of the structure.

555
00:56:25,300 --> 00:56:32,980
I'll show you that again. So this is twisting and also increasing its length.

556
00:56:34,080 --> 00:56:40,620
So that's that's all I had to say, except I have to come back to the homework problem.

557
00:56:41,040 --> 00:56:46,020
Before I do, I want to acknowledge. Fan Fang is a graduate student published pinsky there.

558
00:56:46,140 --> 00:56:51,150
They did a lot of important work in this area very recently. But I particularly want to acknowledge Frank Yu.

559
00:56:51,150 --> 00:56:57,870
So Frank, who is a high school student from Wayzata High School in Minnesota, it's near Minneapolis.

560
00:56:58,540 --> 00:57:01,950
It's and Frank Yu is 16 years old.

561
00:57:02,340 --> 00:57:11,070
And Frank, who knows all the theory I presented, including the mathematical part, and I can show you some nice 3-D printed structures that he made.

562
00:57:11,730 --> 00:57:16,380
Okay. Back to this structure in the Math Institute that have that I've ratio out here.

563
00:57:16,590 --> 00:57:20,990
So it turns out the following is true. There's a four dimensional space.

564
00:57:21,740 --> 00:57:25,090
There are four. And there's a there's a region in that space.

565
00:57:25,100 --> 00:57:28,880
You can define that region absolutely precisely. It's not even very difficult.

566
00:57:29,360 --> 00:57:35,210
And you can take any curve in that four dimensional region, and that will define an origami structure.

567
00:57:35,480 --> 00:57:42,290
So that can be folded a huge number of ways. So, for example, this is one of the ways it can be folded.

568
00:57:45,900 --> 00:57:49,410
You can see it again, I guess. Oop. Okay.

569
00:57:49,410 --> 00:57:55,500
I guess you can't see it, but there's a great many ways that can be folded here.

570
00:57:58,820 --> 00:58:02,820
There it is right there. So.

571
00:58:03,770 --> 00:58:07,240
So I hope this doesn't make you nervous when you leave the room.

572
00:58:07,320 --> 00:58:13,080
You know, maybe you don't want to walk under this glass ceiling, but thank you very much for your your interest.

573
00:58:13,080 --> 00:58:13,350
And.