1 00:00:00,000 --> 00:00:03,838 In the previous lecture we looked at the game of life which was a particular site 2 00:00:03,838 --> 00:00:07,105 or automaton model and in it we saw how we could get just an amazing phenomena, 3 00:00:07,105 --> 00:00:10,958 right, how simple rules can aggregate to produce really sort of complex, novel 4 00:00:10,958 --> 00:00:14,425 outcomes. What we want to do in this lecture is look at an even simpler class 5 00:00:14,425 --> 00:00:17,848 of cellular automaton models, and actually the original cellular automaton models, 6 00:00:17,848 --> 00:00:23,773 and to try and figure out what has to be true about the model in order for it to 7 00:00:23,773 --> 00:00:26,608 produce different types of outcomes. Number one of our core questions was what 8 00:00:26,608 --> 00:00:29,518 kind of outcomes is the system going to produce, is it going to go to equilibrium, 9 00:00:29,518 --> 00:00:32,078 is it going to produce patterns, is it going to be complex, is it going to be 10 00:00:32,078 --> 00:00:35,636 chaotic. And what we want to do is we want to try and understand which of those 11 00:00:35,636 --> 00:00:38,482 things is going to happen. And we're not going to get a definitive answer but again 12 00:00:38,482 --> 00:00:43,601 by using a toy model we are going to get some understanding of what leads to 13 00:00:43,601 --> 00:00:48,415 complex outcomes. Alright, so. First some history. Cellular automata were developed 14 00:00:48,415 --> 00:00:52,093 by a guy named John von Neumann, who is just a brilliant man. Von Neumann built 15 00:00:52,093 --> 00:00:56,251 one of the first computers known as the [inaudible] or the [inaudible]. He also 16 00:00:56,251 --> 00:00:59,707 came up with, was one of the founders of game theory and of growth theory in 17 00:00:59,707 --> 00:01:03,665 economics. So just a brilliant, brilliant mathematical mind. One of the things he 18 00:01:03,665 --> 00:01:06,420 came up with, and this was working with a guy named Stanislaw Ulam, who's a 19 00:01:06,420 --> 00:01:10,406 mathematician, was really the simplest moment he could think of in computation 20 00:01:10,406 --> 00:01:15,925 which is what's going to be called the cellular automata model. His vision, the 21 00:01:15,925 --> 00:01:19,521 cellular automata have been, sort of, studied in gory detail including a recent 22 00:01:19,521 --> 00:01:22,760 book by a guy named Stephen Wolfram who is the developer of Mathamatica called the 23 00:01:22,760 --> 00:01:27,474 New Kind of Science. And in this book, Wolfram explores to really to unbelievable 24 00:01:27,474 --> 00:01:32,015 depth. This is a thousand page book with hundreds and hundreds of illustrations. 25 00:01:32,170 --> 00:01:35,428 How these cellular automata model works. And Wolfram refers to this as a new kind 26 00:01:35,478 --> 00:01:40,140 of science because he is arguing for a computational inductive way of looking at the 27 00:01:40,140 --> 00:01:44,247 world. Okay, so what are these models, what are cellular automaton models? 28 00:01:44,247 --> 00:01:47,367 Well, again, they are exactly what we looked at in the game of life, except for 29 00:01:47,367 --> 00:01:51,077 here instead of being on a two dimensional grid, things are on a one dimensional 30 00:01:51,077 --> 00:01:54,496 line. So you can imagine as before we've had a bunch of cells and they can either 31 00:01:54,496 --> 00:01:59,155 be off, which would be clear, or they can be on. Right, so what we can do is we can 32 00:01:59,155 --> 00:02:03,572 the just then sorta say, okay, how do these things evolve over time. Now the 33 00:02:03,572 --> 00:02:07,901 difference between this and what we did before is that now If I have a cell here, 34 00:02:07,901 --> 00:02:12,177 right, sitting in the center, we are going to assume it only has two neighbors. So 35 00:02:12,177 --> 00:02:16,439 before in the grid world each cell had eight neighbors, now its only got two. Now 36 00:02:16,439 --> 00:02:19,812 the advantage of doing things with only two neighbors is well it's simpler for one 37 00:02:19,812 --> 00:02:24,074 thing, and it also means that we can exhaustively study and that's why 38 00:02:24,074 --> 00:02:28,166 Wolfman's book is so thick. We can study every single one of these rules. So we can 39 00:02:28,166 --> 00:02:32,105 write down every single rule and then ask how do the different rules work. What 40 00:02:32,105 --> 00:02:36,119 behaviors do they produce and that sort of stuff. The other big advantage is that 41 00:02:36,119 --> 00:02:40,232 it's going to be much easier to display these worlds than the other worlds because 42 00:02:40,232 --> 00:02:44,197 we can let time move along this axis. So what I can do is I can have this, here's 43 00:02:44,197 --> 00:02:48,013 the cell at this moment in time, maybe it's filled in, and then I can say what 44 00:02:48,013 --> 00:02:51,977 happens to it at the next period maybe it's off, and then I can say what happens 45 00:02:51,977 --> 00:02:55,843 to it at the next period and maybe it's on. So I can represent time as sort of 46 00:02:55,843 --> 00:02:59,857 moving vertically down the page. Right. So that's the model. Now I've got to decide, 47 00:02:59,857 --> 00:03:04,038 okay, what can the rules look like? Well Here's an example, so let's think about 48 00:03:04,038 --> 00:03:08,765 what a rule would have to look like. So if I think of this cell X, right, right here, 49 00:03:08,765 --> 00:03:13,435 this is the cell X. Now there is, and it's got two neighbors, right? So neighbor one, 50 00:03:13,435 --> 00:03:18,162 neighbor two, or we could call these left and right, if we want. We can ask what are 51 00:03:18,162 --> 00:03:22,717 the possible states those things can be in? W ell, it's possible that all of them 52 00:03:22,717 --> 00:03:27,156 could be off. And it's possible all of them could be on. Or it's possible only 53 00:03:27,156 --> 00:03:31,710 the one to the right is on, or only the cell itself is on, right? So we can think 54 00:03:31,710 --> 00:03:36,205 through and there's basically eight different possibilities. So what would a 55 00:03:36,205 --> 00:03:41,253 rule be? A rule just says �hat do I do in each one of those states?' So it could 56 00:03:41,253 --> 00:03:46,559 say, well, if I'm in the state where we're all currently off, then I'm going to stay 57 00:03:46,559 --> 00:03:51,801 off. And if we're in a state where we're all currently on, then I'm going to go on. 58 00:03:51,801 --> 00:03:57,139 And it could say, these two of us are on I'm also gonna go on. And then what you do 59 00:03:57,139 --> 00:04:00,620 is you think about, okay, here is the cell, we start out with some initial 60 00:04:00,620 --> 00:04:04,493 configuration. We got a Whole bunch of cells and some of hem are colored in and 61 00:04:04,493 --> 00:04:08,367 some of them are not. And then what they do is , each cell says well what are my, 62 00:04:08,367 --> 00:04:12,240 what does my configuration look like? If I'm this cell right here, I notice that 63 00:04:12,240 --> 00:04:15,966 all three of my neighbors are on, so I go to the look up table, see all three 64 00:04:15,966 --> 00:04:19,937 numbers are on and say I am going to be on next period. Okay, so all you do is for 65 00:04:19,937 --> 00:04:23,791 each cell, so like this cell right here. [inaudible] cell right here, it's got its 66 00:04:23,791 --> 00:04:27,722 on but its two neighbors are off so I go up to the look up table and say okay this 67 00:04:27,722 --> 00:04:31,700 is the configuration we are in right here and it might say in that situation go off 68 00:04:31,700 --> 00:04:35,281 [inaudible] in the next period it would stay off. So that's it. Time moves 69 00:04:35,281 --> 00:04:38,961 horizontally and we have these rules that look. Right? Now, one of these that 70 00:04:38,961 --> 00:04:42,640 Wolfman does in his book is he says okay look if you look across all these 71 00:04:42,640 --> 00:04:46,565 different rules you can get all four of these classes of behaviors, right? So you 72 00:04:46,565 --> 00:04:50,147 can get, we talked about this before, you can get fixed points, you can get 73 00:04:50,147 --> 00:04:54,023 alternation, you can get randomness and you can get complexity. And what we want 74 00:04:54,023 --> 00:04:57,948 to understand is why? Why do you get these things? What's true about the rules in 75 00:04:57,948 --> 00:05:01,922 order for this to be true? Okay, in order to get these different types of outcomes. 76 00:05:01,922 --> 00:05:05,848 Okay? Now before we go any further, okay, there's a lot of rules, how do we make ... 77 00:05:05,848 --> 00:05:10,015 Sense of them. How do we keep track of the rules setter. [inaudible] had an ingenious 78 00:05:10,015 --> 00:05:13,835 way of numbering these. So let's think about it. So if I am in this state here: 79 00:05:13,835 --> 00:05:17,952 all off. Well, there are two possibilities here, right? We can be off. Or we could be 80 00:05:17,952 --> 00:05:22,390 on and if I didn't give up this state there could be two possibilities as well 81 00:05:22,390 --> 00:05:25,666 We can be off or we can be on and that's true for every one of these. Two, two, 82 00:05:25,666 --> 00:05:29,027 two, two, two. So there's two different things I can put for each of these things. 83 00:05:29,027 --> 00:05:33,095 So that means there's two to the eighth. Possibilities which means that there are 84 00:05:33,095 --> 00:05:37,825 256 different rules. So now we think holy cow the whole universe of these rules is 85 00:05:37,825 --> 00:05:42,440 of sized 256. There are 256 things that we have to explore. That is why work from 86 00:05:42,440 --> 00:05:46,939 this book runs to one thousand pages. We just give four pages to each rule you 87 00:05:46,939 --> 00:05:51,961 suddenly, you know, used up a thousand pages. Now, Wolfman also comes up with an 88 00:05:51,961 --> 00:05:57,718 ingenious way of numbering this rules. What he does is he says let's just get 89 00:05:57,718 --> 00:06:02,876 used to numbers one, two, four, eight, sixteen, thirty-two, sixty-four, one 90 00:06:02,876 --> 00:06:09,557 twenty eight. And then what he says is, if it's on Right? Then so let's suppose that 91 00:06:09,557 --> 00:06:16,192 our rule, now let me do this a different way. So, suppose that if it's, this is our 92 00:06:16,192 --> 00:06:22,133 rule right here. These three are on. So then [inaudible] we'll call this rule two, 93 00:06:22,133 --> 00:06:26,918 eight, one twenty eight and we'll just add up those numbers to give us 138. So that 94 00:06:26,918 --> 00:06:31,585 will be rule 138. So what we have is the first number with one, the next one with 95 00:06:31,585 --> 00:06:36,136 two, the next one with four, the next one with eight, and so on. And this enables 96 00:06:36,136 --> 00:06:40,337 him to give every rule a unique number between zero and 255. So the rule 97 00:06:40,337 --> 00:06:45,063 everything's off is rule zero the rule where everything is on, we just add up all 98 00:06:45,063 --> 00:06:49,666 these numbers and get 255. So, this is going to give us a numbering system for 99 00:06:49,666 --> 00:06:54,568 the rules. So let's look now at some rules that create some interesting phenomena. 100 00:06:54,568 --> 00:06:59,471 This is rule #30, right, so you have two plus four plus eight plus sixteen and this 101 00:06:59,471 --> 00:07:04,434 rule says if you are currently, if all three of you are off you stay off i f the 102 00:07:04,434 --> 00:07:09,396 one to the right is on or the one to the left is on, right, these two things you go 103 00:07:09,396 --> 00:07:14,178 on. If you are currently on you stay on. And here's a little bit of an asymmetry, 104 00:07:14,178 --> 00:07:18,989 if the one to the right is on you stay on Right. But if you [inaudible] your left is 105 00:07:18,989 --> 00:07:23,257 on over here you go off. So let's think about what happened here. These, this one, 106 00:07:23,257 --> 00:07:27,417 and this one, all have three. All are in this state, right, with all three up. So 107 00:07:27,417 --> 00:07:31,847 they are going to stay up. This one has one to the right on so it is going to come 108 00:07:31,847 --> 00:07:38,477 to life. Right? This one right here, this next one, is currently on with its two 109 00:07:38,477 --> 00:07:44,153 neighbors off, so it's going to stay on. Right? This one right here has the one to 110 00:07:44,153 --> 00:07:48,289 the left on, so it looks like that, so it's going to stay on and the other ones 111 00:07:48,289 --> 00:07:52,543 are all gonna die off. So what we get, we get these three states. Are now on these 112 00:07:52,543 --> 00:07:56,732 three [inaudible]. What happens with the next trade? Well, let's get start, again 113 00:07:56,732 --> 00:08:01,082 the ones to the left are going to stay dead, but this one right here because it's 114 00:08:01,082 --> 00:08:05,486 got one neighbor to the right on is going to come to life, this one because it has 115 00:08:05,486 --> 00:08:09,944 one neighbor to the right on is going to come to life. But this one which is in the 116 00:08:09,944 --> 00:08:13,972 center has three in a row so it is going to die off, so we are going to get 117 00:08:13,972 --> 00:08:18,107 something that looks like that. So what we get is, we get this sort of pattern 118 00:08:18,107 --> 00:08:22,414 spreading out, well again, we are doing this by hand, let's try this... In a more 119 00:08:22,414 --> 00:08:27,449 serious way, using that logo. Okay, so we're going to set this up where there's 120 00:08:27,449 --> 00:08:32,747 one cell that's alive in the center and then we're gonna let it go and we'll see 121 00:08:32,747 --> 00:08:38,591 if we can get those three, right? And now we see is this really interesting pattern 122 00:08:38,591 --> 00:08:44,292 evolving as I move down. And notice how this is creating now we see these 123 00:08:44,292 --> 00:08:50,144 different structures alright we see smaller triangles, bigger triangles and so 124 00:08:50,144 --> 00:08:54,721 on. Right? And one of the things that's been proven about this rule which is sort 125 00:08:54,721 --> 00:08:58,898 of interesting is if I drew a line right down the center like if I picked a 126 00:08:58,898 --> 00:09:03,349 particular cell and drew a line right down the center of its path over time it's 127 00:09:03,349 --> 00:09:07,856 going to be a random sequence of ons and offs so you wouldn't be able to tell, you 128 00:09:07,856 --> 00:09:12,035 wouldn't be able to predict, What's gonna happen next but if you knew what happened 129 00:09:12,035 --> 00:09:16,243 the period before. So, what this is, this is an example rule 30 is an example of a 130 00:09:16,243 --> 00:09:20,391 rule that produces perfect randomness. Alright? Here's the next rule, this is 131 00:09:20,391 --> 00:09:24,981 rule 110. So remember we get the rule the two's on the four's on the eight's on the 132 00:09:24,981 --> 00:09:29,461 thirty-two is on the sixty four is on, so we add those all up we get 110. So think 133 00:09:29,461 --> 00:09:33,830 about this one again, we have three cells over here to the left and these three 134 00:09:33,830 --> 00:09:38,421 cells over here to the right all have no neighbors on so they're all going to stay 135 00:09:38,421 --> 00:09:42,680 off. Now this one has a neighbor to the right on and so it's going to come on. 136 00:09:42,680 --> 00:09:47,698 This one, right here, right? Is currently on but no neighbors on, so it's gonna stay 137 00:09:47,698 --> 00:09:52,344 on. And this cell right here has a neighbor to the left on, right? But notice 138 00:09:52,344 --> 00:09:57,300 how it's gonna then stay off, unlike in the previous case. Well now if I go along 139 00:09:57,300 --> 00:10:02,195 this one is gonna stay off, this one's gonna stay off, but this one, because it's 140 00:10:02,195 --> 00:10:06,785 got a neighbor to the right It's on is in this configuration so it's going to come 141 00:10:06,785 --> 00:10:10,834 to life. This one has two neighbors in a, it has its on and its neighbor to the 142 00:10:10,834 --> 00:10:14,936 right on so it's in this configuration so it's going to stay on, right. But this 143 00:10:14,936 --> 00:10:19,037 cell, right here, the original cell that was on is in this configuration it's on 144 00:10:19,037 --> 00:10:24,969 and the one to it's right is on so it's going to say on as well And then finally, 145 00:10:25,430 --> 00:10:28,903 This cell right here is under configuration as in before where it's 146 00:10:28,903 --> 00:10:33,092 neighbor to the left is on so it stays off and so now we get something that looks 147 00:10:33,092 --> 00:10:35,390 like this where we sort of give this increasing triangle. Now we could, could 148 00:10:35,390 --> 00:10:40,026 ask what happens to rule 110 as we let it run and what we get is we get, this is a 149 00:10:40,026 --> 00:10:44,210 map from Wolfram, we get this really interesting pattern, and this is gonna be 150 00:10:44,210 --> 00:10:48,466 sort of complex we see these particles that sort of move through space and this 151 00:10:48,481 --> 00:10:53,583 rule 110 is classified, is class four by [inaudible] complex rule. Rigth, so what we 152 00:10:53,583 --> 00:10:57,392 got, here is a bette r picture if I start with a random configuration, here is rule 153 00:10:57,392 --> 00:11:00,742 110. And again we see all these sort of interesting particles moving through 154 00:11:00,742 --> 00:11:04,739 space, we see lines moving through, we see things like this interacting and then 155 00:11:04,739 --> 00:11:09,058 causing bigger things, we see all sorts of crunchy interesting stuff. This is 156 00:11:09,058 --> 00:11:14,409 complex, right, is very hard to make sense of. So, what we've seen then which is 157 00:11:14,409 --> 00:11:18,099 interesting, with the simple one dimensional automaton model. It's easy to make 158 00:11:18,099 --> 00:11:21,149 rules where everything just dies. It's easy to make rules where everything gives 159 00:11:21,149 --> 00:11:25,610 blinks. There are some rules where things appear to be random and you actually prove 160 00:11:25,610 --> 00:11:30,711 that they are random, like rule 30. And then there's rules... Like rule 110, 161 00:11:30,711 --> 00:11:35,036 right, to create this complexity. So, what we can do then, is we can ask okay, here's 162 00:11:35,036 --> 00:11:40,851 an interesting question. Why? Right. Why are some rules, why do some rules go to 163 00:11:40,851 --> 00:11:47,205 steady state, some rules blink, some rules random, some rules umm complex. Before we 164 00:11:47,205 --> 00:11:52,011 get to that question of Why, what creates complexity, what creates chaos, what 165 00:11:52,011 --> 00:11:56,572 creates order? Let's just stop for a second and think about how profound these 166 00:11:56,572 --> 00:11:59,811 results are. These are really simple models, much simpler than the game of life 167 00:11:59,811 --> 00:12:05,930 and they can give us anything. And this has led some physicists and mathematicians 168 00:12:05,930 --> 00:12:10,609 to, to suggest: this may be how the world works in some sense. That everything may 169 00:12:10,609 --> 00:12:14,198 come from very simple rules. So all the complex things that we see out there in 170 00:12:14,198 --> 00:12:18,905 the real world, come from very simple binary interactions. So, this has led to the 171 00:12:18,905 --> 00:12:23,490 phrase by the physicist John Wheeler, "It from bid". Now let me quote Wheeler here 172 00:12:23,490 --> 00:12:28,733 because it is really sort of profound. He says it from bit, otherwise every it, 173 00:12:28,733 --> 00:12:32,989 every particle, every field of force, even the space time continuum itself, derives 174 00:12:32,989 --> 00:12:37,854 its function, its meaning, its very existence entirely, even if in some 175 00:12:37,854 --> 00:12:43,274 contexts indirectly, from the apparatus solicited answers, to yes or no questions, 176 00:12:43,274 --> 00:12:49,008 binary choices. Bits, It from bit, symbolizes the idea that every item of the 177 00:12:49,008 --> 00:12:54,281 physical world has at its bottom, a very deep bottom in most i nstances, an 178 00:12:54,281 --> 00:12:59,325 immaterial source of explanation that which we call reality arises in the last 179 00:12:59,325 --> 00:13:03,171 analysis from the posing of yes no questions. And the registering of 180 00:13:03,171 --> 00:13:08,798 equipment evoked responses. In short, that all things physical are information 181 00:13:08,798 --> 00:13:13,380 theoretic in origin and that this is a participatory universe. Okay, that is 182 00:13:13,380 --> 00:13:17,225 Wheeler in 1990. So what Wheeler is basically saying is that it from bit idea is the, 183 00:13:17,225 --> 00:13:22,795 you can actually explain anything... Right by just simple yes from no questions at 184 00:13:22,795 --> 00:13:27,767 the core and so the very, very deep bottom of reality could just be binary switches. 185 00:13:27,767 --> 00:13:32,260 So, it, us, the universe, everything, could literally come from bits. Now that's 186 00:13:32,260 --> 00:13:36,992 a bit of a, you know, that's a big leap from the simple one dimensional cellular 187 00:13:36,992 --> 00:13:41,261 automata model. But you know, the cellular automata model is capable of producing 188 00:13:41,261 --> 00:13:46,305 pretty much anything, so its interesting. Alright, so, let's get to this question of 189 00:13:46,305 --> 00:13:48,918 how does it produce anything. What's going on? Well, Chris Langton, who is a 190 00:13:48,918 --> 00:13:53,346 researcher at the Santa Fe institute, he got his PHD at Michigan, studying these 191 00:13:53,346 --> 00:13:57,640 cellular atomaton, you know, came up with something he calls Langton's Lambda. 192 00:13:57,640 --> 00:14:02,410 And what lamda does is it tells us sort of what the outcomes look like. So, let me 193 00:14:02,410 --> 00:14:06,886 explain what I mean. So, remember the Wolfran number digitals from one to 256. 194 00:14:06,886 --> 00:14:11,656 Langton takes a much simpler approach, he just says look how many things go on? In 195 00:14:11,656 --> 00:14:16,426 this case there's three. So if you think of Langton's lamda as three or as 3/8ths, 196 00:14:16,426 --> 00:14:21,148 either way, it's the percentage of the number of switches that are on. Right? So 197 00:14:21,148 --> 00:14:27,075 this rule would have a, a alpha, a lambda, I'm sorry, of zero or zero over eight. And 198 00:14:27,075 --> 00:14:32,782 this one would have a rule of one over eight. So the, Langton's lambda tells us 199 00:14:32,782 --> 00:14:38,490 the percentage of bits that are on. And this one, remember this was rule thirty. 200 00:14:38,490 --> 00:14:45,277 Right? Would have a Lambda of four over eight. Well, let's go back and look at 201 00:14:45,277 --> 00:14:49,609 these again. This one has a lambda of four over of zero over eight. What's going to 202 00:14:49,609 --> 00:14:53,942 happen? Nothing. Right. Everything is just going to die. Nothing interesting is going 203 00:14:53,942 --> 00:14:57,957 to happe N. What's going to happen to this one that has a one over eight. Well 204 00:14:57,957 --> 00:15:02,131 initially a lot of stuff is going to die off, but then once everything dies off 205 00:15:02,131 --> 00:15:06,622 everything is going to go on but then once everything's on it's all going to die off. 206 00:15:06,622 --> 00:15:11,317 So this thing is going to blink, right? What about rule thirty which has the 207 00:15:11,317 --> 00:15:16,692 lambda four or four over eight? Well, remember this thing was chaotic, right? 208 00:15:16,692 --> 00:15:22,430 This was completely random. And what about rule one ten, right? This was rule one 209 00:15:22,430 --> 00:15:28,002 ten. This has a lambda of five over eight and this thing was complex. Now, what you 210 00:15:28,002 --> 00:15:31,551 can think of then is if you think [inaudible] the bigger lambda gets the 211 00:15:31,551 --> 00:15:35,247 more likely we are to get something interesting. Well that is not quite true 212 00:15:35,247 --> 00:15:38,651 because think about when lambda eight, right, when lambda is eight then 213 00:15:38,651 --> 00:15:42,638 everything automatically goes on. So that is not going to be interesting either. So 214 00:15:42,638 --> 00:15:46,431 what is going to be interesting, it what seem to be, is sort of this in between 215 00:15:46,431 --> 00:15:50,223 region, right, this region where you got sort of either two, three, four, five ,six 216 00:15:50,223 --> 00:15:54,080 things go on, well let's look at it, so here's all the rules... In the, the one 217 00:15:54,080 --> 00:15:58,712 dimensional cellular automata with two neighbors, and if I sum this up I'd get 218 00:15:58,712 --> 00:16:03,581 two hundred fifty six. If I want to know how many class three members this sort of 219 00:16:03,581 --> 00:16:08,450 chaos or random and in that class there's thirty two of them, right. And if we look, 220 00:16:08,450 --> 00:16:12,652 Twenty of them have a lambda equal to four. And they're all in this region 221 00:16:12,652 --> 00:16:16,855 between two and six. Class four is the complex rules, right? And the complex 222 00:16:16,855 --> 00:16:21,512 rules, there's only six of them. And those all happen between three and five, lambda 223 00:16:21,512 --> 00:16:25,941 between three and five. So, here's what's really interesting. Now we want to ask, 224 00:16:25,941 --> 00:16:30,903 what causes chaos and complexity, well Its this region right here. Intermediate 225 00:16:30,903 --> 00:16:36,439 levels of interdependence. Right? So a rule like this which has a lambda of 7/8s 226 00:16:36,439 --> 00:16:40,680 or seven ... Right? Nothing interesting is going to happen. It's just pretty much 227 00:16:40,680 --> 00:16:44,538 going to go to everything being on and then once everything's on, right, it's 228 00:16:44,538 --> 00:16:48,498 going to stay on, so it's going to be stable. S o it's these intermediate levels 229 00:16:48,498 --> 00:16:52,558 where we see the complexity. So if you look at something like, this is the Nikkei 230 00:16:52,558 --> 00:16:56,599 index, where you see these incredibly complex patterns What you'd expect is that 231 00:16:56,599 --> 00:16:59,797 these rules have substantial interdependence. Right? Because that 232 00:16:59,797 --> 00:17:04,094 middle level means that whether I am on or off depends a lot on what other people are 233 00:17:04,094 --> 00:17:07,692 doing. So if there are lot of interdependence in the rules you are going 234 00:17:07,692 --> 00:17:11,589 to see complex patterns like these things. Right? Well what happens in a market. 235 00:17:11,589 --> 00:17:15,587 People's rules depend a lot on what other people are doing. So there is a lot of 236 00:17:15,587 --> 00:17:19,484 interdependence and therefore you get these complex patterns. If there weren't 237 00:17:19,484 --> 00:17:23,532 interdependence, interdependence, right? Then you'd also go on or always go off and 238 00:17:23,532 --> 00:17:27,420 nothing interesting would be happening. So what do we learn from this very, very 239 00:17:27,420 --> 00:17:30,948 simple [inaudible]. First, there again the simple rules can define to [inaudible] 240 00:17:30,948 --> 00:17:35,495 just about anything Incredibly simple rules, second we get the sort of Profound 241 00:17:35,495 --> 00:17:41,304 idea of it from bid and third we get the complexity and randomness its acquired 242 00:17:41,467 --> 00:17:45,097 some intermediate level of interdependency, right? So you can't have 243 00:17:45,097 --> 00:17:49,378 a digit like I always go on or I always go off. You need interdependency in the 244 00:17:49,378 --> 00:17:53,550 actions in order create complex phenomena. Okay, so that's cellular, that's one 245 00:17:53,550 --> 00:17:58,048 dimension of cellular automata. It's a toy model but it gives us a deep insight. And 246 00:17:58,048 --> 00:18:02,545 the deep insight is if we see complexity out there in the world its likely because 247 00:18:02,545 --> 00:18:06,935 people's behavior or the rules that things are following, are interdependent. Okay, 248 00:18:06,935 --> 00:18:07,477 thank you.