1 00:00:02,370 --> 00:00:03,480 Welcome back. 2 00:00:03,480 --> 00:00:06,990 In this video we'll try and explore how to computationally 3 00:00:06,990 --> 00:00:10,770 encode implements of the elementary CA we were just 4 00:00:10,770 --> 00:00:12,450 talking about. 5 00:00:12,450 --> 00:00:15,450 So I'll go to a notebook that I'll be sharing with the group 6 00:00:17,220 --> 00:00:19,320 and we'll want to have some functions 7 00:00:19,320 --> 00:00:21,390 that decode and run cellular automata. 8 00:00:21,390 --> 00:00:24,660 I will be leveraging the binary representation 9 00:00:24,660 --> 00:00:27,120 that I talked about in the introduction video. 10 00:00:27,120 --> 00:00:29,580 So if you're not familiar with binary numbers, 11 00:00:29,580 --> 00:00:31,050 I invite you to pause this video 12 00:00:31,050 --> 00:00:34,230 and make sure to watch this week's bonus video 13 00:00:34,230 --> 00:00:36,090 on binary number. 14 00:00:36,090 --> 00:00:41,090 I'll give one example, this rule 130 function right here, 15 00:00:42,300 --> 00:00:47,130 which implements the CA in sort of a naive way 16 00:00:47,130 --> 00:00:49,980 so that we make the rules as explicit as possible. 17 00:00:49,980 --> 00:00:51,430 So let's start with this one. 18 00:00:52,860 --> 00:00:56,520 So I've written here the rules of the CA, 19 00:00:56,520 --> 00:00:58,920 so the current pattern of the index cell 20 00:00:58,920 --> 00:01:01,429 and its neighborhood, which can be 1 1 1, 21 00:01:01,429 --> 00:01:03,993 1 1 0, 1 0 1, and so on. 22 00:01:06,270 --> 00:01:09,960 And the state of that index cell a step into the future. 23 00:01:09,960 --> 00:01:13,590 So 0 0 0 for the first three rules. 24 00:01:13,590 --> 00:01:17,370 This is rule 30, right, so remember that this 0 0 0 25 00:01:17,370 --> 00:01:22,370 1 1 1 1 0, is the binary representation of rule 30. 26 00:01:23,130 --> 00:01:27,720 Right, so I have 2 plus 4 plus 8 plus 16 27 00:01:27,720 --> 00:01:30,153 plus 0 0 0 equals 30. 28 00:01:31,560 --> 00:01:34,890 And so what this rule says is that if I have 30, 29 00:01:34,890 --> 00:01:36,960 I know what my rule set is 30 00:01:36,960 --> 00:01:41,280 and I'll define this three dimensional array 31 00:01:41,280 --> 00:01:43,800 to help me know, well what are my neighbor to the left 32 00:01:43,800 --> 00:01:47,190 is 0 or 1, 0, I'm 0 or 1, 33 00:01:47,190 --> 00:01:49,020 my neighbor to the right is 0 or 1, 34 00:01:49,020 --> 00:01:50,310 it's a three dimensional array 35 00:01:50,310 --> 00:01:52,773 with only two cells in each dimension. 36 00:01:53,670 --> 00:01:58,087 So really this cube that's gonna give me the new state 37 00:02:01,140 --> 00:02:04,623 of an index cell in a given state. 38 00:02:06,060 --> 00:02:08,910 Or in a given neighborhood, I should say. 39 00:02:08,910 --> 00:02:13,200 And so what we're gonna do is run this one dimensional CA 40 00:02:13,200 --> 00:02:16,290 but use a second dimension to encode time. 41 00:02:16,290 --> 00:02:18,750 Right, so think about a big matrix 42 00:02:18,750 --> 00:02:21,118 where the first row of that matrix 43 00:02:21,118 --> 00:02:23,487 represents your initial conditions, 44 00:02:23,487 --> 00:02:26,858 and the second row represent the update 45 00:02:26,858 --> 00:02:29,100 of those initial conditions 46 00:02:29,100 --> 00:02:30,600 and then the update of that update, 47 00:02:30,600 --> 00:02:31,980 the update of that update, 48 00:02:31,980 --> 00:02:36,870 and so time unfolds and unrolls as you look down the matrix, 49 00:02:36,870 --> 00:02:40,530 right, and it's just an easier way to represent CA 50 00:02:40,530 --> 00:02:43,443 than updating a vector in your face in real time. 51 00:02:44,340 --> 00:02:46,030 So what this code is gonna do 52 00:02:47,430 --> 00:02:50,220 is go over every time step that I'm asking it to do, 53 00:02:50,220 --> 00:02:53,910 so how many rows do I want in my matrix. 54 00:02:53,910 --> 00:02:57,003 For each of these rows it's gonna go over every cell, 55 00:02:58,200 --> 00:03:00,840 and it's gonna check the state of that cell 56 00:03:00,840 --> 00:03:02,880 and the state of its neighborhood. 57 00:03:02,880 --> 00:03:04,530 So who's to your left? 58 00:03:04,530 --> 00:03:07,230 Well, to the left of J is J - 1, 59 00:03:07,230 --> 00:03:11,520 right, to the left of index cell J is cell J - 1. 60 00:03:11,520 --> 00:03:13,560 I'm gonna have this module of function 61 00:03:13,560 --> 00:03:16,500 because if I'm looking to the left of cell 0, 62 00:03:16,500 --> 00:03:20,380 and I hit the minus one, I want to loop back around 63 00:03:21,360 --> 00:03:22,650 to the cell to the right. 64 00:03:22,650 --> 00:03:25,320 So what I'm saying is that I need to specify 65 00:03:25,320 --> 00:03:27,360 how I'm dealing with my boundaries, 66 00:03:27,360 --> 00:03:28,950 and we're gonna see that in the textbook, 67 00:03:28,950 --> 00:03:30,660 there are a lot of ways to do that. 68 00:03:30,660 --> 00:03:34,650 You could fix the state of those boundary cell, 69 00:03:34,650 --> 00:03:38,550 you could have special rules for those boundary cells, 70 00:03:38,550 --> 00:03:42,300 or in this case you could have periodic boundaries. 71 00:03:42,300 --> 00:03:45,540 So really we're saying that our vector is a ring 72 00:03:45,540 --> 00:03:48,303 on which our CA is updating. 73 00:03:49,410 --> 00:03:51,000 The center cell, it's easy, 74 00:03:51,000 --> 00:03:56,000 I don't really need a module here, it's simply index cell J. 75 00:03:56,070 --> 00:03:58,020 And then to the right is J + 1, 76 00:03:58,020 --> 00:04:00,765 I do need a modular operation here again, 77 00:04:00,765 --> 00:04:03,030 because my neighbor to the right 78 00:04:03,030 --> 00:04:04,380 could be the very first neighbor 79 00:04:04,380 --> 00:04:05,760 on the other side of the ring 80 00:04:05,760 --> 00:04:09,963 if I'm the last cell, if J is the last cell. 81 00:04:11,370 --> 00:04:13,800 And then we'll just apply the rule. 82 00:04:13,800 --> 00:04:18,570 So at rule I + 1, next generation for index cell J, 83 00:04:18,570 --> 00:04:19,900 we're gonna apply rule 84 00:04:20,880 --> 00:04:23,910 based on the state of neighbors to the left, 85 00:04:23,910 --> 00:04:27,093 the index cell in the center and neighbor to the right. 86 00:04:28,140 --> 00:04:31,200 Okay, so that's rule 30, I said, 87 00:04:31,200 --> 00:04:34,320 so rule 30 specifies the sequence of 0 or 1, 88 00:04:34,320 --> 00:04:36,570 the rules, the size of the rule set, 89 00:04:36,570 --> 00:04:38,010 and the rules themselves are set, 90 00:04:38,010 --> 00:04:41,580 it's just the outcome that is specified by number 30. 91 00:04:41,580 --> 00:04:42,910 So I'm gonna define that, 92 00:04:42,910 --> 00:04:47,910 and I'll scroll down now to this main code that runs the CA. 93 00:04:49,110 --> 00:04:50,280 So there are a few things, 94 00:04:50,280 --> 00:04:53,190 few parameters we still need to specify, right, 95 00:04:53,190 --> 00:04:54,856 rule is gonna be 30. 96 00:04:54,856 --> 00:04:57,270 And we need specify the size of our ring. 97 00:04:57,270 --> 00:04:59,133 So it's gonna be 64 cells. 98 00:05:00,600 --> 00:05:02,100 The code is a little more general 99 00:05:02,100 --> 00:05:03,480 and I invite you to play with it. 100 00:05:03,480 --> 00:05:05,839 You could specify the number of states, 101 00:05:05,839 --> 00:05:07,980 you could specify the number of neighbors. 102 00:05:07,980 --> 00:05:11,280 So the size of the cardinality of every rule. 103 00:05:11,280 --> 00:05:12,860 So here it's three because you need 104 00:05:12,860 --> 00:05:15,750 to specify the index cell in two neighbors. 105 00:05:15,750 --> 00:05:20,550 And then how many steps will the model run for? 106 00:05:20,550 --> 00:05:21,693 So here's 16. 107 00:05:22,950 --> 00:05:24,750 Initial conditions and we'll see 108 00:05:24,750 --> 00:05:26,280 that especially in two dimensions. 109 00:05:26,280 --> 00:05:30,063 But initial conditions are critical in the case of CA 110 00:05:30,063 --> 00:05:32,670 because really we don't have a lot of parameters. 111 00:05:32,670 --> 00:05:35,253 The rule set is pretty simple, it's deterministic 112 00:05:35,253 --> 00:05:39,030 but different initial condition will lead 113 00:05:39,030 --> 00:05:40,710 to a different time series. 114 00:05:40,710 --> 00:05:44,340 It is a deterministic model that's specified completely 115 00:05:44,340 --> 00:05:46,230 by its rules and initial conditions. 116 00:05:46,230 --> 00:05:48,660 So in this case, we'll keep it super simple. 117 00:05:48,660 --> 00:05:51,720 Every cell is off for generation one or 118 00:05:51,720 --> 00:05:53,280 for our initial conditions 119 00:05:53,280 --> 00:05:58,140 and we'll initialize on cell in the middle of our ring. 120 00:05:59,340 --> 00:06:01,680 So at n divided by two. 121 00:06:01,680 --> 00:06:05,520 If you haven't seen this double division symbol before, 122 00:06:05,520 --> 00:06:06,510 it's a division followed 123 00:06:06,510 --> 00:06:09,780 by a floor operation to make sure it's an integer. 124 00:06:09,780 --> 00:06:12,600 So if n was equal to 1, 125 00:06:12,600 --> 00:06:16,233 I would get 5.5425, right? 126 00:06:17,370 --> 00:06:21,180 Okay, and then I'll run my rule 30 127 00:06:21,180 --> 00:06:23,730 over that tape that's gonna give me this tape subject 128 00:06:23,730 --> 00:06:26,828 which is a matrix and I'll simply plot it 129 00:06:26,828 --> 00:06:29,073 in time to let it update. 130 00:06:31,590 --> 00:06:33,180 16 times steps, right? 131 00:06:33,180 --> 00:06:35,005 And I could go back here 132 00:06:35,005 --> 00:06:38,340 and ask for a little more generations 133 00:06:38,340 --> 00:06:43,020 and I slowly see rule 30 unfolding, right? 134 00:06:43,020 --> 00:06:45,000 And rule 30 I showed you a screenshot 135 00:06:45,000 --> 00:06:48,990 in introduction is a surprisingly interesting rule. 136 00:06:48,990 --> 00:06:51,690 So what's gonna be interesting is to 137 00:06:51,690 --> 00:06:53,310 go explore the other rules and see 138 00:06:53,310 --> 00:06:57,720 whether interesting behavior is the exception or the norm 139 00:06:57,720 --> 00:07:00,630 of these elementary CA. 140 00:07:00,630 --> 00:07:02,670 When you play with this, I invite you to look 141 00:07:02,670 --> 00:07:06,532 at this function at the bottom, which lets you 142 00:07:06,532 --> 00:07:10,440 visualize whatever rule you might be interested in. 143 00:07:10,440 --> 00:07:11,910 So you just specify the number. 144 00:07:11,910 --> 00:07:14,070 If you're interested only in rule 30, 145 00:07:14,070 --> 00:07:16,440 these are the rules we just implemented 146 00:07:16,440 --> 00:07:18,570 as you want to try different numbers, 147 00:07:18,570 --> 00:07:19,860 it might be interesting to be like, 148 00:07:19,860 --> 00:07:22,353 what was the set of rule 18 again? 149 00:07:25,140 --> 00:07:27,120 And just visualize them as you go 150 00:07:27,120 --> 00:07:30,780 because it can be quite hard just from the number to 151 00:07:30,780 --> 00:07:33,420 like decode binary in your head and figure out the rules. 152 00:07:33,420 --> 00:07:36,150 So visualizing them is always useful. 153 00:07:36,150 --> 00:07:38,520 These rule set are essentially, you know, the box 154 00:07:38,520 --> 00:07:40,680 and arrow diagrams of dynamical system. 155 00:07:40,680 --> 00:07:44,160 They specify from this state and neighborhood where 156 00:07:44,160 --> 00:07:48,930 do you go in a very deterministic way, but the box 157 00:07:48,930 --> 00:07:51,570 and arrow representation is still useful as a map 158 00:07:51,570 --> 00:07:52,473 of the system. 159 00:07:54,510 --> 00:07:58,410 Okay, so instead of running this run rule 30 function, 160 00:07:58,410 --> 00:08:01,740 I'm gonna run this general cellular automata cell 161 00:08:01,740 --> 00:08:06,060 auto function, which will allow me to go explore any rule 162 00:08:06,060 --> 00:08:07,620 that I'm interested in. 163 00:08:07,620 --> 00:08:09,723 And so what does this function look like? 164 00:08:11,850 --> 00:08:16,850 Well first of all, this run cell auto is going 165 00:08:18,930 --> 00:08:23,930 to need the rule set for a given number, right? 166 00:08:30,870 --> 00:08:34,530 So I simply give it a number and I say, run the rule set 167 00:08:34,530 --> 00:08:36,603 with number 20, right? 168 00:08:37,980 --> 00:08:41,553 Well, so I have this decode rule function here 169 00:08:44,190 --> 00:08:48,120 and this decode rule function is passed to run cell auto 170 00:08:48,120 --> 00:08:49,530 as a rule set. 171 00:08:49,530 --> 00:08:51,930 I got a little confused 'cause I thought I was defining that 172 00:08:51,930 --> 00:08:54,780 on another line, but it's the argument here. 173 00:08:54,780 --> 00:08:58,050 And so with this decode auto, this decode rule 174 00:08:58,050 --> 00:09:03,050 is gonna do is simply use this list of rules. 175 00:09:04,290 --> 00:09:07,380 So we need an order, we need to specify what is the order 176 00:09:07,380 --> 00:09:09,690 in which we're specifying rules when we're saying 177 00:09:09,690 --> 00:09:11,760 that the numbers to the right here 178 00:09:11,760 --> 00:09:15,586 some to run to number 30, okay? 179 00:09:15,586 --> 00:09:18,720 So the way that's gonna work is using the divide 180 00:09:18,720 --> 00:09:21,360 by two algorithm that you might be familiar with. 181 00:09:21,360 --> 00:09:23,460 So if you don't know what I'm talking about, 182 00:09:23,460 --> 00:09:24,990 I invite you to pause this, 183 00:09:24,990 --> 00:09:26,880 go watch the binary number video. 184 00:09:26,880 --> 00:09:28,530 But essentially it's an algorithm 185 00:09:29,820 --> 00:09:31,590 that takes a number, right? 186 00:09:31,590 --> 00:09:36,437 113 and decomposes it in its binary sequence. 187 00:09:37,560 --> 00:09:39,000 The key to the algorithm 188 00:09:39,000 --> 00:09:42,990 if you don't wanna watch the video is that the final digit 189 00:09:42,990 --> 00:09:46,800 of a binary number essentially encodes plus 0 plus 1. 190 00:09:46,800 --> 00:09:49,080 So it encodes whether you're even or odd number 191 00:09:49,080 --> 00:09:52,680 because all the other binary numbers are powers of 2. 192 00:09:52,680 --> 00:09:54,480 So they have to be even. 193 00:09:54,480 --> 00:09:55,350 And so what the divide 194 00:09:55,350 --> 00:10:00,120 by 2 rule said is are you currently odd or even? 195 00:10:00,120 --> 00:10:03,030 If you're odd, I know the final digit is a plus 1 196 00:10:03,030 --> 00:10:04,200 then it does this divide 197 00:10:04,200 --> 00:10:07,817 by 2 and 4 to go see whether you were 198 00:10:09,960 --> 00:10:11,790 in the bottom or or second half 199 00:10:11,790 --> 00:10:14,943 of the sequence and then repeats the operation iteratively. 200 00:10:17,130 --> 00:10:19,110 And that's how you can create your sequence 201 00:10:19,110 --> 00:10:20,370 of zeros and ones. 202 00:10:20,370 --> 00:10:24,065 So what this decode rule function is doing is simply filling 203 00:10:24,065 --> 00:10:27,900 up these rules by sequence of zeros and one using the divide 204 00:10:27,900 --> 00:10:29,430 by 2 algorithm. 205 00:10:29,430 --> 00:10:32,820 Great, so when I call my run cell auto 206 00:10:32,820 --> 00:10:36,270 with the decode rule, I give it the rule set automatically, 207 00:10:36,270 --> 00:10:38,940 really through the divide by 2 algorithm 208 00:10:38,940 --> 00:10:40,593 for a given rule number. 209 00:10:42,450 --> 00:10:46,201 Then what I need to do in this run cell auto function 210 00:10:46,201 --> 00:10:51,201 right here is the same thing as in the previous version. 211 00:10:51,600 --> 00:10:54,512 For every time step that I'm interested in, for every cell 212 00:10:54,512 --> 00:10:57,914 at every time step, I need to apply the rules. 213 00:10:57,914 --> 00:11:02,914 Well, now the tricky thing is the way I encoded these rules, 214 00:11:03,403 --> 00:11:08,403 I need to know essentially whether 1 0 0 is rule number one 215 00:11:11,910 --> 00:11:14,520 or rule number four in this case, right? 216 00:11:14,520 --> 00:11:17,430 I need to remember the hierarchy of that number. 217 00:11:17,430 --> 00:11:21,510 Well, so these are also encoded from 0 to 8 218 00:11:21,510 --> 00:11:23,790 by their binary sequence, right? 219 00:11:23,790 --> 00:11:27,660 1 1 1 is rule eight, 0 0 0 is rule zero 220 00:11:27,660 --> 00:11:32,660 or I should say this is rule seven and this is rule zero. 221 00:11:33,750 --> 00:11:35,010 So index at zero. 222 00:11:35,010 --> 00:11:39,990 And so all we have to do is ask what is the sequence 223 00:11:39,990 --> 00:11:42,360 of state of your neighborhoods? 224 00:11:42,360 --> 00:11:45,325 The neighbors to the left encodes, whether you know, 2 225 00:11:45,325 --> 00:11:48,840 to the 0 1, 2 to the 2 2 is included. 226 00:11:48,840 --> 00:11:49,680 So that's a four. 227 00:11:49,680 --> 00:11:53,850 If it's on middle neighbor, your own cell 228 00:11:53,850 --> 00:11:57,180 in the cell of interest J, that's a plus 2. 229 00:11:57,180 --> 00:11:58,440 If it's on neighbor to the right, 230 00:11:58,440 --> 00:12:00,000 that's a plus 1 if it's on. 231 00:12:00,000 --> 00:12:05,000 So by looking at the on and off state of neighbors 232 00:12:05,400 --> 00:12:07,680 to the left cells of interest 233 00:12:07,680 --> 00:12:10,200 and neighbors to the right, we can recreate and ask 234 00:12:10,200 --> 00:12:14,760 for rule four in the rule set or rule zero in the rule set. 235 00:12:14,760 --> 00:12:16,860 If you don't know what I was just referring 236 00:12:16,860 --> 00:12:20,340 to by saying like, you know, two to the zero, 237 00:12:20,340 --> 00:12:21,870 two to the one, two to the two, 238 00:12:21,870 --> 00:12:26,700 that's the encoding rule for binary numbers. 239 00:12:26,700 --> 00:12:29,943 And again, that's all in the bonus video for this week. 240 00:12:31,620 --> 00:12:33,750 And so once you have that, this is telling you 241 00:12:33,750 --> 00:12:38,220 whether given a certain neighborhood, cell J 242 00:12:38,220 --> 00:12:42,213 at generation i plus 1 should go to 0 or 2 1. 243 00:12:43,410 --> 00:12:45,390 And so let's run that and we can 244 00:12:45,390 --> 00:12:48,183 run a rule number one, in this case. 245 00:12:49,860 --> 00:12:51,870 We get this cyclic behavior, right? 246 00:12:51,870 --> 00:12:53,220 So we had really interesting behavior 247 00:12:53,220 --> 00:12:55,140 for rule 30 cyclic behavior. 248 00:12:55,140 --> 00:13:00,060 Now, really the classification that we did 249 00:13:00,060 --> 00:13:03,210 in module one for dynamical system, 250 00:13:03,210 --> 00:13:07,470 the idea of time series can be monotonic, cyclic, 251 00:13:07,470 --> 00:13:11,670 asymptotic, chaotic, chaos is a weird concept here 252 00:13:11,670 --> 00:13:15,467 in discrete space, but like, let's call it complexity. 253 00:13:15,467 --> 00:13:18,480 This classification will still apply. 254 00:13:18,480 --> 00:13:21,430 And so we can play with different rules 255 00:13:22,283 --> 00:13:25,020 and here we have a variation of what we just saw 256 00:13:25,020 --> 00:13:26,520 but with a drift to the right. 257 00:13:27,660 --> 00:13:30,993 So sort of a combination of cyclic and monotonic behavior. 258 00:13:34,458 --> 00:13:37,170 Some of the rules that are, whoops, some of the rules 259 00:13:37,170 --> 00:13:38,940 that are known to be interested, for example 260 00:13:38,940 --> 00:13:43,347 are rule 110, which creates a very regular pattern 261 00:13:48,330 --> 00:13:52,890 that you might recognize and that will bring us to the topic 262 00:13:52,890 --> 00:13:56,490 of fractals later this week. 263 00:13:56,490 --> 00:13:58,980 And I have the rules set here and other rules 264 00:13:58,980 --> 00:14:02,220 of interest could be 90, 184. 265 00:14:02,220 --> 00:14:04,320 So let's go check those out. 266 00:14:04,320 --> 00:14:05,973 That was a note to myself. 267 00:14:09,120 --> 00:14:11,598 Again, it's not too surprising 268 00:14:11,598 --> 00:14:14,550 to see these really interesting patterns emerge. 269 00:14:14,550 --> 00:14:16,737 And I say not too surprising, when you look at the rules, 270 00:14:16,737 --> 00:14:18,660 you have no idea what's gonna happen. 271 00:14:18,660 --> 00:14:22,320 This is no doubt emergent behavior, but this pattern 272 00:14:22,320 --> 00:14:27,240 of Sierpiński triangle really, well, it's not too surprising 273 00:14:27,240 --> 00:14:30,810 to sit emerge in a way because we're iteratively 274 00:14:30,810 --> 00:14:33,810 applying the same rules again and again and again. 275 00:14:33,810 --> 00:14:38,810 So there is this potential for self-similar patterns 276 00:14:39,060 --> 00:14:42,330 and you know, I don't wanna, maybe I've been looking at them 277 00:14:42,330 --> 00:14:44,070 for too long. 278 00:14:44,070 --> 00:14:45,990 This is still crazy to me. 279 00:14:45,990 --> 00:14:49,140 It's crazy to me that very simple rules 280 00:14:49,140 --> 00:14:52,740 like these can give rise to interesting behavior like that. 281 00:14:52,740 --> 00:14:55,551 And I say it in all videos this week. 282 00:14:55,551 --> 00:14:59,640 If anything, that's the main story behind CA. 283 00:14:59,640 --> 00:15:02,790 A rule 110 that we just saw 284 00:15:02,790 --> 00:15:06,840 for those interested can actually be shown, 285 00:15:06,840 --> 00:15:10,350 that rule set can be shown to be a touring machine. 286 00:15:10,350 --> 00:15:12,930 There's enough in the rule set, you know, 287 00:15:12,930 --> 00:15:15,780 to give you a default behavior of reading, obviously 288 00:15:15,780 --> 00:15:18,910 but you can also specify conditional statements 289 00:15:19,782 --> 00:15:23,340 and for loops and you know, stopping conditions. 290 00:15:23,340 --> 00:15:25,230 And so you can do computation 291 00:15:25,230 --> 00:15:28,260 with the CA if you have enough memory. 292 00:15:28,260 --> 00:15:30,600 And where's the memory for these weird computers? 293 00:15:30,600 --> 00:15:33,810 It's all in the initial condition, all in the tape. 294 00:15:33,810 --> 00:15:36,990 And so we'll explore that at the very end of this week. 295 00:15:36,990 --> 00:15:39,785 We have to be careful when we play with them 296 00:15:39,785 --> 00:15:43,890 if they're meant to be a model of reality 297 00:15:43,890 --> 00:15:46,680 to test different initial conditions, right? 298 00:15:46,680 --> 00:15:48,630 You don't want to put all your eggs in one basket. 299 00:15:48,630 --> 00:15:49,770 So the same way we have to play 300 00:15:49,770 --> 00:15:52,020 with parameters and dynamical systems, 301 00:15:52,020 --> 00:15:55,581 here we have to play with initial conditions quite a bit. 302 00:15:55,581 --> 00:15:56,880 But we also wanna make sure to 303 00:15:56,880 --> 00:16:00,150 distinguish the power of the rules 304 00:16:00,150 --> 00:16:01,830 with that of the rule set. 305 00:16:01,830 --> 00:16:04,174 So the behavior is set by both 306 00:16:04,174 --> 00:16:07,530 by the rules and the initial conditions. 307 00:16:07,530 --> 00:16:10,620 And the outcome is really set by everything here. 308 00:16:10,620 --> 00:16:13,680 So the analysis of these should go 309 00:16:13,680 --> 00:16:17,340 beyond just saying that this is an incredibly surprising 310 00:16:17,340 --> 00:16:18,810 and interesting pattern. 311 00:16:18,810 --> 00:16:22,023 And so we'll do that slowly over the next few weeks. 312 00:16:23,130 --> 00:16:24,510 But for now, I think that's enough. 313 00:16:24,510 --> 00:16:26,370 I invite you to play with this code 314 00:16:26,370 --> 00:16:28,503 and I'll see you in the next one.