1 00:00:02,540 --> 00:00:03,990 Welcome back. 2 00:00:03,990 --> 00:00:07,620 This video is gonna have somewhat of a different flavor 3 00:00:07,620 --> 00:00:09,000 from the last one. 4 00:00:09,000 --> 00:00:11,040 I thought it'd be a good idea 5 00:00:11,040 --> 00:00:13,350 to explore the concept of fractals, 6 00:00:13,350 --> 00:00:15,723 which has popped up quite a bit in the videos, 7 00:00:17,910 --> 00:00:22,770 if only because rule 30, 110, 90 8 00:00:22,770 --> 00:00:26,460 gave us really interesting spatial structure, 9 00:00:26,460 --> 00:00:31,460 which have this self-similar structure. 10 00:00:31,680 --> 00:00:35,520 So we saw, you know, triangles in those cases, 11 00:00:35,520 --> 00:00:37,020 repeating at all scale. 12 00:00:37,020 --> 00:00:38,250 And I use the word fractal 13 00:00:38,250 --> 00:00:41,850 because I know much like chaos in week five, 14 00:00:41,850 --> 00:00:44,220 it is a word that we hear quite a bit 15 00:00:44,220 --> 00:00:46,200 and that not all of us are familiar 16 00:00:46,200 --> 00:00:48,030 with the formal definition, right? 17 00:00:48,030 --> 00:00:51,750 So chaos is often used to mean something random, 18 00:00:51,750 --> 00:00:52,740 and that's not what it is. 19 00:00:52,740 --> 00:00:55,590 We saw in week five that chaos 20 00:00:55,590 --> 00:00:57,870 implied a deterministic system 21 00:00:57,870 --> 00:01:00,450 and mainly sensitivity to initial conditions 22 00:01:00,450 --> 00:01:01,980 and ergodicity. 23 00:01:01,980 --> 00:01:04,890 And then we saw that there are ways to measure chaos. 24 00:01:04,890 --> 00:01:08,760 It's not just this black box of madness. 25 00:01:08,760 --> 00:01:11,580 We could quantify how chaotic a system was 26 00:01:11,580 --> 00:01:13,890 using the Lyapunov exponent. 27 00:01:13,890 --> 00:01:16,440 So the exponential rate of separation 28 00:01:16,440 --> 00:01:19,713 of two initial conditions that start very close together. 29 00:01:21,330 --> 00:01:23,850 Well, what is something similar for fractality 30 00:01:23,850 --> 00:01:26,030 if we wanna measure how interesting 31 00:01:26,030 --> 00:01:29,940 the outcome of rule 9110 were 32 00:01:29,940 --> 00:01:33,510 in comparison to the outcome of rule two or three 33 00:01:33,510 --> 00:01:35,853 or some of the more boring ones? 34 00:01:37,710 --> 00:01:41,220 Well, it's kind of interesting to think of 35 00:01:41,220 --> 00:01:42,840 these different approaches, 36 00:01:42,840 --> 00:01:44,670 including the one we're gonna see now 37 00:01:44,670 --> 00:01:46,650 as measures of complexities, 38 00:01:46,650 --> 00:01:49,440 measures of complexity, right? 39 00:01:49,440 --> 00:01:52,680 The Lyapunov exponent or measure of complexity 40 00:01:52,680 --> 00:01:57,570 and sensitivity in time of deterministic dynamical system. 41 00:01:57,570 --> 00:01:59,610 And right now, what I wanna explore, 42 00:01:59,610 --> 00:02:01,880 and I'll switch to my board, 43 00:02:01,880 --> 00:02:05,940 is the fractal dimension of an object, 44 00:02:05,940 --> 00:02:09,760 which you could argue is a measure of complexity 45 00:02:09,760 --> 00:02:12,840 in space of that object, right? 46 00:02:12,840 --> 00:02:16,290 With standard one-dimensional, two-dimensional objects, 47 00:02:16,290 --> 00:02:18,840 three-dimensional objects being simple 48 00:02:18,840 --> 00:02:23,840 and everything else being more interesting, complex, right? 49 00:02:24,060 --> 00:02:27,240 And it's kind of a nice turn of event 50 00:02:27,240 --> 00:02:32,240 that when you think of dimensionality as a spectrum, 51 00:02:32,700 --> 00:02:34,890 one, two, three are the exception 52 00:02:34,890 --> 00:02:39,780 and everything in between those fractal dimensions, 53 00:02:39,780 --> 00:02:41,760 we're gonna call more interesting, right? 54 00:02:41,760 --> 00:02:45,810 So if anything, remember that fractal structure 55 00:02:45,810 --> 00:02:47,880 implies a fractal dimension 56 00:02:47,880 --> 00:02:50,220 and that its dimensionality of that object 57 00:02:50,220 --> 00:02:55,220 is 1.5 or 0.5 or 2.15, right? 58 00:02:55,950 --> 00:02:58,833 Just a non-integer dimension. 59 00:02:59,760 --> 00:03:01,800 Really what it refers to 60 00:03:01,800 --> 00:03:05,040 is the way that objects occupy space. 61 00:03:05,040 --> 00:03:06,600 And so we're gonna measure 62 00:03:06,600 --> 00:03:08,220 the way an object occupy the space 63 00:03:08,220 --> 00:03:13,220 using an algorithm or a method called box counting. 64 00:03:13,260 --> 00:03:18,260 And so let's start with very simple objects. 65 00:03:20,620 --> 00:03:22,383 For example, 66 00:03:28,080 --> 00:03:29,130 a grid, right? 67 00:03:29,130 --> 00:03:33,270 So this could be the floor of your room, 68 00:03:33,270 --> 00:03:36,023 if you have a tile floor or whatever it is. 69 00:03:37,290 --> 00:03:40,817 And we wanna measure the dimensionality of an object. 70 00:03:40,817 --> 00:03:42,430 And so what we're gonna do 71 00:03:43,460 --> 00:03:46,710 is cover that object in boxes. 72 00:03:46,710 --> 00:03:49,770 And just to help myself here, 73 00:03:49,770 --> 00:03:54,770 I'm gonna add one row and one column 74 00:03:55,400 --> 00:04:00,400 to my small object to make it fit more nicely in boxes. 75 00:04:04,220 --> 00:04:08,370 And so we could start by covering it 76 00:04:08,370 --> 00:04:11,880 with boxes of size three. 77 00:04:11,880 --> 00:04:13,260 I say size three 78 00:04:13,260 --> 00:04:18,090 because they occupy three rows and columns 79 00:04:18,090 --> 00:04:21,513 in the grid of the paper I'm working with, right? 80 00:04:27,000 --> 00:04:28,860 I'm gonna call that R equal three 81 00:04:28,860 --> 00:04:31,323 for the length of the size of my boxes. 82 00:04:32,340 --> 00:04:35,860 And so here I would need four boxes 83 00:04:37,390 --> 00:04:40,800 to cover my object, to cover my floor. 84 00:04:40,800 --> 00:04:42,600 So what we're gonna be interested in 85 00:04:46,800 --> 00:04:50,680 is splitting the number of boxes needed 86 00:04:50,680 --> 00:04:53,793 as a function of the size of these boxes. 87 00:05:06,860 --> 00:05:10,023 So for boxes of size three, I needed four of them. 88 00:05:12,420 --> 00:05:14,130 Let me switch color here. 89 00:05:14,130 --> 00:05:17,193 We could use boxes of size two. 90 00:05:25,340 --> 00:05:27,450 It's getting a little messy. 91 00:05:27,450 --> 00:05:29,400 It's gonna get messier. 92 00:05:29,400 --> 00:05:31,610 And I needed nine boxes of size two. 93 00:05:31,610 --> 00:05:36,610 So that was R equal two. 94 00:05:43,650 --> 00:05:44,973 And I needed nine of them. 95 00:05:59,370 --> 00:06:02,223 Then we could get even smaller. 96 00:06:03,320 --> 00:06:05,913 I could get boxes of size one. 97 00:06:10,140 --> 00:06:11,880 And here really the point is 98 00:06:11,880 --> 00:06:15,180 we don't even need to do the math. 99 00:06:15,180 --> 00:06:19,290 We sort of know that these green boxes of size one, 100 00:06:19,290 --> 00:06:24,290 well, I can fit four of them in my red boxes of size two. 101 00:06:25,890 --> 00:06:28,480 And I can fit nine of them 102 00:06:28,480 --> 00:06:33,480 in my orange boxes of size three. 103 00:06:34,350 --> 00:06:38,360 And so I know that for each of the nine red boxes, 104 00:06:38,360 --> 00:06:41,180 I now have four green boxes. 105 00:06:41,180 --> 00:06:46,180 So I have nine times four or 36 way up there. 106 00:06:49,200 --> 00:06:51,660 And we sort of know that there is now 107 00:06:51,660 --> 00:06:54,210 this like squared relationship, right? 108 00:06:54,210 --> 00:06:59,210 How many boxes of size two can I fit in boxes of size four? 109 00:07:01,280 --> 00:07:04,590 You can do the math and do two dimension 110 00:07:04,590 --> 00:07:06,480 and get this relationship 111 00:07:06,480 --> 00:07:11,480 where you expect N to go as R to the minus two. 112 00:07:14,220 --> 00:07:16,680 Because if you shrink your boxes by half, 113 00:07:16,680 --> 00:07:18,880 you're gonna need four times more boxes 114 00:07:18,880 --> 00:07:21,423 to cover the same two-dimensional space. 115 00:07:23,160 --> 00:07:25,080 Here's an even simpler object 116 00:07:25,080 --> 00:07:29,723 where that behavior should be easier 117 00:07:31,140 --> 00:07:34,923 or more intuitive to interpret. 118 00:07:36,360 --> 00:07:38,673 One dimensional object. 119 00:07:40,050 --> 00:07:42,093 And we're gonna play the same game. 120 00:07:44,400 --> 00:07:46,630 So if I think boxes of size three 121 00:07:50,970 --> 00:07:54,090 and I made them too big vertically, 122 00:07:54,090 --> 00:07:55,170 but that doesn't matter. 123 00:07:55,170 --> 00:07:57,210 And this is actually a good example. 124 00:07:57,210 --> 00:07:58,590 If there's some object, 125 00:07:58,590 --> 00:08:00,900 some piece of the object remaining 126 00:08:00,900 --> 00:08:02,520 in the box counting method, 127 00:08:02,520 --> 00:08:06,780 you cover it anyway with the box. 128 00:08:06,780 --> 00:08:08,380 So I needed four boxes 129 00:08:14,310 --> 00:08:15,730 of size three, 130 00:08:15,730 --> 00:08:18,310 just like my previous object. 131 00:08:18,310 --> 00:08:19,790 So just like the floor, 132 00:08:19,790 --> 00:08:22,863 four boxes of size three to cover my line. 133 00:08:25,480 --> 00:08:30,003 Now I move to the boxes of size two, 134 00:08:34,080 --> 00:08:37,530 three, four, five. 135 00:08:37,530 --> 00:08:39,040 So five instead of nine. 136 00:08:39,040 --> 00:08:40,983 And for the previous object, 137 00:08:44,520 --> 00:08:46,110 four boxes of size two. 138 00:08:46,110 --> 00:08:51,110 And then I moved to boxes of size one, 139 00:08:52,980 --> 00:08:56,373 one, two, three, four. 140 00:08:57,510 --> 00:08:58,770 Hopefully I'm doing this right. 141 00:08:58,770 --> 00:09:02,973 Five, six, seven, eight, nine, and 10. 142 00:09:04,120 --> 00:09:05,223 Look at that. 143 00:09:06,120 --> 00:09:07,500 Is that surprising? 144 00:09:07,500 --> 00:09:09,063 Let's double it, 10. 145 00:09:12,240 --> 00:09:13,290 Is that surprising? 146 00:09:13,290 --> 00:09:15,480 If I needed five boxes of size two 147 00:09:15,480 --> 00:09:17,760 that I now need the 10 boxes of size one? 148 00:09:17,760 --> 00:09:21,090 No, because I'm covering a line. 149 00:09:21,090 --> 00:09:22,860 So if I shrink my boxes by half, 150 00:09:22,860 --> 00:09:25,530 I should need only twice as many boxes. 151 00:09:25,530 --> 00:09:28,020 In two dimension, I say only twice as many. 152 00:09:28,020 --> 00:09:29,490 Because in two dimension, 153 00:09:29,490 --> 00:09:32,490 it was shrink them by half, 154 00:09:32,490 --> 00:09:33,960 you need four times more 155 00:09:33,960 --> 00:09:36,930 because you need twice as many in both dimensions. 156 00:09:36,930 --> 00:09:40,240 And so really the intuition that I want you to have 157 00:09:41,590 --> 00:09:44,740 is that you can think of these relationships 158 00:09:45,630 --> 00:09:50,630 as a way to measure how an object occupies space. 159 00:09:54,240 --> 00:09:59,240 And so this N going as a radius of the box to the minus one, 160 00:10:02,400 --> 00:10:04,893 really that's saying like dimension one. 161 00:10:06,980 --> 00:10:10,773 And R to the minus two was saying dimension two. 162 00:10:13,890 --> 00:10:16,240 Right, the relationship. 163 00:10:16,240 --> 00:10:20,070 And I'm sorry, some of this might be covered by my video, 164 00:10:20,070 --> 00:10:22,353 mystery of online recording. 165 00:10:25,110 --> 00:10:29,760 But the key notion is that the number of boxes needed 166 00:10:29,760 --> 00:10:31,740 should go as the size of that box 167 00:10:31,740 --> 00:10:34,500 minus the dimensionality of that object. 168 00:10:34,500 --> 00:10:35,790 That's not gonna be perfect. 169 00:10:35,790 --> 00:10:37,890 If the boxes are too big, right? 170 00:10:37,890 --> 00:10:39,900 How many boxes of size 1 million 171 00:10:39,900 --> 00:10:41,160 do I need to cover the line? 172 00:10:41,160 --> 00:10:43,950 One, of size 1,000, one. 173 00:10:43,950 --> 00:10:45,810 Right, so if the boxes are too big, 174 00:10:45,810 --> 00:10:47,730 that relationship doesn't apply. 175 00:10:47,730 --> 00:10:51,840 If the boxes are too small and my object is pixelized, 176 00:10:51,840 --> 00:10:54,030 likewise, it's not gonna quite work. 177 00:10:54,030 --> 00:10:56,460 But there is a regime where that should capture 178 00:10:56,460 --> 00:10:58,590 how that object occupies space. 179 00:10:58,590 --> 00:11:03,270 And that regime hopefully can be expressed 180 00:11:03,270 --> 00:11:06,060 as a power law relationship of this form, 181 00:11:06,060 --> 00:11:09,060 number of boxes falling as a power law 182 00:11:09,060 --> 00:11:11,760 of the size of those boxes. 183 00:11:11,760 --> 00:11:14,010 And when I say size, I mean the side, 184 00:11:14,010 --> 00:11:16,230 the length of the side of those boxes 185 00:11:16,230 --> 00:11:18,663 and the dimensionality should come into play. 186 00:11:20,320 --> 00:11:22,890 Note that for the object in one dimension, 187 00:11:22,890 --> 00:11:24,450 you don't need to think of boxes. 188 00:11:24,450 --> 00:11:27,930 Those can just be like pieces of length 189 00:11:27,930 --> 00:11:29,790 that you put on your object. 190 00:11:29,790 --> 00:11:32,730 And if we were dealing with a three-dimensional object, 191 00:11:32,730 --> 00:11:37,580 we would be like covering it with cubes or actual boxes. 192 00:11:37,580 --> 00:11:42,090 The key part is that because those both dimension one 193 00:11:42,090 --> 00:11:43,890 and two are embedded in two dimensions, 194 00:11:43,890 --> 00:11:48,420 I was able to use like square boxes to cover it up. 195 00:11:48,420 --> 00:11:51,750 But that's a small detail really. 196 00:11:51,750 --> 00:11:54,100 We'll be dealing mostly with two-dimensional objects 197 00:11:54,100 --> 00:11:57,940 and you can think of covering them over pixels. 198 00:11:57,940 --> 00:12:02,010 So saying like, is my object within a box 199 00:12:02,010 --> 00:12:05,490 that starts at this point and end at this other point? 200 00:12:05,490 --> 00:12:07,980 Yes or no, do I keep that box? 201 00:12:07,980 --> 00:12:11,340 That should be the algorithmic view 202 00:12:11,340 --> 00:12:13,440 that you have the box counting method. 203 00:12:13,440 --> 00:12:17,420 So let's put that into action just as an example. 204 00:12:17,420 --> 00:12:19,860 So I showed you this picture before 205 00:12:19,860 --> 00:12:21,930 in the introduction to this week. 206 00:12:21,930 --> 00:12:24,483 This is an outcome of rule 30. 207 00:12:27,180 --> 00:12:28,743 It looks absolutely beautiful. 208 00:12:29,960 --> 00:12:32,700 And we might be interested in measuring 209 00:12:32,700 --> 00:12:34,173 its fractal dimension. 210 00:12:36,660 --> 00:12:40,563 And so we could play the same game and say, 211 00:12:41,880 --> 00:12:46,580 I'm gonna start with sides of dimension three. 212 00:12:46,580 --> 00:12:50,760 That's like one, two, three, one, two, three. 213 00:12:50,760 --> 00:12:54,273 And the game we would play is we start in a corner, 214 00:12:56,800 --> 00:13:01,800 we cover the entire space in these boxes. 215 00:13:04,230 --> 00:13:07,300 And I'm not gonna do the entire space. 216 00:13:07,300 --> 00:13:12,040 And then we go over every box and we ask our code, 217 00:13:12,040 --> 00:13:14,420 is the box in the bottom left corner 218 00:13:14,420 --> 00:13:17,280 necessary to cover my object? 219 00:13:17,280 --> 00:13:18,870 Yes, you keep it. 220 00:13:18,870 --> 00:13:22,470 Is the box above it necessary in covering my object? 221 00:13:22,470 --> 00:13:24,003 No, you remove it. 222 00:13:26,700 --> 00:13:30,033 So I keep the box in the corner, I don't need it. 223 00:13:31,200 --> 00:13:32,730 Is the box above it necessary? 224 00:13:32,730 --> 00:13:34,800 No, I remove that one as well. 225 00:13:34,800 --> 00:13:38,010 And so I end up only keeping the box 226 00:13:38,010 --> 00:13:41,790 that touch or cover my object, right? 227 00:13:41,790 --> 00:13:44,190 And so in this case, it's pretty simple. 228 00:13:44,190 --> 00:13:46,650 You keep everything underneath it 229 00:13:46,650 --> 00:13:48,993 and I'm just gonna do half of my object. 230 00:13:50,170 --> 00:13:53,313 Well, that one was a little too ambitious. 231 00:13:54,750 --> 00:13:56,400 One, two, three. 232 00:13:56,400 --> 00:13:57,233 Look at that, 233 00:14:02,050 --> 00:14:03,467 that one as well. 234 00:14:10,360 --> 00:14:13,300 Going a little fast, playing loose. 235 00:14:13,300 --> 00:14:15,570 You would have very regular box 236 00:14:15,570 --> 00:14:17,283 if you were doing this with code. 237 00:14:19,200 --> 00:14:23,310 Okay, that doesn't look right, but you get the idea. 238 00:14:23,310 --> 00:14:26,010 And we've getting rid of all the boxes 239 00:14:26,010 --> 00:14:27,960 that don't cover the space. 240 00:14:27,960 --> 00:14:29,730 Now, the important part is, 241 00:14:29,730 --> 00:14:32,580 as we make them much and much smaller, 242 00:14:32,580 --> 00:14:36,670 there will come a point where R will be equal, 243 00:14:36,670 --> 00:14:38,760 like three was like based on my grid, 244 00:14:38,760 --> 00:14:42,150 but the pixel size of my image is actually much smaller. 245 00:14:42,150 --> 00:14:46,370 So I could go to that, you know, 0.1. 246 00:14:48,710 --> 00:14:51,900 And then I'd be zooming in here, 247 00:14:51,900 --> 00:14:53,913 going for example in this box, 248 00:14:56,790 --> 00:14:59,313 and I would be covering this with 10 boxes. 249 00:15:03,150 --> 00:15:05,013 And it's not even close to 10 boxes. 250 00:15:06,810 --> 00:15:08,703 So like pretty small boxes. 251 00:15:10,030 --> 00:15:12,300 And you can imagine that you can end up 252 00:15:12,300 --> 00:15:17,250 with boxes that don't cover some of the inner details. 253 00:15:17,250 --> 00:15:19,830 If I move around, like find a big triangle 254 00:15:19,830 --> 00:15:22,200 and like you have a box that fits like this, 255 00:15:22,200 --> 00:15:23,550 it's not needed. 256 00:15:23,550 --> 00:15:25,830 So as you get smaller and smaller boxes, 257 00:15:25,830 --> 00:15:29,070 you're gonna start rejecting some of the inner boxes 258 00:15:29,070 --> 00:15:32,170 that's gonna let you capture the holes 259 00:15:32,170 --> 00:15:34,503 in the structure you're creating. 260 00:15:35,490 --> 00:15:37,930 And so we would like reject this one 261 00:15:37,930 --> 00:15:41,313 and we would end up maybe with boxes like this, 262 00:15:41,313 --> 00:15:44,943 like this, that like go around the object. 263 00:15:48,540 --> 00:15:51,150 So the way you should think of this really, 264 00:15:51,150 --> 00:15:53,730 I like the one example case a little better 265 00:15:53,730 --> 00:15:55,860 because I think they're more intuitive, 266 00:15:55,860 --> 00:16:00,240 is think of it and we'll do that reading as well. 267 00:16:00,240 --> 00:16:03,480 Think of it as measuring the coast of a country, right? 268 00:16:03,480 --> 00:16:06,000 And that comes from a great paper 269 00:16:06,000 --> 00:16:08,430 by Benoit Mandelbeau from the '60s. 270 00:16:08,430 --> 00:16:10,590 How long is the coast of Great Britain? 271 00:16:10,590 --> 00:16:12,810 Well, if you're measuring the coast of a country 272 00:16:12,810 --> 00:16:15,450 with sticks that are a mile long, 273 00:16:15,450 --> 00:16:17,130 well, as you go into bays, 274 00:16:17,130 --> 00:16:21,030 you'll only be able to capture so many details. 275 00:16:21,030 --> 00:16:23,910 Whereas when you make your stick smaller and smaller, 276 00:16:23,910 --> 00:16:26,700 you will be able to measure the coast and go into rivers. 277 00:16:26,700 --> 00:16:30,600 And so the answer you're gonna get is getting excited, 278 00:16:30,600 --> 00:16:33,420 but it's gonna depend on the measuring stick that you have. 279 00:16:33,420 --> 00:16:35,640 And here it depends on the boxes. 280 00:16:35,640 --> 00:16:39,300 And so for the coast, just like for this crazy objects, 281 00:16:39,300 --> 00:16:42,240 we expect that the number of measuring sticks 282 00:16:42,240 --> 00:16:44,610 or boxes that we need to cover an object 283 00:16:44,610 --> 00:16:47,640 should go as a certain power. 284 00:16:47,640 --> 00:16:51,030 And here it could be, it could be minus 1.5. 285 00:16:51,030 --> 00:16:53,730 This object, and I'll let you explore that on its own, 286 00:16:53,730 --> 00:16:56,520 but could have a fractal dimension 287 00:16:56,520 --> 00:17:00,960 which exists between dimension one and two, right? 288 00:17:00,960 --> 00:17:03,840 I know it's less than two because I can represent it 289 00:17:03,840 --> 00:17:06,060 and embed it in 2D, 290 00:17:06,060 --> 00:17:08,670 but I also know it probably less than two 291 00:17:08,670 --> 00:17:11,100 because you have some of these finer details 292 00:17:11,100 --> 00:17:13,440 where I'm gonna be rejecting more boxes 293 00:17:13,440 --> 00:17:16,143 than I would if it was a solid 2D objects. 294 00:17:17,430 --> 00:17:21,630 The question is, is it 1.5 or is it 1.99? 295 00:17:21,630 --> 00:17:23,880 You know, any object that you run on, 296 00:17:23,880 --> 00:17:27,180 like binarize a picture of your cat or your dog 297 00:17:27,180 --> 00:17:29,100 and run box covering method, 298 00:17:29,100 --> 00:17:31,710 you might find a fractal dimension. 299 00:17:31,710 --> 00:17:34,710 The question is whether it's robustly fractal 300 00:17:34,710 --> 00:17:37,590 and whether it's like 1.99 with a bunch of noise, 301 00:17:37,590 --> 00:17:39,660 depending on the size of the boxes. 302 00:17:39,660 --> 00:17:43,320 So it really all boils down to these pictures 303 00:17:43,320 --> 00:17:44,970 of how many boxes are needed 304 00:17:44,970 --> 00:17:48,243 and then trying to represent that as a power law. 305 00:17:50,910 --> 00:17:52,620 I don't wanna go into too much detail. 306 00:17:52,620 --> 00:17:54,300 I think it's an intuitive method 307 00:17:54,300 --> 00:17:57,410 and it's best if you just start implementing it on your own. 308 00:17:57,410 --> 00:18:00,600 And regardless of what you code in space, 309 00:18:00,600 --> 00:18:03,450 it's interesting to think about the dimensionality 310 00:18:03,450 --> 00:18:07,620 of traffic jams or forests 311 00:18:07,620 --> 00:18:11,730 or schools of fish, 312 00:18:11,730 --> 00:18:16,080 whatever it is that you use CA to represent in nature, 313 00:18:16,080 --> 00:18:18,470 you'll be able to think about the dimensionality 314 00:18:18,470 --> 00:18:19,650 of these subjects. 315 00:18:19,650 --> 00:18:22,890 Something we were just not able to do in module one. 316 00:18:22,890 --> 00:18:24,840 And I think that's really interesting. 317 00:18:24,840 --> 00:18:27,090 And so I'll let you play with this box counting method 318 00:18:27,090 --> 00:18:31,860 as you go and play around with creating some CA of your own. 319 00:18:31,860 --> 00:18:33,993 So see you in the next one.