1 00:00:00,000 --> 00:00:05,698 Hi. In this lecture we're going to talk about something called The Game of Life. 2 00:00:05,698 --> 00:00:09,800 Now this is a very simple model of aggregation. Now before I turn to the Game 3 00:00:09,800 --> 00:00:14,062 of Life, I want to preface this lecture a little bit by placing it in context. So 4 00:00:14,062 --> 00:00:18,377 remember why are we taking this course? Well, one to be a more intelligent citizen 5 00:00:18,377 --> 00:00:22,533 of the world, to just understand what's going on around us. Two, to be clear and 6 00:00:22,533 --> 00:00:26,368 better thinkers. Three, to use and understand data and four, to better, you 7 00:00:26,368 --> 00:00:30,221 know design, strategize and decide. So What is, what are we doing here with the 8 00:00:30,221 --> 00:00:33,748 Game of Life? The Game of Life is a very simple model that shows how things 9 00:00:33,748 --> 00:00:37,180 aggregate and it gives us a lot of surprising conclusions. Now, it's a toy 10 00:00:37,180 --> 00:00:40,801 model. It's very simple. It's not really about anything. It's not about climate 11 00:00:40,801 --> 00:00:44,093 change. It's not about the financial system. It's not about eradicating 12 00:00:44,093 --> 00:00:47,902 poverty. It's a model that sort of shows us how complicated aggregation can be. So 13 00:00:47,902 --> 00:00:51,806 the way you want to think of this model is the way if you, it was on piano that you 14 00:00:51,806 --> 00:00:55,521 think about sort of learning your scales or something like that. Or if you play 15 00:00:55,521 --> 00:00:59,424 basketball like I do, you know, practicing your dribbling. This is a model that helps 16 00:00:59,424 --> 00:01:03,228 us practice our thinking, to learn the subtleties of aggregate. [inaudible] Which 17 00:01:03,228 --> 00:01:06,980 can be amazing about The Game Of Life as well as the one dimension cellular 18 00:01:06,980 --> 00:01:10,633 automaton models that we study next, because we're going to see how really 19 00:01:10,633 --> 00:01:14,829 complicated the process of aggregation is. And so when we then go out and look at the 20 00:01:14,829 --> 00:01:18,828 world which involves lots of aggregation, we'll have some deeper appreciation for 21 00:01:18,828 --> 00:01:22,925 why is it that it's so hard to infer by looking at the macro level what's going on 22 00:01:22,925 --> 00:01:26,875 at the micro level, right. And that's one of the things that we saw in Shelling's 23 00:01:26,875 --> 00:01:30,775 model and now we're going to see it, in sort of a more extreme form, in The Game 24 00:01:30,775 --> 00:01:35,128 Of Life. Okay. The Game of Life was developed by a mathematician actually not 25 00:01:35,128 --> 00:01:39,558 that long ago. This was by John Conroy. He's a Cambridge mathematician. He's a 26 00:01:39,558 --> 00:01:44,225 brilliant mathematician who's work has been in group theory and this was just a 27 00:01:44,225 --> 00:01:48,478 game he came up with using just a go board which is a big, you know, grid, 28 00:01:48,478 --> 00:01:53,382 rectangular grid, and has little white and black stones that you place on the board. 29 00:01:53,382 --> 00:01:57,812 So The Game of Life works a lot like Shelling's model. Each cell, right, like 30 00:01:57,812 --> 00:02:02,716 this cell right here, has eight neighbors. And cells can be either alive, which we'll 31 00:02:02,716 --> 00:02:07,262 color dark, or dead, or alternatively on. Or off. And so on is going to be dark. Off 32 00:02:07,262 --> 00:02:11,529 will be light. Now, the rules to The Game of Life are fairly straightforward. If 33 00:02:11,529 --> 00:02:16,124 you're currently off, you can only come on if exactly three of your neighbors are on. 34 00:02:16,124 --> 00:02:20,336 So need exactly three of the people around you to be on. So this cell that is 35 00:02:20,336 --> 00:02:24,494 currently off wouldn't come to life because it only has two neighbors on. If 36 00:02:24,494 --> 00:02:29,034 you're currently on, if there's fewer than two neighbors on, so only zero or one, you 37 00:02:29,034 --> 00:02:33,439 die of boredom because there's nothing going on. Just turn off. If you have more 38 00:02:33,439 --> 00:02:38,090 than three neighbors on, you suffocate because there's too many people around and 39 00:02:38,090 --> 00:02:42,568 they, they're using too many resources, you die off. But if there's two or three 40 00:02:42,568 --> 00:02:47,013 neighbors that are alive, then you can stay alive. So let's formalize that. The 41 00:02:47,013 --> 00:02:52,083 rules are quite simple, right? Cells are either on or off. If you're currently off, 42 00:02:52,083 --> 00:02:56,900 you turn on if exactly three neighbors are on. That's the rule. And if you're 43 00:02:56,900 --> 00:03:01,780 currently on, you can stay on if you've got two or three neighbors on. Okay? So 44 00:03:01,780 --> 00:03:06,464 off, it requires three, on, two or three. Okay? All right. So if you look at this 45 00:03:06,464 --> 00:03:10,610 particular cell here. Write x in the center. What you get is it has three 46 00:03:10,610 --> 00:03:15,101 neighbors. One, two, three. That are on. So that means, the next period we're gonna 47 00:03:15,101 --> 00:03:19,362 get it. It's gonna turn on. Right. So if you look at that cell. The next time. 48 00:03:19,362 --> 00:03:24,074 We'll assume these other ones also stayed on. It's gonna turn on. Alright. Now if 49 00:03:24,074 --> 00:03:28,747 you look at the same cell in this picture, now it has one, two, three, four neighbors 50 00:03:28,747 --> 00:03:33,249 that are alive, so what's going to happen is it's going to turn off, okay. So now 51 00:03:33,249 --> 00:03:37,637 looking at The Game of Life, it's not looking at just individual cells. We can 52 00:03:37,637 --> 00:03:42,424 look at entire configurations of cells. So now here's a starting pattern. So as I see 53 00:03:42,424 --> 00:03:46,641 the world with these two cells, if I look at the one on the left, it has no 54 00:03:46,641 --> 00:03:51,029 neighbors on. And if I look at the one on the right, it has no neighbors on. So 55 00:03:51,029 --> 00:03:55,645 what's going to happen is if I see the world like this, it's just going to end up 56 00:03:55,645 --> 00:04:00,743 dead. Right. Nothing's gonna happen. 'Kay. Now suppose I see the [inaudible] with 57 00:04:00,743 --> 00:04:05,575 three in the row. Well, let's look first at this person on the left. Ian has one 58 00:04:05,575 --> 00:04:10,531 neighbor on. The person in the center has two neighbors on. And the person on the 59 00:04:10,531 --> 00:04:15,240 right has one neighbor on. So what that means is. These two cells on the left. 60 00:04:15,240 --> 00:04:19,342 Right, they're gonna die off. They're gonna turn off. But these and the one on 61 00:04:19,342 --> 00:04:23,769 the right is gonna turn off, but the one in the center is gonna stay alive. But now 62 00:04:23,769 --> 00:04:28,088 there's two other cells we got to worry about, right? Look at this one right here 63 00:04:28,088 --> 00:04:32,191 just on the top. It has three neighbors that are alive. One, two, three as does 64 00:04:32,191 --> 00:04:36,230 this one, one, two, three. So those two right, are gonna come to life. And so if 65 00:04:36,230 --> 00:04:40,476 we let this system, if we let this go in the next period, what we're going to see 66 00:04:40,476 --> 00:04:44,827 is the original one in the center stayed on, right. The one above stayed on and the 67 00:04:44,827 --> 00:04:49,020 one below stayed on. So we now have three in a row that look just like this. Now, 68 00:04:49,020 --> 00:04:53,053 let's let it go one more period. What's going to happen? Well again, as before, 69 00:04:53,053 --> 00:04:57,299 this one in the center is going to stay alive. The one on the top and the bottom 70 00:04:57,299 --> 00:05:01,650 will die off because they have one live neighbor. But now the ones on the left and 71 00:05:01,650 --> 00:05:05,949 the right, right, this one right here and this one right here, they'll come to life 72 00:05:05,949 --> 00:05:09,742 because they each have three live neighbors, okay. All right, so what's 73 00:05:09,742 --> 00:05:14,452 gonna happen is it's gonna go like this. Well now let's let time run. Once we're 74 00:05:14,452 --> 00:05:19,222 like this it'll go like that and once we're like that it'll go like this, and so 75 00:05:19,222 --> 00:05:25,544 what we get is we get a blinker. Right? So. The game of life is interesting 76 00:05:25,544 --> 00:05:29,505 because we started out with these simple rules, right? If you've got, if you're o, 77 00:05:29,505 --> 00:05:33,717 on and two or three of your neighbors are on you stay on, and if you're off you only 78 00:05:33,717 --> 00:05:37,597 come to life if exactly three neighbors are on and what we see is those micro 79 00:05:37,597 --> 00:05:41,037 level rules can create macro level patterns, right, like blinkers. Okay. 80 00:05:41,037 --> 00:05:44,747 Well, lets start with something else. Let's start with something a little, a 81 00:05:44,747 --> 00:05:48,558 little more complicated. Let's look at this one. If we start here, this person, 82 00:05:48,558 --> 00:05:52,419 this cell has two on, this cell has two on, this cell has two on, this cell has 83 00:05:52,419 --> 00:05:56,570 one on. Now if we look around, we can see this cell has three, right? And this cell 84 00:05:56,570 --> 00:06:00,579 has three, and there's no others that have three. [inaudible] happens the next 85 00:06:00,579 --> 00:06:04,589 period, we're gonna get a picture of the [inaudible] like that. So, the game of 86 00:06:04,589 --> 00:06:08,650 life not only can create die off and create blinkers. It can also have systems 87 00:06:08,650 --> 00:06:12,763 that sort of grow. So one of the things we wanna do is we wanna sort of try and 88 00:06:12,763 --> 00:06:16,824 understand. Okay, how does, what can the game of life produce? Well, let's look at 89 00:06:16,824 --> 00:06:20,635 some classic examples. And we'll do this using a program called Net Logo. We're 90 00:06:20,635 --> 00:06:24,299 gonna look at three things. We're gonna look first at the Beacon, which is two 91 00:06:24,299 --> 00:06:28,297 squares of size four. And then we're gonna look at something I call the figure eight, 92 00:06:28,297 --> 00:06:32,057 which is two squares of size nine. And then were gonna look at something called 93 00:06:32,057 --> 00:06:35,817 the F memento, which is just a line of three, with one on the right, and then one 94 00:06:35,817 --> 00:06:39,624 below it on the left, okay? So we're gonna look at these three configurations. But 95 00:06:39,624 --> 00:06:43,337 instead of doing it by hand, 'cause that takes a long time, we're gonna use that 96 00:06:43,337 --> 00:06:47,036 same Net Logo program that we used for [inaudible] model. Okay. So first we're 97 00:06:47,036 --> 00:06:51,033 gonna do the beacon. And what we do there is, we do remember, gonna draw this cells, 98 00:06:51,033 --> 00:06:57,060 we're gonna draw one. Two. One. 2,3,4, and then next to it we've got 1,2,3,4. And now 99 00:06:57,060 --> 00:07:01,993 we can think okay let's press this go once button and what happens is it goes like 100 00:07:01,993 --> 00:07:06,510 that and then it goes like that. Now if you look at the individual rules and 101 00:07:06,510 --> 00:07:11,384 figured out what each cell was to do you'd figure out this is what's gonna happen. 102 00:07:11,384 --> 00:07:16,048 We'll let this go forever. Let me slow this down a little bit. Right, and what we 103 00:07:16,048 --> 00:07:20,489 see is, we get this nice little beacon flashing back and forth. Okay, this is a 104 00:07:20,489 --> 00:07:25,219 lot like the little blinker we had before, where it was going, you know up and down, 105 00:07:25,219 --> 00:07:29,602 sideways, vertical, horizontal, vertical, horizontal and, you know that's a nice 106 00:07:29,602 --> 00:07:34,332 little picture. So let's stop it and now let's make this a little more interesting 107 00:07:34,332 --> 00:07:39,573 and let's draw some more cells and let's make this thing three by three cells. That 108 00:07:39,573 --> 00:07:45,378 one's off one. Here we go. So now these things are, 3x3 blocks and we'll see what 109 00:07:45,378 --> 00:07:50,379 happens here. Okay, now again, each cell gets following those rules from The Game 110 00:07:50,379 --> 00:07:55,189 of Life. So let's let it go once, twice, three times, four times, five times, six 111 00:07:55,189 --> 00:07:59,746 times, seven times, eight times. Okay, that's unbelievable, right? This looks 112 00:07:59,746 --> 00:08:04,747 like an eight on each side, so if you put your head at an angle, it looks like a 113 00:08:04,747 --> 00:08:09,769 figure eight. And if you watch this thing, it's due to him. One, two, three, four, 114 00:08:09,769 --> 00:08:14,968 five, six, seven, eight. So what's really cool with the game of life is that very 115 00:08:14,968 --> 00:08:19,901 simple structures can create these elaborate patterns. And, again, each cell 116 00:08:19,901 --> 00:08:24,634 is only following its own simple rules. What we learn from this is that 117 00:08:24,634 --> 00:08:30,166 aggregation, like, simple things following simple rules can aggregate to form really 118 00:08:30,166 --> 00:08:35,401 complex patterns, okay. So that's. The game of, that's the figure eight. Now 119 00:08:35,401 --> 00:08:41,100 let's do the thing I call the F-famento, numbers that was three things in a row. 120 00:08:41,100 --> 00:08:47,617 Like this. And then one in the center. Off to the left, and one off to the right. Now 121 00:08:47,617 --> 00:08:52,319 let's go slowly through this. It's going to go once, twice, three, four, five, six, 122 00:08:52,319 --> 00:08:56,960 seven, eight, nine. It seems to be taking on a life of its own. Let's let it go. 123 00:08:56,960 --> 00:09:01,479 Forever. And what you see is it's producing things that are like little 124 00:09:01,479 --> 00:09:06,317 gliders. Right, so it's producing things that move out through space, right? So 125 00:09:06,317 --> 00:09:10,080 these things are. It's like almost like it's alive. Now you can start thinking 126 00:09:10,080 --> 00:09:13,653 about some really interesting things. Like think about the human brain, right. The 127 00:09:13,653 --> 00:09:17,137 human brain has these neurons that follow simple rules, and by following these 128 00:09:17,137 --> 00:09:20,711 simple rules, these things are connected in such ways that they can create these 129 00:09:20,711 --> 00:09:24,418 really novel patterns that produce things like memory and thought and cognition and 130 00:09:24,418 --> 00:09:27,947 personality and all that sort of stuff. Well The Game of Life obviously doesn't 131 00:09:27,947 --> 00:09:31,476 explain cognition or anything of this sort, but what it does do is it shows how 132 00:09:31,476 --> 00:09:34,960 simple things following simple rules can create incredibly elaborate patterns. 133 00:09:34,960 --> 00:09:38,982 Remember, because we start out, let's just do it. One more time. We start with an 134 00:09:38,982 --> 00:09:43,634 incredibly simple thing right. We have one, two, three in a row. One up on top 135 00:09:43,634 --> 00:09:48,658 one to the left and then when you watch this thing unfold when I click this each 136 00:09:48,658 --> 00:09:53,310 time what we see is this incredibly elaborate pattern and I'll just go and 137 00:09:53,310 --> 00:09:58,396 show you again right you see this really interesting thing including these things 138 00:09:58,396 --> 00:10:03,297 that glide across the space. They are known as gliders. Here we've got a picture 139 00:10:03,297 --> 00:10:08,197 of, a picture of sentences of simple gliders so here's a time zero and if dealt 140 00:10:08,197 --> 00:10:12,931 in the rows and tables what happens. In the next period where you can use it to 141 00:10:12,931 --> 00:10:17,351 look at sort of each individual cell and you can say okay well, what's gonna 142 00:10:17,351 --> 00:10:21,715 happen. When we look at this cell right here, right. Which is right here in this 143 00:10:21,715 --> 00:10:25,667 thing. It has one, two, three neighbors so it comes to life next time. So if we 144 00:10:25,667 --> 00:10:29,774 follow that through, here's what happens, here's where time T equals zero, and at 145 00:10:29,774 --> 00:10:33,778 time T equals one it looks like this. At time T equals two, it looks like this 146 00:10:33,778 --> 00:10:37,989 because this cell which came to life is now going to be dead because it only has 147 00:10:37,989 --> 00:10:42,200 one live neighbor, right. And then if you follow it around to T equals three and T 148 00:10:42,200 --> 00:10:46,360 equals four, you find that T equals four looks exactly like the look at T equals 149 00:10:46,360 --> 00:10:50,749 one accept for it's moved one cell down. To the right. So this, if I start with 150 00:10:50,749 --> 00:10:55,502 this configuration it's just gonna glide across the space. So this is again, this 151 00:10:55,502 --> 00:10:59,207 example we call an emersion or self-organized patter because this thing 152 00:10:59,207 --> 00:11:03,221 looks like it's moving. If you watch a movie of this particular starting point, 153 00:11:03,221 --> 00:11:07,338 you'll see something just glides across the space. So you might think this thing 154 00:11:07,338 --> 00:11:11,455 is actually flying, but it's not. What's going on is each one of those individual 155 00:11:11,455 --> 00:11:15,392 cells is following a particular rule. Okay. So here's what's really interesting 156 00:11:15,392 --> 00:11:19,359 and about the game of life and one of the reasons why we construct models is to 157 00:11:19,359 --> 00:11:23,425 understand the class of outcome what do we get, right do we get fixed points right. 158 00:11:23,425 --> 00:11:27,391 It is the whose system you sort to get one thing, does it alternate, does it link 159 00:11:27,391 --> 00:11:31,308 right and we saw both those things in the game of life, is it completely random 160 00:11:31,308 --> 00:11:35,324 right, which was see down here, where do we get these complex patterns and now the 161 00:11:35,324 --> 00:11:39,142 interesting thing is been shown in the game of life can give you all four of 162 00:11:39,142 --> 00:11:43,257 these and we saw three of them right we saw that systems that died off and systems 163 00:11:43,257 --> 00:11:47,916 that alternated and systems that were been. You can also think of ESP as almost 164 00:11:47,916 --> 00:11:51,982 completely random, so. You can use it to generate random numbers. So it's 165 00:11:51,982 --> 00:11:56,079 interesting, is very simple rules can aggregate to form, also it's a interesting 166 00:11:56,079 --> 00:11:59,968 macro level phenomenon. Now, one of the things that people gonna ask is what, 167 00:11:59,968 --> 00:12:03,753 what's the limit of what you can get answers. Almost nothing. Anything you 168 00:12:03,753 --> 00:12:07,642 could do with the computer, you can actually do with the game of life, which 169 00:12:07,642 --> 00:12:11,946 is sort of amazing. Here [inaudible] to a configuration, if you want you can plug it 170 00:12:11,946 --> 00:12:16,250 in to Netlogo [inaudible] or text book of work, you got to draw these cells. This is 171 00:12:16,250 --> 00:12:20,295 a glider gun. So what this thing will do is it will pause and it will send out 172 00:12:20,295 --> 00:12:23,923 gliders. And often this lower right direction so what you'll get is a single 173 00:12:23,923 --> 00:12:27,373 just like pulse almost like a heart beat sending out gliders it's really 174 00:12:27,373 --> 00:12:31,342 interesting cuz you got again each cell is just following it's own those same rules 175 00:12:31,342 --> 00:12:34,840 those game of life rules and you're getting this really elaborate pattern. 176 00:12:34,840 --> 00:12:38,396 Okay, so what do we learn from the game of life? A bunch of things. One is, we 177 00:12:38,396 --> 00:12:42,350 [inaudible] what we call self-organization. So these patterns 178 00:12:42,350 --> 00:12:46,120 appear without a designer. So you get these gliders, you get these things that 179 00:12:46,120 --> 00:12:50,357 blink. You get these glider guns. You get all sorts of things. No one designs that 180 00:12:50,357 --> 00:12:53,664 from above. It's individual cells following individual rules, that when 181 00:12:53,679 --> 00:12:57,185 they're placed in certain configurations, they produce these certain patterns. The 182 00:12:57,185 --> 00:13:01,882 patterns appear to self organize. There's also what we call emergence. Now, 183 00:13:01,882 --> 00:13:05,032 emergence means, when those patterns have some sort of functionality. So a glider, a 184 00:13:05,032 --> 00:13:09,930 glider gun, a counter. So you can [inaudible]. Actually counts things, 185 00:13:09,930 --> 00:13:13,540 right, or you can even use it to compute things if you interpret what those cells 186 00:13:13,540 --> 00:13:16,664 mean. So, when those patterns have a functionality we can think of that as 187 00:13:16,664 --> 00:13:20,594 being emergent. So you can think of things like consciousness and cognition, as 188 00:13:20,594 --> 00:13:23,894 being emergent phenomena. Okay, and the game of life produces both these patterns, 189 00:13:23,894 --> 00:13:27,902 and these patterns that it seem to have, can be interpreted as having functions. 190 00:13:27,902 --> 00:13:31,157 The other cool thing about the game of life, right, is it helped us get the logic 191 00:13:31,157 --> 00:13:34,036 right, right. Without running down the model, without running in the computer, 192 00:13:34,036 --> 00:13:38,941 we'd never be able to figure out all the stuff that's going on. And we see how 193 00:13:38,941 --> 00:13:43,258 we can get really complex things from really simple parts. So that's something 194 00:13:43,258 --> 00:13:45,262 that, logically, you might not have anticipated. So, like, I'm sure when I 195 00:13:45,262 --> 00:13:47,935 started this lecture and said, these are these simple rules, you might have gone, 196 00:13:47,935 --> 00:13:50,191 this isn't going to be very interesting, right? And these are just a bunch of 197 00:13:50,191 --> 00:13:54,104 simple rules and it's a checkerboard. But then we see all the amazing stuff that can 198 00:13:54,104 --> 00:13:57,329 come out of the game of life, you start realizing, like, wow, okay. Simple rules 199 00:13:57,329 --> 00:14:01,881 can produce incredible phenomena. That's something that I might not have known, had 200 00:14:01,881 --> 00:14:06,626 I not constructed a simple model. Okay, so that's the game of life, it's a, [cough], 201 00:14:06,626 --> 00:14:12,074 a game that belongs to a class of models called cellular automaton models. Now it's 202 00:14:12,074 --> 00:14:15,702 just one, just one cellular automaton model. What we're gonna do next, in the 203 00:14:15,702 --> 00:14:22,522 next lecture is look at a whole class of even simplier cellular automaton models, 204 00:14:22,522 --> 00:14:25,599 to try and get an understanding of what causes this system, remember that 205 00:14:25,599 --> 00:14:28,688 question, like, what, why does this system got equilibrium, why is it complex, we're 206 00:14:28,688 --> 00:14:31,536 gonna make it a whole class of cellular automaton models to try and get at least 207 00:14:31,536 --> 00:14:36,277 some understanding of why that might be the case. Thank you.