Hi. In this lecture we're going to talk about something called The Game of Life.
Now this is a very simple model of aggregation. Now before I turn to the Game
of Life, I want to preface this lecture a little bit by placing it in context. So
remember why are we taking this course? Well, one to be a more intelligent citizen
of the world, to just understand what's going on around us. Two, to be clear and
better thinkers. Three, to use and understand data and four, to better, you
know design, strategize and decide. So What is, what are we doing here with the
Game of Life? The Game of Life is a very simple model that shows how things
aggregate and it gives us a lot of surprising conclusions. Now, it's a toy
model. It's very simple. It's not really about anything. It's not about climate
change. It's not about the financial system. It's not about eradicating
poverty. It's a model that sort of shows us how complicated aggregation can be. So
the way you want to think of this model is the way if you, it was on piano that you
think about sort of learning your scales or something like that. Or if you play
basketball like I do, you know, practicing your dribbling. This is a model that helps
us practice our thinking, to learn the subtleties of aggregate. [inaudible] Which
can be amazing about The Game Of Life as well as the one dimension cellular
automaton models that we study next, because we're going to see how really
complicated the process of aggregation is. And so when we then go out and look at the
world which involves lots of aggregation, we'll have some deeper appreciation for
why is it that it's so hard to infer by looking at the macro level what's going on
at the micro level, right. And that's one of the things that we saw in Shelling's
model and now we're going to see it, in sort of a more extreme form, in The Game
Of Life. Okay. The Game of Life was developed by a mathematician actually not
that long ago. This was by John Conroy. He's a Cambridge mathematician. He's a
brilliant mathematician who's work has been in group theory and this was just a
game he came up with using just a go board which is a big, you know, grid,
rectangular grid, and has little white and black stones that you place on the board.
So The Game of Life works a lot like Shelling's model. Each cell, right, like
this cell right here, has eight neighbors. And cells can be either alive, which we'll
color dark, or dead, or alternatively on. Or off. And so on is going to be dark. Off
will be light. Now, the rules to The Game of Life are fairly straightforward. If
you're currently off, you can only come on if exactly three of your neighbors are on.
So need exactly three of the people around you to be on. So this cell that is
currently off wouldn't come to life because it only has two neighbors on. If
you're currently on, if there's fewer than two neighbors on, so only zero or one, you
die of boredom because there's nothing going on. Just turn off. If you have more
than three neighbors on, you suffocate because there's too many people around and
they, they're using too many resources, you die off. But if there's two or three
neighbors that are alive, then you can stay alive. So let's formalize that. The
rules are quite simple, right? Cells are either on or off. If you're currently off,
you turn on if exactly three neighbors are on. That's the rule. And if you're
currently on, you can stay on if you've got two or three neighbors on. Okay? So
off, it requires three, on, two or three. Okay? All right. So if you look at this
particular cell here. Write x in the center. What you get is it has three
neighbors. One, two, three. That are on. So that means, the next period we're gonna
get it. It's gonna turn on. Right. So if you look at that cell. The next time.
We'll assume these other ones also stayed on. It's gonna turn on. Alright. Now if
you look at the same cell in this picture, now it has one, two, three, four neighbors
that are alive, so what's going to happen is it's going to turn off, okay. So now
looking at The Game of Life, it's not looking at just individual cells. We can
look at entire configurations of cells. So now here's a starting pattern. So as I see
the world with these two cells, if I look at the one on the left, it has no
neighbors on. And if I look at the one on the right, it has no neighbors on. So
what's going to happen is if I see the world like this, it's just going to end up
dead. Right. Nothing's gonna happen. 'Kay. Now suppose I see the [inaudible] with
three in the row. Well, let's look first at this person on the left. Ian has one
neighbor on. The person in the center has two neighbors on. And the person on the
right has one neighbor on. So what that means is. These two cells on the left.
Right, they're gonna die off. They're gonna turn off. But these and the one on
the right is gonna turn off, but the one in the center is gonna stay alive. But now
there's two other cells we got to worry about, right? Look at this one right here
just on the top. It has three neighbors that are alive. One, two, three as does
this one, one, two, three. So those two right, are gonna come to life. And so if
we let this system, if we let this go in the next period, what we're going to see
is the original one in the center stayed on, right. The one above stayed on and the
one below stayed on. So we now have three in a row that look just like this. Now,
let's let it go one more period. What's going to happen? Well again, as before,
this one in the center is going to stay alive. The one on the top and the bottom
will die off because they have one live neighbor. But now the ones on the left and
the right, right, this one right here and this one right here, they'll come to life
because they each have three live neighbors, okay. All right, so what's
gonna happen is it's gonna go like this. Well now let's let time run. Once we're
like this it'll go like that and once we're like that it'll go like this, and so
what we get is we get a blinker. Right? So. The game of life is interesting
because we started out with these simple rules, right? If you've got, if you're o,
on and two or three of your neighbors are on you stay on, and if you're off you only
come to life if exactly three neighbors are on and what we see is those micro
level rules can create macro level patterns, right, like blinkers. Okay.
Well, lets start with something else. Let's start with something a little, a
little more complicated. Let's look at this one. If we start here, this person,
this cell has two on, this cell has two on, this cell has two on, this cell has
one on. Now if we look around, we can see this cell has three, right? And this cell
has three, and there's no others that have three. [inaudible] happens the next
period, we're gonna get a picture of the [inaudible] like that. So, the game of
life not only can create die off and create blinkers. It can also have systems
that sort of grow. So one of the things we wanna do is we wanna sort of try and
understand. Okay, how does, what can the game of life produce? Well, let's look at
some classic examples. And we'll do this using a program called Net Logo. We're
gonna look at three things. We're gonna look first at the Beacon, which is two
squares of size four. And then we're gonna look at something I call the figure eight,
which is two squares of size nine. And then were gonna look at something called
the F memento, which is just a line of three, with one on the right, and then one
below it on the left, okay? So we're gonna look at these three configurations. But
instead of doing it by hand, 'cause that takes a long time, we're gonna use that
same Net Logo program that we used for [inaudible] model. Okay. So first we're
gonna do the beacon. And what we do there is, we do remember, gonna draw this cells,
we're gonna draw one. Two. One. 2,3,4, and then next to it we've got 1,2,3,4. And now
we can think okay let's press this go once button and what happens is it goes like
that and then it goes like that. Now if you look at the individual rules and
figured out what each cell was to do you'd figure out this is what's gonna happen.
We'll let this go forever. Let me slow this down a little bit. Right, and what we
see is, we get this nice little beacon flashing back and forth. Okay, this is a
lot like the little blinker we had before, where it was going, you know up and down,
sideways, vertical, horizontal, vertical, horizontal and, you know that's a nice
little picture. So let's stop it and now let's make this a little more interesting
and let's draw some more cells and let's make this thing three by three cells. That
one's off one. Here we go. So now these things are, 3x3 blocks and we'll see what
happens here. Okay, now again, each cell gets following those rules from The Game
of Life. So let's let it go once, twice, three times, four times, five times, six
times, seven times, eight times. Okay, that's unbelievable, right? This looks
like an eight on each side, so if you put your head at an angle, it looks like a
figure eight. And if you watch this thing, it's due to him. One, two, three, four,
five, six, seven, eight. So what's really cool with the game of life is that very
simple structures can create these elaborate patterns. And, again, each cell
is only following its own simple rules. What we learn from this is that
aggregation, like, simple things following simple rules can aggregate to form really
complex patterns, okay. So that's. The game of, that's the figure eight. Now
let's do the thing I call the F-famento, numbers that was three things in a row.
Like this. And then one in the center. Off to the left, and one off to the right. Now
let's go slowly through this. It's going to go once, twice, three, four, five, six,
seven, eight, nine. It seems to be taking on a life of its own. Let's let it go.
Forever. And what you see is it's producing things that are like little
gliders. Right, so it's producing things that move out through space, right? So
these things are. It's like almost like it's alive. Now you can start thinking
about some really interesting things. Like think about the human brain, right. The
human brain has these neurons that follow simple rules, and by following these
simple rules, these things are connected in such ways that they can create these
really novel patterns that produce things like memory and thought and cognition and
personality and all that sort of stuff. Well The Game of Life obviously doesn't
explain cognition or anything of this sort, but what it does do is it shows how
simple things following simple rules can create incredibly elaborate patterns.
Remember, because we start out, let's just do it. One more time. We start with an
incredibly simple thing right. We have one, two, three in a row. One up on top
one to the left and then when you watch this thing unfold when I click this each
time what we see is this incredibly elaborate pattern and I'll just go and
show you again right you see this really interesting thing including these things
that glide across the space. They are known as gliders. Here we've got a picture
of, a picture of sentences of simple gliders so here's a time zero and if dealt
in the rows and tables what happens. In the next period where you can use it to
look at sort of each individual cell and you can say okay well, what's gonna
happen. When we look at this cell right here, right. Which is right here in this
thing. It has one, two, three neighbors so it comes to life next time. So if we
follow that through, here's what happens, here's where time T equals zero, and at
time T equals one it looks like this. At time T equals two, it looks like this
because this cell which came to life is now going to be dead because it only has
one live neighbor, right. And then if you follow it around to T equals three and T
equals four, you find that T equals four looks exactly like the look at T equals
one accept for it's moved one cell down. To the right. So this, if I start with
this configuration it's just gonna glide across the space. So this is again, this
example we call an emersion or self-organized patter because this thing
looks like it's moving. If you watch a movie of this particular starting point,
you'll see something just glides across the space. So you might think this thing
is actually flying, but it's not. What's going on is each one of those individual
cells is following a particular rule. Okay. So here's what's really interesting
and about the game of life and one of the reasons why we construct models is to
understand the class of outcome what do we get, right do we get fixed points right.
It is the whose system you sort to get one thing, does it alternate, does it link
right and we saw both those things in the game of life, is it completely random
right, which was see down here, where do we get these complex patterns and now the
interesting thing is been shown in the game of life can give you all four of
these and we saw three of them right we saw that systems that died off and systems
that alternated and systems that were been. You can also think of ESP as almost
completely random, so. You can use it to generate random numbers. So it's
interesting, is very simple rules can aggregate to form, also it's a interesting
macro level phenomenon. Now, one of the things that people gonna ask is what,
what's the limit of what you can get answers. Almost nothing. Anything you
could do with the computer, you can actually do with the game of life, which
is sort of amazing. Here [inaudible] to a configuration, if you want you can plug it
in to Netlogo [inaudible] or text book of work, you got to draw these cells. This is
a glider gun. So what this thing will do is it will pause and it will send out
gliders. And often this lower right direction so what you'll get is a single
just like pulse almost like a heart beat sending out gliders it's really
interesting cuz you got again each cell is just following it's own those same rules
those game of life rules and you're getting this really elaborate pattern.
Okay, so what do we learn from the game of life? A bunch of things. One is, we
[inaudible] what we call self-organization. So these patterns
appear without a designer. So you get these gliders, you get these things that
blink. You get these glider guns. You get all sorts of things. No one designs that
from above. It's individual cells following individual rules, that when
they're placed in certain configurations, they produce these certain patterns. The
patterns appear to self organize. There's also what we call emergence. Now,
emergence means, when those patterns have some sort of functionality. So a glider, a
glider gun, a counter. So you can [inaudible]. Actually counts things,
right, or you can even use it to compute things if you interpret what those cells
mean. So, when those patterns have a functionality we can think of that as
being emergent. So you can think of things like consciousness and cognition, as
being emergent phenomena. Okay, and the game of life produces both these patterns,
and these patterns that it seem to have, can be interpreted as having functions.
The other cool thing about the game of life, right, is it helped us get the logic
right, right. Without running down the model, without running in the computer,
we'd never be able to figure out all the stuff that's going on. And we see how
we can get really complex things from really simple parts. So that's something
that, logically, you might not have anticipated. So, like, I'm sure when I
started this lecture and said, these are these simple rules, you might have gone,
this isn't going to be very interesting, right? And these are just a bunch of
simple rules and it's a checkerboard. But then we see all the amazing stuff that can
come out of the game of life, you start realizing, like, wow, okay. Simple rules
can produce incredible phenomena. That's something that I might not have known, had
I not constructed a simple model. Okay, so that's the game of life, it's a, [cough],
a game that belongs to a class of models called cellular automaton models. Now it's
just one, just one cellular automaton model. What we're gonna do next, in the
next lecture is look at a whole class of even simplier cellular automaton models,
to try and get an understanding of what causes this system, remember that
question, like, what, why does this system got equilibrium, why is it complex, we're
gonna make it a whole class of cellular automaton models to try and get at least
some understanding of why that might be the case. Thank you.
