## Subtitles section Play video

• The game is gonna involve three dots.

• They could be anywhere.

• I said I wasn't going to put them in a triangle

• but that's kind of difficult to avoid.

• Uh, it's not particularly meant to be a an equilateral triangle.

• Let's call this 'A', 'B', and 'C'

• And I'm going to put a starting point

• just randomly on the paper.

• And all we're going to do is roll

• this lovely dice...

• die? dice? (who cares?)

• ...to decide which point to go towards,

• and whatever point we choose

• we're gonna go half way.

• So let's call 'A' 1 or 2.

• If we get one or two, we'll go half way towards 'A'.

• And this is gonna be 3 or 4

• and C can be 5 or 6.

• Uh, so, I just rolled a 2, so I need to go halfway to the point that has been chosen at random

• which is A in this case.

• So, um, we could measure this.

• For the sake of speed I'm just gonna go roughly half way

• I think you could believe that, if you run a line there, we're gonna put a dot there.

• And we're going to keep doing this for a while.

• We're iterating, so we're repeating - we get the last result and work from that.

• But the start point could be anywhere. I could have put the start point outside of the triangle

• and I could still head towards one of these dots.

• Let's just see what happens when I keep rolling the dice.

• I've just rolled a two again, so from-- that was the dot I had before, and I've gone halfway from the previous dot.

• And that's my halfway point. So if I roll this one again...

• One! Uh, I wish I'd rigged this now.

• From this dot (the last dot I had) halfway towards the 1, so it's gonna-- we're just aiming towards 1.

• This is unlikely to keep going, but, I dunno. Place your bets now.

• Four! Four is the B dot, so I've got to go, kinda from the previous dot, halfway on this line is about there.

• And I've jumped a long way this time.

• One, back towards A, so we're going there. And I'm gonna just keep doing this.

• So, you got the idea I think now. From the previous dot, we go towards one of these.

• Six. Excellent, we've gone all the way.

• This was a dot-- kind of halfway over there.

• Six.

• One.

• Back up here...

• Two.

• Five. All the way down here.

• I'm going to do this a lot now, Brady. Maybe this would be worth speeding up.

• [music]

• Ah, I'm bored. [Laughs]

• But, it's not that exciting. The interesting thing that's happening is that it created kind of some lines,

• but basically I've just made a lot of dots on the paper.

• It's possible that something interesting would happen if I could do this quick enough

• that I wouldn't get so bored that I have to stop.

• Uh, but maybe we want a computer to do this.

• We have a problem with the computer though on random numbers.

• Like, uh, generating random numbers, uh, it's difficult when a computer has to do what it's told.

• But, let's, uh, park that issue for another video, perhaps.

• If I could get a computer to roll the dice for me and put the dot on,

• then we could see what happens quickly,

• and, let's do that.

• This screen has, just as we had,

• we've got three dots, they're not even in a nice equilateral triangle,

• they can be anywhere. This thing says 'trace point',

• it's just going to trace where that goes,

• and the computer is, with its magic, gonna choose A, B or C,

• and move that point halfway -- accurately this time.

• So you see that it's leaving a mark anywhere it goes.

• So it's gone to A several times in a row, jumped to B,

• and then back to A, and then to C.

• But, I mean, doing a running commentary on this for any longer than I have done

• is probably not going to make anyone's day.

• So let's speed it up, ah, go a little bit faster and you begin to see something similar to the

• structure I had. The outlines get defined quite well.

• But I'm just going to speed up a little bit more because the outcome of this,

• I genuinely think is slightly surprising, but also a little tiny bit frightening,

• because you start to wa-- wait... is that structure I'm seeing...

• am I imagining that?

• How is it dodging those little weird patches in the middle?

• Why is it never going in those?

• They're definitely triangles now.

• And then I start to see a shape which actually if you've done any sort of

• investigating bits of recreational maths, this is a familiar shape.

• It's full of triangles.

• It looks like that trace point, wherever it's going, which is definitely going randomly,

• I haven't rigged this, this is doing it differently every time we run it,

• but it is dodging those triangles and it's making

• a shape, which I think everyone can see now.

• This is called the Sierpenski Gasket. It turns up all over the place.

• This is a fractal thing you get, actually you can see now,

• I think we've got enough to see that the big triangle,

• uh, has a black triangle in the middle, and...

• three copies of itself around the outside of that black bit.

• But then each copy of itself has a black bit in the middle and three copies around itself.

• And that's what a fractal is, the self similar thing, you zoom in and you see a little copy of itself.

• But the fact this is genuinely coming from rolling a dice on a computer and doing it quickly...

• Makes me... sss-- I don't know, just slightly disturbed about reality.

• Brady: I would have thought every point... had-- had a chance of being filled at some stage,

• but there are lots of areas that will just never get touched.

• Right, and so I mean-- And you can rig it so, I-- I said you could start anywhere, right?

• So I could start with my point right in the middle.

• And then we definitely have a point right in the middle, in that black.

• But what happens if you iterate enough, it veers away from that.

• So, the long-term behavior is what's interesting, is that even if you start in a black thing...

• the-- the random going half way moves you away from that black area and you...

• Actually, the technical term for this shape is called an attractor.

• Um, some people, you may have heard of these things called strange attractors in chaos theory.

• Well it's kind of advanced maths in it, but this shape is the attraction and it's also just, it's just pretty.

• The fact that random behavior produces something which is very structured, it is a nice outcome.

• Well, we had three points. I mean, we could have four, we could have five, we could, pick a number, um,

• uh, probably a whole number, otherwise counting the points is difficult.

• We could have four points and say, well, let's roll a dice, or a four-sided dice (a d4) to go to one of those.

• What pattern would you get? You could say, instead of going half way, go a third of the way,

• or 3/4 the way, what will happen?

• Do any of them make patterns? Do they go chaotic?

• People have spent a long time fiddling with the rules.

• It's nice to find out for yourself, but there's one rule I'll show you which does something

• which nobody really expected. So I-- I'm not going to go into details. This is worth looking up.

• But I've got a red shape and a blue shape, and they're kind of just summarizing...

• two possible outcomes of the rule. Like...

• we'll start with a point and with some probability would go to one of the corners of the red triangle...

• um, but with a small probability will-- will flip over to the blue triangle.

• And every time you kind of roll a dice to see where to go...

• you-- you follow this slightly more complicated set of rules but just, they're just random.

• Like, you go here with a certain probability, and then you flip back to the red one with a certain probability.

• And...

• if you look up, uh, this particular rule, you get this picture after a while you start to see these

• green dots appearing in what looks like

• a very structured way in fact it becomes

• very obvious if you run it long enough

• that the picture you're getting looks

• like something real. It looks like a fern.

• That's crazy!

• This is called Barnes Lee's fern

• compare it with a real fern it's

• surprisingly accurate. This is a nice

• example of how you don't have to draw

• something natural by hand. All the

• information for this fern is captured in

• a few lines of probability like the

• rules of the game are the fern or are a

• picture of a fern. It's actually really

• useful if you want to program video game graphics.

• You don't want to draw every tree,

• every fern. So if you can just kind

• of store all the trees and ferns as a rule

• and say whene- whenever you want a fern you

• just do that for a while. That is a much

• easier way to sort of ca... And lots of

• natural things turn out to have this fractal

• structure, self similar things which

• means you can generate them by iterating stuff.

• Genuinely good graphics coming

• from the fractal generated graphics

• rather than hand-painted. To get the

• structure of a tree on repeat if you

• just add one tree and you copied it,

• it becomes really obvious when you're

• running around the forest like it's just

• a computer forest. But if it looks,

• has this natural sort of variation with

• some randomness in it, you get much more realistic looking

• trees and ferns and things.

• Our thanks to The Great Courses Plus for making this extra video possible.

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• Go to thegreatcoursesplus.com/numberphile to check them out.

• There's also a link in the video description.

• I'd recommend this one about the Tibetan Plateau and the Himalayas,

• bit of an interest of mine. It's got some really cool stuff

• about how mountains are formed and, would you believe it,

• it's part of a 36 part series of lectures

• called "The World's Greatest Geological Wonders".

• Have a look at that, that's seriously interesting stuff.

• or the link in the description to start you off with that free trial.

The game is gonna involve three dots.

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B1 fern triangle halfway point probability fractal

# Chaos Game - Numberphile

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林宜悉 posted on 2020/03/27
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