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  • 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.

  • Two. Your dice is rigged Brady.

  • 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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  • 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

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

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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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