This Fractal Is More Complex Than the Mandelbrot Set

[00:00] Well, that's a joke. I mean I'm squared, but it wasn't accidental. I don't know how to point at anything

[00:13] No time to explain now - I need to unsquare myself and very gradually I'll reduce that power until... Ah I lied again! I'm now square rooting myself.

[00:28] Up until now I've been always using the original pixels from every frame and then moving them I've moved them. I'm going to move them again,

[00:42] I'm going to repeat that again, and again, and again. And again. Look at me now, I'm a fractal! I'm the "Matt"-elbrot set. Wonderful fractal. Hugely overrated. [Mandelbrot set makes deflating sound]

[00:57] Forget it, we're going to get away from that. There are much better fractals out there and I was one of them. I was a Julia set, but I was the boring one - the one right in the middle.

[01:09] Instead, this time: square root myself, duplicate. But, before we do it again slide to the left; or the right, your left (I forget which way I did it in the code.) Slide somewhere, do it again. Slide, do it again. Repeat, repeat, repeat, repeat.

[01:26] Look at me now! It's Julia set Matt. Check that out. Uh here's a fun game, follow the mug. Where is it now? Could be anywhere. I apologise for making you see this.

[01:39] [Stand-Up Maths theme] but first we have to deal with the coordinates of where a pixel is in a video frame.

[01:52] It's measured from the top-left corner. Ridiculous. I'm gonna put a point on the screen in a moment - I don't know where because I'll run the code to put this in afterwards. Okay, I said I was going to mark a point on the screen. Argh! Did I say where? Ah, it doesn't matter. Put it anywhere.

[02:09] I assume I put it carefully in the, uh, top-left corner over here. (Ah, of course!) So once you've got a point with its coordinates  I can convert that into a more useful coordinate system. No! I mean, yes I'm going to do that

[02:25] but then I'm also going to turn them into a complex coordinate. is equal to two. And complex numbers, they're so great.

[02:37] You don't need to know about them for the purposes of this video, but the point is you can manipulate them like normal numbers. So, every coordinate in the frame has kind of just got one number instead of two, and if we square them, they move in an unusual way

[02:52] in the 2D space that is this frame; ah, so there. Here's a bunch of points - I square them. Look at them go! take its coordinates, turn them into a complex number, square that complex number,

[03:08] see where that corresponds to as a new location and then move the pixel from the original place to the new place. to this. To get a sense of complex squaring we can move complex number z around and you can see it's square, z-squared, over here

[03:25] move around correspondingly. And as you move z around you get all sorts of crazy behaviour. Now on the real number line that reverts to just regular square numbers. So if I cross two, two squared is four - that makes sense.

[03:39] as I go around the unit circle z-squared's getting away. It's all the way back over there. I have to do another lap with regular z to catch back up again. but whenever z crosses the unit circle - watch this -

[03:53] z-squared crosses at exactly the same time. Look at that. So anywhere out here, z-squared's further away from the origin than z is and that doesn't matter. Like we can be wildly apart, opposite sides and still when I come in this way they cross simultaneously.

[04:13] And then inside, doesn't matter where z is, z-squared's always closer to the origin. so inside the unit circle squaring  gets you towards zero and if you were to repeat this square and square and square and square and  square you'd end up right in the middle

[04:26] and then outside if you square it gets further away and  further away and further away. That's all you really need to know about squaring complex  numbers. Well, the Mandelbrot set is basically a  game of complex number sumo wrestling,

[04:41] where the point has to stay inside the circle despite being  repeatedly squared. Although, as we already saw, if you're already inside the circle and you keep  getting squared you just go to the middle.

[04:53] If you step outside though, then you fall away but what's  going to make you step outside? Well, your opponent is "plus c", that is a constant which is also a  complex number. So every time you're squared you get bumped by "plus c"

[05:08] the game now is: can you stay inside the circle? And whether or not you're successful entirely depends on what that "plus c" is. [Wilhelm scream as wrestler falls from ring]

[05:21] So in this game one wrestler is  at the origin, which we're going to have as a complex number, and here comes their opponent "plus c" equals 0.35 plus 0.21i. Now the very first move, when plus c comes into the ring, because squaring  zero does nothing;

[05:37] the first thing that will happen is a shove in the "plus c" direction, which is why we're now at 0.35, 0.21; we square that result, we add on plus c again, we move to a new location.

[05:49] Now, I'm not going to animate them moving every single time we update the position. Well, it just stays inside the circle. the other wrestler wins.

[06:03] Time for a new game with a new "plus c" opponent. And, actually, this time "plus" c will win. The point will get pushed far, far away from the arena. And I have cheated slightly by referring to the arena as a circle,

[06:17] it's actually a more complicated shape and that's  the shape we're working towards: it's the Mandelbrot set. It's all the points such that the game goes  one way or the other. And my mate, Ben Sparks, did a Numberphile video about this a while ago. In it he has a phenomenal GeoGebra file

[06:33] "Thanks Matt. The basic idea, as you know, is pretty simple: if you start at the origin and square that number, you'll get zero. But, if you add a constant then you'll get  that constant;

[06:47] you'll get that point, if you square that point and add the constant, you'll get that point. And if you keep going you'll follow this nice spirally path, in this case. Obviously depends which constant you add  each time and that's why I can drag the c around

[07:02] and you see the different paths start happening: so we get a tighter spiral, or we get different sort of spirals over here depending on what c you  use in iteration. But sometimes it's unstable, it doesn't come back. Stable - unstable. And that's what these splodges in the background are doing.

[07:16] Black for stable, blue for unstable - where you've kind of  pushed it out of the arena. Uh, and we can just find out which is which, so I'm just colouring in  the screen. I'm actually going to try and see if I can find the boundary between stable and unstable,

[07:29] Um, and, I mean, if I can go slowly you can see the  spirals, inside, and the unstable bits, outside. We're getting a nice little visualisation of a shape,

[07:43] which I think is familiar to a lot of  mathematicians, or at least it is now. But imagine being the first person to see this shape, this is The Mandelbrot Set. The points inside are  stable and the points outside are unstable:

[07:59] stable, unstable, stable, unstable, stable, unstable. Nice." It's called a set because it's the set of all the values of c, such that  you lose or win the complex sumo game.

[08:15] It depends slightly on how you define things and to explain  that, future Matt is going to write some more code to visualize it. I'm not going to do that, my code  is so slow this would take forever.

[08:27] Instead, check this out this is a very low-resolution Mandelbrot set, but there's a slider at the top there this is just for dramatic reveal: so as we reduce the grid size we get a better and better resolution Mandelbrot set.

[08:42] And I'm going to refer to this as a The Mandelbrot Set. I use the word set a lot; but I'm saying set  to mean all of the inside. If you're wondering where the colours come from, they're technically artificial.

[08:56] - You can add colours as a measure of how fast does the game end, like how fast can you get to the middle or how fast can you get thrown out of the complex sumo ring. whoever's made it has put them on there.

[09:11] In this case it's my friend, Christian Lawson Perfect, who coded this up such that it responds as I move that slider around. So, if that's The Mandelbrot Set, what are Julia sets?

[09:25] The Mandelbrot Set assumes you start at  the origin, at value zero, then you play the game. A Julia set is exactly the same idea, but you can  start anywhere you want on the complex plane.

[09:39] So, your player starts at some complex value; then you  play complex sumo see if you end up in or out of the circle eventually. So, this is the trivial Julia  Set, this is all the places where you can start

[09:53] and safely remain within the circle when you're  adding a "c" value of zero. And I can change the "c" value, thanks to Christian's incredible coding, by just moving the mouse around. So wherever the mouse is, that's the new starting "c" value

[10:08] and all the points you can see on the screen are the places where, if you start there, you'll stay within the circle forever. And I occasionally mix these up when I'm saying them because the Mandelbrot Set are all the "c" values for the one starting central point -

[10:24] as opposed to the Julia Set, which  is a fixed "c" value showing you all the starting points, Now, for some "c" values, you can see that all the starting points that are safe and you'll win the game

[10:36] are in one big connected, albeit a fractally, region; but for other "c" values, like out here (look!) They become these disconnected... Like, fractal dust. Oh, that! I love the ones here like the spiral effects - look at that, isn't that phenomenal;

[10:52] actually, if I zoom in a bit... Okay, I'm going to see if I can get that middle bit so we watch it join, so the two ends come in, scoop around. Oh! look at that isn't that phenomenal?

[11:06] I could spend hours exploring this and you can too - because we'll link it in the description below. And if you find any fun ones, as you can see in the bottom  lefthand corner, you can read off what "c" value you're looking at

[11:19] and you can then share those! So put them in the comments below and, other people, you can just type them in Ah! Julia Sets they're incredible! Exploring by moving the mouse around is a lot of  fun; but, what if we want to see a couple different Julia Sets at once and categorize them in a sensible way?

[11:36] Well, we could plot the pictures according to what their starting value of "z" is: we could then add on the starting positions +1 and -1,

[11:49] We could add in +i and -i, that works as well. so you've got "1 + i", "-1 + i" and so on... We could push that by going out  to two in each direction.

[12:04] We could increase how many we can see at once by - let's say we do the half values, [In fact, we can do loads! And if we want we can change the grid size to fit more in at once.

[12:16] Yes it's me again, in the future, messing with Christian's phenomenal code. I'm going to make the grid size smaller and smaller so we see more and more of them at once. We still got the boring circle in the middle but look at all these other interesting ones coming in,

[12:29] and if you make them incredibly small it's the Mandelbrot set! It's true: the limit of plotting  Julia sets according to their "c" values is, indeed, the Mandelbrot Set!

[12:46] But the Mandel Set, of course, was just a plot of "c" values. And it kind of makes sense because, if you're starting at the origin and  you're adding "c", So the Mandelbrot Set's, it's not all "the Julia Sets,"

[13:01] but basically; you know what, let's-let's increase the samples per grid, make it a bit bigger, let's actually fade across a bit and this is, um, actually the image I was showing you before...

[13:14] The Julia Sets are just the Mandelbrot Set that we're trying to get away from, In fact, I've squared myself!]

[13:26] Now, if these fractals are all about squaring a complex number; why, when I fractaled myself on camera at the beginning of the video did I use a square root? Why was I going in the other direction?

[13:41] Well, I was literally playing  the game of complex sumo in reverse! Because, to win the game, to be in the set, you have to stay inside  the unit circle.

[13:53] and then play the game backwards to see where you could have been the move before? Or the move before that? Or the move before that? And still ultimately end up in the circle.

[14:05] That's what I was doing. This is what it looks like if you just apply it to a circle. This video was partly inspired by a Matt Henderson  tweet, from 2020, that I saw where they've got this phenomenal animation showing, basically, that effect.

[14:20] I was like, "well I want to do that but I want to apply it to video footage." So I sat down and wrote  my own code, which is why it's terrible. Humankind have put a phenomenal amount of effort into this stuff because that's how we do visual effects

[14:36] I was doing it so inefficiently. I mean, there are shaders and all sorts of things you can use, and that's how Christian Lawson Perfect was able to make such a great interactive Julia set explorer -

[14:49] highly recommend you check that out. But I wanted to do it myself. Much like the sponsor of this video: Jane Street. They use mathematics to solve complex problems in The  Financial World,

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[15:56] They've been designed with people in mind and, between you and me, very mathsy people in mind. So if you're interested in finding out more - do check out the link in the description below.

[16:09] the many, many opportunities you could have to work at Jane Street. And huge thanks to Jane Street for sponsoring this,

[16:21] The video is over but I'm still here because there's one more cool thing that Christian made for me that I wanted to share with you. Before, I moved the mouse around the Julia Set itself to change the "c" value.

[16:35] Christian did a second one, where I moved the mouse around The Mandelbrot set and that changes the "c" value of the Julia Set. And that means you can see how the "c" value for each Julia Set And if I stay inside the Mandelbrot,

[16:50] if I go outside Mandelbrot, it's now infinitely many disconnected fractal bits of dust. and the moment I step back inside, even this ridiculous bit of the fractal set,

[17:07] But, because it's one of the ridiculous bits it's like a crazy connected... I've been meaning to make this video for a very long time five or six videos another video came  out covering similar topics

[17:21] But it's been, it's been several years since that came out (I'll link to it below, more great things about the  link between Julia Sets and The Mandelbrot Set, It's just so cool! I, like I say I really could spend ages exploring all of this.

[17:40] and a reminder I'm going to the  Edinburgh Festival Fringe doing all 23, or whatever, shows And then I'm on tour around the UK.

[17:52] A few people have asked, 'cause they've seen on some, but not all an upgrade to-to VIP, which is a bit of a  misnamed type of ticket. come in. watch the sound check when I'm setting up the stage.

[18:09] and you get some exclusive merch. And I'll do the kind of meet and greet for you before the show: So, if you're someone who wants to see the sound check and wants to leave straight after the show you can do that.

[18:26] I look forward to seeing you all there hopefully, we'll announce more dates soon. And finally, I'd like to apologise for starting the video with squared Matt as a jump scare. He's behind me, isn't he? "Hi!

[18:42] Why would you do that? "I want to say like and subscribe!" Just... Oh my goodness, haunting. Good! I'm sorry.