What's so special about the Mandelbrot Set? - Numberphile

maybe this is Mandelbrot set back to basics I need your help though can you think of a number 7 good can you square that number 49 can you do this while filming you've done it correctly so far okay next question you're gonna get upset Square 49 no okay good answer the thing is I don't really care about the answer what I'm caring about is an iteration I can't keep asking you to do the same thing again and again and if I ask you to pick a number and keep squaring it you already know what's gonna happen right it's gonna get big blowout is it square one okay okay good now square half sometimes it blows up and actually that happens when it's bigger than one and it becomes quite obvious when you try it but one doesn't blow up so once different and numbers less than one like 1/2 actually become a quarter when you square them and that becomes a 16th and so on but then numbers much less than one like minus six and that one's blown up as well so what I'm really talking about and what iteration is quite often talking about is when iterations are stable when they sort of head towards somewhere you can see and when they're not and it's best to see this on a diagram so have a look at my screen here you picked the number seven which is kind of off my screen here but these red arrows we're moving them around this is not the greatest diagram let me get this clear in this to start with it's just a line but the arrows are head in that way because when I square this number I'm moving around they go that way and they never come back much like your son did but if I go less than this number one they don't go that way anymore they go that way in fact they head towards zero and they're stable there so for the sake of moving it around it's kind of nice to see the difference is a really sharp line then it completely changes behavior but that way they're unstable that way they're stable stable unstable stable unstable they want stable unstable you get the idea in fact if you go negative they still stable their head positive because we need to go a number that's negatively get positive still head toward zero until you hit minus 1 and then unstable stable stable unstable stable now despite all my commentary it's not Maddox I think yet and if you know anything about the theory of iterations and the consequences of them or if you've watched any other video about the man that you know that we're not just talking about real numbers are your numbers on and the line I've just been showing you we're talking about numbers in two dimensions so complex numbers and I'm not gonna give you an introduction to complex numbers here because I don't think you need to hear it and there are other places you can get them Holly on the numberphile Channel has done excellent videos about how this happens but I want to show you the same comments that she's made on brown paper I'd like to show you moving here's my two-dimensional number line here is the number that you get to pick randomly we can still do it in one dimension just by sliding along here and the other dot that's moving around this tiny one is the square of the other dot so one squared is about one bigger than one it heads off that way lower than one goes that way but I can move off the line now and it has a weirdly circular flavor to it which I kind of like it's like chasing the number around it moves quicker than you expect and that's already quite pleasing if you're a nerd like me but what I really interested is the iteration if you keep doing the interruption square it square a-square we saw what happened on the real line in two dimensions it looks different so it looks like this so this line indicates if we start there it goes to there when you square it and then to there and there and so if I move this around the picture is kind of pleasing kind of wobbles around but it's always heading towards the center which I'll say is stable and there's a reason there's a circle on the screen because anytime I'm inside this circle it looks like it's stable even if it's kind of pretty as it moves but if I go near the edge of the circle it's less obviously stable it takes a lot longer to get to the center and then if I go outside the circle unstable stable unstable stable stable unstable stable this is a really nice demonstration of what complex numbers are helpful in the instead of your entire image of what's happening being on a line you've got picture but it gets better right instead of just squaring it what would happen if you did square it add something square it and something square add something now we can go back to the real line and see it happening on just one dimension but I feel like we should just go straight into two dimensions this is just the square square root square a-square if this letter C is indicating a number I could add every time so let me just slide it away from zero the moment it's not adding anything but if I add this number this red number which is a complex number is two dimensional you can see something very different has happened if I start with this number and I keep squaring it and adding that number square add that number square at that number Square add that number this is the path that goes and it makes a lovely star shape what's interesting though is not just a star shape although I kind of like it is the stability so all around here is stable it always gives you the same sort of stability like a starry stability even if it's expanding or shrinking but you kind of get the feeling that sometime it's gonna blow up and it's not obvious where because if I go near the circle you think that's kind of like it's gonna blow but doesn't oh so there with a moment there so outside the circle still stable but somewhere else outside the circle unstable and in fact it's not obvious where this happening stable unstable stable stable unstable no stables do and but over here inside the circle always stable and I go there I keep saying the circle as if it's important it's really not important or the boundary is clearly not circled but it's kind of nice to move this around I think that some places are unstable that some are stable what I'd really like to see and I hope you were too is the boundary of stability so let's see that in fact let's look at the boundary if the number I choose to add each time is 0 you get the circle that's like the simple iteration if you choose to add something different by the way I'm going I'm gonna write something down here what I'm capturing is this iteration which always looks a bit technical and if I cover that bit up it's what we did at the beginning I said square a number and that's your new number square it square it square it but instead of just where you square it add something there you see four it's a constant but it's a complex number now we're ready to look at that if C is zero you get a circle if it's not zero I see where my brain melts a little tight fit if C is not a zero the shapes become these shapes and I always find this a little bit of a shock so there may be some familiar looking shapes oh no sometimes now even connected together they're like dusty little particles and these regions particular when there's obviously a region which is not a circle necessarily they're called Julia sets and again Holly's done a really lovely video about Julia sets little bit of history about Julia sets them found by Gaston Julia hence the name who was a guy French mathematician I think early twentieth century he did it without a computer like he wasn't able to just do this and move on judge Abreu and see the boundary he was doing this by hand and realizing there was some beautiful structure like I mean every time I tell the story I'm like best play he also is really interesting in that he had no nose genuine you gonna find a picture of Gaston Julia he has a leather thing across his face his nose was damaged in I think the First World War before I leave Julia entirely the sets that you can see give you a little bit of a hint about what's coming so let me show you two things about the Julia says first of all the simplest one is a circle is not that exciting but they are exciting but they're not always obvious region sometimes they look like they're separate regions these don't look like they're connected the way I'm animating it here they're all slightly dusty looking that's just the way I'm making it happen quickly but sometimes they're definitely not joined together and sometimes they look like things you might recognize the outline it's looking vaguely like something I've hinted about this video may be looking like but there is something important that Holly mentions in some of her videos about which ones are joined together connected mathematically and which ones are disconnected separate like dust let me show you what they end up describing so if we go back to this diagram when I did the iteration we were originally starting by you picking a number and a squaring it what happened in 1979 is a guy called Benoit now is his first name didn't get better for me her name was Mandelbrot is very famous figure in mathematics these days but it was 1979 nineteen eighty it's kind of recently in the grand scale of mathematics and while I was like hmm I wonder what happens if I always start a zero instead of me saying pick a number I would say like pick a number as long as it's zero and then all I really care about is what's the constant I at each time and which ones of those give me stable and which ones don't now he didn't have a good computer he was doing it on an IBM working for IBM at the time and having to print out on a dot matrix or probably even pre-dawn matrix printer but we can kind of shortcut that so let me show you what it looks like always start a zero and move the constant around each time and if you do that all you'll see is sometimes it's stable like this and sometimes it's different stable I just really love this I'm just going to move around they're kind of predictable only at no predict all you get spirally things you get spider everything sometimes it's not stable it's unstable sometimes it's got like how many arms is that seven arms sometimes it's got three arms sometimes it's got something else and sometimes it's just rubbish squaring and adding these what they call these are the orbits of for every constant up pick which you mentioned I picked one arbitrarily happen to be up in the top left there this one in fact that's its orbit but any other concert you pick any one on the screen has an orbit and some of them are boring they're just gone and some of them are not they're stable and some of them are not quite sure what the hell's go excuse my English they're the the idea that maybe some points are special and someone not is what benoît mandelbrot first began to investigate other people were investigating it but it ended up with this idea getting his name on it so let me simulate what he first saw remember he didn't have a monitor on his computer he had a printer he like type some code in if you want a picture you print it like ten minutes later after you've sold all the printer problems which still occur weird if you that has got to go and look what he's like haven't have I broken the printer or haven't messed my code up or is what I'm seeing actually real and what he found was that what he saw the technicians got their fur and they were like he's got it wrong again and they cleaned up his images and took all this dog dust but they assumed were printer artifacts they're like there's this smudgy bit here tell you not matter be there we'll just like chop that off or teleprinter and he eventually had to go the technician and say my prints keep coming back different from what I expected like we're just cleaning them up he's like stop cleaning them I think what I'm trying to see is a weird messy dusty looking thing you know like oh you want Matt I don't know exactly what happened there's a really good book by James glide called chaos which kind of tells that story worth a read but I'm gonna simulate the dusty saw I'm sorry about the lack of brown paper but there's a reason for the color this is the same orbit thing I was showing you just now can you see there's a lock it's spiral III there but I'm even a black trail because I'm telling George Abra to color the screen black if it's stable and if it's unstable like that one it goes blue black blue stable unstable and I'm just gonna scribble on the screen now this is not gonna make a good picture but I wonder I want any kind of to feel what Mandelbrot felt like so he was seeing black and blue dots or in this case probably just white and black dots black for stable and my case blue for unstable and he started to wonder why there's a black region here this that's not over there it's not symmetrical I really think it must have felt something like this when you have a rubbish printer and you don't trust your code and you wonder what the hell are you saying so this is going to take ages right I'm going to shortcut the process what he eventually saw in higher resolution than he saw is this picture and the black regions like I was coloring earlier are stable you can see anywhere I go in the black region you get a nice stable pattern with with lovely orbits and that's the thing I love seeing move but if I go over the edge unstable stable unstable stable unstable so it's even stable in this region it goes into like two piles but if I go outside the region disappeared if I go up here you get into three piles and in fact Holly's done a lovely video on how many piles you start making depending on which of these little bulbs you go into but the first time the Mandelbrot saw this yet no idea about any of that structure he just saw a shape which shocked him first of all it's beautiful it's kind of weird but getting a viewer to display it like I've got on the screen now you can go and watch any number of YouTube videos and zoom in on it but you can see what Mandelbrot couldn't see that he saw this little blob and thought I wonder what that is and he had no idea until he wrote his code better but it was another version not quite the same and you can just zoom in forever and now I don't think YouTube needs more videos of me zooming in on the Mandelbrot set however I'm going to do it anyway so the original Mandelbrot set that Mandelbrot saw was two colors stable unstable black white or whatever I did black and blue the colors you're seeing in this are an arbitrary decision but they're not as arbitrary as you might think so let's go back to the original which is out here black for stable but the colors indicate that it's unstable but the color indicates how unstable and so if you think back to the very first question I asked you I said Square it square it square it and within two iterations you like no because it's big and that's an indication that you don't need to ask any more iterations to know what it's going to do with complex numbers it's less obvious what it's going to do is much harder to predict but eventually we proved mathematicians as I say we not me mathematicians proved that if you go outside a circle radius - it's never coming back but if you inside you can't guarantee it it might be bouncing around near the edge and it might stabilize again and you see some of the orbits are complicated so all you do with the computer is you ask it to check let's say 200 times and if you're still inside radius to circle off 200 you like let's color it black it's probably stable but after like 50 iterations if you've just crossed outside you like or now I know it's gone maybe the next time you check it takes sixty iterations and well that's less unstable maybe I could color that differently that was a long way of saying the different colors on the image are the different levels of instability really it's how many times you checked before you knew they were going to explode and what matters when you zoom in is that the colors change really really quickly so all the colors on the screen here indicating that a tiny movement of the original point gives you massively different behavior and that's become what we call the sort of hallmark of chaos theory now is where a tiny change gives you fundamentally different behavior on the matter what set your move your chain every point you made is depending on which scene you pick but Julia set you fix one and I can turn this software into Julia set mode I'll do it slightly first for press J on here there's the circle which is the original Julia set and that's in the middle here but if I move the mouse around you see the Julia sets with different values of C start changing beautifully like I mean it's just a quick sketch animation but it's already much more nice than the outlines you saw earlier and what I love about this is the Mandelbrot set is kind of like a map of Julia sets so if you go and find a bit of the Mandelbrot set down here this is called seahorse Valley because it kind of looks like you've got loads of seahorses spiraling around but if I switch to a Julia set around this point it looks the same but am i zoom out the Julia set you don't degenerate into Mandelbrot you just get the seahorse Julia set there it is that's as big as you get so if you zoom in on the Mandelbrot set you get little sort of like regions which are like the Julia set from that area it's like a map it's a geography or or vitter ative stability when you say it like that it sounds complicated but you just do something again and again and find out what happens in the long term and the fact that it makes something so beautiful I mean it's it's become so cliche from a petitions to get excited about but I still get excited about it because it's lovely I didn't design it it's just there we can explore it I mean just sittin stare hypnotize at it for hours and you can program it in ten seconds on a spreadsheet if you want so as you said it's not in the Mandelbrot set right and what that means so let's call this number say well I've already used C but let's call it C anyways

Channel: Numberphile

Views: 652,536

Rating: 4.9717703 out of 5

Keywords: numberphile, mandelbrot, julia set, mandelbrot set

Id: FFftmWSzgmk

Length: 16min 52sec (1012 seconds)

Published: Thu Apr 18 2019

Reddit Comments

[u/SisRob]

Great video! By the way, the app he uses for zooming is called XaoS. Mind blowing piece of software (especially if you ever tried to implement mandelbrot set renderer yourself)

[u/Shelleen]

I love zoomed in Julia sets, the weird spiral tentacles looks like you are peeking into another universe. I have no idea how many hours I've put into entering different values on my Amiga and waiting hours for the image to show up one pixel at a time.

[u/j0nthegreat]

that was a really cool video

[u/eaglessoar]

best video on youtube about mandelbrot sets: https://www.youtube.com/watch?v=AGUlJus5kpY

/s but man, this was like entertainment in college for me lol, i watched so many like 15 minute long deep zooms and other videos