The Mandelbrot Set

what up ding dollars my name is Michael and today we will be listing off the 8 breeds of cats that might be able to learn to play the banjo number 5 might surprise you this is Michael's toys thank you for tuning in to the dog channel it has been a wild ride this year I've had a lot of fun I think last year in total I made like 4 videos but this year because of dog I've made like more than 20 we're talking relativity Morse code homonyms density balls I built every strictly convex Delta hydrant I also listed off prime numbers for three hours that was a treat for me and I really hope it was for you now should those videos all have gone on Vsauce 1 maybe I don't feel like they were big enough but they were certainly fun and without the dog channel I wouldn't have made them so thanks for tuning in here exciting news is that minefield is out on Vsauce 1 right now and for this season I was able to convince YouTube to make two of this season's episodes free for everyone to view the first one about the cognitive trade-off hypothesis which I hope you've already seen and the 4th episode which is about the Stanford Prison Experiment pretty exciting stuff but we're here today to talk about the Mandelbrot set it is a famous set of complex numbers named after mathematician Benoit Mandelbrot I got together with our designer John laser and we designed this poster it is a beautiful thing it imagines the Mandelbrot set as an island of finite land and infinite coasts it has all kinds of information about the set and it imagines all the different features of the set as actual geographic features point is what is the Mandelbrot set well today we're going to talk about that first of all you may have noticed that I said complex numbers what is a complex number well it's little bit complex just kidding it's not really we all know and love the number line it's a way to visualize all of the real numbers on a line a one-dimensional line like this where zero can be placed somewhere like that and then numbers like you know one might be here two might be here over here we might have negative one oh I can imagine three being there and four there this contains all real numbers not just the integers but all rational numbers for instance right here well that's gonna be a half well it's actually gonna be one and a half one point five the ratio of every circles circumference to its diameter in Euclidean space is about right now I don't know there that's PI 3.14159 it's a real number it's on here it's just one one times one is one but what isn't here well is a lot of stuff for example where is the square root of negative one where on this real number line is that number well here's the problem any real number times itself is positive negative 1 times negative 1 is just 1 negative 2 times negative 2 is 4 so the square root of negative 1 may not exist but mathematicians have found it very useful very powerful to say that the square root of negative 1 is a number it's a number called I I for Imaginary and probably someone said yeah but I is a letter not a number and they were like now you can look as hard as you want at the real number line you will never find I on that line but we can put I on the line by drawing an axis that is perpendicular to the real number line this way instead of saying well is I to the right or left of 0 we can just say how about I is above 0 there's I which means that the same distance in the other direction from 0 we have negative I and then right here we've got 2 odd three I and so on what we now have is not just a real number line but a complex number plane a complex number is a combination of a real number and a complex component that involves I actually every real number is a complex number here's an example of a complex number just to get us started if I pick any point on this plane it will be a complex number this one right here this is to the real number two plus I it's one up from zero this is 2 plus I a number like this one right here at that point that point is negative 1 that's the real component and the imaginary component is 3i negative 1 plus 3i is this number right there every number here is a complex number and it is these numbers that either belong or don't belong in the Mandelbrot set by picking all the numbers that belong and plot plotting them on a complex plane what we wind up with is a beautiful shape known as the Mandelbrot set visualize like this if you look at our poster this right here is the imaginary axis and this is the real axis how do you know if a number complex number belongs to the Mandelbrot set or not whether it is a piece of land on this island or if it's like out here in the ocean not part of the set I could spend a bunch of time talking about Julia sets and we could start more generally but I want to go right to Mandelbrot so let's talk about functions a function is just a mapping from numbers to some other numbers and the function that we're going to be using to determine whether a number belongs to the Mandelbrot set or not looks like this it's a function of a variable Z such that its value at Z equals Z squared plus C where C is any complex number C is the number that we have located on this plane and we want to know whether it belongs in the set or not what you have to do though is you have to start this with Z equal to zero and you need to iterate this function and if the value that we keep getting out just grows bigger and bigger without bound the number is not in the set if however the number sort of stays somewhat small and doesn't grow to infinity then the number belongs in the set let's do some examples we begin by applying this function to Z where Z equals zero and let's say that our complex number is going to be the number one how about that we'll just try this river one is the real component but there's this kind of invisible imaginary component because we can imagine that one is just equal to one plus zero times I anyway what is f of Z well we have to begin with Z equal to zero that's the rule for creating the Mandelbrot set Z squared zero squared all right and then we add 1 which is C in this case we want to know if one is in the set so 0 squared plus 1 perfect what is that equal well 0 squared is 0 plus 1 is 1 wonderful now we iterate this function meaning our result becomes our new value of Z so now Z is equal to 1 so what is this function at 1 well if if we need Z squared that means we're now using 1 squared and we're adding C which is 1 and the answer we get is 1 squared 1 plus 1 which is 2 now this is our value for Z 2 all right so what is f of 2 well that's 2 squared plus 1 which is 4 500 all right let's plug this one back in now Z is equal to 5 f of 5 equals 5 squared plus 1 5 squared is 25 plus 1 26 my gosh you guys this is just gonna keep getting bigger and bigger and bigger it is going to grow without bound and so one is not part of the Mandelbrot set but now let's try this for the complex number negative one which can also be thought of as negative one plus zero I now we begin with F of Z where Z equals zero zero squared alright cool and then we add C which is negative one so we're adding negative one what does this give us well zero squared is zero and then we're adding a negative one which means we're subtracting one so we get a negative one okay so now our new value of Z is negative one it originally was zero now it's negative one so what is f of negative one well it is negative one squared since we need to do the Z squared and then we need to of course add C which is in this case still negative one negative 1 squared is 1 plus negative 1 1 minus 1 equals 0 oh now I think you can already see what's about to happen we now need to plug in 0 as our next value of Z but we've already did that we already did that we know that 0 squared plus a negative 1 equals negative 1 we're going to oscillate back and forth between negative 1 and 0 negative 1 and 0 forever this will not grow without bound therefore I like that infinity symbol it probably annoyed you that I drew it like this I just did that for a kicks because on Michaels toys you never know what's gonna happen ok so we can see that because at negative 1 this function does not grow without bound as we iterate negative 1 is part of the Mandelbrot set and sure enough if we look on our little graph here not graph our map negative 1 which is actually the very center of the main disk here that certainly part of the Mandelbrot set now let me point out a couple of the features I love here right here there's a point at the real component is negative 3/4 and the imaginary component is zero this is a single point it is the only point that belongs to the set that is along this line that intersects the real axis right at negative three quarters that I call a point bridge it is an infinitely thin bridge perfect for brief romantic walks cuz it's just a single point okay anyway we've got other things like the seahorse coast beautiful seahorse shapes adorn this side of the main cardioid this is a cardioid this wonderful heart shape you can find cardioids all over for instance on some of the baby brats the baby brats are very similar looking to the entire Mandelbrot set the main disc okay we talked about that but on the seahorse coast side we also have this main disc spiral coastline it's a beautiful place I really wish it was real a really cool thing that this poster points out is that while the perimeter of Mandelbrot Island may be infinite but its area is not we're not exactly sure what its area is we do know that a circle can completely enclose the set that is centered at the origin with a radius of two but right now our best estimate for its area is let me see here about one point five zero six four eight four square units so there you go maybe someday we will find a way to find the exact area it may not be rational in fact it might not be I don't know boy it's so much more to learn such an inspiring poster really really love it thanks to John for helping me design this I love this poster if you want it well you got to subscribe to the curiosity box this poster comes in the latest box box ten we only have a few hundred left if you're a subscriber you got it or it's coming if you haven't subscribed we have a few left we always make a few thousand more then we have subscribers but the only way to be guaranteed that you will get our latest inventions and discoveries and favorite math and science objects you need to be a subscriber I am loving working on this box so far we have raised nearly $150,000 for all research it's awesome lots more facts on this poster a lot of cool things in his most recent box including this shirt which I wanted to wear in the episode but Jake took mine this is his which is a small and I'm not gonna try to fit into this just kidding this is me wearing the small hey you like that a Kraken Kraken mayor lunar park there's a moon of Saturn that has liquid hydrocarbon lakes that literally have waves they're only like a fraction of a centimeter large but I mean you could surf that right so we designed this shirt it's like a vintage national parks poster imagining that Kraken mayor the the one of the big lakes on the UM Titan Saturn's moon was a was a part which could actually happen in the future sometime I think this really tight shirt makes gravity stronger or it just makes me cooler it's like look at me I'm like I don't care sit the way I want also a puzzle now the inventor of this puzzle called it I think Foursquare I believe I got a copy I loved it and the point of course is to get these pieces out but after I sort of you know solved it and learned its Wiles I said woah we need to print ien on the bottom iea you see that why is that printed on the bottom of this puzzle it won't make any sense until you solve the puzzle and even then it might not I'm excited to see what you guys think about that puzzle there's obviously many more things in the box including oh boy I can't show you this actually this is a prototype of a product coming out in the next box box 11 can i I don't want to give away what it is I can do whatever I want but I choose not to that's nice this is the only clue I'll give you you hear that all right hopefully that titillated you I don't know I don't want to show it off because I think the final product is gonna be a little bit different but the point is that the Mandelbrot set is beautiful we actually had a giant version of this poster made to put up in our office and it's a beaut you can celebrate mathematics and new discoveries and put this up in your room or dorm room or keep it in your car for reference during when you're in traffic I don't know your life point is I do love you and I'm glad you are here watching today as always [Music]

Channel: D!NG

Views: 963,100

Rating: 4.9165559 out of 5

Keywords: vsauce, michael stevens, mandelbrot, mandelbrot set, benoit mandelbrot, math, maths, mathematics, fractals, education, learning, set theory, complex numbers, the number i, ding, d!ng, dingsauce

Id: MwjsO6aniig

Length: 15min 30sec (930 seconds)

Published: Wed Dec 12 2018

Reddit Comments

[u/MaxxBreak]

I've been obsessed with the Mandelbrot set and fractals like it for the past year. I just finished a Google map of the Mandelbrot set which can be viewed here: mandelmap.deadbeef.codes

Note, this is calculated and drawn in real time, it would have taken approximately 19,000TB and probably over a year to calculate and store all the tiles required to make up the map.

[u/CyborgJunkie]

For those that haven't seen it, here is what it looks like when you zoom in on the edge.

[u/Jaged1235]

I gotta say, the Mandelbrot set is one badass fucking fractal.

[u/MrDominu]

Michael here!

[u/XiPingTing]

Fun fact: The boundary of the Mandelbrot set is entirely connected (you could draw the whole thing without lifting your pen), but it is so squiggly that this boundary is actually two-dimensional.

Grossly oversimplifying the concept of fractal dimension: If you make a square x2 as big, its perimeter is x2 as big, but its area is 4x as big. If you make the Mandelbrot set to 2x as big, its area is 4 times as big but its perimeter is also 4x as big.

[u/jethro-cull]

Numberline contains all rational numbers

Okay

Pi is over here

Alright

​

[u/apirateiwasmeanttobe]

Made my first Mandelbrot on the c64. First frame over - 2-2i to 2+2i took hours to render.

[u/theY4Kman]

Whoa, that was an entertaining and educational product placement.

[u/GanZheng]

Very Knowledgeable Video. Thanks for sharing.