I used to be in a jazz band called The Mandelbrot Set. Our music was very simple yet very detailed. Audiences complained that our live sets seemed never-ending.
But having never ending live-sets would set you outside the Mandelbrot Set, would it not? Also, I would expect that a musical group, which put a literal fractal into its name, would be more criticized for being "repetitive", if you catch my drift;-)
This is one of the greatest math related videos I've ever seen online. You just made me appreciate and understand a complex math concept better than any teacher spanning a year's of taking math classes.
If only I was smart enough to really understand. Still so captivating though. I´m glad there is bright minds out there that really can appreciate this beauty.
@@TheAffeMaria The first thing is to tackle math problems in a way that you don't judge yourself; whether as a genius or a "Not genius." Neither attitude is helpful. There are ways to learn this stuff; it's more a matter of your curiosity.
i programmed the mandelbrot on my first amiga. But it is the first time, that someone explains this context to me. And :-) i am ashamed. To take a deeper look at the formular... Thank you for this Experience.
Yes, and believe it or not it also explains the meaning behind some Christian Biblical references relating to Hebrew math, and the Abraham, Isaac and Jacob characters. How exciting. Dump the preachers and go to the math and physics guys for some final answers.
@Resource Room Before I read the full of your comment, I thought you were referring to mathematicians by those names (Abraham de Moivre, Sir Isaac Newton, and Jacob Bernoulli)
@Jayna Lynn You would have to demonstrate there is a fractal describing the universe. I'm not saying it's impossible, just that there is no evidence for this statement.
@@d.sherman8563 You would only have to show that it is infinitely "rough." Fractals don't necesarily need to be described by simple equations. Newer physical theories seem to suggest that on the smallest level the universe is made of either discrete chunks of space or smoothish manifolds, eliminating the possiblity of it being infinitely rough. However, on most scales above the subatomic, the universe is a pretty good aprroximation of a fractal
@@bearsoundzMusic fortunately all explanations went above my head. But I've been fascinated by mandelbrot for decades. From the moment I saw the first one calculated as a screensaver over a network of apollo domain computers 35 years ago.
+Mathologer - Hello! Sorry, I'm just now seeing your comment. Actually, construction hasn't even started. I got a huge tree removed in preparation, but construction had to be delayed a few more months. But it's getting closer to that time. I will definitely let you know when I'm back up and running. For the past 5 months I've only done projects and videos that I can do in my new living room.
Lol every trip where I try to unravel the mysteries takes me on a strange rabbit hole of tool songs/analysis, math videos, philosophy videos, and adult swims off the air. Every time it’s a loop I’ve noticed
Go take a look at the bifurcation of the logistic map, then how it gets applied to the mandelbrot set, you will get a 3D map of the mandelbrot... Its absolutely stunning and fits in perfectly with whats being discussed here
@@effekt4 are you talking about Veritasium's video? because it's absolutely stunning, the way the bifurcation diagram fits, combined with this video.. oh man, mandelbrot set is really something special.
@@milanstevic8424 not specifically but that one is very good. Numberphile also goes into further detail. That video has the same visual chart in this video but on a diffeeent axis, so you get a top down view
Even after developing several applications that involved the Mandelbrot-set and variations on it, you actually managed to give me a deeper understanding of how the shapes of the Mandelbrot-set came to be, in less than 16 minutes! That's one more subscriber for you :).
+Smonjirez Great :) I actually did get a similar comment from someone else with a background similar to yours. Having said that, judging by all the other comments you two were the only people who watched this video who were really able to appreciate it for what it does.
That was very helpful, again. I've found that the numbers around roughly X=-1.8 are excellent for teaching the inner workings of Mandelbrot's set, as it is next to impossible to intuitively get a feeling for where it will land if just above zero on the Y. I think I got that from you and your -0.75 a few years back from when I watched this the first time. Impactful.
Math ist just WOW! Das Teil habe ich meinen Atari schon vor fast 30 Jahren errechnen (und mit eigenem "Grafikdruckertreiber" sogar drucken!) lassen und später "Primzahlwolken" (Linie mit Punkten für jede Zahl und Abknicken um teilweise auch dynamische Winkel bei jeder Primzahl) auf meinem ersten 386iger in der Hoffnung gebaut, Muster zu erkennen... Es wird echt Zeit, dass wir diese Art von im Universum "eingebauten" Phänomenen verstehen. Kanäle wie dieser hier sorgen dafür, dass sich mehr Leute mit sowas beschäftigen und irgendwer vielleicht den Sinn von allem aufdeckt ;) Danke, #mathologer!
Thank you so much for making this amazing video! I have loved fractals for almost three decades and this is the most wonderful explanation of why they are what they are I have ever seen.
Never has such a good explanation of the Mandelbrot set! Thank you sir! I finally get how we obtain the image, AND I had fun doing so! You are are truly a formidable educator.
Great video, I loved the Star Wars angle! I did my master's research in this area and it was fascinating. Plus you get to make lots of pretty pictures :D
things that make this one of, if not THE most geek/nerd video on youtube are the following: -lightsaber pointer -star wars references / star wars shirt -talking about math love it.
Danke für die fantastischen Videos. Sehr schön visualisiert. Man lernt nie aus. Wenn man aus einer anderen Mathe-Richtung kommt, ist das echt interessant.
The Mandelbrot set was the first chaos math set I programed into my micro back in the late 80's. Took half a day to render. I since found the Logistic map to be far more fascinating - specially when dealing with the point of accumulation. The Mandelbrot contains the Logistic and all the Julia sets. Veritasium explains the Logistic in the Mandelbrot quite well. Worth a look for those who are interested?
The "buddhabrot" is particularly interesting to render as it's so much more compute-intensive, and requires atomic memory operations to parallelize easily since any given iteration could potentially read-modify-write any pixel in the image, and has to do so on every iteration of the inner loop. You also need a huge number of samples, far higher than the number of pixels.
Great! Because of this Video i wasted a whole Day write a Software that generates buddhabrot. And let it run with a depth of 10million iterations. Calculations took 2 hours.
Wow, great video. I wrote a Mandelbrot program myself and have never seen stepping along the parabola like that. It's a really good visualisation. Trying to picture how to do the same thing with the complex numbers too!
15 min 21 s. there are Mandelbrot's numbered: 5, 8, 3, 7, 4. So if 8 is bigger then 7,5,4,3 maybe its worth calculating distance between them using scale meassure? And see what we see?
Am I missing something? Because the Buddha-Brot assigns density to the points within the Mandelbrotset, yet they never escape to infinity... so whats happening there? I get the later one with discs thats well explained, but the "Buddha-Brot" doesn't have discs.
The Buddah-Brot is done with the itérations (the successive points, that will eventually go to infinity) of the points outside of mandelbrot. Some of them will go inside before going out to infity as with the blue point at 6:00
I know why b/c you can't have Eternal Life thru Buddha ONE way that is through the Son .. Life Eternal (infinity) John 17:3 And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent.
15:13 with that procedure, you can actually find the fibonancy sequence in the mandelbrod set. It's just amazing how so many things in maths are related
For those interested in exploring the Buddhabrot set a bit more, I have a 16 gigapixel version that you can explore in your browser here: nebula.scottandmichelle.net/nebula/index.html#bbrot
This was awesome. Having coded up one of these from the base math, and made it so you could fly thru it, I didn't think there was much I didn't know about the Mandelbrot set... but there was quite a bit here new to me!
I'm so glad I watched this, there were some good videos from other uploaders but there was just something I had yet to understand, and I thank you for explaining it to me in layman terms :)
watching this, numberphiles video, and veritasiums video on the mandelbrot set really brings different aspects of the madelbrot set together and slowly connects them all
Thank you for this video. It's very illuminating. I'm eagerly awaiting your next video, explaining what happens with the complex numbers where the imaginary part is not 0.
It would be *much* harder to zoom into the Buddhabrot set because unlike the Mandelbrot set, even when only showing a small portion of it, you still need to calculate all the pixels of the entire set.
drawing a y=x line and "bouncing" it with the x^2+c works because if you have a height of say n, since y=x, where the y=n line meets y=x, x=n, from which you go up (or down) to meet the quadratic again. Sorry if that made no sense
+Yuji Okitani made sense to me, basically relying on the "put the number you get back in" reiterative process y=x is a nice line that lets us chuck our result into the x for the next step.
+Yuji Okitani Yuji, you're a genius! you synthesize Yutaka Nishiyama, Hamilton et Perelman, Kurzweil et Henstock, and this map a '2d' sequence onto a ricci-flow 'spheroid' surface! what an intriguing topology you hint at! you hint at bouncing in more than 'i,j,k'... intriguing! share also this on math-stack-exchange! imagine if the topology also undulate - if the mapped topology move as the set move... it is the gap between a type of set - it become a verge on lie group theory, set theory etc.. I wonder how you would map to flexagon, given we can embed image into flexagon via technique as photooptic moment or as 'euler disc' etc, as well as transparent overlay. can you find/generate for wall-sun-sun prime et proof?
Loved it! Thank you. I'Ve written (copy/pasted) several Mandelbrot simulators over the years and never really understood the modulus operation that makes the colors. Your video enlightened me.
+i.made.a.universe Great, why don't you link to some of your simulators (links always seem to get flagged as spam by TH-cam but I always approve them as soon as I see them :)
Lovely video prof. Polster. The nature and beauty of mathematics, a subject of yours I did in undergrad in 2013. I still think about the concepts today.
+Ariyan Adabzadeh He said in another video that he is using a projector so that he can see it on the wall behind him. He then overlays the projected images onto the footage so it doesnt look crappy.
I've learnt a lot about the Mandelbrot set, of course including how the halo is determined by how many steps it takes to explode to infinity. I never knew it was determined by that set circle. Thanks for that :)
The green line is the y=x line. Whenever a line meets the parabola, we wanna convert the y value it to the x value , to make the output to the input. This happens on the y =x line, by going horizontally from the y point. Then we draw a line from the point to the parabola, and repeat. See 3b1b's video on the power tower for more info
By the way, did you know that if you alternate between three different number systems (complex, split-complex where you have a root j²=1, j!=1 and dual where you have e²=0, e!=0), you get something that very much looks like something belonging to the darkside? orig02.deviantart.net/8dbb/f/2009/190/1/0/battlebrot_by_kram1032.png I can't recall the order though - these images are very sensitive to the exact order. I think it was split-complex -> dual -> complex but I'd have to retry to really know. Haven't played around with this in a while but there are some fun things you can do by mixing up the "standard" Mandelbrot Set formula.
Buffoon1980 if you know complex numbers, what I did isn't that big a change. So I assume you do know them. Then you know that multiplication of any two complex numbers is defined as: (a+b i)(c+d i) = a c + a d i + b c i + b d i² = a c + i (a d + b c) + b d i² and here the definition of i comes into play: i²=-1 So: a c - b d + i (a d + b c) Now what I did amounts to changing the definition of i to either be i²=+1 or i²=0 And to avoid confusion, I renamed "i" in each of those cases. So I define: j²=1, e²=0 and I get: (a+b j)(c+d j) = a c + a d j + b c j + b d j² = a c + j (a d + b c) + b d j² = | j²=1 a c + b d + j (a d + b c) or (a+b e)(c+d e) = a c + a d e + b c e + b d e² = a c + e (a d + b c) + b d e² = | e²=0 a c + e (a d + b c) And basically, which of those variations I do, I vary on each step. Of course, the actual Mandelbrot iteration is: z -> z²+c which, if z=x+iy and c=a+ib, expands to: x-> x²-y² + a y-> 2 x y + b But if I instead go: z=x+jy, I get: x -> x²+y² + a y -> 2 x y + b And finally, if I use z=a + eb: x -> x² + a y -> 2 x y + b So it's just a small modification of my iteration. Each of those three variants obviously give very different pictures if you plot their orbits. But I didn't just use each of them separately. Instead, I alternated between them. There are many ways you could do this but I chose a sequence where all three variants are called in the same order. Of this there still are six variants (ije,jie,iej,jei,eij,eji). I'm not entirely sure which one of those I picked to produce the above image but I think it was jei. So my final algorithm, I think, looks like this: x1 = x0² + y0² + a y1 = 2 x0 y0 + b x2 = x1² + a y2 = 2 x1 y1 + b x3 = x2² - y2² + a y3 = 2 x2 y2 + b and from there it'd repeat, so: x4 = x3² + y3² + a y4 = 2 x3 y3 + b etc. I know this can seem like much at first, but if you invest just a few minutes into this - maybe just manually carry out a couple of these, as was done in the video, to see what happens, you should get a sense for this. It's really not too difficult. The largest barrier is that it's a new, unfamiliar concept. __________ Technical note (this is completely unnecessary to understand the above, so feel free to ignore): Actually, come to think of it, it might be that I actually, "technically" did the iteration eij instead, depending on how you pick the starting value: Usually, these images are initialized with z0=0, which means that the first iteration, no matter which of the above you start with, will give you z1 = a + b _ where _ stands for e, i or j, depending on your current iteration. For the above scheme, z1 = a + b e But there is nothing from stopping you to initialize z0 = a + b _ in which case you'll get a picture as if the whole iteration was done one later. In a variant of the algorithm you actually start with z0 randomly. This, then, gives the so-called "Buddhagram". For the normal Buddhabrot rendering of the Mandelbrot Set that mostly means some extra fuzziness. But for something like the above alternated scheme, it might mean something rather different. I should really try that some time...
Kram1032 Aw man, I reeeeally hope you didn't type all that solely for my benefit, because it's going to be 99.9% lost on me. I mean, I'll give it a look, but since you start off by saying you assume I know complex numbers, I could be in trouble... because I pretty much don't :P I could maybe give you the dictionary definition, but... there's a pretty good chance I might be thinking of irrational numbers. Or imaginary numbers. Or grandiloquent numbers, which as far as I know is something I just made up, but may actually exist. That's how ignorant I am :P But, I appreciate the effort!
anyone ever noticed approaching the k-hole on ketamine feels a hell of a lot what a mandelbrot being zoomed into infinity looks like? timestamp 10:30 for the peak
Towards the end of your presentation, where the light rays were bouncing around in the circle, I saw an illustration of spherical aberration. The outer edges of the mirror on the Hubble were supposed to be ground parabolic, to keep the reflected rays "on the focal point". The grind on the outer edges ended up being "too" circular..... hence the slight drift off focus, or the aberration. Thank you, I'm pleased that I stumbled onto your site...... supposed to be asleep right now ( blew that idea out of the water! ).
Darth may be disappointed, but I thought this was pretty neat. I've never seen anyone talk about the interior structure of the Mandelbrot set before, and I've known about it since the '80s.
+Sierra yup orig12.deviantart.net/3468/f/2010/038/f/d/crown_of_the_elves_by_kram1032.png z_(n+1)=z_n^n+c (It's noisy because this is rather slow to calculate)
***** well, an official name? Iunno. I called it crown of the elves back then because the top structure looked like a crown to me and, well, it's green. Nothing particularly clever :) I guess it's technically an "iterated power mandelbrot set"? - or, well, a buddhabrot variant of that? Something like that. I haven't seen it anywhere else but it's very possible that others had that same idea and made it too. - My dA page is filled mostly with my experiments. It's been a while that I did anything new though. But this video inspired me to try it again for once and I actually have a new one cooking up right now! In general what higher powers do is they up the symmetry of the set. So while power 2 has a single mirror symmetry, power 3 has two mirror axes as well as a 180° rotational symmety. Power 4 has 3 mirror axes and a 3-fold rotational symmetry and this continues forever. However, that only applies to having constant powers. Crown of the Elves, I'm pretty sure, is constrained to a single mirror symmetry because all those symmetries are actually aligned - like, at least one of the main antennas of a power-set (there are as many as rotational symmetries) will always point the same way. So that direction is the only symmetry that's stable throughout all iterations. All the others, if you keep piling on higher and higher powers, essentially vanish away. But you can try completely arbitrary functions. However, not all of them work well. For instance, I tried an exponential function too but that basically didn't work at all. But that might have been a result of the bailout condition. Like, with the simple power sets, i.e. z->z^n+c where n in a fixed Integer, they all have the same bailout condition: If an iteration becomes larger than 2, it will inevitably escape. But with the exponential function, this is wrong. Instead, I think, an entire half plane would be escaping. I didn't quite get that right yet though.
@@jackciscoe8027 there is more important things than logic, if u only see logical aspects and make them your foundation of what u think reality is , u wont grow beyond yourself, for you limit yourself with exactly this mindset.
I've always loved the Mandelbrot set. This video demonstrated some nice attributes I wasn't aware of. Any time you want to do another video exploring it will be a good day for me! 😁
I have shapeshifted from Buddah to Darth Vader before and five. You touch my soul, Mathologer. It's you and Nancy Pi, 2's in mirrors Sew much love in Jedi frequency. Gratitude.
Hey, Mathologer, this may interest you. So I made Mandelbrot images where the pixels are colored by lines connecting each z0 to z1 and z1 to z2 and so on. So in a sense drawing the actual path taken by the number c, not just the end points of each iteration. Here: imgur.com/a/36shf#0. And then I experimented with outher techniques and also just made some Buddhabrots: imgur.com/a/NVpIO#0. And finally, I made some extra images, some showing how what I made compares with the Buddhabrot set: roshan106.imgur.com/all/ What do you think?
I'm glad to see the picture I created ages ago. It's the one you explain in the end with the diffetent colors inside corresponding to the cycle length.
Hey, there was something I've been wondering about the Mandelbrot set, but unfortunately my ignorance and mathematical illiteracy prevents me from expressing it clearly, so I hope anyone reading this will bear with me :P But this seemed like a good place to ask. So, at 1 iteration, you get a shape, which is (I think) a perfect circle, right? It's the yellow circle shown at 4:30 in this video, yeah? The one he says contains the whole Mandelbrot set. And that circle has a circumference and an area, which I imagine can be easily calculated (although the set uses i, would that complicate things?). What would the area and circumference be? And then, at 2 iterations, you get a shape which is a kind of oval, smaller than the circle at 1 iteration, and with a smaller circumference (can you even use that word for ovals?) and area, which I also imagine could be easily calculated. What would the circumference and area of the oval be? And as you go to 3 iterations, then 4, then 5, then more and more, the shapes get more complex, but the area within each one must necessarily be smaller than the one before. And at some point, the distance around the edge of the shapes must get larger than the circumference of the circle (somewhere between 5-10 iterations I guess). I assume, possibly unwisely, that the area must approach some quantifiable amount (looks like around one quarter of the size of the circle at 1 iteration?) and the distance around the edge must approach infinity. Am I right so far? Have I made some hideous error? Or am I expressing myself too poorly to be understood at all? Anyway, what I'm really interested to know is if there's any way of calculating the area and the distance around the edge of high iterations. Like, at a million iterations, how long is the edge of that shape compared to the one-iteration circle? Is it possible to calculate? I'm guessing it would be billions or trillions times longer... or maybe you would have to multiply it by a number with a billion zeroes after it... or more... maybe much more... Or less! I honestly don't know. But I'm intrigued to find out! So have people looked into this sort of thing? I'm assuming so. Did they find anything interesting? Is there any interesting pattern in the increasing distances, or the decreasing patterns? Have I just been spouting meaningless gibberish? A distinct possibility :P But if anyone bothers reading this, much less replying, you have my gratitude.
+Buffoon1980 You're basically right in that the area is finite. (1.5065918849 ± 0.0000000028) The perimeter however is infinite, just like a real island has a finite area but the length of its shoreline is not a well-defined quantity.
Melinda Green Sure, I understand that (kind of :P). But that's only true when the set is iterated infinitely, isn't it? I wish I could explain more clearly. With two iterations, for example, we get an oval, with (I think!) all the numbers that take more than two iterations to escape to infinity (or that never escape to infinity) within the oval. Is it possible to calculate the area of that oval, and the distance around its edge? I'm sorry, is that any clearer?
+Buffoon1980 You actually get the limit area of the "infinitely" iterated set by looking how the area changes in each step, as far as I know. The very first circle is super easy to calculate: So the formula to get the Mandelbrot-Set is z->z²+c. Now if you try plugging in random values into c, you'll perhaps notice (and this is also provable rigorously), that if you pluck in any value where the absolute value is greater than 2, it will ALWAYS diverge (i.e. if you keep going, eventually your value will grow unbounded) . For smaller values, it varies, some values stay small, others eventually become larger than 2 and thus will inevitably escape. The very largest number (in magnitude) which won't escape is -2. This means that what you really are looking at for "the first iteration" is "when is |z| smaller than 2?" or asked differently, "when is |z|² smaller than 2²=4?" You can easily check for the real numbers that, if you have x², the values won't ever go beyond 4 if x is greater than -2 and smaller than 2. It turns out that the same is true for complex numbers. (if you have a complex number x+iy, its absolute value is defined by sqrt(x²+y²) - you might recognize this as the Pythagorean relationship z²=x²+y² - it's the exact same idea) So then the initial circle has a radius of 2 and thus an area of 4pi. The area becomes more and more complex at each iteration but in the end it won't ever be larger than 4pi. What's more, it actually drops at each iteration. Look, for instance, at this table (the text isn't that important if you don't know German. This sadly isn't on the English Wikipedia. But the images suffice: de.wikipedia.org/wiki/Mandelbrot-Menge#Galerie_der_Iteration ) In these images, white means the value is 0 at this point. Black means infinite. The colors tell you "in what direction" you go - note, on the real line you can only go left or right. But the complex plane is a plane. You can go in any direction on a circle. The colors tell you which exact direction you pick, just like the hue slider on a color wheel. Note that, in the very first step, there is no black. That's because, at that point, nothing has become very large yet. Nevertheless, there are values in that image which correspond to values greater than 2. The more you iterate, the sharper the image gets. - starting values close and closer to the inside of the set will eventually escape into blackness. At each step, points are discarded (larger than 2) - they are never added back in again, so the area shrinks. So each iteration will have an area between 4pi and +Melinda Green's (1.5065918849 ± 0.0000000028). An exact formula, but this is much more technical, can be found here: mathworld.wolfram.com/MandelbrotSet.html - search for "area"
Kram1032 Thanks, that's on the right track of what I was looking for :) Unfortunately it doesn't answer much of what I was most interested to know (sorry!) I already knew that the area would drop with each iteration, since each shape is contained entirely within the one before it could be no other way. I'll put my next questions in simple points, for clarity. a) Is there any way to calculate the area for any given iteration? Obviously the circle's easy enough, but how about for much larger iterations? I saw the calculations on wolfram for the first 3 lemniscates (not a word of heard of before...), but I honestly can't tell what those calculations actually determine, whether it's the shape of the curve, the area, the length, or what. Help! :P b) Is there any mathematical pattern in the decrease in area as the iterations get higher? For starters, it would seem reasonable to think that the decrease gets less as the iterations get higher, but is that correct? c) Is there any way to calculate the length of each curve? Once again, with the circle it's easy, but it would get more complex pretty rapidly. This is what I'm particularly interested in, since I imagine it would grow to fantastically large numbers quite quickly. Anyway, thanks again for taking an interest, and trying to assist my poor feeble brain. Sorry if my ignorance is starting to irritate!
Buffoon1980 ok, so those equations below the lemniscates are what's known as implicit equations of those lemniscates. You can see there that it quickly becomes more and more complex. In principle you could calculate an area for all of them. I don't think there is a closed form equation to get to that though. What you gotta do is, for each of those curves, determine the integral over the interior of the curve. - Formally, integration is what you do to get to the area of things. One way in which this is done is via Monte Carlo Integration. Basically, you define an area (say the circle of radius 2 in which the Mandelbrot Set definitely fits) and then you randomly throw darts at at area. Then, for each dart, you determine whether it has landed inside the Mandelbrot Set or outside of it. You keep track of that. And in the end, after millions of darts, you get two numbers: The number of darts you threw (all of them landed in the circle but not necessarily in the Mandelbrot set which is contained in the circle) and the number of darts that actually landed in the set. Now you simply divide those two numbers darts_in_set/darts_total and that tells you what fraction of the darts landed in the set. This you simply multiply by the area of the circle (which in our case is known to be 4pi) and thus you arrive at an approximate but, as the number of darts increases, ever better approximation. There are a couple of ways to make it a bit smarter (the easiest by far is to shrink the area in which the darts may land - you'll notice that the circle of radius 2 is quite a bit larger than the actual Mandelbrot set, with lots of wasted space), but this is how a large class of problems like the one you ask for are generally solved. If you do this for fixed iterations, i.e. "does this point escape after 5 steps? No? Declare it in the set!", you'll get the areas of the lemniscates of those iterations (in my example iteration 5) Of course, this will only ever give you approximations. Calculating an exact value becomes much more involved very quickly and is eventually impossible. (Unless there are some clever methods nobody knows yet) Now to get the curve length of one of the Mandelbrot lemniscate at any given iteration, you gotta take the lemniscate and do a path integration along it. This also becomes more and more involved as the iterations increase and eventually this diverges to infinity which, I assume, is the main reason why that isn't usually done. I can only assume that you are not familiar with this kind of math but a TH-cam comment is definitely too much to explain it. You'll need calculus for this. Specifically, integrals, path integrals and you'll probably also need to understand derivatives and coordinate transforms. Try looking those things up. Furthermore, look up implicit curve and parametric curve and how to convert between them. Oh and btw, Wikipedia generally isn't wrong about those topics but its language for mathsy things usually is overly complicated. Try using other resources for it. I don't know how far Khan Academy is with these topics thus far but it surely is a good starting point. Finally, on the exact drop of the area: mathworld.wolfram.com/images/eps-gif/MandelbrotSetAreas_1000.gif This graph suggests that it falls roughly exponentially towards the limit. But I don't know exact statistics about that. The graph shows the area at a give iteration.
I've been looking into how fractal art is made because I find them so fascinating. All the videos I've watched were WAY over my head; didn't understand a word they said (math was my worst and most disliked subject in school, LOL!). I was totally able to understand your explanation because you taught visually and using Star Wars in the explanations totally helped, ha, ha! Reminds me of when we were homeschooling our son and used Star Wars to help teach about history and Hitler. Makes it a LOT more fun! Thank you so much for this video ... absolutely LOVED it, and you're a FANTASTIC teacher!!!
Man, I love these videos. They make me feel both really smart and really stupid at the same time. I spent ages trying to figure out basically how the Mandelbrot set works, it hurt my brain. I wish I'd had this video then. Have to admit you kind of lost me with this stuff about tractor beams... definitely gonna have to rewatch that. A while back I was trying to describe the Mandelbrot set in its most basic sense to my girlfriend. I just couldn't find a way to do it. Eventually I figured that maybe I should show her the set at 1 iteration (ie a very basic shape) then 2, 3, 10, 20, whatever, so she'd get the idea that in one sense it's basically a set of mathematically derived shapes nestled within each other, growing more and more complex (soooooooooo complex :P) as they went. Unfortunately... by that point she'd got bored and refused to listen to me any more. Then a bit later we broke up. I don't think it was Mandelbrot related, but... it probably didn't help :P Anyway, thanks for this video. I'd heard of the Buddhabrot but had no idea what it actually was until now.
+Buffoon1980 Glad you like the videos and thank you very much for saying so. I'd say give the tractor beam bit another go, that's where the real "meat" of the video is hiding. Always hard to get the balance right when it comes to being as accessible as possible and at the same time really explain some genuinely deep stuff :)
Mathologer Oh, cheers, I definitely intend to give it another go :) Seriously, you do a fantastic job with being as accessible as possible, I didn't mean to imply the fault was yours at all. I was just a bit distracted when you were explaining how those red lines were derived, which turned out to be crucial :P
Someone explain to me (6:00 - 6:17) why a point within the Mandelbrot set doesn't go to infinity if you start there, but does if you got there starting from another point outside.
This is my take, more visually. The answer is no, it's not finite because the "perimeter" is the self replicating equation *itself* that adds and multiplies. So put a pencil mark on the upper most tip of one of the lightning bolt hairs. Now try to put a pencil mark above/on the next tip of a bolt to the right of your mark, not one that you can see, but the actual next hair in line....your pencil will never move because the next hair beside the one you can see that you would LIKE to put the next mark above, actually has a *smaller* hair to the left of that one, and that one has a smaller hair with an even smaller set of hairs next to it. So you would never be able to put a pencil mark next to the starting mark because you can always "zoom in" and discover there is something closer to your starting point, you just couldn't see it without magnification. This is the basis of the "Monster Set" dilema, which led to the Julia Set, which lead to the Mandlebrot Set. Monster Set = make 3 lines of equal proportion side by side with a space ____ ____ ____ Now, below, reduce everything by thirds, but completely leave out the middle line altogether. You will find everything can reduce to quarks/quantum....looks like one line but if you zoom in, you'll see it's thirds minus the middle bar. Mandlebrot increases, not decrease and graphs the equation into plot points.
i don't understand what's going on around 6:20. The formula used is iterative. So lets say i set my bailout to 500 iterations. If is start with a point and after 5 iterations i get a point inside the set, should that not mean my starting point must be inside the set as well ?
Back in like 89 we used to let the highly advanced..(!) i386 of our lab run a Mandelbrot iteration. Next morning when we arrived, peeps from all over the institute gathered around the 'amazing' 16 color monitor to gaze at this nights output. Some times we had been over ambitious, and only 1/3 or less was drawn :p amazing to see these days zooms in the set. @Mathologer Best Mandelbrot explanation ever! Thank you -liked
I noticed that if you follow the bulbs at the border like the fibbonaci series, you reach a spot where the bulbs quickly gain incredibly many twigs on their ends. So you would look at the largest, then largest again, then the big circle on the left, then the one at the top, then the 5, 8, 13, 21 etc. , always picking the largest bulb between your 2 last bulbs.
The older I get the more I realize that mathematicians are the only people with a decent chance of fully understanding the true nature of reality, or at the very least, recognizing and understanding the consistent patterns in our universe.
I used to be in a jazz band called The Mandelbrot Set. Our music was very simple yet very detailed. Audiences complained that our live sets seemed never-ending.
But having never ending live-sets would set you outside the Mandelbrot Set, would it not? Also, I would expect that a musical group, which put a literal fractal into its name, would be more criticized for being "repetitive", if you catch my drift;-)
It had a lot of repeats in the arrangements.
haha thats cool
Q: What does the B stand for in Benoit B. Mandelbrot?
A: Benoit B. Mandelbrot.
+multimotyl Nice one :)
+multimotyl This is actually a little known fact, but the B actually stands for Blorb
multimotyl CHRIS BENOIT ISN'T DEAD HE IS A MANDELBROT SET
Mandelbrot === -1/12
multimotyl da BEARS!
dude went from Buddha to Darth Vader in like 5 seconds
Cam dude is 'NUTS'
I was just waiting for a Pink Floyd reference.
What stars really look like what???
@ have you seen him? :))))))))
The power of marijuana.
This is one of the greatest math related videos I've ever seen online. You just made me appreciate and understand a complex math concept better than any teacher spanning a year's of taking math classes.
Great, mission accomplished :)
If only I was smart enough to really understand. Still so captivating though. I´m glad there is bright minds out there that really can appreciate this beauty.
@@TheAffeMaria The first thing is to tackle math problems in a way that you don't judge yourself; whether as a genius or a "Not genius." Neither attitude is helpful.
There are ways to learn this stuff; it's more a matter of your curiosity.
for the first time in my life i can say : I understand how this shape is computed!
Really well explained!
+mr_os Great, mission accomplished then :)
i programmed the mandelbrot on my first amiga. But it is the first time, that someone explains this context to me. And :-) i am ashamed. To take a deeper look at the formular... Thank you for this Experience.
Yes, and believe it or not it also explains the meaning behind some Christian Biblical references relating to Hebrew math, and the Abraham, Isaac and Jacob characters. How exciting. Dump the preachers and go to the math and physics guys for some final answers.
@Resource Room
Before I read the full of your comment, I thought you were referring to mathematicians by those names
(Abraham de Moivre, Sir Isaac Newton, and Jacob Bernoulli)
That's not how it's computed rofl 🤣🤣🤣🤣
When i saw 'Homework' i got scared shitless for a second...
I know. I came here to hide from my homework responsibilities and now I'm getting reminded of my worst fears. :/
the answer to his HW question is because adding RGB to the graph added a 3rd dimension
When I saw "Homework" I was like "what?"...
What had I gotten myself into, I slowly move away from screen and walk out of the room with cold perspiration on my forehead
why, just don't do it
"Trust me I'm a Jedi" *Is holding a red lightsaber teaching me about the dark side"
Confirmed Sith
Seems legit to me
Sounds like a Jedi.
Definitely not lying...
@@asheep7797 you sound like quite the trustworthy sheep. I'll take your word for it.
Bhuddabrot actually looks kinda like a nebula.
It looks like the Orion nebula. I've got a 1 m^2 composite of the Buddhabrot and the Orion nebula on my wall!
@Jayna Lynn You would have to demonstrate there is a fractal describing the universe. I'm not saying it's impossible, just that there is no evidence for this statement.
So that it has a non integer Dimensionalität?
D. Sherman I would argue that there is a fractal that describes the universe ....it’s the universe lol
@@d.sherman8563 You would only have to show that it is infinitely "rough." Fractals don't necesarily need to be described by simple equations.
Newer physical theories seem to suggest that on the smallest level the universe is made of either discrete chunks of space or smoothish manifolds, eliminating the possiblity of it being infinitely rough. However, on most scales above the subatomic, the universe is a pretty good aprroximation of a fractal
1 minute: interesting
5 mins: desperately trying to comprehend
The presenters explanation is among the best i can remember i have seen, it is so elegant
Make Wavez
, I mostly don't understand Mr Mathologer's mathematicals but I do love his cosmic patterns ...
This is why we have computers. It would literally take a man's lifetime to calculate all the points possible in the Mandelbrot set.
@@bearsoundzMusic fortunately all explanations went above my head. But I've been fascinated by mandelbrot for decades. From the moment I saw the first one calculated as a screensaver over a network of apollo domain computers 35 years ago.
8 minutes: lost 🙃
Never a dull moment! I loved the video. You've got such a great way of explaining and visualizing things.
+Wood 'n' Stuff w/ Steve French How have you been? Did you finish your move to your new workshop ?
+Mathologer - Hello! Sorry, I'm just now seeing your comment. Actually, construction hasn't even started. I got a huge tree removed in preparation, but construction had to be delayed a few more months. But it's getting closer to that time. I will definitely let you know when I'm back up and running. For the past 5 months I've only done projects and videos that I can do in my new living room.
Cool, all under control then :)
Wood 'n' Stuff w/ Steve French ii
Good thing he didn't invent it. Making something basically simple into something more complex doesn't help imo.
The "B" in Benoit B. Mandelbrot stands for Benoit B. Mandelbrot.
:)
fractal geometric name ;-)
the little copies of the mandelbrot set are called mandelbrötchen. :3
ha ha... :- |)
MagicMatt93 xDdss
I started tripping acid around 1am today. It is now almost 7 and I am somehow here getting a math lesson.
😂😂
2 many I balls for me lol
Lol every trip where I try to unravel the mysteries takes me on a strange rabbit hole of tool songs/analysis, math videos, philosophy videos, and adult swims off the air. Every time it’s a loop I’ve noticed
good.
You're a male with a negative pregnancy. It's the kind in you that wanna get out. Have fun.
I thought I knew a lot about the Mandelbrot set and couldn't be suprised with a video destined to a large audience. I' so happy I was wrong.
The mad thing about this is that it is probably infinitely surprising, depending on what "this" is...
@@myeffulgenthairyballssay9358 my surprise bails out at 500
Go take a look at the bifurcation of the logistic map, then how it gets applied to the mandelbrot set, you will get a 3D map of the mandelbrot... Its absolutely stunning and fits in perfectly with whats being discussed here
@@effekt4 are you talking about Veritasium's video? because it's absolutely stunning, the way the bifurcation diagram fits, combined with this video.. oh man, mandelbrot set is really something special.
@@milanstevic8424 not specifically but that one is very good. Numberphile also goes into further detail. That video has the same visual chart in this video but on a diffeeent axis, so you get a top down view
Even after developing several applications that involved the Mandelbrot-set and variations on it, you actually managed to give me a deeper understanding of how the shapes of the Mandelbrot-set came to be, in less than 16 minutes! That's one more subscriber for you :).
+Smonjirez Great :) I actually did get a similar comment from someone else with a background similar to yours. Having said that, judging by all the other comments you two were the only people who watched this video who were really able to appreciate it for what it does.
+Mathologer
I often do have a feeling that quite a few people do not truly appreciate the mathematical beauty of this kind of stuff :)
This is fantastic. I've never seen anyone tackle the obvious questions about the set like this video does.
That was very helpful, again. I've found that the numbers around roughly X=-1.8 are excellent for teaching the inner workings of Mandelbrot's set, as it is next to impossible to intuitively get a feeling for where it will land if just above zero on the Y. I think I got that from you and your -0.75 a few years back from when I watched this the first time. Impactful.
Math ist just WOW!
Das Teil habe ich meinen Atari schon vor fast 30 Jahren errechnen (und mit eigenem "Grafikdruckertreiber" sogar drucken!) lassen und später "Primzahlwolken" (Linie mit Punkten für jede Zahl und Abknicken um teilweise auch dynamische Winkel bei jeder Primzahl) auf meinem ersten 386iger in der Hoffnung gebaut, Muster zu erkennen...
Es wird echt Zeit, dass wir diese Art von im Universum "eingebauten" Phänomenen verstehen.
Kanäle wie dieser hier sorgen dafür, dass sich mehr Leute mit sowas beschäftigen und irgendwer vielleicht den Sinn von allem aufdeckt ;)
Danke, #mathologer!
I've been studying this since I was 12 and I'm 42. Learned some previously unknown properties. Nice.
Great, that's what I love to hear :)
Thank you so much for making this amazing video! I have loved fractals for almost three decades and this is the most wonderful explanation of why they are what they are I have ever seen.
Never has such a good explanation of the Mandelbrot set! Thank you sir! I finally get how we obtain the image, AND I had fun doing so! You are are truly a formidable educator.
I never cease to be amazed by the Mandelbrot set!
The best and cleanest and easiest explanation there on the mandlebrot set. Thank you!
I thought I had seen everything concerning the M set over the decades. I was wrong. You showed me things I had not seen before. Thank you very much.
That's great :)
Great video, I loved the Star Wars angle! I did my master's research in this area and it was fascinating. Plus you get to make lots of pretty pictures :D
how many of you were hoping he was gonna zoom into the black and it would reveal some interesting goodies?
You have to Go there yourself 😂
Noooo ! Go into the light !!
Well he did, by showing the Buddha one. There's a lot there.
I know I was hoping and I just realized upon reading this comment that it never zoomed once in the video 😭😭😭
things that make this one of, if not THE most geek/nerd video on youtube are the following:
-lightsaber pointer
-star wars references / star wars shirt
-talking about math
love it.
Danke für die fantastischen Videos. Sehr schön visualisiert. Man lernt nie aus. Wenn man aus einer anderen Mathe-Richtung kommt, ist das echt interessant.
The Mandelbrot set was the first chaos math set I programed into my micro back in the late 80's. Took half a day to render.
I since found the Logistic map to be far more fascinating - specially when dealing with the point of accumulation.
The Mandelbrot contains the Logistic and all the Julia sets.
Veritasium explains the Logistic in the Mandelbrot quite well. Worth a look for those who are interested?
The "buddhabrot" is particularly interesting to render as it's so much more compute-intensive, and requires atomic memory operations to parallelize easily since any given iteration could potentially read-modify-write any pixel in the image, and has to do so on every iteration of the inner loop. You also need a huge number of samples, far higher than the number of pixels.
Great! Because of this Video i wasted a whole Day write a Software that generates buddhabrot. And let it run with a depth of 10million iterations. Calculations took 2 hours.
... pics, or it didn't happen. >:-]
You must have a supercomputer!
Share github repo please 😍
@@thomasstarzynski6787 yes
share git thanks
I really liked that! Possibly the single most interesting video about the Mandelbrot set that I have ever seen. Thanks!
2:10 "trust me I'm a Jedi" while holding a sith blade. 👌
Killing younglings with surprise homework
Because he’s talking about the dark side.
He *was* a jedi
Wow, great video. I wrote a Mandelbrot program myself and have never seen stepping along the parabola like that. It's a really good visualisation. Trying to picture how to do the same thing with the complex numbers too!
What in the world is the Mandelbrot Set used for??
@@ViveLaIsrael not in the world*
Great video, as always!
Yeah
15 min 21 s. there are Mandelbrot's numbered: 5, 8, 3, 7, 4. So if 8 is bigger then 7,5,4,3 maybe its worth calculating distance between them using scale meassure? And see what we see?
Butterbrot XD
(bread & butter in german, uploader and some here will understand)
Yes, in fact, about 10% will understand :)
#Deutsch
i get that
but was it worth getting?
Same in Russian I think
Your homework is to figure out how all of this works.
Sir I don't even know how I got here.
Best interpretation I ever saw! Thank you! How deeply connected everything is...
Am I missing something? Because the Buddha-Brot assigns density to the points within the Mandelbrotset, yet they never escape to infinity... so whats happening there?
I get the later one with discs thats well explained, but the "Buddha-Brot" doesn't have discs.
The Buddah-Brot is done with the itérations (the successive points, that will eventually go to infinity) of the points outside of mandelbrot. Some of them will go inside before going out to infity as with the blue point at 6:00
edited, yes you're correct.
I know why b/c you can't have Eternal Life thru Buddha ONE way that is through the Son .. Life Eternal (infinity) John 17:3 And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent.
I think you missed the point of the conversation.
did you actually watch this video?
15:13 with that procedure, you can actually find the fibonancy sequence in the mandelbrod set.
It's just amazing how so many things in maths are related
And the magic if you look at the set on the xz or yz axis
Years later and this is still the only video I found that explains this so well
For those interested in exploring the Buddhabrot set a bit more, I have a 16 gigapixel version that you can explore in your browser here: nebula.scottandmichelle.net/nebula/index.html#bbrot
+seligman99 Wow, this is really beautiful. Thank you very much for contributing this rendering :)
This was awesome. Having coded up one of these from the base math, and made it so you could fly thru it, I didn't think there was much I didn't know about the Mandelbrot set... but there was quite a bit here new to me!
Looking for someone in the comments who drops acid and does this math
lol... it was my first experience with magic mushrooms when i was 13 that sparked my interest in math and science. Good stuff.
I'm a person who does math and doesn't need acid because of it. :)
So if you stopped doing math you'd need acid?
+Simon It was on a psychedelic forum that I learned of the Mandelbrot set.
love me some lsd. and love me some math
I'm so glad I watched this, there were some good videos from other uploaders but there was just something I had yet to understand, and I thank you for explaining it to me in layman terms :)
:)
watching this, numberphiles video, and veritasiums video on the mandelbrot set really brings different aspects of the madelbrot set together and slowly connects them all
Like... connecting all the dots?
12:55 if veritasium is correct then the chaotic behaviour is caused by the logistic map which is part of the mandelbrot set. :)
The "real cross section" of the Mandelbrot set that I am talking about in this video is exactly the logistic map :)
Thank you for this video. It's very illuminating. I'm eagerly awaiting your next video, explaining what happens with the complex numbers where the imaginary part is not 0.
It would be *much* harder to zoom into the Buddhabrot set because unlike the Mandelbrot set, even when only showing a small portion of it, you still need to calculate all the pixels of the entire set.
drawing a y=x line and "bouncing" it with the x^2+c works because if you have a height of say n, since y=x, where the y=n line meets y=x, x=n, from which you go up (or down) to meet the quadratic again. Sorry if that made no sense
+Yuji Okitani Makes sense enough to me (but maybe not to others reading this :)
+Yuji Okitani made sense to me, basically relying on the "put the number you get back in" reiterative process
y=x is a nice line that lets us chuck our result into the x for the next step.
yeah
+AwxAngel It's like the process of feedback (putting back the result) is represented by bouncing it off the y=x line :P
+Yuji Okitani Yuji, you're a genius!
you synthesize Yutaka Nishiyama, Hamilton et Perelman, Kurzweil et Henstock,
and this map a '2d' sequence onto a ricci-flow 'spheroid' surface!
what an intriguing topology you hint at!
you hint at bouncing in more than 'i,j,k'... intriguing!
share also this on math-stack-exchange!
imagine if the topology also undulate -
if the mapped topology move as the set move...
it is the gap between a type of set -
it become a verge on lie group theory, set theory etc..
I wonder how you would map to flexagon, given we can embed image into flexagon via technique as photooptic moment or as 'euler disc' etc, as well as transparent overlay.
can you find/generate for wall-sun-sun prime et proof?
Bravo, the only explanation I have seen which clearly lays out this concept.
amazing video! dont let brady know, but i prefer this to the numberphile videos on the mandelbrot set! keep it up!
+heyits- alex Won't tell him :)
Loved it! Thank you. I'Ve written (copy/pasted) several Mandelbrot simulators over the years and never really understood the modulus operation that makes the colors. Your video enlightened me.
jeez...pun NOT intended.
+i.made.a.universe Great, why don't you link to some of your simulators (links always seem to get flagged as spam by TH-cam but I always approve them as soon as I see them :)
5:01 Speaking of which, what's the area of the Mandelbrot set?
im guessing its an infinite decimal less than 4
√(6π-1)-e
@@traso56 That is an approximation to the area, not the actual area.
Lovely video prof. Polster. The nature and beauty of mathematics, a subject of yours I did in undergrad in 2013. I still think about the concepts today.
DO A BEHIND THE SCENES VIDEO. I DONT KNOW HOW YOU LOCATE THE PICTURES WITH ACCURACY!!!!
+Ariyan Adabzadeh He said in another video that he is using a projector so that he can see it on the wall behind him. He then overlays the projected images onto the footage so it doesnt look crappy.
ok thanks!!
I've learnt a lot about the Mandelbrot set, of course including how the halo is determined by how many steps it takes to explode to infinity. I never knew it was determined by that set circle. Thanks for that :)
A lot of other interesting new knowledge in this video.
1:01 it looks like a nebula in space that looks just like a Mandelbrot
Thats really interesting, I never knew that! :)
If you spend enough time studying the shapes, you'll start getting freaked out when you realize you've seen everything before. ;)
The green line is the y=x line. Whenever a line meets the parabola, we wanna convert the y value it to the x value , to make the output to the input. This happens on the y =x line, by going horizontally from the y point. Then we draw a line from the point to the parabola, and repeat. See 3b1b's video on the power tower for more info
By the way, did you know that if you alternate between three different number systems (complex, split-complex where you have a root j²=1, j!=1 and dual where you have e²=0, e!=0), you get something that very much looks like something belonging to the darkside?
orig02.deviantart.net/8dbb/f/2009/190/1/0/battlebrot_by_kram1032.png
I can't recall the order though - these images are very sensitive to the exact order. I think it was split-complex -> dual -> complex but I'd have to retry to really know.
Haven't played around with this in a while but there are some fun things you can do by mixing up the "standard" Mandelbrot Set formula.
+Kram1032 I... don't understand many of those words :P But that looks awesome!
+Kram1032 That looks very cool :)
+Kram1032
I'd say it looks more like Yoda... nice one!
Buffoon1980 if you know complex numbers, what I did isn't that big a change.
So I assume you do know them. Then you know that multiplication of any two complex numbers is defined as:
(a+b i)(c+d i) =
a c + a d i + b c i + b d i² =
a c + i (a d + b c) + b d i²
and here the definition of i comes into play:
i²=-1
So:
a c - b d + i (a d + b c)
Now what I did amounts to changing the definition of i to either be i²=+1 or i²=0
And to avoid confusion, I renamed "i" in each of those cases. So I define: j²=1, e²=0 and I get:
(a+b j)(c+d j) =
a c + a d j + b c j + b d j² =
a c + j (a d + b c) + b d j² = | j²=1
a c + b d + j (a d + b c)
or
(a+b e)(c+d e) =
a c + a d e + b c e + b d e² =
a c + e (a d + b c) + b d e² = | e²=0
a c + e (a d + b c)
And basically, which of those variations I do, I vary on each step. Of course, the actual Mandelbrot iteration is:
z -> z²+c
which, if z=x+iy and c=a+ib, expands to:
x-> x²-y² + a
y-> 2 x y + b
But if I instead go: z=x+jy, I get:
x -> x²+y² + a
y -> 2 x y + b
And finally, if I use z=a + eb:
x -> x² + a
y -> 2 x y + b
So it's just a small modification of my iteration.
Each of those three variants obviously give very different pictures if you plot their orbits.
But I didn't just use each of them separately. Instead, I alternated between them.
There are many ways you could do this but I chose a sequence where all three variants are called in the same order. Of this there still are six variants (ije,jie,iej,jei,eij,eji). I'm not entirely sure which one of those I picked to produce the above image but I think it was jei.
So my final algorithm, I think, looks like this:
x1 = x0² + y0² + a
y1 = 2 x0 y0 + b
x2 = x1² + a
y2 = 2 x1 y1 + b
x3 = x2² - y2² + a
y3 = 2 x2 y2 + b
and from there it'd repeat, so:
x4 = x3² + y3² + a
y4 = 2 x3 y3 + b
etc.
I know this can seem like much at first, but if you invest just a few minutes into this - maybe just manually carry out a couple of these, as was done in the video, to see what happens, you should get a sense for this. It's really not too difficult. The largest barrier is that it's a new, unfamiliar concept.
__________
Technical note (this is completely unnecessary to understand the above, so feel free to ignore):
Actually, come to think of it, it might be that I actually, "technically" did the iteration eij instead, depending on how you pick the starting value:
Usually, these images are initialized with z0=0, which means that the first iteration, no matter which of the above you start with, will give you z1 = a + b _
where _ stands for e, i or j, depending on your current iteration. For the above scheme, z1 = a + b e
But there is nothing from stopping you to initialize z0 = a + b _ in which case you'll get a picture as if the whole iteration was done one later.
In a variant of the algorithm you actually start with z0 randomly. This, then, gives the so-called "Buddhagram". For the normal Buddhabrot rendering of the Mandelbrot Set that mostly means some extra fuzziness. But for something like the above alternated scheme, it might mean something rather different. I should really try that some time...
Kram1032 Aw man, I reeeeally hope you didn't type all that solely for my benefit, because it's going to be 99.9% lost on me. I mean, I'll give it a look, but since you start off by saying you assume I know complex numbers, I could be in trouble... because I pretty much don't :P I could maybe give you the dictionary definition, but... there's a pretty good chance I might be thinking of irrational numbers. Or imaginary numbers. Or grandiloquent numbers, which as far as I know is something I just made up, but may actually exist.
That's how ignorant I am :P But, I appreciate the effort!
Very good explanation... Probably the best I've seen so far
anyone ever noticed approaching the k-hole on ketamine feels a hell of a lot what a mandelbrot being zoomed into infinity looks like?
timestamp 10:30 for the peak
Towards the end of your presentation, where the light rays were bouncing around in the circle, I saw an illustration of spherical aberration. The outer edges of the mirror on the Hubble were supposed to be ground parabolic, to keep the reflected rays "on the focal point". The grind on the outer edges ended up being "too" circular..... hence the slight drift off focus, or the aberration. Thank you, I'm pleased that I stumbled onto your site...... supposed to be asleep right now ( blew that idea out of the water! ).
⭐️⭐️⭐️⭐️⭐️THAT WAS FOR THE LACK OF A BETTER WORD: BRILLIANT! Thank you, Mathologer! 😀 👍
Glad you liked it :)
This is even better than the Numberphile Mandelbrot video! Great job!
Darth may be disappointed, but I thought this was pretty neat. I've never seen anyone talk about the interior structure of the Mandelbrot set before, and I've known about it since the '80s.
+Martin Heermance That was the mission :)
+Sierra yup orig12.deviantart.net/3468/f/2010/038/f/d/crown_of_the_elves_by_kram1032.png
z_(n+1)=z_n^n+c
(It's noisy because this is rather slow to calculate)
***** well, an official name? Iunno. I called it crown of the elves back then because the top structure looked like a crown to me and, well, it's green. Nothing particularly clever :)
I guess it's technically an "iterated power mandelbrot set"? - or, well, a buddhabrot variant of that? Something like that. I haven't seen it anywhere else but it's very possible that others had that same idea and made it too. - My dA page is filled mostly with my experiments. It's been a while that I did anything new though. But this video inspired me to try it again for once and I actually have a new one cooking up right now!
In general what higher powers do is they up the symmetry of the set. So while power 2 has a single mirror symmetry, power 3 has two mirror axes as well as a 180° rotational symmety. Power 4 has 3 mirror axes and a 3-fold rotational symmetry and this continues forever.
However, that only applies to having constant powers. Crown of the Elves, I'm pretty sure, is constrained to a single mirror symmetry because all those symmetries are actually aligned - like, at least one of the main antennas of a power-set (there are as many as rotational symmetries) will always point the same way. So that direction is the only symmetry that's stable throughout all iterations. All the others, if you keep piling on higher and higher powers, essentially vanish away.
But you can try completely arbitrary functions. However, not all of them work well. For instance, I tried an exponential function too but that basically didn't work at all. But that might have been a result of the bailout condition.
Like, with the simple power sets, i.e.
z->z^n+c
where n in a fixed Integer, they all have the same bailout condition: If an iteration becomes larger than 2, it will inevitably escape. But with the exponential function, this is wrong. Instead, I think, an entire half plane would be escaping. I didn't quite get that right yet though.
This was my favourite video of yours, very well done.
I really love mathematics. I love how everything is so logical. I really wish I studied it more while in school. It's so interesting.
You’re still alive! Go for it
@@jackciscoe8027 there is more important things than logic, if u only see logical aspects and make them your foundation of what u think reality is , u wont grow beyond yourself, for you limit yourself with exactly this mindset.
I've always loved the Mandelbrot set. This video demonstrated some nice attributes I wasn't aware of. Any time you want to do another video exploring it will be a good day for me! 😁
What would a fractal with the equation Z*i0=C²+Z*i0+C³ look like?
I THOROUGHLY enjoyed your premise and teaching style making a 15 min lesson on furthering my understanding of this mathematical/artistic/divine wonder
I was waiting to see where Mandelbrot tells Luke he's his uncle on his mother's side.
I have shapeshifted from Buddah to Darth Vader before and five.
You touch my soul, Mathologer. It's you and Nancy Pi, 2's in mirrors
Sew much love in Jedi frequency.
Gratitude.
Hey, Mathologer, this may interest you. So I made Mandelbrot images where the pixels are colored by lines connecting each z0 to z1 and z1 to z2 and so on. So in a sense drawing the actual path taken by the number c, not just the end points of each iteration. Here: imgur.com/a/36shf#0. And then I experimented with outher techniques and also just made some Buddhabrots: imgur.com/a/NVpIO#0. And finally, I made some extra images, some showing how what I made compares with the Buddhabrot set: roshan106.imgur.com/all/ What do you think?
+Roshan Sharma These look great. Thanks for linking to these pictures :)
+Roshan Sharma neat techniques! That last link doesn't seem to work though. It says your images aren't publicly available. Very nice experiments!
Kram1032 Oh, oops, here's a link that'll hopefully work. imgur.com/a/yoa6d
Those look insane! neat!
How did you make these images?
I'm glad to see the picture I created ages ago. It's the one you explain in the end with the diffetent colors inside corresponding to the cycle length.
Mandelbrot REALLY actually scares me somehow. It just doesn't stop when it really needs to.
Like an uncle who just keeps talking?
Hey, there was something I've been wondering about the Mandelbrot set, but unfortunately my ignorance and mathematical illiteracy prevents me from expressing it clearly, so I hope anyone reading this will bear with me :P But this seemed like a good place to ask.
So, at 1 iteration, you get a shape, which is (I think) a perfect circle, right? It's the yellow circle shown at 4:30 in this video, yeah? The one he says contains the whole Mandelbrot set. And that circle has a circumference and an area, which I imagine can be easily calculated (although the set uses i, would that complicate things?). What would the area and circumference be?
And then, at 2 iterations, you get a shape which is a kind of oval, smaller than the circle at 1 iteration, and with a smaller circumference (can you even use that word for ovals?) and area, which I also imagine could be easily calculated. What would the circumference and area of the oval be?
And as you go to 3 iterations, then 4, then 5, then more and more, the shapes get more complex, but the area within each one must necessarily be smaller than the one before. And at some point, the distance around the edge of the shapes must get larger than the circumference of the circle (somewhere between 5-10 iterations I guess). I assume, possibly unwisely, that the area must approach some quantifiable amount (looks like around one quarter of the size of the circle at 1 iteration?) and the distance around the edge must approach infinity.
Am I right so far? Have I made some hideous error? Or am I expressing myself too poorly to be understood at all?
Anyway, what I'm really interested to know is if there's any way of calculating the area and the distance around the edge of high iterations. Like, at a million iterations, how long is the edge of that shape compared to the one-iteration circle? Is it possible to calculate? I'm guessing it would be billions or trillions times longer... or maybe you would have to multiply it by a number with a billion zeroes after it... or more... maybe much more... Or less! I honestly don't know. But I'm intrigued to find out!
So have people looked into this sort of thing? I'm assuming so. Did they find anything interesting? Is there any interesting pattern in the increasing distances, or the decreasing patterns?
Have I just been spouting meaningless gibberish? A distinct possibility :P But if anyone bothers reading this, much less replying, you have my gratitude.
+Buffoon1980 You're basically right in that the area is finite. (1.5065918849 ± 0.0000000028) The perimeter however is infinite, just like a real island has a finite area but the length of its shoreline is not a well-defined quantity.
Melinda Green Sure, I understand that (kind of :P). But that's only true when the set is iterated infinitely, isn't it? I wish I could explain more clearly.
With two iterations, for example, we get an oval, with (I think!) all the numbers that take more than two iterations to escape to infinity (or that never escape to infinity) within the oval. Is it possible to calculate the area of that oval, and the distance around its edge?
I'm sorry, is that any clearer?
+Buffoon1980 You actually get the limit area of the "infinitely" iterated set by looking how the area changes in each step, as far as I know.
The very first circle is super easy to calculate:
So the formula to get the Mandelbrot-Set is z->z²+c. Now if you try plugging in random values into c, you'll perhaps notice (and this is also provable rigorously), that if you pluck in any value where the absolute value is greater than 2, it will ALWAYS diverge (i.e. if you keep going, eventually your value will grow unbounded) . For smaller values, it varies, some values stay small, others eventually become larger than 2 and thus will inevitably escape.
The very largest number (in magnitude) which won't escape is -2.
This means that what you really are looking at for "the first iteration" is "when is |z| smaller than 2?"
or asked differently, "when is |z|² smaller than 2²=4?"
You can easily check for the real numbers that, if you have x², the values won't ever go beyond 4 if x is greater than -2 and smaller than 2.
It turns out that the same is true for complex numbers.
(if you have a complex number x+iy, its absolute value is defined by sqrt(x²+y²) - you might recognize this as the Pythagorean relationship z²=x²+y² - it's the exact same idea)
So then the initial circle has a radius of 2 and thus an area of 4pi.
The area becomes more and more complex at each iteration but in the end it won't ever be larger than 4pi. What's more, it actually drops at each iteration.
Look, for instance, at this table (the text isn't that important if you don't know German. This sadly isn't on the English Wikipedia. But the images suffice: de.wikipedia.org/wiki/Mandelbrot-Menge#Galerie_der_Iteration )
In these images, white means the value is 0 at this point. Black means infinite. The colors tell you "in what direction" you go - note, on the real line you can only go left or right. But the complex plane is a plane. You can go in any direction on a circle. The colors tell you which exact direction you pick, just like the hue slider on a color wheel.
Note that, in the very first step, there is no black. That's because, at that point, nothing has become very large yet. Nevertheless, there are values in that image which correspond to values greater than 2.
The more you iterate, the sharper the image gets. - starting values close and closer to the inside of the set will eventually escape into blackness. At each step, points are discarded (larger than 2) - they are never added back in again, so the area shrinks.
So each iteration will have an area between 4pi and +Melinda Green's (1.5065918849 ± 0.0000000028).
An exact formula, but this is much more technical, can be found here:
mathworld.wolfram.com/MandelbrotSet.html - search for "area"
Kram1032 Thanks, that's on the right track of what I was looking for :) Unfortunately it doesn't answer much of what I was most interested to know (sorry!) I already knew that the area would drop with each iteration, since each shape is contained entirely within the one before it could be no other way. I'll put my next questions in simple points, for clarity.
a) Is there any way to calculate the area for any given iteration? Obviously the circle's easy enough, but how about for much larger iterations? I saw the calculations on wolfram for the first 3 lemniscates (not a word of heard of before...), but I honestly can't tell what those calculations actually determine, whether it's the shape of the curve, the area, the length, or what. Help! :P
b) Is there any mathematical pattern in the decrease in area as the iterations get higher? For starters, it would seem reasonable to think that the decrease gets less as the iterations get higher, but is that correct?
c) Is there any way to calculate the length of each curve? Once again, with the circle it's easy, but it would get more complex pretty rapidly. This is what I'm particularly interested in, since I imagine it would grow to fantastically large numbers quite quickly.
Anyway, thanks again for taking an interest, and trying to assist my poor feeble brain. Sorry if my ignorance is starting to irritate!
Buffoon1980 ok, so those equations below the lemniscates are what's known as implicit equations of those lemniscates. You can see there that it quickly becomes more and more complex. In principle you could calculate an area for all of them. I don't think there is a closed form equation to get to that though. What you gotta do is, for each of those curves, determine the integral over the interior of the curve. - Formally, integration is what you do to get to the area of things.
One way in which this is done is via Monte Carlo Integration. Basically, you define an area (say the circle of radius 2 in which the Mandelbrot Set definitely fits) and then you randomly throw darts at at area. Then, for each dart, you determine whether it has landed inside the Mandelbrot Set or outside of it.
You keep track of that. And in the end, after millions of darts, you get two numbers: The number of darts you threw (all of them landed in the circle but not necessarily in the Mandelbrot set which is contained in the circle) and the number of darts that actually landed in the set.
Now you simply divide those two numbers darts_in_set/darts_total and that tells you what fraction of the darts landed in the set. This you simply multiply by the area of the circle (which in our case is known to be 4pi) and thus you arrive at an approximate but, as the number of darts increases, ever better approximation.
There are a couple of ways to make it a bit smarter (the easiest by far is to shrink the area in which the darts may land - you'll notice that the circle of radius 2 is quite a bit larger than the actual Mandelbrot set, with lots of wasted space), but this is how a large class of problems like the one you ask for are generally solved.
If you do this for fixed iterations, i.e. "does this point escape after 5 steps? No? Declare it in the set!", you'll get the areas of the lemniscates of those iterations (in my example iteration 5)
Of course, this will only ever give you approximations. Calculating an exact value becomes much more involved very quickly and is eventually impossible. (Unless there are some clever methods nobody knows yet)
Now to get the curve length of one of the Mandelbrot lemniscate at any given iteration, you gotta take the lemniscate and do a path integration along it. This also becomes more and more involved as the iterations increase and eventually this diverges to infinity which, I assume, is the main reason why that isn't usually done.
I can only assume that you are not familiar with this kind of math but a TH-cam comment is definitely too much to explain it.
You'll need calculus for this. Specifically, integrals, path integrals and you'll probably also need to understand derivatives and coordinate transforms.
Try looking those things up.
Furthermore, look up implicit curve and parametric curve and how to convert between them.
Oh and btw, Wikipedia generally isn't wrong about those topics but its language for mathsy things usually is overly complicated. Try using other resources for it. I don't know how far Khan Academy is with these topics thus far but it surely is a good starting point.
Finally, on the exact drop of the area:
mathworld.wolfram.com/images/eps-gif/MandelbrotSetAreas_1000.gif
This graph suggests that it falls roughly exponentially towards the limit. But I don't know exact statistics about that. The graph shows the area at a give iteration.
I've been looking into how fractal art is made because I find them so fascinating. All the videos I've watched were WAY over my head; didn't understand a word they said (math was my worst and most disliked subject in school, LOL!). I was totally able to understand your explanation because you taught visually and using Star Wars in the explanations totally helped, ha, ha! Reminds me of when we were homeschooling our son and used Star Wars to help teach about history and Hitler. Makes it a LOT more fun! Thank you so much for this video ... absolutely LOVED it, and you're a FANTASTIC teacher!!!
仏像のスケルトンの作り方ですが、どのようなプログラミングを組んでますか？色の設定など、細かい注目点はあるはずです。
Man, I love these videos. They make me feel both really smart and really stupid at the same time. I spent ages trying to figure out basically how the Mandelbrot set works, it hurt my brain. I wish I'd had this video then. Have to admit you kind of lost me with this stuff about tractor beams... definitely gonna have to rewatch that.
A while back I was trying to describe the Mandelbrot set in its most basic sense to my girlfriend. I just couldn't find a way to do it. Eventually I figured that maybe I should show her the set at 1 iteration (ie a very basic shape) then 2, 3, 10, 20, whatever, so she'd get the idea that in one sense it's basically a set of mathematically derived shapes nestled within each other, growing more and more complex (soooooooooo complex :P) as they went. Unfortunately... by that point she'd got bored and refused to listen to me any more. Then a bit later we broke up. I don't think it was Mandelbrot related, but... it probably didn't help :P
Anyway, thanks for this video. I'd heard of the Buddhabrot but had no idea what it actually was until now.
+Buffoon1980 Glad you like the videos and thank you very much for saying so. I'd say give the tractor beam bit another go, that's where the real "meat" of the video is hiding. Always hard to get the balance right when it comes to being as accessible as possible and at the same time really explain some genuinely deep stuff :)
Mathologer Oh, cheers, I definitely intend to give it another go :) Seriously, you do a fantastic job with being as accessible as possible, I didn't mean to imply the fault was yours at all. I was just a bit distracted when you were explaining how those red lines were derived, which turned out to be crucial :P
There aren't just Mandelbrot sets on the outside of the Mandelbrot set, they're inside the dark part too.
We need insane zoom-in videos of that
First fractal program, discover on Amiga computer years 80'!
Amiga and news AmigaOS4 ruleeez! 👏✌️👌
WOW! This just blew my mind and rebuilt it in many senses. This just put some major pieces together for me, now I'm off on some neat iterations
Wow, this explained it really well. Thanks
Someone explain to me (6:00 - 6:17) why a point within the Mandelbrot set doesn't go to infinity if you start there, but does if you got there starting from another point outside.
Is the perimeter of the Mandelbrot Set finite?
Gazpacho King that's a very good question! Any mathematicians care to answer?
My wild ass guess is yes.
Gazpacho King No it's not. It's called non-measurable curve.
This is my take, more visually. The answer is no, it's not finite because the "perimeter" is the self replicating equation *itself* that adds and multiplies. So put a pencil mark on the upper most tip of one of the lightning bolt hairs. Now try to put a pencil mark above/on the next tip of a bolt to the right of your mark, not one that you can see, but the actual next hair in line....your pencil will never move because the next hair beside the one you can see that you would LIKE to put the next mark above, actually has a *smaller* hair to the left of that one, and that one has a smaller hair with an even smaller set of hairs next to it. So you would never be able to put a pencil mark next to the starting mark because you can always "zoom in" and discover there is something closer to your starting point, you just couldn't see it without magnification. This is the basis of the "Monster Set" dilema, which led to the Julia Set, which lead to the Mandlebrot Set. Monster Set = make 3 lines of equal proportion side by side with a space ____ ____ ____ Now, below, reduce everything by thirds, but completely leave out the middle line altogether. You will find everything can reduce to quarks/quantum....looks like one line but if you zoom in, you'll see it's thirds minus the middle bar. Mandlebrot increases, not decrease and graphs the equation into plot points.
Thanks for making this, I've never been interested in math back in school but your videos are fun and actually exciting to watch!
Mission accomplished :)
666K views! The dark side is strong in this one...
I liked the bit at the end with the prongs, never thought about that before
“There is no spoon.”
Best TH-cam video in the mathematical field so far.
Can you do a sequel, like you said in the end, about the oddities of this graph?
Thank you!
The use of parabolas as a visual demonstration of what is happening , was beautiful !
The dark side of the almond bread set *lmao*
影 ShadowZZZ
i don't understand what's going on around 6:20.
The formula used is iterative. So lets say i set my bailout to 500 iterations. If is start with a point and after 5 iterations i get a point inside the set, should that not mean my starting point must be inside the set as well ?
So, the Fig Tree is connected to the Almond Bread.
Back in like 89 we used to let the highly advanced..(!) i386 of our lab run a Mandelbrot iteration.
Next morning when we arrived, peeps from all over the institute gathered around the 'amazing' 16 color monitor to gaze at this nights output. Some times we had been over ambitious, and only 1/3 or less was drawn :p
amazing to see these days zooms in the set.
@Mathologer
Best Mandelbrot explanation ever!
Thank you -liked
Excellent video, congratulations. I wish it was much longer and I wouldn't mind if it were a bit more technical.
The channel is meant to be as accessible as possible, which means relatively short videos that use simple terms.
I noticed that if you follow the bulbs at the border like the fibbonaci series, you reach a spot where the bulbs quickly gain incredibly many twigs on their ends. So you would look at the largest, then largest again, then the big circle on the left, then the one at the top, then the 5, 8, 13, 21 etc. , always picking the largest bulb between your 2 last bulbs.
Love this channel :)
The older I get the more I realize that mathematicians are the only people with a decent chance of fully understanding the true nature of reality, or at the very least, recognizing and understanding the consistent patterns in our universe.