The dark side of the Mandelbrot set

Join the Mathologer and his guest Darth Vader as they explore the Dark Side of the Mandelbrot set. Featuring an introduction to how the Mandelbrot set and the halo surrounding it is conjured up, an ingenious way to visualise what's really going on inside the Mandelbrot set, as well as an appearance of the amazing Buddhabrot fractal.
Special thanks to Melinda Green who discovered the Buddhabrot fractal in 1993 for letting us use her Buddhabrot pictures in this video. Check out her website for more information about this fractal as well as 4d Rubik's cubes, stereophotography, etc.: superliminal.com
Enjoy!

[@multimotyl]

Q: What does the B stand for in Benoit B. Mandelbrot? A: Benoit B. Mandelbrot.

[@Mathologer]

+multimotyl Nice one :)

[@SJ23982398]

+multimotyl This is actually a little known fact, but the B actually stands for Blorb

[@kasperdomagala4544]

multimotyl CHRIS BENOIT ISN'T DEAD HE IS A MANDELBROT SET

[@dangnabbit1379]

Mandelbrot === -1/12

[@mikeymadnez]

multimotyl da BEARS!

[@Guy_de_Loimbard]

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.

[@CZpersi]

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;-)

[@rudolphguarnacci197]

It had a lot of repeats in the arrangements.

[@QuindaliusBarnacleJonesJr.]

haha thats cool

[@cam4617]

dude went from Buddha to Darth Vader in like 5 seconds

[@lapinchiloca]

Cam dude is 'NUTS'

[@memeking9926]

I was just waiting for a Pink Floyd reference.

[@jellymaynemitch]

What stars really look like what???

[@cucicearoland5949]

@What stars really look like have you seen him? :))))))))

[@Kickex]

The power of marijuana.

[@Nachtopus]

I started tripping acid around 1am today. It is now almost 7 and I am somehow here getting a math lesson.

[@hanomano8361]

😂😂

[@timchandler4427]

2 many I balls for me lol

[@acidcatheter8645]

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

[@salvinorindoge3811]

good.

[@andreighira6870]

You're a male with a negative pregnancy. It's the kind in you that wanna get out. Have fun.

[@abc-ks8mm]

When i saw 'Homework' i got scared shitless for a second...

[@Voltaic314]

I know. I came here to hide from my homework responsibilities and now I'm getting reminded of my worst fears. :/

[@codyknapp2683]

the answer to his HW question is because adding RGB to the graph added a 3rd dimension

[@groszak1]

When I saw "Homework" I was like "what?"...

[@imambaybars3405]

What had I gotten myself into, I slowly move away from screen and walk out of the room with cold perspiration on my forehead

[@_shadow_1]

why, just don't do it

[@tomc.5704]

"Trust me I'm a Jedi" *Is holding a red lightsaber teaching me about the dark side"

[@dickstryker]

Confirmed Sith

[@xSenwar]

Seems legit to me

[@asheep7797]

Sounds like a Jedi. Definitely not lying...

[@xSenwar]

@@asheep7797 you sound like quite the trustworthy sheep. I'll take your word for it.

[@bluedog28]

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.

[@Mathologer]

Great, mission accomplished :)

[@TheAffeMaria]

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.

[@bigbluebuttonman1137]

@@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.

[@alligatorboy2000]

Bhuddabrot actually looks kinda like a nebula.

[@Dalroc]

It looks like the Orion nebula. I've got a 1 m^2 composite of the Buddhabrot and the Orion nebula on my wall!

[@d.sherman8563]

@@SmashedByMUNKEEz 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.

[@TobiasWeg]

So that it has a non integer Dimensionalität?

[@Dirtbiker463]

D. Sherman I would argue that there is a fractal that describes the universe ....it’s the universe lol

[@Adraria8]

@@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

[@osenseijedi]

for the first time in my life i can say : I understand how this shape is computed! Really well explained!

[@Mathologer]

+mr_os Great, mission accomplished then :)

[@unfinishedbusiness4088]

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.

[@Pinkmacmleod]

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.

[@shambosaha9727]

@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)

[@Rueyful]

That's not how it's computed rofl 🤣🤣🤣🤣

[@MattTheMagician23]

The "B" in Benoit B. Mandelbrot stands for Benoit B. Mandelbrot.

[@Mathologer]

:)

[@netabolt6546]

fractal geometric name ;-)

[@sofia.eris.bauhaus]

the little copies of the mandelbrot set are called mandelbrötchen. :3

[@anirudhachakri4295]

ha ha... :- |)

[@llauoylliklliwi970]

MagicMatt93 xDdss

[@makewavez2005]

1 minute: interesting 5 mins: desperately trying to comprehend

[@bearsoundzMusic]

The presenters explanation is among the best i can remember i have seen, it is so elegant

[@MsGnor]

Make Wavez , I mostly don't understand Mr Mathologer's mathematicals but I do love his cosmic patterns ...

[@patrickbodine6010]

This is why we have computers. It would literally take a man's lifetime to calculate all the points possible in the Mandelbrot set.

[@bloepje]

@@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.

[@wilma8326]

8 minutes: lost 🙃

[@SteveFrenchWoodNStuff]

Never a dull moment! I loved the video. You've got such a great way of explaining and visualizing things.

[@Mathologer]

+Wood 'n' Stuff w/ Steve French How have you been? Did you finish your move to your new workshop ?

[@SteveFrenchWoodNStuff]

+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.

[@Mathologer]

Cool, all under control then :)

[@rohitjain1455]

Wood 'n' Stuff w/ Steve French ii

[@w.hoffman3308]

Good thing he didn't invent it. Making something basically simple into something more complex doesn't help imo.

[@JamesSpeiser]

I've been studying this since I was 12 and I'm 42. Learned some previously unknown properties. Nice.

[@Mathologer]

Great, that's what I love to hear :)

[@moraigna66]

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.

[@myeffulgenthairyballssay9358]

The mad thing about this is that it is probably infinitely surprising, depending on what "this" is...

[@milanstevic8424]

@@myeffulgenthairyballssay9358 my surprise bails out at 500

[@effekt4]

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

[@milanstevic8424]

@@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.

[@effekt4]

@@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

[@DigiFootageFX]

This is fantastic. I've never seen anyone tackle the obvious questions about the set like this video does.

[@Smonjirez]

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 :).

[@Mathologer]

+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.

[@rhiannonz85]

+Mathologer I often do have a feeling that quite a few people do not truly appreciate the mathematical beauty of this kind of stuff :)

[@Odolwa2]

how many of you were hoping he was gonna zoom into the black and it would reveal some interesting goodies?

[@tinaboennemann9805]

You have to Go there yourself 😂

[@donaldhamilton5345]

Noooo ! Go into the light !!

[@NatsGhost]

Well he did, by showing the Buddha one. There's a lot there.

[@MichelleRosewood]

I know I was hoping and I just realized upon reading this comment that it never zoomed once in the video 😭😭😭

[@elschalo]

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!

[@dcterr1]

I never cease to be amazed by the Mandelbrot set!

[@captainTubes]

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.

[@ozwhistles]

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?

[@jmiller6066]

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.

[@theadmiralyt1574]

2:10 "trust me I'm a Jedi" while holding a sith blade. 👌

[@nemou4985]

Killing younglings with surprise homework

[@k8cav77]

Because he’s talking about the dark side.

[@vizaloron9242]

He *was* a jedi

[@claymarzobestgoofy]

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.

[@MrWave58]

Best interpretation I ever saw! Thank you! How deeply connected everything is...

[@masterhaemi]

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.

[@irrelevant_noob]

... pics, or it didn't happen. >:-]

[@andrewchan5153]

You must have a supercomputer!

[@thomasstarzynski6787]

Share github repo please 😍

[@kingbob851]

@@thomasstarzynski6787 yes

[@LiwaySaGu]

share git thanks

[@wd5jlr]

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.

[@Mathologer]

That's great :)

[@TomAtkinson]

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.

[@250v8]

I really liked that! Possibly the single most interesting video about the Mandelbrot set that I have ever seen. Thanks!

[@evanoman5526]

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

[@WarpRulez]

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.

[@Kugelschrei]

Thanks for making this, I've never been interested in math back in school but your videos are fun and actually exciting to watch!

[@Mathologer]

Mission accomplished :)

[@craig_z]

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!

[@BuggaUgga]

Danke für die fantastischen Videos. Sehr schön visualisiert. Man lernt nie aus. Wenn man aus einer anderen Mathe-Richtung kommt, ist das echt interessant.

[@RobertCWebb]

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!

[@ViveLaIsrael]

What in the world is the Mandelbrot Set used for??

[@official-obama]

@@ViveLaIsrael not in the world*

[@m0nde]

This was my favourite video of yours, very well done.

[@techstuff2237]

The best and cleanest and easiest explanation there on the mandlebrot set. Thank you!

[@handyMath]

Great video, as always!

[@oliver8500]

Yeah

[@dadutchboy2]

1:24 damn... mandelbrot looking kinda thicc

[@DesA-hj7ms]

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.

[@JustChristianSt.M4.10]

I THOROUGHLY enjoyed your premise and teaching style making a 15 min lesson on furthering my understanding of this mathematical/artistic/divine wonder

[@xanderzero3000]

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

[@billenglish3003]

Like... connecting all the dots?

[@riftmusic5232]

Wow, this explained it really well. Thanks

[@adamotfsechler6098]

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

[@tolkienfan1972]

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! 😁

[@NorwayT]

⭐️⭐️⭐️⭐️⭐️THAT WAS FOR THE LACK OF A BETTER WORD: BRILLIANT! Thank you, Mathologer! 😀 👍

[@Mathologer]

Glad you liked it :)

[@mheermance]

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.

[@Mathologer]

+Martin Heermance That was the mission :)

[@Kram1032]

+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)

[@Kram1032]

***** 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.

[@benshapiro2wt298]

Years later and this is still the only video I found that explains this so well

[@gaufill]

Loved this video! I would love to see more like this. Thanks!

[@Mathologer]

+Gordon Aufill This one was a killer to put together. I think I need a holiday. Maybe something lighthearted for pi day before I tackle some more serious stuff again :)

[@princetamrac1180]

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

[@hillaryclinton2415]

And the magic if you look at the set on the xz or yz axis

[@AlexTrusk91]

Butterbrot XD (bread & butter in german, uploader and some here will understand)

[@Mathologer]

Yes, in fact, about 10% will understand :)

[@cyancoyote7366]

#Deutsch

[@taureon_]

i get that

[@bachelorofstuff]

but was it worth getting?

[@50p35]

Same in Russian I think

[@peterbentleyhk]

FANTASTIC ! Thanks for this beautifully clear explanation

[@marcelotemer]

Thank you! Thank you! Beautiful step by step illustration!

[@JubeProductions]

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.

[@jackciscoe8027]

You’re still alive! Go for it

[@kijuubi]

@@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.

[@heyitsalex99]

amazing video! dont let brady know, but i prefer this to the numberphile videos on the mandelbrot set! keep it up!

[@Mathologer]

+heyits- alex Won't tell him :)

[@jesuslovespee]

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.

[@jesuslovespee]

jeez...pun NOT intended.

[@Mathologer]

+i.made.a.universe Great, why don't you link to some of your simulators (links always seem to get flagged as spam by KZbin but I always approve them as soon as I see them :)

[@gambet0007]

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 :)

[@Mathologer]

:)

[@ariyanadabzadeh5486]

DO A BEHIND THE SCENES VIDEO. I DONT KNOW HOW YOU LOCATE THE PICTURES WITH ACCURACY!!!!

[@Tumbolisu]

+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.

[@ariyanadabzadeh5486]

ok thanks!!

[@CelentAle]

First fractal program, discover on Amiga computer years 80'! Amiga and news AmigaOS4 ruleeez! 👏✌️👌

[@DiogoSiqueira]

Man, what an incredible video! I have lots of trouble with numbers, but when I do understand how they work in nature I can see how they're amazing. Thanks for the explanation.

[@Mathologer]

That's great :)

[@johnwhetstone9158]

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.

[@yujiokitani4492]

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

[@Mathologer]

+Yuji Okitani Makes sense enough to me (but maybe not to others reading this :)

[@awxangel6781]

+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.

[@yujiokitani4492]

yeah

[@truefaceofevil]

+AwxAngel It's like the process of feedback (putting back the result) is represented by bouncing it off the y=x line :P

[@JohnSmith-cl3ez]

+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?

[@DodderingOldMan]

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.

[@Mathologer]

+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 :)

[@DodderingOldMan]

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

[@davidm3210]

Bravo, the only explanation I have seen which clearly lays out this concept.

[@aaronk2250]

Very good video. Well explained in about as understandable as such a complicated subject could be. He was like that science teacher in school that actually cared.

[@Skellborn]

Love this channel :)

[@anonymous.t6649]

Mandelbrot REALLY actually scares me somehow. It just doesn't stop when it really needs to.

[@Jupiter-T]

Like an uncle who just keeps talking?

[@dcjunkieful]

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.

[@jamesg4987]

Very good explanation... Probably the best I've seen so far

[@Pr1meOne]

1:01 it looks like a nebula in space that looks just like a Mandelbrot

[@pendergastj]

Thats really interesting, I never knew that! :)

[@memeking9926]

If you spend enough time studying the shapes, you'll start getting freaked out when you realize you've seen everything before. ;)

[@seligman99]

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

[@Mathologer]

+seligman99 Wow, this is really beautiful. Thank you very much for contributing this rendering :)

[@dinoscythe6335]

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 :)

[@dinoscythe6335]

A lot of other interesting new knowledge in this video.

[@hoehue6362]

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.

[@Kram1032]

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.

[@DodderingOldMan]

+Kram1032 I... don't understand many of those words :P But that looks awesome!

[@Mathologer]

+Kram1032 That looks very cool :)

[@Radonatos]

+Kram1032 I'd say it looks more like Yoda... nice one!

[@Kram1032]

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...

[@DodderingOldMan]

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!

[@Iv_john_vI]

666K views! The dark side is strong in this one...

[@RockHanger]

‘Absolutely incredible presentation!

[@richardray7976]

Beautiful breakdown!

[@simonpranzi7898]

Looking for someone in the comments who drops acid and does this math

[@ianjackson7810]

lol... it was my first experience with magic mushrooms when i was 13 that sparked my interest in math and science. Good stuff.

[@DJGenki]

I'm a person who does math and doesn't need acid because of it. :)

[@treegone]

So if you stopped doing math you'd need acid?

[@sintheticsounds1686]

+Simon It was on a psychedelic forum that I learned of the Mandelbrot set.

[@syntheticsynesthete2542]

love me some lsd. and love me some math

[@BusterKitten]

I was waiting to see where Mandelbrot tells Luke he's his uncle on his mother's side.

[@barrankobama4840]

Very nicely done!

[@AdamFerrari64]

This video helped me understand The Logistic Map a lot better- WOW! Thank you sir

[@georgelionon9050]

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.

[@cons8501]

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

[@georgelionon9050]

edited, yes you're correct.

[@HearTruth]

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.

[@noahegler9131]

I think you missed the point of the conversation.

[@LeadenMarshmallow]

did you actually watch this video?

[@archersterling2901]

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

[@colinjava8447]

I liked the bit at the end with the prongs, never thought about that before

[@felipemercadolopez5151]

Amazing video!! Thank you so much for sharing

[@n.l.4025]

“There is no spoon.”

[@Cosine_Wave]

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?

[@Mathologer]

+Roshan Sharma These look great. Thanks for linking to these pictures :)

[@Kram1032]

+Roshan Sharma neat techniques! That last link doesn't seem to work though. It says your images aren't publicly available. Very nice experiments!

[@Cosine_Wave]

Kram1032 Oh, oops, here's a link that'll hopefully work. imgur.com/a/yoa6d

[@Kram1032]

Those look insane! neat!

[@tune_m]

How did you make these images?

[@Sarika428]

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

[@jackmack1061]

Best site of its type I know of. Many thanks.

[@oreodog]

SO COOL HEHE I love your videos

[@hindigente]

Excellent video, congratulations. I wish it was much longer and I wouldn't mind if it were a bit more technical.

[@adamweishaupt3733]

The channel is meant to be as accessible as possible, which means relatively short videos that use simple terms.

[@vampireducks1622]

Great presentation! Thanks!

[@nathangloversr]

Your comedy was so refreshing :))))) great video, very accurate and explained very well :))))

[@riccardodellorto4267]

Can you do a sequel, like you said in the end, about the oddities of this graph? Thank you!

[@NawnyaBusinaz]

No offense but I enjoy the artistic side of the Mandelbrot over the mathematical.

[@bradybenson8018]

I'm offended

[@nemou4985]

- there is no artistic side without the mathematical - Also, the buddhabrot is pretty artistic

[@silmeria1984]

This is fascinating. Thank you.

[@gregcollins7602]

very cool. love the visuals.

[@MrMe-yu7ve]

What would a fractal with the equation Z*i0=C²+Z*i0+C³ look like?

[@JulioSanchez-kf6gk]

Im so high on acid

[@ericwoods6705]

Julio Sanchez stay trippy my hippie

[@MarkosKapox]

still high at acid?

[@bruno-tt]

This is amazingly well explained - I have little knowledge about complex numbers, but was able to completely understand this.

[@Mathologer]

That's great :)

[@johngaltshank]

Top-shelf work.