The dark side of the Mandelbrot set
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- čas přidán 5. 09. 2024
- 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!
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!
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.
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.
"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.
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
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!
@@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.
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
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.
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
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.
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 :)
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
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 🙃
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 :)
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!
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.
This is fantastic. I've never seen anyone tackle the obvious questions about the set like this video does.
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 😭😭😭
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
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 never cease to be amazed by the Mandelbrot 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?
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.
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?
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*
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 :)
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
1:24
damn... mandelbrot looking kinda thicc
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!!
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.
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.
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.
The best and cleanest and easiest explanation there on the mandlebrot set. Thank you!
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
Danke für die fantastischen Videos. Sehr schön visualisiert. Man lernt nie aus. Wenn man aus einer anderen Mathe-Richtung kommt, ist das echt interessant.
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
Mandelbrot REALLY actually scares me somehow. It just doesn't stop when it really needs to.
Like an uncle who just keeps talking?
Great video, as always!
Yeah
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!
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?
I really thought this would be a video of a Mandelbrot Set Zoom with The Dark Side of the Moon playing in the background
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
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.
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.
At what point is the foci of the cardioid that you can see in the biggest part of the mandelbrot set?
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. ;)
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.
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 :)
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.
⭐️⭐️⭐️⭐️⭐️THAT WAS FOR THE LACK OF A BETTER WORD: BRILLIANT! Thank you, Mathologer! 😀 👍
Glad you liked it :)
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.
What would a fractal with the equation Z*i0=C²+Z*i0+C³ look like?
It's really just x^2 + y^2 with a limit set at 4. They make it sound complicated by using "imaginary" numbers.
First fractal program, discover on Amiga computer years 80'!
Amiga and news AmigaOS4 ruleeez! 👏✌️👌
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.
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
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.
Best interpretation I ever saw! Thank you! How deeply connected everything is...
I liked the bit at the end with the prongs, never thought about that before
666K views! The dark side is strong in this one...
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.
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 :)
12:00 that's interesting, this looks very similar to what happens with the logistic map formula and its bifurcation/eventual chaos
they are similar formulas, though, but it's interesting that this works with inputs from -2 to 1/4 instead of 0 to 4
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!!!
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
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.
Years later and this is still the only video I found that explains this so well
I was waiting to see where Mandelbrot tells Luke he's his uncle on his mother's side.
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! ).
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
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
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.
Hilarious trick to teach me the Mandelbrot!! Job done teacher!!! Thanks. (it's taken years for me!!!).
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.
4:51 Escape radius = x
x^2 - 2 = x
Therefore x = 2
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 is an interesting variant of the julia-set! I don't know, if it already exists, but whatever.
You take the normal iteration and change the constant c. (With every iteration you increase i)
So for example:
c = 0 if i = 0 Mod 3
c = -0.75 if i = 1 Mod 3
c = 0.5 if i = 2 Mod 3
Is this a know variant or did I discover this?
Can you do a sequel, like you said in the end, about the oddities of this graph?
Thank you!
For anyone interested in playing with Mandelbrots I recommend Tierazon 2. Still runs fine even on Windows 10, plenty of formula and customisation options. Pity about the singlethreading (SMP was only barely a thing back in 98) but it does utilise the available power of a single core very well, so even moderately deep zooms are fast to render.
“There is no spoon.”
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?
Wow, this explained it really well. Thanks
Buddhabrot almost sounds like "Butterbrot". But it is - wow - very beautiful!
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?
Out of all that, out of all my life dealing with mathematically *offset* people, what stuck, is that x^2+(-1) was stated as x^2-1, beautiful. absolutely. I have endured years of people insisting that since you're adding something it's therefore easier, and they refuse to try subtraction. it's wonderful to see hope for the world in the form of someone automatically saying x-1 without trying to put it that way, its just natural.
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.
It isn't often described, but when you imagine rotating the set like a rotisserie chicken, you can see how the chaos occurs and how even that is mapped into the 'order' of the set. So even paradoxes belong to the Mandelbrot set... >boom< >pop< >whizz< Don't mind me, just hitting my chaos points as my mind blows. No Death star laser required.
No offense but I enjoy the artistic side of the Mandelbrot over the mathematical.
I'm offended
- there is no artistic side without the mathematical
- Also, the buddhabrot is pretty artistic
I wouldn't call those "tractor beams" (aren't those star TREK anyway?) but tractor points - you see their behavior when you draw the sequence for a point (below a hovering mouse pointer for example) - like three points of conversion inside the upper and lower bulb .
+Xario Withoutalastname Sure, just trying to stay in "character". But definitely there is a tractor beam in action in the first Star Wars movie. Time to watch it again :)
Mathologer
You've got it!
So, the Fig Tree is connected to the Almond Bread.
Can you make a video updating on the most interesting mathematical discoveries about the Buddha?
Big like on this one. Those relationships are gorgeous!
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!
The dark side of the almond bread set *lmao*
影 ShadowZZZ
I am not a mathematician, but I didn't find this completely incomprehensible. Thank you.
I feel like it’s a patrallel universe.
My original reason for owning a computer. Just so I could generate mandelbrot sets and zoom in.
Indeed... Yes, you are a true " JEDI MATH MASTER " !!!
The way the number of periods on the parabola doubles more and more quickly, becomes chaotic, and has islands of order reminds me a lot of the function that produces the Feigenbaum constant. Does that appear here too?