The Mandelbrot Set: Inside Math's Famous Fractal
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- Опубліковано 20 тра 2024
- The Mandelbrot set is a special shape, with a fractal outline. Use a computer to zoom in on the set’s jagged boundary and no matter how deep you explore, you’ll always see near-copies of the original set - an infinite, dizzying cascade of self-similarity and novel features. The Mandelbrot set is a perfect example of how a simple mathematical rule can produce incredible complexity.
This video covers how the Mandelbrot set is constructed by iterating a quadratic function on the complex plane. It also delves into the connections between Mandelbrot and Julia sets while explaining the mechanics of how they both work. We also retrace the history of the discovery and exploration of these important sets, including current research on solving the key Mandelbrot Locally Connected conjecture (MLC).
Read the full article: www.quantamagazine.org/the-qu...
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Specials thanks for their Mandelbrot Set visualizations:
Maths.town / @mathstown
Movie Vertigo / @movievertigo
Dynamic Maths www.dynamicmath.xyz/mandelbro...
Fractally luckgrib.com/fractally/
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Recommend in-depth videos about the Mandelbrot set:
What's so special about the Mandelbrot Set? - Numberphile • What's so special abou...
The Mandelbrot Set - Numberphile: • The Mandelbrot Set - N...
Beyond the Mandelbrot set, an intro to holomorphic dynamics - 3Blue1Brown: • Beyond the Mandelbrot ...
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Chapters:
00:00 What is the Mandelbrot set?
00:58 How an iterated quadratic function defines the Mandelbrot set
01:30 The field of complex dynamical systems
01:54 Julia sets explained
04:06 The discovery of the Mandelbrot set
05:03 Constructing Mandelbrot sets vs Julia sets
05:53 Why mathematicians study the boundary regions
06:22 Mandelbrot Locally Connected conjecture, MLC
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Quanta Magazine is an editorially independent publication supported by the Simons Foundation: www.simonsfoundation.org/ - Наука та технологія
There is so much more to learn about the Mandelbrot set. Read the full Quanta article "The Quest to Decode the Mandelbrot Set, Math’s Famed Fractal" www.quantamagazine.org/the-quest-to-decode-the-mandelbrot-set-maths-famed-fractal-20240126/
Thank you! ^.^
Make a video on examples of np-complete problems
Amazing article, thank you!
Mandelbrot zooms are the gateway drug to math UA-cam
lmao
For me it was conways game of life
They were the gateway drug to me going to back to school to finish my B.S. in math
Superb comment!
lol 😅
Who would've known that some of the most beautiful structures in the universe come about from something simple as quadratics
Hey! Don’t forget the constant term!
@@olbluelipsI think by a quadratic they mean an entire quadratic function, which includes the constant term. And you're right as we can not forget the constants' contributions!!!!
Harmonic oscillator, which is described by quadratic potential, is the most widely used concept in physics, all the way to quantum field theories
I wrote a Julia and Mandelbrot visualizer on my Texas Instruments TI-83 calculator back in school. It took half a school day and almost a full set of batteries to make a very low resolution, low precision fractal. But it was still very cool to me 😊
Done the sane with Python in more recent times. It's so neat that you can reproduce something so profound, the infinity captured on the screen.
i made one in p5 js and one in scratch, fun
@@NonviableOfficial A is just a typo A made 😎
Reminder that the "B." in Benoît B. Mandelbrot stands for "Benoît B. Mandelbrot", more or less apocryphally speaking.
Bruhhh that's a good one.
Krispen wah!
The visualizations Quanta Magazine's videos are amazing
There is a link in the description Dynamic Mathematics with interactive applets to play around.
Great introductory video. Love the explanations with visuals and animations. I am also thrilled one of my applets was used for a few seconds. Thanks for the mention in the credits. ∞🙏
I had a dream….a rich warm lucid dream that I was beginning to emerge from, but was conscious enough to hold on to it as I began to awake. The focal point as it was ending, was of buildings bathed in afternoon sunshine. I hung onto the image and watched it as if on a screen on the back of my eyelids as it slowly dissolved into a moving Mandelbrot set. I was awake with my eyes closed to witness this stunning denouement until it finally gave way. The experience left me feeling that I had just been given an insight in to how our dreams are built on this framework. Almost like the architecture of consciousness is built upon it. It was a wonderful, transformative moment and I shall always treasure it.
Woah, that sounds fascinating
well you had to awake before knowing it's a dream right? Now what if your everday awake consiousness is...never mind.
Thanks for sharing the Mandelbrot love!
This is absolutely beautiful. I love Inner Worlds Outer Worlds and the references to fractals. I loved hearing it in more detail.
Mandelbrot gave a lecture once in my college!
What did they speak about
@@estelleg-f6069 all the brots that he's mandeled
What I find most incredible about these things is that the area defined by the set is finite but the line defining its perimeter is infinitely long... and any segment of the perimeter is also infinitely long. You can cut the perimeter into an infinite number of pieces and each piece is infinitely long 😮
Something similar to poincare model of hyperbolic space.
This gives me a strong feeling that our universe is created by a simple set of equations similar to the Mandelbrot set.
Well general relativity and the standard model can be written out conscisely in few lines
@@whataboutthis10 The full standard model langrangian is anything but concise. The concise version just condenses the various sections of it together. Even then it's nowhere near as elegant as F=ma or E=mc2. Not easy to calculate either.
@@ArawnOfAnnwn You're right, maybe it seems to be short in longitude. But it is still incredibly complicated.
Beautiful, ty for sharing!
Man, math connects SOOO much to itself - Fibbonaci sequence, golden ratio, Mandelbrot set all have a thing in common
absolutely breathtaking
I saw this when i first took acid in 89.. looked up at the stars and watched how the universe exploded and is expanding, and how it is all connected via some magnetic electric fabric.
and how the trees and rock i was laying on was just an extension of my body, and is spinning around the sun, and sun around and the centre of our galaxy..and could feel the size and distance between the stars and myself.. i was blown away how if i could jump and fly, that theres nothing between the lens of my eye and the celestial bodies i was looking at .
the feeling still persists..love it
I bought a computer 1982 just to explore the fractal world. I programmed a matrice dot printer to in detail do the printouts. It took for ever, but the results were amazing. So many organic and other natural structures seems to be fractal in principle.
The Mandelbrot set got me into IT, after having found a book in the library with code examples of how to draw them in QBASIC. Thanks, Benoit!
I remember back in high school the computer would take all night to draw one still image of the fractal expression. I almost cry seeing it in motion and in color
If math is God's language, Mandelbrod set is His humanely-accessible signature.
extremely exciting
I really hope I can contribute one day to new discoveries in this field, its probably my favorite
Great video 👍
I remember having dreams of fractals since early childhood... Sometimes they were visually beautiful, but also gave me anxiety because I'd had trouble waking up from a loop
I really dig Quanta Mag offerings. Good, intelligent, entertaining, and fundamentally reliable science.
I love this Quanta Magazine youtube channel
There are a lot of funny properties like Fibbonaci sequence.
I feel the video and the title are unfocused. It takes a while to review a well known concept, but the main point of the video is that there's research if it's a connected set (which isn't "full understanding") which is given only a passing mention. The linked article actually gives an answer to that.
while "cool", i don't understand why it would be anything beyond that. it's an algorithm which requires repetition (action) to produce what we know as the Mandelbrot set or fractal image, thus it's really just as finite physically as our ability to perform operations on it and potentially infinite as the set of basic numbers is - only cooler looking when drawn on a surface. potentially infinite isn't same as physically infinite, which means there's no mystery or "hidden world" in it - nothing beyond what we ourselves operate exists behind it.
Well, numbers are really invisible, unless one writes them down.
Does anyone know the coordinates of the center of the first video? I really liked the video. Thanks
Reverse 1999,37,mentioned this in her first encounter voice line .
quite interesting.
and very special game.
I wonder if we might learn how to use this future fractal solution and apply it to energy creation somehow? Amazing video as always!
I really like the animations when the complexity (== iteration limit?) gets gradually increased over time. I think, I didn't see this kind of animation before. It really highlights how the iteration limit and ultimate complexity of the picture are related. Before seeing this animated, I only had a fuzzy sense of that connection. Like - qualitative: "if you want to see more details, you need more iterations". Now it's much clearer. Before seeing this, I somehow assumed, with less iterations, you would just have "more noise" or something. Now I see that you actually don't get a "noisy" version but a "smoothed" one when using less iterations.
Where does color come from? A point can either belong to the set, or not belong to the set - hence the black and white graphs from the 80s. What does the color encode in the flashy modern zooms?
The Julia and Newtons fractal are also very interesting sets to analyze
Aaah memories... As a yougun' in the eighties I found an article in French (Science et Vie) about Julia sets and despite the difficulties of reading french, managed to program my own Julia generator. One black and white picture took several days to be drawn on my 8 bit computer...
Nice. I've used/modelled the Mandelbrot set and the fractal to unify the quantum with cosmology. A perfect match; published in The International Journal of Quantum Foundations, Speculations. My work is an invitation for more work for others with greater minds.
Math truly is a wonderful, beautiful thing.
If you should happen to *accidentally* do any illicit psychedelic drugs, definitely watch a mandelbrot zoom. You'll get lost in its beauty (or so I've heard, anecdotally)
Can confirm from personal experience. Although on shrooms and nitrous my whole reality turned into fractals and now here I am studying theoretical physics partly because of that experience.
@@krumkutsarov618jo im studying chemstry and i had a pantheistic god experience with pschydelics and nitrous.
Fractals showed up.
And im also looking in chaos theory now to understand why this Happens.
Nice to see an alley
The Mandelbrot set is a door to infinity ♾️
I have a video series on building a digital circuit that renders the mandelbrot set :)
I just checked it out and it's great. I watched part 0 and then skipped ahead to part 5 and it's pretty impressive. How easy do you think it would be to go from this circuit to a real physical circuit in hardware? I've kind of always wanted to play around with FPGAs but never really had the excuse to do it.
@@aspuzling it shouldn't be too difficult to get it in hardware. The tough thing is finding an FPGA with enough ram lol
go alex!!!
i think it's all about prespective. if it were to be visualised in a depth center it would help us greatly understand how the crucual points of distance would form, then have an iris depth and distortion prespective to it. idk leave me alone
I think my favorite fact about Mandelbrot and Julia sets is that if you zoom in on a part of the Mandelbrot set and choose a starting point to make a Julia set for and do the same zoom on that Julia set they will look very similar. But as you zoom out they might look entirely different
So what if it is connected, what understanding would this lead to?
Seems a visualisation of Fibonacci sequence, constructal law, flow, life in the third dimension, and the holographic universe all just swimming in my head at the moment with only more questions and ruminations.
Lots of fractal zoom videos on UA-cam, some are even in 4K for extra crispy detail.
It's one the keys into unlocking the perceived universe with our own human sense.
Can I get in on this?
can you just tell me if there is something new in this video or if this is just another mandelbrot set video because i've seen about enough of them
Output turns out to be an input!
I always ask myself, how would/could this ever be reverse-engineered, going from graphics to an equation? And yet it seems to be a really simple algorithm in the end producing such an infinity/eternity.
Now take our universe and try to reverse engineer it into a formula. How simple will it be in the end?
How do you get the colors?
Usually, number of iterations to escape to infinity.
One vital thing this video neglected to make clear is that the colored parts are NOT part of the the Mandelbrot set. Only the black parts have not been positively excluded from the set. The colors usually represent how far away from the set a point is. Simplest version of coloring is to have the color represent how quickly the absolute value became greater than a value where it has been proven that it will always go off to infinity. If it happens quickly it's "far" and the more iterations it takes the closer it is. And since we can't keep iterating forever there's a cutoff number where you give up and say it's probably part of the set... Probably
Imagine hiding some valuable stuff way down inside where no one will ever find it.
I’m also a methemetician and I too know that a fine tooth comb is disconnected upon closer investigation
2:40 They gloss over this but it's something from quantum mechanics. In order to get the math to work for wave functions to work on any given 2d plane imaginary numbers are needed. Since we live in a 3d universe, to get any real world application of this math you must multiply two imaginary equations together making all imaginary numbers cancel out.
as far as I know for quantum mechanics the probability is just the squared magnitude of the complex number, aka squared distance from origin
this would produce a 1d value (real number) from a 2d input (complex number)
what exactly do you mean by "getting real number by multiplying complex numbers and making complex parts zero out" and how is 3d space relevant?
why do we use the complex plane and not just the regular x and y axis?
Complex numbers can be multiplied, resulting in a complex number.
Multiplication is best visualized in 'polar coordinates', it is both stretching and rotation, hopefully this inspires looking up the complex numbers)
In math lingo the complex numbers form an algebra, while a real plane (x, y) only forms a vector space
What is the Mandelbrot Locally Connected Conjecture?
Quanta Magazine ily
By the way if you can Read Benoit Mandelbrot auto biography, is more about surviving 2nd WW, and being a rebel scientist
Why don't you set infinite on 8 plains instead of + graphs?
One imagines the pedagogy of mapping Mandelbrot complexity, might (one day) extend to unravel the mystery; of how the same (identical) DNA in the 46 chromosomes, in the nucleus of every cell in your body; selectively activate to express the diversity of cells in your body, at just the right time; in just the right places to biosynthesize you as an individual.
4:32 I remember not so crude or poorly made paintings in the 80's. I remerber the august 1985 scientific american. I think that he is talking about 70's
An 8-bit desktop computer from the early 1980s can make a decent image overnight...if you don't zoom in very much. In order to calculate with the precision needed to zoom in, you need a computer that can handle higher precision. A 16. It, 32 bit, or 64 bit machine like is common today.
❤❤❤
our experience of consciousness is probably analogous to computing a fractal point within the ultra complex state-space of our experience (brain, body and surroundings)
Something tells me Pi is behind all of these sets
Pi is behind anything involving the complex plane, including the gaussian distribution from statistics.
Disregarding that which reaches infinity may be more practical. However disregarding such information will never provide the true answers.
I'm no expert, but I just have a feeling that our human propensity to believe that uncovering the mathematical formula of how these things work will explain nature. And I'm not sure, the mathematics is just an expression, a "description" of what's happening. But we already know enough to draw some conclusions... if nature wants to build complex things, it has to start with simpler things. Simple things growing in a pattern is how complex structures arise. The math of it all is surely very interesting and useful in many domains, but from a phhilosophy of science angle... learning greater and greater expressions to describe the process in mathematical terms won't get us to why nature does it that way. Will it? I dunno.
Off-the-cuff example... cosmological constants. Like the speed of light, or Planck units. Sure, we know them to fascinating degrees of precision. Does any of that really give us the mechanical "why"? Not really. I guess it becomes an ontological debate in the end.
random vs nature selection ? both of them is same.... without random, there is no nature selection... random is important to any patent of life... we also need change that life is a similarly like a car, car is a move because of momentum of energy, same as us, human also move because of momentum of energy... but the explosion is happening is very very very smal scale.... this is why the temperature exist on human...
Its analogous to our universal construct. Once humans get to an extremely high technology level, it is inevitable we will create extreme high detail universal simulations ourselves with the building blocks of our own universal construct as a base template complete with full nerve induction, no different from our own. We will allow it to run its course. We can speed it up as needed, slow it down, or reverse it. We can terminate it, or use an entropic model the way our own appears at the moment. Imagine this new simulated universe eventually doing the same, etc, etc, etc. Energy manipulation and eternities of this. This video reminded me of this, I think its completely as natural as the math, as natural as 60 atom units self-organizing to create every single thing we see in our universal construct.
Do Sims play computer games and have they already developed like a Sim^2 game on their Sim computers;?
Will never understand why is this better than Koch"s snowfalke. I know that there must be some proof this is fractal (obviously), but it doesnt even look like one! ANd also so weird name....
von Koch's curve/snowflake is identical (up to rotation and scaling) at all magnifications. Likewise, the Mandelbrot set has self similarity (almost everywhere if you zoom in), but also infinite diversity, so it is much richer to study.
@@FadkinsDiet yeah but its just so weird! Not pretty at all! And the name too! 😆
@@kexcz8276 So you have a problem with Jewish names specifically? Or just any non-American name? Either way it seems like you're telling on yourself
Thank you for sharing.
God is amazing. He is a God of numbers. Numbers is a Biblical book in the Torah.
The first time I saw the Mandlebrot Set was the amazing vid by Dr Jason Lisle.
It reminds me of the toy Spirograph. Anyone remember Spirograph! I had it in the 60's, it was awesome !
I don't understand step #1
epic
This feel like a Tool video
Zooming out or in..... same same
a full understanding 🤔if connectedness conjecture is proven true for mandelbrot set
ummm
there is more to it
Geeze I feel like an idiot.
The bg music is intrusive.
Maybe it's a bit too loud
Arthur C Clarke in his later years mistakenly claimed such as "God's thumbprint." He didn't realize that this extremely basic math was precisely evidence of the contrary....
Couldn't the universe be fractal whether one zooms in or out?
😍😍
This is how you travil in space chums.
… I think that the idea of fractals and of strange attractors, etc., is something that’s being ignored in today’s ideas of how to do neural networks, artificial intelligence. You ask, well, how do you incorporate such ideas into AI? I don’t know, it’s just an intuition I have; something to be looked at by people who are much smarter than I.
Generated using TalkBud
📝 Summary of Key Points:
📌 The Mandelbrot set is a mathematical concept that represents a form of modern algorithmic exploration. It is generated by iterating a quadratic equation and showcases the complexity that can arise from a simple rule.
🧐 Julia sets are intricate shapes produced by iterating a function of complex numbers. The Mandelbrot set is closely related to Julia sets and is constructed by iterating the same quadratic equations used to produce Julia sets.
🚀 The Mandelbrot set has a complex fractal boundary that reveals intriguing features when zoomed in. Mathematicians are particularly interested in the local connectivity of the set, which is a 40-year-old problem known as the Mandelbrot locally connected conjecture.
💡 Additional Insights and Observations:
💬 "The Mandelbrot set acts as an atlas, cataloging the different kinds of Julia sets."
📊 No specific data or statistics were mentioned in the video.
🌐 The video does not reference any external sources or references.
📣 Concluding Remarks:
The Mandelbrot set is a fascinating mathematical concept that showcases the complexity and beauty that can arise from a simple rule. It is closely related to Julia sets and has a complex fractal boundary. The local connectivity of the set remains a longstanding problem in mathematics, but even with a complete understanding, the fascination with the Mandelbrot set would not diminish.
The idea of "fully understanding the mandlebrot set" is laughable. Utter hubris.
Why we are not taught these concepts in our schools ? Even a cursory understanding will ignite the interest
0 nor infinity neither are reachable
You are just obaying the tru laws of mathimatics. Wht it looks kinda like space.
Fingerprint of God.
He has funny shape fingers
We can't even comprehend or understand what he looks like. @@DanceySteveYNWA
Rethinking how I should store my crypto 😂
Uahhhhh
I have cloaked geometry models in blender
BTW there’s no necessity of complex numbers for any of this, you can do it in the xy plane just as well.
Haha Z
Yeah. Pronouncing 'c' and 'z' in the same way will not lead to any confusion at all... 🙄
Unvoiced vs voiced. Not the same.
First
Almond bread /mandelbrot. (bróót). You can not pronounce foreign names. That makes it difficult to understand these video's.
background music completely unnecessary and extremely annoying
Nothing compared to 3d fractals....
Interesting choice of people chosen to explain yet another achievement of White Christan Men...
Mandelbrot was Jewish