The Mandelbrot Set: Inside Math's Famous Fractal

Рет қаралды 182,942

Күн бұрын

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...
----------
Specials thanks for their Mandelbrot Set visualizations:
Maths.town / @mathstown
Movie Vertigo / @movievertigo
Dynamic Maths www.dynamicmath.xyz/mandelbro...
Fractally luckgrib.com/fractally/
----------
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 ...
----------
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
----------
- VISIT our website: www.quantamagazine.org
- LIKE us on Facebook: / quantanews
- FOLLOW us Twitter: / quantamagazine
Quanta Magazine is an editorially independent publication supported by the Simons Foundation: www.simonsfoundation.org/

Пікірлер: 160

@QuantaScienceChannel 3 ай бұрын

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/

@TheMemesofDestruction 3 ай бұрын

Thank you! ^.^

@meenakshikalaskar6786 3 ай бұрын

Make a video on examples of np-complete problems

@johannesdeboeck 3 ай бұрын

Amazing article, thank you!

@Mutual_Information 3 ай бұрын

Mandelbrot zooms are the gateway drug to math KZfaq

@neutron417 3 ай бұрын

lmao

@rickybloss8537 3 ай бұрын

For me it was conways game of life

@rachel_rexxx 3 ай бұрын

They were the gateway drug to me going to back to school to finish my B.S. in math

@benjaminthishandleistaken 3 ай бұрын

Superb comment!

@OfficialGOD 3 ай бұрын

lol 😅

@TheAliencreeper13 3 ай бұрын

Who would've known that some of the most beautiful structures in the universe come about from something simple as quadratics

@olbluelips 3 ай бұрын

Hey! Don’t forget the constant term!

@antagonisticalex401 3 ай бұрын

@@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!!!!

@whataboutthis10 3 ай бұрын

Harmonic oscillator, which is described by quadratic potential, is the most widely used concept in physics, all the way to quantum field theories

@antivanti 3 ай бұрын

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 😊

@DawidUliczny-ro7eo 3 ай бұрын

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.

@anikinmartinez4726 2 ай бұрын

i made one in p5 js and one in scratch, fun

@antivanti 24 күн бұрын

@@NonviableOfficial A is just a typo A made 😎

@minhuang8848 3 ай бұрын

Reminder that the "B." in Benoît B. Mandelbrot stands for "Benoît B. Mandelbrot", more or less apocryphally speaking.

@picco_only 3 ай бұрын

Bruhhh that's a good one.

@usernameluis305 3 ай бұрын

Krispen wah!

@Salted_Potato 3 ай бұрын

The visualizations Quanta Magazine's videos are amazing

@BeautyInMath 3 ай бұрын

There is a link in the description Dynamic Mathematics with interactive applets to play around.

@rossmcleod7983 3 ай бұрын

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.

@clockwork8548 3 ай бұрын

Woah, that sounds fascinating

@wildmanmike100 3 ай бұрын

well you had to awake before knowing it's a dream right? Now what if your everday awake consiousness is...never mind.

@BeautyInMath 3 ай бұрын

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. ∞🙏

@markus_park 3 ай бұрын

Mandelbrot gave a lecture once in my college!

@estelleg-f6069 3 ай бұрын

What did they speak about

@zulqarnain9955 3 ай бұрын

@@estelleg-f6069 all the brots that he's mandeled

@caaanita 3 ай бұрын

This is absolutely beautiful. I love Inner Worlds Outer Worlds and the references to fractals. I loved hearing it in more detail.

@MathsTown 3 ай бұрын

Thanks for sharing the Mandelbrot love!

@dpie4859 3 ай бұрын

This gives me a strong feeling that our universe is created by a simple set of equations similar to the Mandelbrot set.

@whataboutthis10 3 ай бұрын

Well general relativity and the standard model can be written out conscisely in few lines

@ArawnOfAnnwn 3 ай бұрын

@@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.

@samuelrojas3766 3 ай бұрын

@@ArawnOfAnnwn You're right, maybe it seems to be short in longitude. But it is still incredibly complicated.

@Elena-jl5jv 20 күн бұрын

Beautiful, ty for sharing!

@gl0bal7474 3 ай бұрын

absolutely breathtaking

@EquaTechnologies 3 ай бұрын

Man, math connects SOOO much to itself - Fibbonaci sequence, golden ratio, Mandelbrot set all have a thing in common

@psycox8758 3 ай бұрын

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 😮

@venkteshv377 3 ай бұрын

Something similar to poincare model of hyperbolic space.

@vedantnipane2268 18 күн бұрын

I love this Quanta Magazine youtube channel

@dewaard3301 3 ай бұрын

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!

@DanielWSonntag 3 ай бұрын

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

@opfwkyod 3 ай бұрын

extremely exciting I really hope I can contribute one day to new discoveries in this field, its probably my favorite

@MathewSan_ 3 ай бұрын

Great video 👍

@ebptube 3 ай бұрын

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.

@norb6492 3 ай бұрын

I really dig Quanta Mag offerings. Good, intelligent, entertaining, and fundamentally reliable science.

@bee1707 3 ай бұрын

If math is God's language, Mandelbrod set is His humanely-accessible signature.

@williesco760 2 ай бұрын

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

@tuams 3 ай бұрын

I wonder if we might learn how to use this future fractal solution and apply it to energy creation somehow? Amazing video as always!

@mikejohnston9113 3 ай бұрын

Does anyone know the coordinates of the center of the first video? I really liked the video. Thanks

@MusicEngineeer 3 ай бұрын

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.

@Rxke 3 ай бұрын

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...

@arilegall2001 3 ай бұрын

The Julia and Newtons fractal are also very interesting sets to analyze

@dubstarm511 Ай бұрын

Reverse 1999,37,mentioned this in her first encounter voice line . quite interesting. and very special game.

@Danboi. 3 ай бұрын

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

@Eye_Exist 25 күн бұрын

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.

@Elena-jl5jv 20 күн бұрын

Well, numbers are really invisible, unless one writes them down.

@reddmst Ай бұрын

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?

@markoboychuk 3 ай бұрын

The Mandelbrot set is a door to infinity ♾️

@danraz0ralice340 3 ай бұрын

It's one the keys into unlocking the perceived universe with our own human sense.

$@fractalnomics$

@fractalnomics 3 ай бұрын

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.

@Smoth48 3 ай бұрын

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)

@krumkutsarov618 3 ай бұрын

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.

@ZAck56100 3 ай бұрын

@@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

@WildEngineering 3 ай бұрын

I have a video series on building a digital circuit that renders the mandelbrot set :)

@aspuzling 3 ай бұрын

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.

@WildEngineering 3 ай бұрын

@@aspuzling it shouldn't be too difficult to get it in hardware. The tough thing is finding an FPGA with enough ram lol

@mr8966 3 ай бұрын

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.

@Amonimus 3 ай бұрын

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.

@antivanti 3 ай бұрын

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

@SoCalFreelance 3 ай бұрын

Lots of fractal zoom videos on KZfaq, some are even in 4K for extra crispy detail.

@euclid3044 3 ай бұрын

go alex!!!

@dronefootage2778 3 ай бұрын

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

@NavyFountain 3 ай бұрын

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

@SnoopyDoofie 3 ай бұрын

Imagine hiding some valuable stuff way down inside where no one will ever find it.

@Snowflake_tv 3 ай бұрын

Output turns out to be an input!

@ConnoisseurOfExistence 3 ай бұрын

So what if it is connected, what understanding would this lead to?

@jc4418 3 ай бұрын

I’m also a methemetician and I too know that a fine tooth comb is disconnected upon closer investigation

@carrellwashington4007 3 ай бұрын

Can I get in on this?

@mirrkkoo 3 ай бұрын

How do you get the colors?

@alistairmackintosh9412 3 ай бұрын

Usually, number of iterations to escape to infinity.

@antivanti 3 ай бұрын

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

@gabrielreinert8458 28 күн бұрын

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?

@samsonsoturian6013 3 ай бұрын

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.

@creativenametxt2960 3 ай бұрын

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?

@uriabinenshtok 3 ай бұрын

why do we use the complex plane and not just the regular x and y axis?

@whataboutthis10 3 ай бұрын

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

@Snowflake_tv 3 ай бұрын

What is the Mandelbrot Locally Connected Conjecture?

@ArchiRuban 3 ай бұрын

Quanta Magazine ily

@y4lnux 3 ай бұрын

By the way if you can Read Benoit Mandelbrot auto biography, is more about surviving 2nd WW, and being a rebel scientist

@antonychipman3088 3 ай бұрын

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.

@jamesmoran7511 2 ай бұрын

Disregarding that which reaches infinity may be more practical. However disregarding such information will never provide the true answers.

@Mrcometo 3 ай бұрын

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

@FadkinsDiet 3 ай бұрын

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.

@mrbengg 3 ай бұрын

Something tells me Pi is behind all of these sets

@FadkinsDiet 3 ай бұрын

Pi is behind anything involving the complex plane, including the gaussian distribution from statistics.

@hsaneener9292 3 ай бұрын

Why don't you set infinite on 8 plains instead of + graphs?

@kexcz8276 3 ай бұрын

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....

@FadkinsDiet 3 ай бұрын

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.

@kexcz8276 3 ай бұрын

@@FadkinsDiet yeah but its just so weird! Not pretty at all! And the name too! 😆

@FadkinsDiet 3 ай бұрын

@@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

@hotmeish 3 ай бұрын

❤❤❤

@liamcore7203 3 ай бұрын

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.

@whataboutthis10 3 ай бұрын

Do Sims play computer games and have they already developed like a Sim^2 game on their Sim computers;?

@mercster 3 ай бұрын

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.

@mercster 3 ай бұрын

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.

@Sweetiepiepizza 2 ай бұрын

Zooming out or in..... same same

@jkem3311 3 ай бұрын

Couldn't the universe be fractal whether one zooms in or out?

@willben44b49 3 ай бұрын

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)

@djayjp 3 ай бұрын

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....

@jac585 Ай бұрын

This is how you travil in space chums.

@pass179 3 ай бұрын

Geeze I feel like an idiot.

@Elena-jl5jv 20 күн бұрын

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 !

@HatiKeseorangan 3 ай бұрын

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...

@OneBentMonkey 3 ай бұрын

This feel like a Tool video

@jpphoton 2 ай бұрын

a full understanding 🤔if connectedness conjecture is proven true for mandelbrot set ummm there is more to it

@TeslaElonSpaceXFan 3 ай бұрын

😍😍

@onehatmedia 3 ай бұрын

The idea of "fully understanding the mandlebrot set" is laughable. Utter hubris.

@agrimkumar9623 3 ай бұрын

Why we are not taught these concepts in our schools ? Even a cursory understanding will ignite the interest

@justmesayinghai 11 күн бұрын

epic

@masonwalker1301 3 ай бұрын

I don't understand step #1

@Streetsy 3 ай бұрын

The bg music is intrusive.

@xoiyoub 3 ай бұрын

Maybe it's a bit too loud

@jac585 Ай бұрын

You are just obaying the tru laws of mathimatics. Wht it looks kinda like space.

@Snowflake_tv 3 ай бұрын

Uahhhhh

@nowsc 3 ай бұрын

… 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.

@1leiffearn 3 ай бұрын

Rethinking how I should store my crypto 😂

@user-yc6xn5ze6h 3 ай бұрын

Fingerprint of God.

@DanceySteveYNWA 3 ай бұрын

He has funny shape fingers

@deMocskonyi 3 ай бұрын

We can't even comprehend or understand what he looks like. @@DanceySteveYNWA

@marcalimarian 3 ай бұрын

0 nor infinity neither are reachable

@ericchastain1863 3 ай бұрын

I have cloaked geometry models in blender

@abdelkaioumbouaicha 3 ай бұрын

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.