Decoding The Mandelbrot Set: Math’s Famed Fractal

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).

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Specials thanks for their Mandelbrot Set visualizations:
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Recommend in-depth videos about the Mandelbrot set:
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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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[Mutual_Information]

Mandelbrot zooms are the gateway drug to math YouTube

[TheAliencreeper]

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

[antivanti]

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 😊

[Salted_Potato]

The visualizations Quanta Magazine's videos are amazing

[BeautyInMath]

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. ∞🙏

[minhuang]

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

[EquaTechnologies]

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

[caaanita]

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

[MathsTown]

Thanks for sharing the Mandelbrot love!

[rossmcleod]

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.

[markus_park]

Mandelbrot gave a lecture once in my college!

[psycox]

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 😮

[ebptube]

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.

[dpie]

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

[opfwkyod]

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

[DanielWSonntag]

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

[bee]

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

[Danboi.]

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

[dewaard]

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!

[williesco]

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