Structure from Simplicity: A Look at the Mandelbrot Set
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- Опубліковано 2 сер 2021
- Math might just be a formal game we play, but fractals seem to point toward something higher, something hauntingly Platonic.
This video shows a part of the Mandelbrot set, a famously simple equation which tirelessly unfolds into an infinitely elaborate landscape.
As you remember from algebra, the real numbers can be represented on a number line. Complex numbers, which have a real part and an imaginary part, can be represented on a number plane, with the real part on the x axis and the imaginary part on the y axis. “Real” and “imaginary” are misnomers; both flavors of number are equally real (or equally imaginary, if you’re a skeptic), so complex numbers should just be thought of as 2D entities that have all the properties required to enable counting, plus some more general directional properties which play an important role in algebra.
The Mandelbrot set is calculated by repeatedly squaring every number in the complex plane, then adding it to itself, squaring it again, adding it to its original self again, and so on, in an iterative process. When you do this, small enough numbers will stay finite, large numbers will grow quickly toward infinity, and medium-sized numbers will grow at different rates relative to each other. The white patterns in this video are areas where the complex numbers are growing slowly relative to their neighbors.
It took my laptop a few hours to calculate this. Each frame has 4 million complex numbers, with a few hundred iterations at each point, for a few billion calculations per frame. And it’s a 60 fps video, so in total it’s a few dozen trillion repetitions of that simple formula.
The pattern’s structural richness arises in part from the huge number of calculations involved, but in essence this is just the flowering of a tiny seed which has been given ample computational nutrition. And that seed’s beautiful potential, which has waited implicitly and invisibly in logical abstraction long before anyone drew it out, hints that math may be more than just a formulaic story we tell; the Mandelbrot set, at least, seems to have roots which extend beyond time. What else is in that soil?
Thank you for this video, the description, the previous comment's replies.
And thank you for the Amazon review that led us to this work. :)
You’re welcome! :) Thanks for enjoying the video and the review.
Are you the one that posted that long ass review about the tungsten cube?
Yeah lol
@@RichBehiel is it true you cured your mortality by lifting the cube?
@@yonyem6724 yes, that is correct.
@@RichBehiel You now have an eternity to gaze into the fractal abyss, tungsten heavily in hand.
"And if thou gaze long into an abyss, the abyss will also gaze into thee."
But even an abyss dare not gaze into a cube of tungsten...
@@drangusgrangus those who dare to gaze into tungsten, and who can absorb its gaze in return, will be blessed with limitless wisdom and unstoppable power. I once tried to stare at my tungsten cube. Couldn’t last more than a second. Though I have now become accustomed to the density of the cube, I cannot yet handle its gaze. There’s something timeless about it, some ineffable aspect of its appearance that sits still and yet moves deeply, something that remembers, knows, and predicts all that ever was, is, and ever will be. This is no ordinary metal.
All the people here who bought this wireless tungsten cube to admire its surreal heft have precisely the wrong mindset. I, in my exalted wisdom and unbridled ambition, bought this cube to become fully accustomed to the intensity of its density, to make its weight bearable and in fact normal to me, so that all the world around me may fade into a fluffy arena of gravitational inconsequence. And it has worked, to profound success. I have carried the tungsten with me, have grown attached to the downward pull of its small form, its desire to be one with the floor. This force has become so normal to me that lifting any other object now feels like lifting cotton candy, or a fluffy pillow. Big burly manly men who pump iron now seem to me as little children who raise mere aluminum.
I can hardly remember the days before I became a man of tungsten. How distant those days seem now, how burdened by the apparent heaviness of everyday objects. I laugh at the philistines who still operate in a world devoid of tungsten, their shoulders thin and unempowered by the experience of bearing tungsten. Ha, what fools, blissful in their ignorance, anesthetized by their lack of meaningful struggle, devoid of passion.
Nietzsche once said that a man who has a why can bear almost any how. But a man who has a tungsten cube can bear any object less dense, and all this talk of why and how becomes unnecessary.
Schopenhauer once said that every man takes the limits of his own field of vision for the limits of the world. Tungsten expands the limits of a man’s field of vision by showing him an example of increased density, in comparison to which the everyday objects to which he was formerly accustomed gain a light and airy quality. Who can lament the tragedy of life, when surrounded by such lightweight objects? Who can cry in a world of styrofoam and cushions?
Have you yet understood? This is no ordinary metal. In this metal is the alchemical potential to transform your world, by transforming your expectations. Those who have not yet held the cube in their hands and mouths will not understand, for they still live in a world of normal density, like Plato’s cave dwellers. Those who have opened their mind to the density of tungsten will shift their expectations of weight and density accordingly.
To give this cube a rating of anything less than five stars would be to condemn life itself. Who am I, as a mere mortal, to judge the most compact of all affordable materials? No. I say gratefully to whichever grand being may have created this universe: good job on the tungsten. It sure is dense.
I sit here with my tungsten cube, transcendent above death itself. For insofar as this tungsten cube will last forever, I am in the presence of immortality.
😂
Almost made me tear up at how beautiful it is, maybe that makes me a bit of a nerd lol. Great work and can’t wait for your next uploads.
Thanks, glad you enjoyed the video! :)
And no it doesn’t make you a nerd, it makes you someone who is interested in our beautiful world and its abstract extensions. That’s a noble character trait!
@@RichBehiel❤❤ thank you so much!! I really do appreciate the affirmation.. you’re right!! 😊
Thanks for this marvelous work , also when you stop Zooming you can find different fields inside Mandelbrot Boundaries, also when change the color mapping in different color scale
I’m glad you enjoyed it! :)
Yeah, in retrospect I wish I would have switched over to a colormap halfway through the video, to really bring it to life.
wow that looks awesome, are you programming this? how?
Thanks! :) this was programmed in Python, with matplotlib. Unfortunately I lost the code, but basically I used the equation in the video description and rendered a bunch of frames while narrowing and shifting the domain.
@@RichBehiel looks amazing man, salute
Hey tungsten guy!!!
I wrote this using chatgpt , very interesting result I must say .
I have recently procured a tungsten cube, and I must say, it has proven to be a most efficacious remedy for my faint mortality. The cube's heft and density serve as a constant reminder of the fleeting nature of existence, and its metallic gleam serves as a symbol of the resilience and perseverance of the human spirit.
Upon first handling the cube, I was struck by its sublime craftsmanship. The edges were smoothed to perfection and the surface was polished to a mirror-like sheen, evocative of the gleaming armor of a knight of yore. And yet, despite its delicate appearance, the cube exudes an aura of invincibility, as if it were forged in the fires of Mount Doom itself.
In my daily meditations, I have found that the cube serves as a tangible embodiment of the concept of "carpe diem", a reminder to seize the day and make the most of one's time on this mortal coil. Its presence on my desk serves as a constant prompt to live fully and with intention.
Furthermore, the cube's tungsten composition lends it a certain gravitas, evocative of the timeless wisdom of ancient sages. It is as if the cube were imbued with the accumulated knowledge and experience of centuries past, and serves as a conduit for this wisdom to flow into the present.
Oh no, I’ve been replaced by AI 😭
E questo è l' andamento del Primo Vento, che si dice Folata di fiandre . E sono 💯 baci😘