Since the Mandelbrot iteration happens on the complex plane, is there any recommended reading / research about its relation to scientific fields where the complex plane is applied, like mechanical oscillatory systems, quantum mechanics and electromagnetism?
The quest to decode the Mandelbrot set
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Re: The quest to decode the Mandelbrot set
#22If you enjoyed learning about the Hashlife algorithm for Conway's Game of Life, the bilinear approximation algorithm for computing whether a point is in the mandelbrot set has the same je ne sais quoi. No one has done a really accessible writeup of it yet, but this blog post and the linked forum thread are a good start: https://mathr.co.uk/blog/2022-02-21_deep_zoom_theory_and_pra...
I did some research on this on the side for my dissertation, but never published it. The fact centers approximate the boundary generalises to almost any point in the plane as a consequence of normality of some sequences, and generalises to most families of complex iteration under very mild conditions. I’ve had a preprint that I never felt like finishing for something like 15 years lying around.
Re: The quest to decode the Mandelbrot set
#23A very interesting cast of underdog characters appears in the article.
You’ve got one guy with a relentless spirit to continue with mathematics in is spare time after being blacklisted from mainstream academia because of antisemitism.
Another is a childhood prodigy, who set the record for the youngest American IMO team member, but got burned out as an adult and went into finance but found his way back through the mentorship of another mathematician.
And a third was a biology major. After graduating, he worked as a baker. But he wanted a career change so he entered a master’s program in math and proved an impressive result.
Re: The quest to decode the Mandelbrot set
#24Making a program to render a view of the Mandelbrot set is a fun exercise, I recommend it. I found a viewer that works in the browser: > Mandelbrot Viewer is a universal (desktop and mobile) vanilla JS implementation of a plain Mandelbrot set renderer – supporting mouse, touch and keyboard interaction. https://mandelbrot.silversky.dev/
Here's my attempt done a long time ago, not very polished: https://brazzy.de/en/Mandelbrot.php Even for that, the actual Mandelbrot calculations were the smaller part. It's really amazing how such a trivially simple formula spawns such endless complexity.
Complexity is trivial. That's the lesson here. It's human perception that's special. It's special in that it has an uncommonly low bar on what's considered impressively complex.
Which implies something important, no doubt.
Re: The quest to decode the Mandelbrot set
#25The Mandelbrot set is quite well known. Yet something I learned recently blew my mind. It's about the uncanny resemblance between the images generated by the Mandelbrot set, and among all things, the popular image of Buddha. For example: https://en.wikipedia.org/wiki/Buddhabrot Even when looking at the 2D Mandelbrot set renderings, I can't help but wonder whether the similarity of the "bulbs" to the rather unique Bud…
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Re: The quest to decode the Mandelbrot set
#26> the word evoked the notion of a new kind of geometry — something fragmented, fractional or broken. But that's not what "fractal" means; it means "fractional dimension". To say the word "fractal" evoked something is subjective - evoked it for whom?
For people who know similar-sounding words. Words can evoke things well outside their etymology or denotation, one of the many reasons people like and use words.
Re: The quest to decode the Mandelbrot set
#27> the word evoked the notion of a new kind of geometry — something fragmented, fractional or broken. But that's not what "fractal" means; it means "fractional dimension". To say the word "fractal" evoked something is subjective - evoked it for whom?
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Re: The quest to decode the Mandelbrot set
#28Since the Mandelbrot iteration happens on the complex plane, is there any recommended reading / research about its relation to scientific fields where the complex plane is applied, like mechanical oscillatory systems, quantum mechanics and electromagnetism?
I had a couple of related questions. To what extent does complex dynamics map to physical phenomena? And in the opposite direction, how is renormalisation used outside of quantum physics?
Re: The quest to decode the Mandelbrot set
#29What is the conjectured topology of the Mandelbrot set if MLC is true?
My understanding is that there's a certain number of bulbs, each centred around a point which becomes periodic with period p after k steps. But how do they all stick together?
Re: The quest to decode the Mandelbrot set
#30> the word evoked the notion of a new kind of geometry — something fragmented, fractional or broken. But that's not what "fractal" means; it means "fractional dimension". To say the word "fractal" evoked something is subjective - evoked it for whom?
For people who don't know what a fractional dimension is. And mostly don't care.
Fractal art of all kinds was part of a certain trend in 80s/90s culture, which also influenced the look of the early Internet. And electronic dance music, clubs, and raves.
It was maximalist, psychedelic, colourful, busy, inclusive, recursive, and complex.
Whatever the math was doing, it was a very popular signifier of certain kinds of experience.
I suspect it's not a surprise that if faded into the background when the Internet began to commercialise and blandify in the later 90s.