3 ms·

I'm not sure if every fractal can be expressed as an iterative formula f(z,c). In 2012 I found a fractal by using a fundamentally different approach. It arises

by mg 1y ago

I'm not sure if every fractal can be expressed as an iterative formula f(z,c). In 2012 I found a fractal by using a fundamentally different approach. It arises when you colorize the complex plane by giving each pixel a grey value that corresponds to the percentage of gaussian integers that it can divide: https://www.gibney.org/does_anybody_know_this_fractal https://www.gibney.org/does_anybody_know_this_fractal

akunzler 1y ago
Good point, this site then supports every (as far as I know) fractal you make with iterations of complex numbers and constant cutoff values, mandelbrot style. There are surely infinitely many more ways to generate other families of fractals though
akomtu 1y ago
What's the heck is gaussian integers? I've tried to parse your article, but still confused.
- Sharlin 1y ago
  Simply the complex numbers where the real and imaginary parts are both integers. Eg. 0, 3+i, 123-45i, -7+8i. Same as the 2D grid of integer Cartesian coordinates.
- mg 1y ago
  You can think of them as the complex equivalents to normal integers. Complex numbers have two components. If both are integers, the complex number is a Gaussian integer.
- bmacho 1y ago
  https://www.google.com/search?q=gaussian+integer https://www.google.com/search?q=gaussian+integer
zahlman 1y ago
> I'm not sure if every fractal can be expressed as an iterative formula f(z,c). It's also unclear to me that every iterative f(z,c) formula will produce something visually interesting, or indeed that meets the definition of "fractal".
badosu 1y ago
You can make a fractal out of the state graph of a double pendulum: https://www.youtube.com/watch?v=dtjb2OhEQcU https://www.youtube.com/watch?v=dtjb2OhEQcU I don't doubt there could be an iterative formula that maps to it, but I'd be very surprised.
- hoseja 1y ago
  Isn't that trivially an iterative formula? You know, the physics solver iterating dT after dT.

by mg 1y ago

akunzler 1y ago

Good point, this site then supports every (as far as I know) fractal you make with iterations of complex numbers and constant cutoff values, mandelbrot style. There are surely infinitely many more ways to generate other families of fractals though

akomtu 1y ago

What's the heck is gaussian integers? I've tried to parse your article, but still confused.

Sharlin 1y ago
Simply the complex numbers where the real and imaginary parts are both integers. Eg. 0, 3+i, 123-45i, -7+8i. Same as the 2D grid of integer Cartesian coordinates.
mg 1y ago
You can think of them as the complex equivalents to normal integers. Complex numbers have two components. If both are integers, the complex number is a Gaussian integer.
bmacho 1y ago
https://www.google.com/search?q=gaussian+integer https://www.google.com/search?q=gaussian+integer

zahlman 1y ago

> I'm not sure if every fractal can be expressed as an iterative formula f(z,c). It's also unclear to me that every iterative f(z,c) formula will produce something visually interesting, or indeed that meets the definition of "fractal".

badosu 1y ago

You can make a fractal out of the state graph of a double pendulum: https://www.youtube.com/watch?v=dtjb2OhEQcU https://www.youtube.com/watch?v=dtjb2OhEQcU I don't doubt there could be an iterative formula that maps to it, but I'd be very surprised.

hoseja 1y ago
Isn't that trivially an iterative formula? You know, the physics solver iterating dT after dT.