Title: On the superposition of mandel-like sets
Post by: Hiato on December 26, 2012, 03:47:55 PM
Hello all, I have been watching this forum for some time now and I figured I may as well register so that one day I might contribute. To that end, allow me to create my first post based on something (I feel) has not been explored terribly much.
Consider the classic Mandelbrot iterator: z->z^2+c. The question to ask here is "what does z->z^2 + z^3 +c look like?" I fiddled around with forms such as those arriving at interesting results but nothing worth mentioning. However, the problem was that, after the superposition of sufficiently many such sets, the fractal became unbounded (that is, by and large the entire plane was in the set). I convinced myself that this was due to the fact that higher powers dominated a tad much and messed things up, so I began to divide these powers by constants only to realise that, for clever choices of such constants I was just looking at truncated power series. Thus, I jumped straight to it and explored what a handful of the closed forms of such power series did - and it was fairly pretty. I now present to you some quick renders and associated formulas (each starts with a*z^2 and goes up in powers of z from there). Interestingly, all of these seem to contain (somewhere) the entire Mandelbrot set [visually verified] thus making them all at least as interesting as the Mandelbrot set itself :P
The following formulae were explored (due to their pleasing series expansions):
They represent "mandelolX.png" where X is 0..5 respectively, all these files (including the fragmentarium/GLSL code) can be found here (http://hiato.uctleg.net/mandelol/).
Enjoy :)
Consider the classic Mandelbrot iterator: z->z^2+c. The question to ask here is "what does z->z^2 + z^3 +c look like?" I fiddled around with forms such as those arriving at interesting results but nothing worth mentioning. However, the problem was that, after the superposition of sufficiently many such sets, the fractal became unbounded (that is, by and large the entire plane was in the set). I convinced myself that this was due to the fact that higher powers dominated a tad much and messed things up, so I began to divide these powers by constants only to realise that, for clever choices of such constants I was just looking at truncated power series. Thus, I jumped straight to it and explored what a handful of the closed forms of such power series did - and it was fairly pretty. I now present to you some quick renders and associated formulas (each starts with a*z^2 and goes up in powers of z from there). Interestingly, all of these seem to contain (somewhere) the entire Mandelbrot set [visually verified] thus making them all at least as interesting as the Mandelbrot set itself :P
The following formulae were explored (due to their pleasing series expansions):
They represent "mandelolX.png" where X is 0..5 respectively, all these files (including the fragmentarium/GLSL code) can be found here (http://hiato.uctleg.net/mandelol/).
Enjoy :)
Title: Re: On the superposition of mandel-like sets
Post by: Alef on December 26, 2012, 05:42:58 PM
Instead of our chief;D
Well, z=z*sinh(z)-c^2 fas my first formula ever. It entered Fractal Explorer. There is some property of mandelbrot set, universality, if I remember correctly, that it appears with many formulas. It even is hard to get rid of it. Probably if you make something what have all the properties of mandelbrot set, but not a mandelbrot set, you'll eternalise yourself in wikipedia;)
hello and welcome to the forums
Well, z=z*sinh(z)-c^2 fas my first formula ever. It entered Fractal Explorer. There is some property of mandelbrot set, universality, if I remember correctly, that it appears with many formulas. It even is hard to get rid of it. Probably if you make something what have all the properties of mandelbrot set, but not a mandelbrot set, you'll eternalise yourself in wikipedia;)