Hi all,
Calcyman had an interesting idea regarding the 3D Mandelbulb. His original post is here:
http://www.fractalforums.com/index.php?topic=2354.msg10836#msg10836His idea was to include all the "higher orders" when calculating the Mandelbrot (Mandelbulb) set, and he came up with a clever way to do it that can be calculated efficiently. His example started with:
... which is the basic form of the Mandelbulb. Then he added in higher orders, like so:
The idea is for each term to contribute less as the exponent gets higher. So, each term is divided by a factorial, which gets large very quickly, and the really high-order terms become negligible. But the clever thing about this is that this series is actually just the MacLaurin series expansion for
. So, his formula can be calculated just like the original with one additional multiplication by a cosine term.
When I saw his post, the question that came to my mind was whether this had been studied in the 2D realm of the original Mandelbrot. I started some very basic research and found some interesting things lurking within that magical complex plane.
I decided to generalize his idea to be "z raised to any power, multiplied by any function of z, plus z
0."
First, let me paste the UltraFractal formula that I used, in case anyone else is interested in experimenting further. The formula is so simple that I'm nearly certain this has been done before, to some extent.
MandelTimesFunc {
init:
z = 0
loop:
z = z^@power * @Function(z) + #pixel
bailout:
|z| < @bail
default:
title = "Complex Power Times Complex Function"
float param power
caption = "Power"
hint = "The power to which Z is raised. Set to zero to reduce calculation to just Function(Z)."
default = 2.0
endparam
float param bail
caption = "Bailout"
default = 1e6
endparam
}
That will allow you to choose any function you like (as long as it's supported by UltraFractal) and multiply it by z raised to any power, in other words:
As the "hint" suggests, you can also set the power to zero, reducing the calculation to simply:
So, I began by looking at just sin(z) and cos(z), without multiplication by a power of z.
sin(z) + z
0 looks like this: