Click on any picture to see a full-sized version.
The fractal exists in the complex plane. 3)
A 2D image, with Red and Green values
for Real and Imaginary numbers, suffices.
This equation is applied to each point:
Zn+1 = Zn2 + C
where C is the point, and Z0 = C.
With real X and Y values,
the equivalent equations are:
Xn+1 = Xn2 - Yn2 + X0
Yn+1 = 2 * Xn * Yn + Y0
If a C's Zn is farther than 2 from 0+0i,
then its later Zs are all far from 0+0i,
and that C is not a part of the fractal.
Cs that remain have blue added.
Here are all of the nodes
to calculate on iteration:
Nodes are grouped together
for organization and density.
In the end, each shade of blue
is turned into a different color, 4)
and the center is turned to black.
The image is sharpened
by calculating repeatedly.
This gives a pretty result.
With more iterations,
the color bands get
harder to tell apart.
Multiplying the normalized steps
and moduloing them back to 0-to-1
makes the colors easier to distinguish.
After figuring this much out,
I found out regerogarc did, too.
I am working on better versions
and neat tricks for this project.
I will post updates when they're done.