Fractal Coordinate Library

In future I will expand this into a database, however for the time being this page will suffice.
I created this page to index the positions of the interesting fractals that appear using recursive operations on complex numbers.
The most famous complex-number fractals are certainly the Julia Sets and the Mandelbrot Set.
I’ll be sure to cover the Mandelbrot and Julia sets, but since many others are doing exactly this I will try to focus on less known equations.

This page will get pretty massive, so I might put in some collapsible divs once I add a couple more functions.
The plan is to switch it to a database fairly soon so I don’t want to develop this page too much.

z=1/(1-z^2)+c

c=0.652-0.306i
c=0.565-0.2i
c=0.478-0.094i
c=0.391+0.012i
c=0.304+0.118i
c=0.478-0.01i
c=0.3772+0.044i
c=0.1181+0.1914i

z=c/(1-z^3)-c

c=0.55-1.1i
c=0.574-0.9507i
c=0.838+0.692i
c= 0.8476+0.75173i
c= 0.8548+0.7965i

z=z^z-c/(1-z^2)

Contains beautiful chaotic patterns, predominantly on the left hand-side, for some c values these stretch a very long distance, but become larger the further left you go.

c=0.9-0.3325i. View-port is a rectangle from (-1,-1) to (1,1)
c=1-0.41i. View-port is a rectangle from (-10, -10) to (4,10)
c=0.91+0.2875i. View-port is a rectangle from (-10, -10) to (4,10)
0.88+0.52i. View-port is a rectangle from (-10, -10) to (4,10)

As the imaginary value trends higher the details become confined closer to (0,0)

c= -0.68-0.3i. Viewport is a rectangle from (-1,-1) to (1,1)
c=-0.89-0.3i. Viewport is a rectangle from (-1,-1) to (1,1)

For imaginary values that aren’t too far from 0, as the real component of c drops below -1 the shapes become closed and their boundaries a lot more clear.

c=-1.24-0.3i

Keeping the imaginary value fixed on -.3i, the internal fractals disappear around -1.25-0.3i.
This may be a point of interest for those who like to zoom-in and explore the deeper details.

c= -1-0.25i. Zoomed in on fine details. Viewport from (-.65,-.65,-.4,-.4)
C= -1.04-0.26i Zoomed in on fine details. View-port just down to the left of the previous image.
By the time you reach c=-1.08-0.27i the inner shapes have become separated islands.
c=0.9268816885147892+0.5937094448073679i

The ideal range seems like

-1.1 to -0.8 or 0.8 to 1.1 for the real values, and,
-1.2 to -.2 imaginary value when the real value is negative, or,
positive or negative .18 to .6 as an imaginary value when the real value is positive.

z=z^2+c (Julia-Sets & Mandelbrot Set)

Recursions of Two Complex Numbers

Usually each Julia-Set, or any related fractals made with recursions on the complex-plane, only use a single, usually complex, “Magic Number” commonly identified as c.

The following functions use two magic numbers which I will label c and d respectively.

z=z**4/(z+c)/(z+d)+(c+d)/2

c=1.395597-0.09951i
d=0.8541587+1.67580i
c=1.22462068-0.381012614i
d=0.43161579+0.6802431i
c=0.557524-0.48924045i
d=1.75854753+1.466066i
c=1.3355104-0.834i
d=0.69686-0.558667i
c=1.75529645-1.3637857i
d=0.00183825+0.2405473i
c=0.6750699+0.4840647i
d=1.323942-1.02991i

c/(1-z^2)+d

c=0.2242062-1.15087i
d=-0.74911173+1.137821i
c=-0.005891-1.4712599i
d=-0.8388+0.39069i
c=1.774681+1.8248235i
d=-0.445322-1.896184i
c=-0.143582-1.98998i
d=-0.075189+1.8490056i
c=-0.52036732+1.66432i
d=-0.34655476-0.6482633i
c=-0.2811821-1.887599i
d=0.6521721+0.664937i
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