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
![](https://trogramming.com/blog/wp-content/uploads/2022/06/B0004.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/B0005.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/B0006.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/B0007.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/B0008.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/B0000.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/B0006-1.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/B0483-1024x576.png)
z=c/(1-z^3)-c
![](https://trogramming.com/blog/wp-content/uploads/2022/06/C0000-1024x576.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/C0002-1024x576.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/C0000-1-1024x576.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/C0004-1024x576.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/C0007-1024x576.png)
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.
![](https://trogramming.com/blog/wp-content/uploads/2022/06/coltest0000-1024x576.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/JuliaColTest0000.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/JuliaColTest0009.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/JuliaColTest0013.png)
As the imaginary value trends higher the details become confined closer to (0,0)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/JuliaColTest0004.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/JuliaColTest0007.png)
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.
![](https://trogramming.com/blog/wp-content/uploads/2022/06/JuliaColTest0012.png)
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.
![](https://trogramming.com/blog/wp-content/uploads/2022/06/FractalZoom1-1024x564.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/FractalZoom2-1024x564.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/FractalZoom3-1024x576.png)
![](https://trogramming.com/blog/wp-content/uploads/2022/06/JuliaColTest0002.png)
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
![](https://trogramming.com/blog/wp-content/uploads/2022/06/Test0002-1024x576.png)
d=0.8541587+1.67580i
![](https://trogramming.com/blog/wp-content/uploads/2022/06/Test0022-1024x576.png)
d=0.43161579+0.6802431i
![](https://trogramming.com/blog/wp-content/uploads/2022/06/Test0016-1024x576.png)
d=1.75854753+1.466066i
![](https://trogramming.com/blog/wp-content/uploads/2022/06/Test0090-1024x576.png)
d=0.69686-0.558667i
![](https://trogramming.com/blog/wp-content/uploads/2022/06/Test0788-1024x576.png)
d=0.00183825+0.2405473i
![](https://trogramming.com/blog/wp-content/uploads/2022/06/Test0595-1024x576.png)
d=1.323942-1.02991i
c/(1-z^2)+d
![](https://trogramming.com/blog/wp-content/uploads/2022/06/Ac-0.2242061506943367-1.1508702940961602j-d-0.74911172923422021.1378210417074026j0005-1024x576.png)
d=-0.74911173+1.137821i
![](https://trogramming.com/blog/wp-content/uploads/2022/06/c-0.0058910995785042-1.4712599217498576j-d-0.83880232136526310.390689627942546j0000-1024x576.png)
d=-0.8388+0.39069i
![](https://trogramming.com/blog/wp-content/uploads/2022/06/c1.77468096119163741.8248235185464474j-d-0.4453217176454811-1.8961842091298577j0000-1024x576.png)
d=-0.445322-1.896184i
![](https://trogramming.com/blog/wp-content/uploads/2022/06/c-0.1435819980231856-1.989982500958876j-d-0.075189324476951481.8490056051925605j0000-1024x576.png)
d=-0.075189+1.8490056i
![](https://trogramming.com/blog/wp-content/uploads/2022/06/c-0.52036732160230191.6643248178277252j-d-0.34655476054377043-0.6482632676736078j0000-1024x576.png)
d=-0.34655476-0.6482633i
![](https://trogramming.com/blog/wp-content/uploads/2022/06/c-0.2811821134157171-1.887598835539305j-d0.6521720850816330.6649369380347778j0000-1024x576.png)
d=0.6521721+0.664937i