(For experts: The character of the fixed points is structured by Newton’s method to find the zero of a factor (z-z_f)^p.
In this way the ordinary Julia sets of the quadratic iteration are obtained by putting one fixed point with p=-0.5 and varying a second around p=0.5.)
But you don`t have to know any mathematical details. Just play around with position number and character of fixed points and try to understand intuitively their impact on the obtained picture.
In this way you can draw really beautiful self similar structures and discover intuitively results which are partially not understood and may be not even known by actual mathematical research.
To get an idea how this works just press the "+" button at the beginning (you know have 3 fixed points). Then choose "make figure symmetric" in the menu (In this way you obtain the famous fractal for finding the zeros of z^3-1 with Newton’s method). By varying the position of the cross in the lower window you change the character of one of the fixed points and you can find an astonishing variety of fractal figures. Especially the black cusps at the border of the Mandelbrot set are leading to interesting structures.
The colorgradient is changed with the volume up/down buttons of your phone.
In addition you can explore the Mandelbrot and Julia sets of the quadratic, cubic and sine fractal.
Very remarkable is the possibility of exploring the Collatz fractal which is connected to the famous Collatz conjecture.
Search for small parts in the Mandelbrot set of the Collatz fractal where you can find copies of the well known Mandelbrot set of the quadratic iteration. Then search for a small region of the Julia set which is connected to the copy of the Mandelbrot set.
You will find that the small Mandelbrot sets are embedded in an astonishing variety.
All of this is produced by the iteration of f(z)=c*z+1/4-((c-1/2)*z+1/4)*cos(pi*z) !!
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| Downloads: 8 | ||||
| Updated At: 2026-06-18 10:22:54 | ||||
| Publisher: Florian Jasch | ||||
| Operating System: Android | ||||
| License Type: Free |