Sites with 100 pictures:    Ratio    RatioField    Landscape    Quaternion    Attractor

Fractal programs

Mandelbrot and Julia sets

for Windows

Detailed account of the theory

Try this little Mandelbrot program - which can make pictures of poster size (new version July 2010). Place the cursor and left/right-click to zoom in/out. The program is closed by Esc. The colour scale is chosen by the number keys and adjusted by scrolling. After pressing key A and S, the scrolling is aimed at the density of the colours and displacement of the scale, respectively. Key L produces lighting effect, to be adjusted by keys D, F and G and scrolling. Key J to switch from the Mandelbrot set to the Julia sets, and vice versa. The vertical arrows displace the section. See this site for more possibilities, for large pictures and for an explanation of the theory: Mandelbrot.


(click on the pictures to see them in their original size)

Press on these pictures to see them being drawn

takes 30 secs. - closed by pressing mouse or key:


Sites with programs and pictures (no theory):

       Julia (a simplified program to begin with)

       Ratio (the main program)





Julia sets

For an iteration of the form z → f(z) + ρ, where f(z) is a complex function and ρ is a complex number (and therefore corresponds to a point in the plane) a Julia set is associated in the plane. The set is infinitely thin, a sort of curve, but usually not a true curve, since it can be twisted and ramified in a manner that is incompatible with the idea of a curve. It is self-similar in the sense that by the iteration it is mapped onto itself, and this means that every part has the same structure as the whole. The Julia set can be made visible by plotting the points whose distance from it is lesser than a given small number. The domain outside the Julia set is composed of one or more Fatou domains, and these can be coloured by various methods (potential, distance, field lines, ...). The Julia sets can be very different for the different parameters ρ, and in order to find the values giving the most interesting or attractive Julia sets, a Mandelbrot set must be constructed (by means of two critical points for the function f(z)). A Mandelbrot set has an interior (often coloured black), an exterior (which can be coloured in the same ways as the Julia sets) and a boundary which locally has a structural relationship with the Julia sets, but which as a whole is not self-similar. For a point ρ on the boundary of the Mandelbrot set, the structure of the Julia set for the iteration z → f(z) + ρ resembles the structure of the Mandelbrot set at this locality. When a program is started, the function has to be typed in and the critical points chosen, and then the Mandelbrot set appears, and in this you can choose section by zooming and go to the Julia set for a point.

A detailed account of the theory of the making of pictures, can be found in this Wiki-book:

       Pictures of Julia and Mandelbrot sets

- the book in htm-format:

       Pictures of Julia and Mandelbrot sets

This fractal project is explained in the article:

       The Sad Fate of the Fractals.

If you want to see the uncompiled programs, possibly in order to make alterations, see the site:

       The uncompiled fractal programs of

If you are user of the Ultra Fractal program, see the site:

       Make attractive pictures of Mandelbrot and Julia sets with Ultra Fractal.

Produced by Gert Buschmann

Established: 2003   Updated: June 2020

This site is a subsection of

(an article on aesthetics)