It is mathematically described as the connected set of points c within the complex plane, where
2
|Z | with Z = Z + c and Z = (0,0)
| n| n n+1 0
stays bounded for infinite growing n. The iteration of course, which would be necessary to check this formula must be stopped after a finite number of steps in practice. Therefore a maximum iteration depth has to be defined. For the pictures on this CD we used a maximum of upto 65000. If the starting point is not excluded until the maximum iteration depth is reached, it will be said to belong to the mandelbrot set and given the color black in the resulting image. The whole mandelbrot set looks like a man made of spheres or apples - therefore it is often called the apple man. This apple man may be found in many of the pictures due to the property of self replication.
The exclusion of points is based on a property of the iteration formula which says that |Zn| is infinitely growing if the point lies outside a circle of a definite radius around the origin. For the images on this CD this radius was set to be 10, i.e. the iteration stops if the point Xn lies outside of that circle - the starting point of this iteration will then be supposed not to belong to the mandelbrot set.
The pictures are generated by computing the iterations for each point within a chosen rectangular area and resolution. The points which are not belonging to the mandelbrot set, are colored corresponding to the iteration depth at which the exclusion was discovered. The mandelbrot set itself is shown as a black area while areas of other colors are showing the edge of the set. In some cases the color black was also used at the edge for a better optical effect.
All pictures except of frac0000, which shows an overview of the whole mandelbrot set, are magnified sub areas of other pictures. So we finally get a hierarchy of sub areas with frac0000 as the root. The numbering of the pictures is generated by topological sorting of all sub areas. The file mappings contains the relation between numbers and names, which allow to conclude the position of the pictures within this hierarchy. The name 0 denotes the overview of the whole mandelbrot set, a is the name of the first sub area, b is the second sub area etc. The name aa denotes the first sub area of the picture a, ab means the second one. Therefore acab is the second sub area of sub area a of the third sub area of the first sub area of the overview picture. For a better readability sequences of identical characters are replaced by this character with a preceeding numerical counter, e.g. the long name abaacbbb will be replaced by the short form ab2ac3b.