Scott R. Nelson's Fractals

Some favorite fractal images

How are Mandelbrot images drawn?

The Mandelbrot set is defined by the recursive equation Z = Z^2 + C, where Z and C are complex values. The starting value for Z is 0.0. If for some C, Z approaches infinity, C is considered not to be a member of the set. That is, with Z starting at zero, and the operation Z = Z^2 + C performed over and over forever, if Z at some point approaches infinity, then C is not a member of the Mandelbrot set. It can be shown that if the absolute value of Z has become greater than 2.0, the value of Z will increase towards infinity.


The number of iterations performed before the absolute value of Z exceeds 2.0 is called the height or potential associated with a point C. Coloring the points based on their heights yields beautiful patterns.

A region in the Mandelbrot set is defined by specifying a real and imaginary value for C and an extent to specify what range is to be viewed. With the real and imaginary values both set to 0.0 and the extent set to 4.0, the region from -2.0 to 2.0 in each direction will be plotted. This forms a perfect circle with the the "standard" Mandelbrot pattern inside it. Because all numbers greater than distance 2.0 from the center will immediately head off towards infinity, there is nothing interesting in that outer region.

The Julia set is drawn like the Mandelbrot set, but instead of varying the Z value based on the screen pixel position, Z remains constant (representing one point in the Mandelbrot set) and C is varied based on the screen position.

A sample zoom in one point of the Mandelbrot set

The following set of images are all centered about the point:
    Real = -1.1887039461406583296, Imaginary = 0.30338463100781276616
The actual computation was performed using 64-bit double-precision math, so you may not be able to reproduce the exact same zoom in a Mandelbrot program without a few more digits of accuracy.

The following sequence of images zoom from an extent of 20 down to an extent of 3.6e-11 where the limits of the computation method begin to show their effect. Note that the number of iterations is limited to 256 at first to make the "black" part draw faster. This limit is increased as needed until it reaches a limit of 65535. There are a few images at the same position with just different limits on the number of iterations to show more detail.

The image sequence:

More Fractals

The Mandelbrot repeats itself infinitely. It can be quite interesting to look at the many variations of the main shape. They can be found at many different orientations. Depending on where they are found, there are many very interesting patterns at the edges.

Here are a few interesting examples of the main shape:

Examples of the interesting patterns around the main shape:

The most interesting patterns are usually found near the edges of the main shape, expecially near the "cracks". Here are some examples:

Here are a few more interesting patterns:

You can make the image as big as you want and it just gets more interesting:

A few Julia set images: