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The Mandlebrot set is a famous fractal which shows the beauty and chaos
that emerge when complex numbers are repeatedly subjected to polynomial
transformations.
It is named after Benoit Mandlebrot, a Polish-born mathematician
who poineered the field of 'fractal geometry' and the use
of computers to generate fractal images. He discovered the
Mandlebrot set in 1979.
The algorithm that generates this canonical fractal is:
Z(i + 1) = (Z(i))^2 + k
where k is the value of a point on the complex plane. Images of this
fractal are generated by calculating repeated iterations of the above
function, and testing to see if the value of Z(i) has escaped from a
predetermined boundary. Typically the boundary is set to abs(Z(i)) < 4,
since numbers greater than 4 will grow quickly when squared. Complex
numbers less than 4, however do not all diverge under this function,
since the 'imaginary' component may square to a negative 'real' value.
The image displayed is generated by applying this calculation to each
point on the complex plane, and coloring the corresponding pixel based
on how many iterations of the function it takes to escape the boundary.
Points that remain bounded are in the set. Points that escape will be
assigned a color depending on the number of iterations it takes to
diverge.
How to use this Fractalscope
The 'max iterations' setting determines the maximum number of
iterations of the Mandlebrot equation to which each pixel will be
expanded. Warning: higher settings for this value will slow rendering
down significantly. If rendering seems to lag, lower this value.
Use the buttons on the control panels below the image to zoom in
out and pan around the Mandlebrot image. Keyboard controls are also
available. The 'z' key zooms in on the image, and the 'x' key zooms
out. They arrow keys allow you to slide the image up, down, left, and
right. The 'r' key will reset the image to it's original position.
To modify the color scheme, click the 'add color' button in the 'color
picker' panel to the right. Choose a color by clicking on the color box and
using the color picker. The 'iteration width' determines width of that
colors 'band' in the resulting imgage. Be sure to select at least
two colors default color scheme will continue to show! If fewer than
two colors are selected, a default color sheme will be rendered.
The 'rotate colors' button will activate color rotation and bring the
image to life.
