MathyBits
Mandelbrot Set
To see more detail, click two points to draw a rectangle and zoom in (the down and up arrow keys also zoom in and out).
The Mandelbrot Set is defined by a test: each point in the plane is subjected to a geometric transformation over and over again. If the resulting sequence of points all stay close to the origin, no matter how many times the transformation is applied, then the original point is in the Mandelbrot Set. If the points eventually move away from the origin, then the original point is instead part of the "escape set."
Select a point and use these buttons to see how it behaves in the test.
The transformation has three steps:
1) A rotation doubles the angle of the point with the positive horizontal axis.
2) A dilation squares the distance of the point from the origin.
3) A translation moves the point based on the location of the original point.
Using complex numbers, the transformation is written as zn+1 = zn2 + c, where c represents the original point and z0 = 0.
The images of the Mandelbrot Set are created by testing each point (pixel). Points in the Mandelbrot Set are colored black. Points that escape quickly are colored with a deep shade of the selected color. Points that escape more and more slowly are colored with lighter and lighter shades.