An
image that retains its detail and pattern regardless of how deeply one "zooms" into it. The
self-similarity of the image is usually due to a
recursive mathematical formula which is used to create the image. The
Mandelbrot set is one example, but a lesser-known example is
Newton's method, which generates
Julia set fractals:
Newton's method is a
function that is used to estimate the
roots of a polynomial of any degree. It is of the form:
x
n+1 = x
n - d(f(x) / f'(x))
where d is a dampening factor (explained below), f(x) is the polynomial you want the roots of, and f'(x) is the
derivative of that polynomial. When running the function for the first time, set xn to a number that is close to the root you want to find (this arbitrary number is called a "guess"). Newton's method won't give exact answers, but will take a certain number of iterations to approach the root: you pick when to stop. d is a number between 0 and 1; using a smaller value makes the function take longer to get to a root, but the function will be more stable.
Images generated by Newton's method are available. In the image, each
pixel is used as a guess into Newton's method for whatever the polynomial happens to be. The pixel is colored depending on the root it converged to and the number of iterations it took (darker = more iterations). Start with:
http://www.flash.net/~nav2/z6minus1.gif which is Newton's Method on the polynomial z^6 - 1. Then:
http://www.flash.net/~nav2/z6minus1_C_0+I.gif (same function, centered at 0+I), then:
http://www.flash.net/~nav2/z6minus1_Z_2000.gif (centered at 0+I, zoomed in 2000x), and finally:
http://www.flash.net/~nav2/z6minus1_D_onefourth.gif (same function, centered at 0, d = 1/4).