Newtons method for mandelbrot and julia sets

 

 

Newton's method takes a function  f  and returns another function

 

                      f(z)

	Nf(z) = z - -------

	             f'(z)

 

called the Newton iteration function which is a rational function

when  f  is a polynomial.  For example, when  f(z) = z^3 - 1 ,

 

                 2z^3 + 1

	Nf(z) = ---------- .

	           3z^2

 

Now, iterate  Nf  over a grid in the complex plane not unlike

the algorithm used to compute a Julia set.  The orbit of any

given point will tend to find one of the three roots of  f . 

Color the point a different color (say, red, green, or blue)

depending on which of the three roots it converges to, and a

different shade of that color depending on the speed of conver-

gence.  If the point does not converge after a predetermined

number of maximum iterations, then color the point black. 

 

Increase this maximum value and watch the black region diminish

in size until all that remains is a fractal basin boundary inter-

spersed with marvelous shades of red, green, and blue.  In the

limit, as the maximum number of iterations gets large, this

algorithm solves the following somewhat perplexing riddle:

 

        Use three colors to paint a sheet of paper,

        with the only restriction being that wherever

        two of the colors meet, all the colors have

        to meet.

 

despite one's first impression that the riddle is insoluable.

 

-- 

Tom Scavo

scavo@cie.uoregon.edu