newtbasin

    Based on the Newton formula for finding the roots of z^p - 1.
    Pixels are colored according to which root captures the orbit.

      z(0) = pixel
      z(n+1) = ((p-1)*z(n)^p + 1)/(p*z(n)^(p - 1)).

    Two parameters: the polynomial degree p, and a flag to turn
    on color stripes to show alternate iterations.


The Newton formula is an algorithm used to find the roots of polynomial
equations by successive "guesses" that converge on the correct value as
you feed the results of each approximation back into the formula. It works
very well -- unless you are unlucky enough to pick a value that is on a
line BETWEEN two actual roots. In that case, the sequence explodes into
chaos, with results that diverge more and more wildly as you continue the
iteration.

This fractal type shows the results for the polynomial Z^n - 1, which has
n roots in the complex plane. Use the <t>ype command and enter "newtbasin"
in response to the prompt. You will be asked for a parameter, the "order"
of the equation (an integer from 3 through 10 -- 3 for x^3-1, 7 for x^7-1,
etc.). A second parameter is a flag to turn on alternating shades showing
changes in the number of iterations needed to attract an orbit. Some
people like stripes and some don't; as always, xmfract gives you a choice!

The coloring of the plot shows the "basins of attraction" for each root of
the polynomial -- i.e., an initial guess within any area of a given color
would lead you to one of the roots. As you can see, things get a bit weird
along certain radial lines or "spokes," those being the lines between
actual roots. By "weird," we mean infinitely complex in the good old
fractal sense. Zoom in and see for yourself.

This fractal type is symmetric about the origin, with the number of
"spokes" depending on the order you select.

See also: Newton