Julia Sets and the Mandelbrot Set

Universality of the Mandelbrot set

For the details of the Curry-Garnett-Sullivan experiment, first note that for all values of c, z = 1 is a root of the polynomial f_c(z) = z³ + (c - 1)z - c.

fc(1)	= 13 + (c - 1)*1 - c
	= 1 + (c - 1) - c
	= 0

Always starting with z₀ = 0, Curry, Garnett, and Sullivan painted c

black if Newton's method converged to 1,

white if it converged to some other root of f_c(z), and

grey if it converged to a cycle of at least 2 points.

Here are their results, with vertical and horizontal ranges -2 to 2.

On the left below is a magnification of the blob on the top of the picture; on the right, a magnification of the little grey region.

It's the Mandelbrot set, yet again.

Keep in mind the function being iterated is Newton's method for a cubic polynomial. It is not at all like z² + c. Yet here's the Mandelbrot set, yet again.

Return to Universality of the Mandelbrot set.