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These are based on idea by gamma that started in another thread....

The formula is the standard third-order Newton's method, but with a twist -- each pixel's loop begins with the exponent set to 3, then at each iteration of the loop, the exponent increases by a fixed amount.

For the first three image, the exponent increment is 0.01, 0.5, and 1.0. The fourth image has an imaginary increment (0,1).

Aggressive 9x9 anti-aliasing was used because this gets severely noisy in the negative real half-plane.

First image: exponent increment = 0.01. You can see the effect beginning to develop in the portion of the Julia set that lies on the negative real axis.


Second image: exponent increment = 0.5.


Third image: exponent increment = 1.0.


Fourth image: exponent increment is imaginary (0,1). The coloring (unlike the others in this series) is based on the histogram of the counts with a 2X repeat of the underlying gradient. The gradient starts at white and ends at black, so there is a discontinuity from white to black when it wraps around. I felt this thematically matched the presence of the branch cuts induced by the complex exponents.