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Embossed Spirals



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A familiar Seahorse Valley Mandelbrot with an unusual coloring: the "grad" of the smoothed iterations. Black = the smoothed iteration value increases in the positive real direction, violet = positive imaginary, white = negative real, pinkish = negative imaginary.

This coloring method, when used with a cyclic gradient with strong luminance contrast, produces an embossed, three-dimensional look.

A second layer with normal smoothed iterations darkens the seahorse cores and turns the edge of the minibrot yellow-brown.

Mathematically, the calculations resemble those for distance estimator. The derivative dz_n/dc is maintained, starting as 0 (since z₀ = 0 is a constant independent of c) and being updated by new-der = 2 times old-der, times z_n, plus 1. So far, so identical to distance estimator.

The difference is in what's done at the end of the iteration. For distance estimator, the modulus of the escaped z is multiplied by its own logarithm and by half, and divided by the modulus of the derivative. For this, I multiply the escaped z (not its modulus) by half and divide by the derivative (not its modulus), leaving the logarithmic term of the product out entirely and yielding a complex number. That's the grad. It's the argument of this complex number that is used for the coloring: the direction of the gradient.