Composite Math

Most traditional fractals use a single complex formula such as "z^2+c."  The formula is iterated until conditions for escaping iteration are reached or the limiting number of iterations are done.  Iteration consists of starting with an initial z, z0, and evaluating the formula, then using the result as the next z, zn, to evaluate the formula, etc. Pixels are colored according to escape "time", or some relationship with the potential of z (last z, average z, etc.) or other data accumulated during iteration.

"Dreamer" is an example of a pipe.  In this case the output of a prefunction (z^c) replaces initial z in the formula ln(z)+k(zn-1)+j (a variation of the phoenix formula -- z^2+k(zn-1)+j.)  The image shows fragmentation due to the prefunction, and spirals from the "phoenix" formula.

In "Necklace", two formulas are "anded" together.  The phoenix formula is iterated first to its escape point, then the results of this iteration are used as the starting conditions for the convergent formula (z+j)(z+k)(z^2+1).  The detail in the bands of the image come from the convergent formula, while the phoenix formula tends to dictate the overall shape of the bands.  Note: neither formula produces an image quite like the composite image when iterated by itself.  

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