Here is a variant on Mandelbrot and Julia fractals that popped into my head a while back, althoughI don't recall the motivation. Regardless, the results are nifty.
Consider the function, , iterated on its own result to generate a Mandelbrot set. For a polynomial, treat the coefficients as elements of a vector.
With each iteration of , apply some transformation to that vector. Here, for the usual quadratic Mandelbrot, where the coefficients vector is, and with every iteration some linear transformation, , is applied to .
The inner loop then looks like,
Not surprisingly, the identity matrix yields the traditional Mandelbrot set:
The exterior color represents the number of iterations before the orbit meets the escape criteria. The interior color is a function of the convergence value: the hue shows the argument; the saturation shows the magnitude.
Unless otherwise indicated, the following images use the same display parameters as the above. They are all shown over the same part of the complex plane, centered on with a width of four units.
