Sequence Fractals Part IV #3


si = 1, -1.598, (repeating)
Center:1.0+0.0i , Zoom = 0.5

Another repeating two step sequence. Again, if we look at every other iteration, we are basically looking at quartic polynomials.

Back in Sequence Fractals Part III #6 I was musing about the different ways to set up the parameter spaces for quartic polynomials. The full parameter space always has three complex (six real) dimensions. I am going to skip the details for now, but if you write down any degree four polynomial in z with parameters a,b,c sprinkled into the coefficients, you will almost always define a parameterization that captures almost all of the affine-conjugate classes.

If we choose to parameterize the quartic space with z^4+2cz^2+bz+c^2+ac. (Not that anyone would do this unless they were playing with two-step sequence fractals based on z^2+s_i*c.) Then today’s picture arises from taking a c-slice of the parameter space, fixing a = -1.598 and b = 0.

In general a c-slice with b=0 is equivalent to the sequence fractal with two-step sequence 1,a.

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