Sequence Fractals Part III #6

313.jpg

z^2+c+s_i
si = a,-a,…; a = -0.125i
Center: 0; Zoom = 0.4

I want to follow up on my comment in Sequence Fractals Part III #4 that sequence fractals are a special class, more than just a subset of the quartics. The comments are general, not specific to today’s picture. Start by asking what does the quartic parameter space look like?

The general quartic polynomial is a_1z^4+a_2z^3+a_3z^2+a_4z+a_5. But this gets reduced to three parameters.

Convention is to group the polynomials into affine-conjugate classes. Sounds complicated right, math words are so cool. An affine transformation is a simple function of the form b_1z+b_0. Affine transformations are the only complex functions that are differentiable and invertible.

It turns out that f_1(z)=g^{-1}(f(g(z)) behaves just like f. This, g^{-1}fg, is affine conjugacy. It is why by convention we only look at z^2+c and not the general case for quadratics. Each belongs to a different affine-conjugate class, and collectively all classes are represented.

The first step is to select one representative function from each affine conjugate equivalence class. There are many ways to do this. But in all cases there are three complex parameters, a,b,c, not just c as with the quadratics.

For example we could drop one of the middle terms. (Like we do for quadratics.)

z^4+az^2+bz+c
z^4+az^3+bz+c
z^4+az^3+bz^2+c

The first case is called monic (first coefficient is 1) and centered (second coefficient is 0). The critical points sum is 0 for monic, centered polynomials. In the third case, 0 is always a critical point. Calculating the other critical points is difficult.

Knowing the critical points is handy for generating fractals (as well as for complex dynamics in general). It is useful to work backwards from the critical points. Consider

4(\int z(z-a)(z-b))+c critical points 0, a, b
4(\int(z-a)(z-b)(z+a+b))+c critical points a, b, -a-b
4\int(z-a)(z-b)(z-c) critical points a, b, c

Here ∫ does not indicate an actual iteration, I am using it to symbolize the anti-derivative as in a first calculus course. It makes is easy to see the critical points. The first one in the usual format is z^4+\frac{4}{3}(a+b)z^3+2abz^2+c. Expanding the others are equally messy and just obscures the critical points. The second one is monic-centered.

I have a feeling that these are just the tip of the iceberg. Many other reasonable formulations possible.

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