Sequence Fractals Part I #1


si = 1, -1, 1, -1, …
Center: -1.8+0i, Zoom = 0.4.

The description “combine fractal and sequence” is intentionally vague. Part I will look at the simplest way to combine fractals and sequences, namely “just add”. The sequence in this case is simple, alternating 1 and -1.

Define the follow functions:
f_0(z,c) = z^2+c+1.
f_1(z,c) = z^2+c-1.
f_c(z) = f(z,c) = f_1(f_0(z,c),c) = (z^2+c+1)^2+c-1.
f0 is the first step, of the iteration, f1 is the second step. The iteration alternates between the two, f0, f1, f0, f1 … .
fc(z) = f(z,c) combines the first two steps, and the even orbits are found by iterating f.

For now, we will hide the odd orbit points, and the fact that we have two different functions, and just look into f. f is a degree 4 polynomial, we can leverage some of the experience from z2+c.

The derivative with respect to z is f'_c(z) = 4z(z^2+c+1), so z=0 is a critical point (f’=0) for all values of c. z_0=0 was used as the starting point for this all similar these pictures unless otherwise stated. A degree 4 polynomial has three critical points, the other two are the subject of a later post.

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