More Cubic Observations

In a previous post I had some observations regarding the full Cubic Mandelbrot formula i.e.

z_n+1 = αz_n³ + βz_n² + γz_n+ c

where α, β and γ are constant real, imaginary or complex numbers.

Varying the values of α, β and γ and using the appropriate critical value results in a variety of fractals containing power 2 Mandelbrots often distorted, such distorted Mandelbrots appear as islands in the standard Mandelbrot at the blunt end.

The first formula is:

z_n+1 = z_n³ + (0 + i)z_n + c

the critical values are 0.408248290464 – 0.408248290464i and -0.408248290464 + 0.408248290464i, the first value produces pictures one way up and the second produces the pictures the other way up.

Formula 1 Example 1

Formula 1 Example 2

Formula 1 Example 3

Formula 1 Example 4

Formula 1 Example 5

Formula 1 Example 6

Formula 1 Example 7

The second formula is:

z_n+1 = z_n³ + z_n + c

the critical values are 0.577350269189i and -0.577350269189i, the first value produces picture one way up and the second produces the pictures the other way up. When I first considered this second post of cubic observations I was to deal with formulae that had no z component, these are generally easier to deal with as one of the critical values is always zero, I have discovered something that surprised me and that is that a formula without a z component can produce a fractal that matches a fractal produced using a formula lacking a z squared component (ignoring size and location).

The formula:

z_n+1 = (1/3)z_n³ + (0 + i)z_n³ + c

with critical values 0 and -2i produces the same fractal. Oddly, I discovered that an initial value of i produces the same picture as -2i even though it isn’t a solution to f'(z) = 0 (see the page on critical values), I have no idea why.

Formula 2 Example 1

Formula 2 Example 2

Formula 2 Example 3

Formula 2 Example 4

Formula 2 Example 5

Finally a formula with a complex parameter:

z_n+1 = z_n³ + (1 + i)z_n + c

the critical values for this formula were determined to be 0.26746253501 – 0.634325568665i and -0.26746253501 + 0.634325568665i.

At first glance everything looks OK, zooming in things are not quite as expected, the Mandelbrot islands are distorted but more than that they appear to have been nibbled.

Formula 3 Example 1

Formula 3 Example 2

Formula 3 Example 3

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