Pentation Fractals

[bo198214]

You mean, you iterate \(f(z)={^z}(\sqrt{2})\) with initial value \(z_0=c\).
But a pentation fractal (in your terminology) would be to iterate \(f(z)={^z}c\) with initial value \(z_0=c\) or even \(z_0=0\) or \(z_0=1\).
Do you agree?

\(z_0=1\) because \({^0}(\sqrt{2})=1\).

Daniel, a fractal is made up of pixels, each representing a complex value \(c\), for each pixel the color of the pixel is associated with the escape behaviour of the iteration of a function \(f_c\). If that function does not depend on \(c\) and the initial value for the iteration also does not depend on \(c\) - like when you suggest that \(f_c(z)={^z}(\sqrt{2})\) and \(z_0=1\)  - then the fractal will have the same color for each pixel. Clearly this is not the case in the pictures given by you, hence the function and initial value can not be as you gave them.
So what is really used in the fractal?