Pentation Fractals

[Daniel]

Or more precisely: If \(c\) is the pixel value as a complex number, then the "tetration" fractal would be obtained by the iteration of \(f_c(z)=c^z\), so the "pentation" fractal would be obtained by the iteration of the function \(f_c(z)={}^zc\) while your fractal is obtained by iteration of which function?

I'm iterating \[^z(\sqrt{2})\].

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

What I have done here could be done with a number of different methods. The fractals could provide insight as to with methods are equivalent.
\({\sqrt{2}\uparrow^4 n} = \underbrace{^{^{^{^{^1{\sqrt{2}}}\cdot}\cdot}\sqrt{2}}\sqrt{2}}_{n \sqrt{2}}\)