03/18/2023, 05:16 AM
Tetrational Geometry
The equation \[f(exp(z)) = exp(f(z))\] tells us we want to be looking at tetration. Now add the constraint \[f(f(z))=z\] in the context of tetration. This means we are looking at period 2 tetration. The orange disk in the following image and the red disk in the image after is the area where tetration displays period 2 behavior. See Pseudocircle The number 0 lies in the disk and is period 2 under tetration,
\[0^0=1, 0^1=0\]
![[Image: StandardGrid.gif]](https://tetration.org/original/Fractals/Atlas/StandardGrid.gif)
![[Image: MOONJ002.gif]](https://tetration.org/original/Fractals/Atlas/Pseudocircle/MOONJ002.gif)
\[f(exp(z)) = exp(f(z)) \implies f(z) = \, ^{z+n}e\] which is inconsistent with the complex base being within the pseudocircle.
The question could be generalized to \[f(a^z) = a^{f(z)} \textrm{ and } f(f(z))=z \textrm{ where } a \textrm{ is in the pseudocircle.}\]. Since \[a \ne e\] the question has no solution beyond the trivial solution.
The equation \[f(exp(z)) = exp(f(z))\] tells us we want to be looking at tetration. Now add the constraint \[f(f(z))=z\] in the context of tetration. This means we are looking at period 2 tetration. The orange disk in the following image and the red disk in the image after is the area where tetration displays period 2 behavior. See Pseudocircle The number 0 lies in the disk and is period 2 under tetration,
\[0^0=1, 0^1=0\]
\[f(exp(z)) = exp(f(z)) \implies f(z) = \, ^{z+n}e\] which is inconsistent with the complex base being within the pseudocircle.
The question could be generalized to \[f(a^z) = a^{f(z)} \textrm{ and } f(f(z))=z \textrm{ where } a \textrm{ is in the pseudocircle.}\]. Since \[a \ne e\] the question has no solution beyond the trivial solution.
Daniel