[Regular tetration] [Iteration series] norming fixpoint-dependencies

[Gottfried]

ok that makes some sense.

now explain me what you are doing in this thread and WHY ?

Using base b= sqrt(2)

• what is the height-representation of 3 ?
  By Kousnetzov? By Robbins slog? By regular-tetration using fixpoint 2? using fixpoint 4?
  

Besides the difficulty to determine "the" height of 3 by the above methods - there is still room to find an optimal solution for tetration. My idea gives one which is only depending on the integer heights, thus for instance independent of the selection of the fixpoint.

and WHY is f(z) periodic ???

Code:

´
f(z)         = ... z^^-3 + z^^-2 - z^^-1 + z - z^^1 + z^^2 - z^^3 +...

let y=b^b^z = z^^2
f(y)        = ... y^^-3 + y^^-2 - y^^-1 + y    - y^^1 + y^^2 - y^^3 +...
            = ... z^^-1 + z     - z^^ 1 + z^^2 - z^^3 + ...
            = f(z)

f(z) and f(y)=f(z^^2) are equal => f(z) is periodic in terms of the height-parameter

how to compute its period ??

f(z) = f(z^^2) where z^^2 = b^b^z ==> period-length is delta_h = 2

It is more difficult to find the amplitude