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My name is Daniel Geisler, welcome to my web site dedicated to promoting research into the question of what lies beyond exponentiation - tetration, the hyperoperators and the Ackermann function.

The original site is now at

Extending tetration and the Ackermann function to the complex numbers fractals


Tetration is iterated exponentiation, written as . A major math problem that many amateur mathematicians find their way to is the question of how to extend tetration from the whole numbers , to reals and complex values . The expression is easy to compute, but expressions like and becomes very difficult, but not impossible.

Iterated Functions

Ackermann Function

Exponentiation , tetration , pentation and on with the higher hyperoperators.

Tetration Escape Fractal

Fractional iteration with code


, where is a fixed point

the Lyapunov multiplier, denoted , with .

Mathematica code

Flow[f_, t_, x_, L_, order_ : 3] := Module[{},
   H[0] = L;
   H[1] = f'[L]^t ;
    H[max] = 
     First[r[t] /. 
       RSolve[{r[0] == 0, 
         r[t] == Sum[
            Derivative[k][f][L] BellY[max, k, 
              Table[H[j] /. t -> t - 1, {j, max}]], {k, 2, max}] + 
           f'[L] r[t - 1]}, r[t], t]],
    {max, 2, order}];
   Sum[1/k! H[k] (x - L)^k, {k, 0, order}]

Discrete and continuous tetration compared