Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. The arithmetic operator consisting of repeated exponentiation, by analogy with exponentiation being repeated multiplication and multiplication being repeated addition, ba denoting a to the power of a to the power of ... to the power of a, in which a appears b times. Notation: ba, or
Etymologies
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Examples

In such a way, the name tetration indicates, that this operation is fourth (id est, tetra) in the hierarchy of operations after summation, multiplication, and exponentiation.

So off I go to Google largest numbers and we are learning about googol, googolplex, Graham's number, large number notation, exponentiation, tetration.

Basic properties of tetration are determined by the base ln (b) are plotted in figure 3.

Requirement of definition of tetration determine the asymptotic of the solution.

Q corresponds to the asymptotic behavior of tetration.

Q tetermines periodicity of quasiperiodicity of tetration.

As the asymptoric of tetration is crutually depend of base

Intermediate levels are shown with thin lines., tetration is asymptotically periodic.

At b = exp (1 / e), tetration is not periodic, and no exponential asymptotic exist.

At corresponds to tetration, decaying to the asymptotic value
shevek commented on the word tetration
As exponentiation is to multiplication, tetration is to exponentiation. Quite an interesting operation, because no one has yet found a well behaved way to generalize tetration beyond the natural numbers. Tetration is notated as a superscript before the number to be tetrated.
July 11, 2008