Tétration, également connu sous le nom de hyper4, superpuissance, superexponentiation, powerlog , ou power tower, est un opérateur mathématique binaire (c'est-à-dire un avec seulement deux entrées), défini comme avec y copies de x. En d'autres termes, la tétration est une exponentiation répétée. Formellement, c'est  où n est un entier non négatif.

La tétration est le quatrième hyperopérateur, et le premier hyperopérateur qui n'apparaît pas dans les mathématiques classiques. Lorsqu'elle est répétée, elle est appelée pentation.

Si c est une constante non triviale, la fonction croît à un rythme similaire à dans la hiérarchie de croissance rapide‏‎.

## Base

L'addition est définie comme un comptage　(fonction successeur) répété : La multiplication est définie comme une addition répétée : L'exponentiation est définie comme une multiplication répétée : Par analogie, la tétration est définie comme une exponentiation répétée : ### Notations

Tetration was independently invented by several people, and due to lack of widespread use it has several notations:

• In notation des puissances itérées de Knuth it is $$x \uparrow\uparrow y$$, nowadays the most common way to denote tetration.
• $$^yx$$ is pronounced "to-the-$$y$$ $$x$$" or "$$x$$ tetrated to $$y$$." The notation is due to Rudy Rucker, and is most often used in situations where none of the higher operators are called for.
• Robert Munafo uses $$x^④y$$, the hyper4 operator.
• In notation des flèches chaînées de Conway it is $$x \rightarrow y \rightarrow 2$$.
• In liner array notation of BEAF it is $$\{x, y, 2\}$$.
• In notation hyper-E it is E[x]1#y (alternatively x^1#y).
• In star notation (as used in the Big Psi project) it is $$x *** y$$.
• An exponential stack of n 2's was written as E*(n) by David Moews, the man who held Bignum Bakeoff.

## Properties

Tetration lacks many of the symmetrical properties of the lower hyper-operators, so it is difficult to manipulate algebraically. However, it does have a few noteworthy properties of its own.

### Power identity

It is possible to show that $${^ba}^{^ca} = {^{c + 1}a}^{^{b - 1}a}$$:

${^ba}^{^ca} = (a^{^{b - 1}a})^{(^ca)} = a^{^{b - 1}a \cdot {}^ca} = a^{^ca \cdot {}^{b - 1}a} = (a^{^ca})^{^{b - 1}a} = {^{c + 1}a}^{^{b - 1}a}$

For example, $${^42}^{^22} = {^32}^{^32} = 2^{64}$$.

## Generalization

### For non-integral $$y$$

Mathematicians have not agreed on the function's behavior on $$^yx$$ where $$y$$ is not an integer. In fact, the problem breaks down into a more general issue of the meaning of $$f^t(x)$$ for non-integral $$t$$. For example, if $$f(x) := x!$$, what is $$f^{2.5}(x)$$? Stephen Wolfram was very interested in the problem of continuous tetration because it may reveal the general case of "continuizing" discrete systems.

Daniel Geisler describes a method for defining $$f^t(x)$$ for complex $$t$$ where $$f$$ is a holomorphic function over $$\mathbb{C}$$ using Taylor series. This gives a definition of complex tetration that he calls hyperbolic tetration.

### As $$y \rightarrow \infty$$

One function of note is infinite tetration, defined as

$^\infty x = \lim_{n\rightarrow\infty}{}^nx$

If we mark the points on the complex plane at which $$^\infty x$$ becomes periodic (as opposed to escaping to infinity), we get an interesting fractal. Daniel Geisler studied this shape extensively, giving names to identifiable features.

## Examples

Here are some small examples of tetration in action:

• $$^22 = 2^2 = 4$$
• $$^32 = 2^{2^2} = 2^4 = 16$$
• $$^23 = 3^3 = 27$$
• $$^33 = 3^{3^3} = 3^{27} = 7 625 597 484 987$$
• $$^42 = 2^{2^{2^2}} = 2^{2^4} = 2^{16} = 65,536$$
• $$^35 = 5^{5^5} \approx 1.9110125979 \cdot 10^{2,184}$$
• $$^52 = 2^{2^{2^{2^2}}} \approx 2.00352993041 \cdot 10^{19,728}$$
• $$^310 = 10^{10^{10}} = 10^{10,000,000,000}$$
• $$^43 = 3^{3^{3^3}} \approx 10^{10^{10^{1.11}}}$$

When given a negative or non-integer base, irrational and complex numbers can occur:

• $$^2{-2} = (-2)^{(-2)} = \frac{1}{(-2)^2} = \frac{1}{4}$$
• $$^3{-2} = (-2)^{(-2)^{(-2)}} = (-2)^{1/4} = \frac{1 + i}{\sqrt{2}}$$
• $$^2(1/2) = (1/2)^{(1/2)} = \sqrt{1/2} = \frac{\sqrt2}{2}$$
• $$^3(1/2) = (1/2)^{(1/2)^{(1/2)}} = (1/2)^{\sqrt{2}/2}$$

Functions whose growth rates are on the level of tetration include:

## Super root

Let $$k$$ be a positive integer. Since ka is well-defined for any non-negative real number a and is a strictly increasing unbounded function, we can define a root inverse function $$sr_k \colon [0,\infty) \to [0,\infty)$$ as:

$$sr_k(n) = x \text{ such that } ^kx = n$$

### Numerical evaluation

The second-order super root can be calculated as:

$$\frac{ln(x)}{W(ln(x))}$$

where $$W(n)$$ is the Lambert W function.

Formulas for higher-order super roots are unknown.

## Pseudocode

Below is an example of pseudocode for tetration.

function tetration(a, b):
result := 1
repeat b times:
result := a to the power of result
return result