Wiki Googologie
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Wiki Googologie

Tétration, également connu sous le nom de hyper4, superpuissance, superexponentiation, powerlog , ou power tower,[1] 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 \(\{x, y, 2\}\)[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 \(x *** y\).[3]
  • 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:

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

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.[5]

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

Sources

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