View source for Tetration

{{distinguish|titration}}
[[Image:TetrationComplexColor.png|thumb|border|268px|[[Domain coloring]] of the [[holomorphic function|holomorphic]] tetration <math>{}^{z}e</math>, with [[hue]] representing the function [[Argument (complex analysis)|argument]] and [[brightness]] representing magnitude]]
[[Image:TetrationConvergence2D.svg|thumb|<math>{}^{n}x</math>, for {{math|1=''n'' = 2, 3, 4, …}}, showing convergence to the infinitely iterated exponential between the two dots]]

In [[mathematics]], '''tetration''' (or '''hyper-4''') is an [[operation (mathematics)| operation]] based on [[iterated]], or repeated, [[exponentiation]]. It is the next [[hyperoperation]] after [[exponentiation]], but before [[pentation]]. The word was coined by [[Reuben Louis Goodstein]] from [[tetra-]] (four) and [[iterated function|iteration]]. 

Under the definition as repeated exponentiation, the notation <math>{^{n}a}</math> means <math>{a^{a^{\cdot^{\cdot^{a}}}}}</math>, where ''{{Mvar|n}}'' copies of ''{{Mvar|a}}'' are iterated via exponentiation, right-to-left, I.e. the application of exponentiation <math>n-1</math> times. ''{{Mvar|n}}'' is called the "height" of the function, while ''{{Mvar|a}}'' is called the "base," analogous to exponentiation. It would be read as "the {{mvar|n}}th tetration of {{mvar|a}}". 

Tetration is also defined recursively as 
:<math>{^{n}a} := \begin{cases} 1 &\text{if }n=0 \\ a^{\left(^{(n-1)}a\right)} &\text{if }n>0 \end{cases} </math>,
allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers.

The two inverses of tetration are called the '''super-root''' and the '''[[super-logarithm]]''', analogous to the nth root and the logarithmic functions. None of the three functions are [[elementary function|elementary]].

Tetration is used for the [[Large numbers#Standardized system of writing very large numbers|notation of very large numbers]].

==Introduction==

The first four [[hyperoperation]]s are shown here, with tetration being considered the fourth in the series. The [[unary operation]] [[Successor function|succession]], defined as <math>a' = a + 1</math>, is considered to be the zeroth operation.

#[[Addition]]
#:<math>a + n = a + \underbrace{1 + 1 + \cdots + 1}_n</math>
#::{{Mvar|n}} copies of 1 added to {{Mvar|a}}.
#[[Multiplication]]
#:<math>a \times n = \underbrace{a + a + \cdots + a}_n</math>
#::''{{Mvar|n}}'' copies of ''{{Mvar|a}}'' combined by addition.
#[[Exponentiation]]
#:<math>a^n = \underbrace{a \times a \times \cdots \times a}_n</math>
#::''{{Mvar|n}}'' copies of ''{{Mvar|a}}'' combined by multiplication.
#Tetration
#:<math>{^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n</math>
#::''{{Mvar|n}}'' copies of ''{{Mvar|a}}'' combined by exponentiation, right-to-left.<ref name="mathvault">{{Cite web|url=https://mathvault.ca/derivative-tetration-hyperexponentiation/|title=Derivative of $x^x$, $x^{x^x}$, and a Venture Into Tetration and Hyper-Exponentiation|date=2016-01-01|website=Math Vault|language=en-US|access-date=2019-07-25}}</ref>

Succession, {{math|(''a′'' {{=}} ''a'' + 1)}}, is the most basic operation; while addition ({{math|''a'' + ''n''}}) is a primary operation, for addition of natural numbers it can be thought of as a chained succession of {{mvar|n}} successors of {{mvar|a}}; multiplication {{math|(''a'' × ''n''}}) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving {{mvar|n}} numbers of {{mvar|a}}. Exponentiation can be thought of as a chained multiplication involving {{mvar|n}} numbers of {{mvar|a}} and tetration (<math>^{n}a\!</math>) as a chained power involving {{mvar|n}} numbers {{mvar|a}}. Each of the operations above are defined by iterating the previous one;<ref name="uwu">Neyrinck, Mark. [http://skysrv.pha.jhu.edu/~neyrinck/extessay.pdf An Investigation of Arithmetic Operations.] Retrieved 9 January 2019.</ref> however, unlike the operations before it, tetration is not an [[elementary function]].

The parameter {{mvar|a}} is referred to as the '''base''', while the parameter {{mvar|n}} may be referred to as the '''height'''. In the original definition of tetration, the height parameter must be a natural number; for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration; for any positive [[real number|real]] <math> a > 0 </math> and non-negative [[integer]] <math> n \ge 0 </math>, we can define <math>\,\! {^{n}a} </math> recursively as:<ref name="uwu"/>
:<math>{^{n}a} := \begin{cases} 1 &\text{if }n=0 \\ a^{\left(^{(n-1)}a\right)} &\text{if }n>0 \end{cases} </math>
The recursive definition is equivalent to repeated exponentiation for [[Natural number|natural]] heights; however, this definition allows for extensions to the other heights such as <math>^{0}a</math>, <math>^{-1}a</math>, and <math>^{i}a</math> as well – many of these extensions are areas of active research.

== Terminology ==
There are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale.

* The term ''tetration'', introduced by Goodstein in his 1947 paper ''Transfinite Ordinals in Recursive Number Theory''<ref>{{cite journal |author=R. L. Goodstein |title=Transfinite ordinals in recursive number theory |jstor=2266486 |journal=Journal of Symbolic Logic |volume=12 |year=1947 |issue=4 |doi=10.2307/2266486 |pages=123–129}}</ref> (generalizing the recursive base-representation used in [[Goodstein's theorem]] to use higher operations), has gained dominance. It was also popularized in [[Rudy Rucker]]'s ''[[Infinity and the Mind]]''.
* The term ''superexponentiation'' was published by Bromer in his paper ''Superexponentiation'' in 1987.<ref>{{cite journal |author=N. Bromer |title=Superexponentiation |journal=Mathematics Magazine |volume=60 |issue=3 |year=1987 |pages=169–174 |jstor=2689566}}</ref> It was used earlier by Ed Nelson in his book Predicative Arithmetic, Princeton University Press, 1986.
* The term ''hyperpower''<ref>{{cite journal |author=J. F. MacDonnell |title=Somecritical points of the hyperpower function <math>x^{x^{\dots}}</math> |journal=International Journal of Mathematical Education |year=1989 |volume=20 |issue=2 |pages=297–305 |mr=994348 |url=http://www.faculty.fairfield.edu/jmac/ther/tower.htm |doi=10.1080/0020739890200210}}</ref> is a natural combination of ''hyper'' and ''power'', which aptly describes tetration. The problem lies in the meaning of ''hyper'' with respect to the [[hyperoperation]] sequence. When considering hyperoperations, the term ''hyper'' refers to all ranks, and the term ''super'' refers to rank 4, or tetration. So under these considerations ''hyperpower'' is misleading, since it is only referring to tetration.
* The term ''power tower''<ref>{{MathWorld |urlname=PowerTower |title=Power Tower}}</ref> is occasionally used, in the form "the power tower of order {{mvar|n}}" for <math>{\ \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop n}</math>. This is a misnomer, however, because tetration cannot be expressed with iterated ''power'' functions (see above), since it is an iterated ''exponential'' function.

Owing in part to some shared terminology and similar [[Mathematical notation|notational symbolism]], tetration is often confused with closely related functions and expressions. Here are a few related terms:
:{|class="wikitable"
! Form
! Terminology
|-
|<math>a^{a^{\cdot^{\cdot^{a^a}}}}</math>
|Tetration
|-
|<math>a^{a^{\cdot^{\cdot^{a^x}}}}</math>
|Iterated exponentials
|-
|<math>a_1^{a_2^{\cdot^{\cdot^{a_n}}}}</math>
|Nested exponentials (also towers)
|-
|<math>a_1^{a_2^{a_3^{\cdot^{\cdot^\cdot}}}}</math>
|Infinite exponentials (also towers)
|}

In the first two expressions {{mvar|a}} is the ''base'', and the number of times {{mvar|a}} appears is the ''height'' (add one for {{mvar|x}}). In the third expression, {{mvar|n}} is the ''height'', but each of the bases is different.

Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean [[iterated function|iterated]] [[power (mathematics)|powers]] or iterated [[exponential function|exponentials]].

== Notation ==
There are many different notation styles that can be used to express tetration. Some notations can also be used to describe other [[hyperoperation]]s, while some are limited to tetration and have no immediate extension.
:{|class="wikitable"
! Name
! Form
! Description
|-
| Rudy Rucker notation
| <math>\,{}^{n}a</math>
| Used by Maurer [1901] and Goodstein [1947]; [[Rudy Rucker]]'s book ''[[Infinity and the Mind]]'' popularized the notation.
|-
| [[Knuth's up-arrow notation]]
| <math>a {\uparrow\uparrow} n</math>
| Allows extension by putting more arrows, or, even more powerfully, an indexed arrow.
|-
| [[Conway chained arrow notation]]
| <math>a \rightarrow n \rightarrow 2</math>
| Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain
|-
| [[Ackermann function]]
| <math>{}^{n}2 = \operatorname{A}(4, n - 3) + 3</math>
| Allows the special case <math>a=2</math> to be written in terms of the Ackermann function.
|-
| Iterated exponential notation
| <math>\exp_a^n(1)</math>
| Allows simple extension to iterated exponentials from initial values other than 1.
|-
| Hooshmand notations<ref name="uxp">
 {{cite journal
  | author=M. H. Hooshmand,
  | year=2006
  | title=Ultra power and ultra exponential functions
  | journal=[[Integral Transforms and Special Functions]]
  | volume=17
  | issue=8
  | pages=549–558
  | doi= 10.1080/10652460500422247
 }}</ref>
| <math>\begin{align}
  &\operatorname{uxp}_a n \\[2pt]
  &a^{\frac{n}{}}
\end{align}</math>
| Used by M. H. Hooshmand [2006].
|-
| [[Hyperoperation]] notations
| <math>\begin{align}
  &a [4] n \\[2pt]
  &H_4(a, n)
\end{align}</math>
| Allows extension by increasing the number 4; this gives the family of [[hyperoperation]]s.
|-
| Double caret notation
| <code>a^^n</code>
| Since the up-arrow is used identically to the caret (<code>^</code>), tetration may be written as (<code>^^</code>); convenient for [[ASCII]].
|-
| [[Bowers's operators]]
| {{ubl
 | {a,b,4}
 | {a,b,4,1}
 | a {4} b
 }}
| Similar to the hyperoperation notations.
|}

One notation above uses iterated exponential notation; this is defined in general as follows:

:<math>\exp_a^n(x) = a^{a^{\cdot^{\cdot^{a^x}}}}</math> with {{mvar|n}} {{mvar|a}}s.

There are not as many notations for iterated exponentials, but here are a few:
:{| class="wikitable"
! Name
! Form
! Description
|-
| Standard notation
| <math>\exp_a^n(x)</math>
| [[Leonhard Euler|Euler]] coined the notation <math>\exp_a(x) = a^x</math>, and iteration notation <math>f^n(x)</math> has been around about as long.
|-
| Knuth's up-arrow notation
| <math>(a{\uparrow})^n(x)</math>
| Allows for super-powers and super-exponential function by increasing the number of arrows; used in the article on [[large numbers]].
|-
| Text notation
| {{code|2=tex|exp_a^n(x)}}
| Based on standard notation; convenient for [[ASCII]].
|-
| J Notation
| {{code|2=j|x^^:(n-1)x}}
| Repeats the exponentiation. See [[J (programming language)]]<ref>{{cite web|title=Power Verb|url=http://www.jsoftware.com/help/dictionary/d202n.htm|work=J Vocabulary|publisher=J Software|accessdate=28 October 2011}}</ref>
|}

== Examples ==
Because of the extremely fast growth of tetration, most values in the following table are too large to write in scientific notation. In these cases, iterated exponential notation is used to express them in base 10. The values containing a decimal point are approximate.

:{| class="wikitable"
! <math>x</math>
! <math>{}^{2}x</math>
! <math>{}^{3}x</math>
! <math>{}^{4}x</math>
! <math>{}^{5}x</math>
|- align=right
| 1
| 1
| 1
| 1
| 1
|- align=right
| 2
| 4
| 16
| [[65536 (number)|65,536]]
| 2<sup>65,536</sup> or (2.00353 × 10<sup>19,728</sup>)
|- align=right
| 3
| 27
| 7,625,597,484,987
| <math>\exp_{10}^3(1.09902)</math> (3.6 × 10<sup>12</sup> digits)
| <math>\exp_{10}^4(1.09902)</math>
|- align=right
| 4
| 256
| 1.34078 × 10<sup>154</sup>
| <math>\exp_{10}^3(2.18726)</math> (8.1 × 10<sup>153</sup> digits)
| <math>\exp_{10}^4(2.18726)</math>
|- align=right
| 5
| 3,125
| 1.91101 × 10<sup>2,184</sup>
| <math>\exp_{10}^3(3.33928)</math> (1.3 × 10<sup>2,184</sup> digits)
| <math>\exp_{10}^4(3.33928)</math>
|- align=right
| 6
| 46,656
| 2.65912 × 10<sup>36,305</sup>
| <math>\exp_{10}^3(4.55997)</math> (2.1 × 10<sup>36,305</sup> digits)
| <math>\exp_{10}^4(4.55997)</math>
|- align=right
| 7
| 823,543
| 3.75982 × 10<sup>695,974</sup>
| <math>\exp_{10}^3(5.84259)</math> (3.2 × 10<sup>695,974</sup> digits)
| <math>\exp_{10}^4(5.84259)</math>
|- align=right
| 8
| 16,777,216
| 6.01452 × 10<sup>15,151,335</sup>
| <math>\exp_{10}^3(7.18045)</math> (5.4 × 10<sup>15,151,335</sup> digits)
| <math>\exp_{10}^4(7.18045)</math>
|- align=right
| 9
| 387,420,489
| 4.28125 × 10<sup>369,693,099</sup>
| <math>\exp_{10}^3(8.56784)</math> (4.1 × 10<sup>369,693,099</sup> digits)
| <math>\exp_{10}^4(8.56784)</math>
|- align=right
| 10
| 10,000,000,000
| 10<sup>10,000,000,000</sup>
| <math>\exp_{10}^3(10)</math> (10<sup>10<sup>10</sup></sup>+1 digits)
| <math>\exp_{10}^4(10)</math>
|}

==Properties==
Tetration has several properties that are similar to exponentiation, as well as properties that are specific to the operation and are lost or gained from exponentiation. Because exponentiation does not [[commutative law|commute]], the product and power rules do not have an analogue with tetration; the statements <math display="inline">{}^a \left({}^b x\right) = \left({}^{ab} x\right)</math> and <math display="inline">{}^a \left(xy\right) = {}^a x {}^a y</math> are not necessarily true for all cases.<ref>Alexander Meiburg. (2014). [http://web.math.ucsb.edu/~padraic/ucsb_2013_14/mathcs103_s2014/mathcs103_s2014_alex_presentation.pdf Analytic Extension of Tetration Through the Product Power-Tower] Retrieved November 29, 2018</ref>

However, tetration does follow a different property, in which <math display="inline">{}^a x = x^{\left({}^{a-1} x\right)}</math>. This fact is most clearly shown using the recursive definition. From this property, a proof follows that <math>\left({}^b a\right)^{\left({}^c a\right)} = \left({}^{c+1} a\right)^{\left({}^{b-1} a\right)}</math>, which allows for switching ''b'' and ''c'' in certain equations. The proof goes as follows:
: <math>\begin{align}
      &\left({}^b a\right)^{\left({}^c a\right)}         \\
  ={} &\left(a^{{}^{b-1} a}\right)^{\left({}^c a\right)} \\
  ={} &a^{\left({}^{b-1} a\right)\left({}^c a\right)}    \\
  ={} &a^{\left({}^c a\right)\left({}^{b-1} a\right)}    \\
  ={} &\left({}^{c+1} a\right)^{\left({}^{b-1} a\right)}
\end{align}</math>

When a number {{mvar|x}} and 10 are [[coprime]], it is possible to compute the last {{mvar|m}} decimal digits of <math>\,\!\ ^{a}x</math> using [[Euler's theorem]], for any integer {{mvar|m}}.

===Direction of evaluation===
When evaluating tetration expressed as an "exponentiation tower", the exponentiation is done at the deepest level first (in the notation, at the apex).<ref name="mathvault"/> For example:
:<math>\,\!\ ^{4}2 = 2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{\left(2^4\right)} = 2^{16} = 65,\!536</math>

This order is important because exponentiation is not [[associative]], and evaluating the expression in the opposite [[Bracket (mathematics)#Algebra|order]] will lead to a different answer:
:<math>\,\! 2^{2^{2^2}} \ne \left({\left(2^2\right)}^2\right)^2 = 4^{2 \cdot2} = 256</math>

Evaluating the expression the left to right is considered less interesting; evaluating left to right, any expression <math>^{n}a\!</math> can be simplified to be <math>a^{\left(a^{n-1}\right)}\!\!</math>.<ref name="tetration extensions">{{citeweb |url=http://www.mpmueller.net/reihenalgebra.pdf |last=Müller |first= M. |title= Reihenalgebra: What comes beyond exponentiation? |accessdate= 12 December 2018}}</ref> Because of this, the towers must be evaluated from right to left (or top to bottom). Computer programmers refer to this choice as [[right-associative]].

==Extensions==
Tetration can be extended in two different ways; in the equation <math>^na\!</math>, both the base {{mvar|a}} and the height {{mvar|n}} can be generalized using the definition and properties of tetration. Although the base and the height can be extended beyond the non-negative integers to different [[domain of a function|domain]]s, including <math>{^n 0}</math>, complex functions such as <math>{}^{n}i</math>, and heights of infinite {{mvar|n}}, the more limited properties of tetration reduce the ability to extend tetration.

===Extension of domain for bases===
====Base zero====
The exponential <math>0^0</math> is not consistently defined. Thus, the tetrations <math>\,{^{n}0}</math> are not clearly defined by the formula given earlier. However, <math>\lim_{x\rightarrow0} {}^{n}x</math> is well defined, and exists:<ref>{{Cite web|url=https://math.blogoverflow.com/2015/01/05/climbing-the-ladder-of-hyper-operators-tetration/|title=Climbing the ladder of hyper operators: tetration « Stack Exchange Mathematics Blog|website=math.blogoverflow.com|access-date=2019-07-25}}</ref>
:<math>\lim_{x\rightarrow0} {}^{n}x = \begin{cases}
  1, & n \text{ even} \\
  0, & n \text{ odd}
\end{cases}</math>

Thus we could consistently define <math>{}^{n}0 = \lim_{x\rightarrow 0} {}^{n}x</math>. This is analogous to defining <math>0^0 = 1</math>.

Under this extension, <math>{}^{0}0 = 1</math>, so the rule <math>{^{0}a} = 1</math> from the original definition still holds.

====Complex bases====
[[Image:Tetration period.png|thumbnail|Tetration by period]]
[[Image:Tetration escape.png|thumbnail|Tetration by escape]]

Since [[complex number]]s can be raised to powers, tetration can be applied to ''bases'' of the form {{math|''z'' {{=}} ''a'' + ''bi''}} (where {{mvar|a}} and {{mvar|b}} are real). For example, in {{math|<sup>''n''</sup>''z''}} with {{math|''z'' {{=}} ''i''}}, tetration is achieved by using the [[principal branch]] of the natural logarithm; using [[Euler's formula]] we get the relation:

:<math>i^{a+bi} = e^{\frac{1}{2}{\pi i} (a + bi)} = e^{-\frac{1}{2}{\pi b}} \left(\cos{\frac{\pi a}{2}} + i \sin{\frac{\pi a}{2}}\right)</math>

This suggests a recursive definition for {{math|<sup>''n''+1</sup>''i'' {{=}} ''a′'' + ''b′i''}} given any {{math|<sup>''n''</sup>''i'' {{=}} ''a'' + ''bi''}}:
:<math>\begin{align}
  a' &= e^{-\frac{1}{2}{\pi b}} \cos{\frac{\pi a}{2}} \\[2pt]
  b' &= e^{-\frac{1}{2}{\pi b}} \sin{\frac{\pi a}{2}}
\end{align}</math>

The following approximate values can be derived:
{| class="wikitable"
|-
! <math display="inline">{}^{n}i</math>
! Approximate value
|-
| <math display="inline">{}^{1}i = i</math>
| {{math|''i''}}
|-
| <math display="inline">{}^{2}i = i^{\left({}^{1}i\right)}</math>
| {{math|0.2079}}
|-
| <math display="inline">{}^{3}i = i^{\left({}^{2}i\right)}</math>
| {{math|0.9472 + 0.3208''i''}}
|-
| <math display="inline">{}^{4}i = i^{\left({}^{3}i\right)}</math>
| {{math|0.0501 + 0.6021''i''}}
|-
| <math display="inline">{}^{5}i = i^{\left({}^{4}i\right)}</math>
| {{math|0.3872 + 0.0305''i''}}
|-
| <math display="inline">{}^{6}i = i^{\left({}^{5}i\right)}</math>
| {{math|0.7823 + 0.5446''i''}}
|-
| <math display="inline">{}^{7}i = i^{\left({}^{6}i\right)}</math>
| {{math|0.1426 + 0.4005''i''}}
|-
| <math display="inline">{}^{8}i = i^{\left({}^{7}i\right)}</math>
| {{math|0.5198 + 0.1184''i''}}
|-
| <math display="inline">{}^{9}i = i^{\left({}^{8}i\right)}</math>
| {{math|0.5686 + 0.6051''i''}}
|}

Solving the inverse relation, as in the previous section, yields the expected {{math|<sup>0</sup>''i'' {{=}} 1}} and {{math|<sup>−1</sup>''i'' {{=}} 0}}, with negative values of {{mvar|n}} giving infinite results on the imaginary axis. Plotted in the [[complex plane]], the entire sequence spirals to the limit {{math|0.4383 + 0.3606''i''}}, which could be interpreted as the value where {{mvar|n}} is infinite.

Such tetration sequences have been studied since the time of Euler, but are poorly understood due to their chaotic behavior. Most published research historically has focused on the convergence of the infinitely iterated exponential function. Current research has greatly benefited by the advent of powerful computers with [[fractal]] and symbolic mathematics software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.{{Citation needed|date=July 2019}}

===Extensions of the domain for different heights===

====Infinite heights====
[[Image:Infinite_power_tower.svg|thumb|<math>\textstyle \lim_{n\rightarrow \infty} {}^nx</math> of the infinitely iterated exponential converges for the bases <math>\textstyle \left(e^{-1}\right)^e \le x \le e^{\left(e^{-1}\right)} </math>]]
[[Image:TetrationConvergence3D.png|thumbnail|The function <math>\left| \frac{\mathrm{W}(-\ln{z})}{-\ln{z}} \right|</math> on the complex plane, showing the real-valued infinitely iterated exponential function (black curve)]]

Tetration can be extended to [[Infinity|infinite]] heights;<ref>{{cite web | url=http://math.blogoverflow.com/2015/01/05/climbing-the-ladder-of-hyper-operators-tetration/ | title=Climbing the ladder of hyper operators: tetration | publisher=George Daccache | date=January 5, 2015 | accessdate=18 February 2016}}</ref> i.e., for certain {{mvar|a}} and {{mvar|n}} values in <math>{}^{n}a</math>, there exists a well defined result for an infinite {{mvar|n}}. This is because for bases within a certain interval, tetration converges to a finite value as the height tends to [[infinity]]. For example, <math>\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}}</math> converges to 2, and can therefore be said to be equal to 2. The trend towards 2 can be seen by evaluating a small finite tower:

:<math>\begin{align}
  \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.414}}}}}
    &\approx \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.63}}}} \\
    &\approx \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.76}}} \\
    &\approx \sqrt{2}^{\sqrt{2}^{1.84}} \\
    &\approx \sqrt{2}^{1.89} \\
    &\approx 1.93
\end{align}</math>

In general, the infinitely iterated exponential <math>x^{x^{\cdot^{\cdot^{\cdot}}}}\!\!</math>, defined as the limit of <math>{}^{n}x</math> as {{mvar|n}} goes to infinity, converges for {{math|''e''<sup>−''e''</sup> ≤ ''x'' ≤ ''e''<sup>1/''e''</sup>}}, roughly the interval from 0.066 to 1.44, a result shown by [[Leonhard Euler]].<ref>Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." ''Acta Acad. Scient. Petropol. 2'', 29–51, 1783. Reprinted in Euler, L. ''Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae''. Leipzig, Germany: Teubner, pp. 350–369, 1921. ([http://math.dartmouth.edu/~euler/docs/originals/E532.pdf facsimile])</ref> The limit, should it exist, is a positive real solution of the equation {{math|1=''y'' = ''x''<sup>''y''</sup>}}. Thus, {{math|1=''x'' = ''y''<sup>1/''y''</sup>}}. The limit defining the infinite tetration of {{mvar|x}} fails to converge for  {{math|''x'' > ''e''<sup>1/''e''</sup>}} because the maximum of {{math|''y''<sup>1/''y''</sup>}} is {{math|''e''<sup>1/''e''</sup>}}.

This may be extended to complex numbers {{mvar|z}} with the definition:
:<math>{}^{\infty}z = z^{z^{\cdot^{\cdot^{\cdot}}}} = \frac{\mathrm{W}(-\ln{z})}{-\ln{z}} ~,</math>
where {{math|W}} represents [[Lambert's W function]].

As the limit {{math|1=''y'' = <sup>∞</sup>''x''}} (if existent, i.e. for {{math|''e''<sup>−''e''</sup> < ''x'' < ''e''<sup>1/''e''</sup>}}) must satisfy {{math|1=''x''<sup>''y''</sup> = ''y''}} we see that {{math|1=''x'' ↦ ''y'' = <sup>∞</sup>''x''}} is (the lower branch of) the inverse function of {{math|1=''y'' ↦ ''x'' = ''y''<sup>1/''y''</sup>}}.

====Negative heights====
We can use the recursive rule for tetration,
:<math> {^{k+1}a} = a^{\left({^{k}a}\right)}, </math>

to prove <math>{}^{-1}a</math>:
:<math> ^{k}a = \log_a \left(^{k+1}a\right);</math>

Substituting −1 for {{mvar|k}} gives
:<math> {}^{-1}a = \log_{a} \left({}^0 a\right) = \log_a 1 = 0</math>.<ref name="tetration extensions"/>

Smaller negative values cannot be well defined in this way. Substituting −2 for {{mvar|k}} in the same equation gives
:<math> {}^{-2}a = \log_{a} \left( {}^{-1}a \right) = \log_a 0 </math>

which is not well defined. They can, however, sometimes be considered sets.<ref name="tetration extensions"/>

For <math> n = 1 </math>, any definition of <math>\,\! {^{-1}1} </math> is consistent with the rule because
:<math> {^{0}1} = 1 = 1^n </math> for any <math>\,\! n = {^{-1}1} </math>.

====Real heights====
{{more footnotes|date=July 2019}}

At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of {{mvar|n}}. There have, however, been multiple approaches towards the issue, and different approaches are outlined below.

In general, the problem is finding — for any real {{math|''a'' > 0}} — a ''super-exponential function'' <math>\,f(x) = {}^{x}a</math> over real {{math|''x'' > −2}} that satisfies
*<math>\,{}^{-1}a = 0</math>
*<math>\,{}^{0}a = 1</math>
*<math>\,{}^{x}a = a^{\left({}^{x-1}a\right)}</math>for all real <math>x > -1.</math><ref>{{cite web |url=https://www.researchgate.net/profile/Dmitrii_Kouznetsov/publication/265066072_5_methods_for_real_analytic_tetration/links/544e57db0cf26dda08900c3d.pdf|title=5+ methods for real analytic tetration|last=Trappmann|first=Henryk|last2=Kouznetsov|first2=Dmitrii||date=June 28, 2010|access-date= 5 December 2018}}</ref>

To find a more natural extension, one or more extra requirements are usually required. This is usually some collection of the following:
*A ''continuity'' requirement (usually just that <math>{}^{x}a</math> is continuous in both variables for <math>x > 0</math>).
*A ''differentiability'' requirement (can be once, twice, {{mvar|k}} times, or infinitely differentiable in {{mvar|x}}).
*A ''regularity'' requirement (implying twice differentiable in {{mvar|x}}) that:
:<math>\left( \frac{d^2}{dx^2}f(x) > 0\right)</math> for all <math>x > 0</math>

The fourth requirement differs from author to author, and between approaches. There are two main approaches to extending tetration to real heights; one is based on the ''regularity'' requirement, and one is based on the ''differentiability'' requirement. These two approaches seem to be so different that they may not be reconciled, as they produce results inconsistent with each other.

When <math>\,{}^{x}a</math> is defined for an interval of length one, the whole function easily follows for all {{math|''x'' > −2}}.

=====Linear approximation for real heights=====
[[Image:Real-tetration.png|thumbnail|<math>\,{}^{x}e</math> using linear approximation.]]
A [[linear approximation]] (solution to the continuity requirement, approximation to the differentiability requirement) is given by:
:<math>{}^{x}a \approx \begin{cases}
  \log_a\left(^{x+1}a\right) &  x \le -1 \\
                       1 + x & -1 < x \le 0 \\
    a^{\left(^{x-1}a\right)} &  0 < x
\end{cases}</math>

hence:
{| class="wikitable"
! Approximation
! Domain
|-
|  <math display="inline">{}^x a \approx x + 1</math>
| for {{math|−1 < ''x'' < 0}}
|-
|  <math display="inline">{}^x a \approx a^x</math>
| for {{math|0 < ''x'' < 1}}
|-
|  <math display="inline">{}^x a \approx a^{a^{(x-1)}}</math>
| for {{math|1 < ''x'' < 2}}
|}

and so on. However, it is only piecewise differentiable; at integer values of {{mvar|x}} the derivative is multiplied by <math>\ln{a}</math>. It is continuously differentiable for <math>x > -2</math> if and only if <math>a = e</math>. For example, using these methods <math>{}^\frac{\pi}{2}e \approx 5.868...</math> and <math>{}^{-4.3}0.5 \approx 4.03335...</math>

A main theorem in Hooshmand's paper<ref name="uxp"/> states: Let <math>0 < a \neq 1</math>. If <math>f:(-2, +\infty)\rightarrow \mathbb{R}</math> is continuous and satisfies the conditions:

*<math>f(x) = a^{f(x-1)} \;\; \text{for all} \;\; x > -1, \; f(0) = 1,</math>
*<math>f</math> is differentiable on {{open-open|−1, 0}},
*<math>f^\prime</math> is a nondecreasing or nonincreasing function on {{open-open|−1, 0}},
*<math>f^\prime \left(0^+\right) = (\ln a) f^\prime \left(0^-\right) \text{ or } f^\prime \left(-1^+\right) = f^\prime \left(0^-\right).</math>

then <math>f</math> is uniquely determined through the equation
:<math>f(x) = \exp^{[x]}_a \left(a^{(x)}\right) = \exp^{[x+1]}_a((x)) \quad \text{for all} \; \; x > -2,</math>

where <math> (x) = x - [x] </math> denotes the fractional part of {{mvar|x}} and <math> \exp^{[x]}_a </math> is the <math> [x] </math>-[[iterated function]] of the function <math> \exp_a </math>.

The proof is that the second through fourth conditions trivially imply that {{mvar|f}} is a linear function on {{closed-closed|−1, 0}}.

The linear approximation to natural tetration function <math>{}^xe</math> is continuously differentiable, but its second derivative does not exist at integer values of its argument. Hooshmand derived another uniqueness theorem for it which states:

If <math> f: (-2, +\infty)\rightarrow \mathbb{R}</math> is a continuous function that satisfies:

*<math> f(x) = e^{f(x-1)} \;\; \text{for all} \;\; x > -1, \; f(0) = 1,</math>
*<math>f</math> is convex on {{open-open|−1, 0}},
*<math>f^\prime \left(0^-\right) \leq f^\prime \left(0^+\right).</math>

then <math>f = \text{uxp}</math>. [Here <math>f = \text{uxp}</math> is Hooshmand's name for the linear approximation to the natural tetration function.]

The proof is much the same as before; the recursion equation ensures that <math>f^\prime (-1^+) = f^\prime (0^+),</math> and then the convexity condition implies that <math>f</math> is linear on {{open-open|−1, 0}}.

Therefore, the linear approximation to natural tetration is the only solution of the equation <math>f(x) = e^{f(x-1)} \;\; (x > -1)</math> and <math>f(0) = 1</math> which is [[convex function|convex]] on {{open-open|−1, +∞}}. All other sufficiently-differentiable solutions must have an [[inflection point]] on the interval {{open-open|−1, 0}}.

===== Higher order approximations for real heights =====
[[File:Approximations of 0.5 tetratrated to the x.png|thumb|a comparison of the linear and quadratic approximations (in red and blue respectively) of the function <math>^{x}0.5</math>, from {{math|1=''x'' = −2}} to {{math|1=''x'' = 2}}.]]
Beyond linear approximations, a [[quadratic approximation]] (to the differentiability requirement) is given by:
:<math>{}^{x}a \approx \begin{cases}
  \log_a\left({}^{x+1}a\right) & x \le -1 \\
  1 + \frac{2\ln(a)}{1 \;+\; \ln(a)}x - \frac{1 \;-\; \ln(a)}{1 \;+\; \ln(a)}x^2 & -1 < x \le 0 \\
  a^{\left({}^{x-1}a\right)} & x >0
\end{cases}</math>

which is differentiable for all <math>x > 0</math>, but not twice differentiable. For example, <math>{}^\frac{1}{2}2 \approx 1.45933...</math> If <math>a = e</math> this is the same as the linear approximation.<ref name="uwu"/>

Because of the way it is calculated, this function does not "cancel out", contrary to exponents, where <math>\left(a^\frac{1}{n}\right)^n = a</math>. Namely,
:<math>
  {}^n\left({}^\frac{1}{n} a\right)
  = \underbrace{
      \left({}^\frac{1}{n}a\right)^{
        \left({}^\frac{1}{n}a\right)^{
          \cdot^{\cdot^{\cdot^{\cdot^{
            \left({}^\frac{1}{n}a\right)
          }}}}
        }
      }
    }_n
  \neq a
</math>.

Just as there is a quadratic approximation, cubic approximations and methods for generalizing to approximations of degree {{mvar|n}} also exist, although they are much more unwieldy.<ref name="uwu"/><ref name=SolveAnalyt>Andrew Robbins. [https://web.archive.org/web/20090201164821/http://tetration.itgo.com/paper.html Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm]. The extensions are found in part two of the paper, "Beginning of Results".</ref>

====Complex heights====
[[File:Tetration analytic extension.svg|400px|right|thumb|Drawing of the analytic extension <math>f = F(x+{\rm i}y)</math> of tetration to the complex plane. Levels <math>|f| = 1, e^{\pm 1}, e^{\pm 2}, \ldots</math> and levels <math>\arg(f) = 0, \pm 1, \pm 2, \ldots</math> are shown with thick curves.]]
<!--
The existence of an analytic extension of <math>{}^{z}a</math> to complex values of <math>z</math> is not yet established. For <math>a = e</math>, it could be a solution of the [[functional equation]] <math>F(z + 1) = \exp(F(z))</math> with the additional conditions that <math>F(0) = 1</math> and <math>F(z)</math> remains finite as <math>z\to\pm{\rm i}\infty</math>.
Update: The existence of such an analytic extension of <math>{}^{z}a</math> has now been established. 
!-->

It has now been proven<ref name="PAU10">{{cite journal
|author=W. Paulsen and S. Cowgill
|title=Solving <math>F(z+1) = b^{F(z)}</math> in the complex plane
|journal=Advances in Computational Mathematics
|pages=1–22
|date=March 2017
|url=http://myweb.astate.edu/wpaulsen/tetration2.pdf
|doi=10.1007/s10444-017-9524-1
|volume=43
}}</ref> that there exists a unique function {{mvar|F}} which is a solution of the equation {{math|1=''F''(''z'' + 1) = exp(''F''(''z''))}} and satisfies the additional conditions that {{math|1=''F''(0) = 1}} and {{math|''F''(''z'')}} approaches the [[fixed point (mathematics)|fixed points]] of the logarithm (roughly {{math|0.318 ± 1.337''i''}}) as {{mvar|z}} approaches {{math|±''i''∞}} and that {{mvar|F}} is [[holomorphic function|holomorphic]] in the whole complex {{mvar|z}}-plane, except the part of the real axis at {{math|''z'' ≤ −2}}. This proof confirms a previous [[conjecture]].<ref name="MOC09">{{cite journal
|author=D. Kouznetsov
|title=Solution of <math>F(z + 1) = \exp(F(z))</math> in complex <math>z</math>-plane
|journal=[[Mathematics of Computation]]
|volume=78
|issue=267
|pages=1647–1670
|date=July 2009
|url=http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/S0025-5718-09-02188-7.pdf
|doi=10.1090/S0025-5718-09-02188-7
}}</ref> The complex map of this function is shown in the figure at right. The proof also works for other bases besides ''e'', as long as the base is bigger than <math>e^\frac{1}{e}</math>. The complex double precision approximation of this function is available online.{{citation needed|date=February 2016}}<!-- This function is not [[entire function|entire]], as there are singularities of <math>F(z)</math> on the real axis at the points <math>z=-2,-3,-4,\ldots</math>.!-->

The requirement of the tetration being holomorphic is important for its uniqueness. Many functions {{mvar|S}} can be constructed as
: <math>S(z) = F\!\left(~z~
    + \sum_{n=1}^{\infty} \sin(2\pi nz)~ \alpha_n
    + \sum_{n=1}^{\infty} \Big(1 - \cos(2\pi nz)\Big) ~\beta_n
  \right)
</math>

where {{mvar|α}} and {{mvar|β}} are real sequences which decay fast enough to provide the [[Limit (mathematics)#Limit of a sequence|convergence of the series]], at least at moderate values of {{math|Im ''z''}}.

The function {{mvar|S}} satisfies the tetration equations {{math|1=''S''(''z'' + 1) = exp(''S''(''z''))}}, {{math|1=''S''(0) = 1}}, and if {{math|''α<sub>n</sub>''}} and {{math|''β<sub>n</sub>''}} approach 0 fast enough it will be analytic on a neighborhood of the positive real axis. However, if some elements of {{math|{''α''}<nowiki/>}} or {{math|{''β''}<nowiki/>}} are not zero, then function {{mvar|S}} has multitudes of additional singularities and cutlines in the complex plane, due to the exponential growth of sin and cos along the imaginary axis; the smaller the coefficients {{math|{''α''}<nowiki/>}} and {{math|{''β''}<nowiki/>}} are, the further away these singularities are from the real axis.

The extension of tetration into the complex plane is thus essential for the uniqueness; the [[real-analytic]] tetration is not unique.

== Non-elementary recursiveness ==
Tetration (restricted to <math>\mathbb{N}^2</math>) is not an [[ELEMENTARY|elementary recursive function]]. One can prove by induction that for every elementary recursive function {{mvar|f}}, there is a constant {{mvar|c}} such that
:<math>f(x) \leq \underbrace{2^{2^{\cdot^{\cdot^{x}}}}}_c.</math>
We denote the right hand side by <math>g(c, x)</math>. Suppose on the contrary that tetration is elementary recursive. <math>g(x, x)+1</math> is also elementary recursive. By the above inequality, there is a constant {{mvar|c}} such that <math>g(x,x) +1 \leq g(c, x)</math>. By letting <math>x=c</math>, we have that <math>g(c,c) + 1 \leq g(c, c)</math>, a contradiction.

== Inverse operations ==
[[Exponentiation]] has two inverse operations; [[Nth root|roots]] and [[logarithm]]s. Analogously, the [[Inverse function|inverse]]s of tetration are often called the '''''super-root''''', and the ''super-logarithm'' (In fact, all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function <math>{^3}y=x</math>, the two inverses are the cube super-root of {{mvar|y}} and the super logarithm base&nbsp;{{mvar|y}} of {{mvar|x}}.

===Super-root===
{{redir|Super-root|the directory supported by some Unixes|super-root (computing)}}
The super-root is the inverse operation of tetration with respect to the base: if <math>^n y = x</math>, then {{mvar|y}} is an {{mvar|n}}th super root of {{mvar|x}} (<math>\sqrt[n]{x}_s</math> or <math>\sqrt[n]{x}_4</math>).

For example,
:<math>^4 2 = 2^{2^{2^{2}}} = 65{,}536</math>

so 2 is the 4th super-root of 65,536.

==== Square super-root ====
[[Image:The graph y = √x(s).png|thumb|right|The graph <math>y = \sqrt{x}_s</math>.]]
The ''2nd-order super-root'', ''square super-root'', or ''super square root'' has two equivalent notations, <math>\mathrm{ssrt}(x)</math> and <math>\sqrt{x}_s</math>. It is the inverse of <math>^2 x = x^x</math> and can be represented with the [[Lambert W function]]:<ref name = "Corless">
{{cite journal
  | last1 = Corless  | first1 = R. M.
  | last2 = Gonnet | first2 = G. H.
  | last3 = Hare | first3 = D. E. G.
  | last4 = Jeffrey | first4 = D. J.
  | last5 = Knuth | first5 = D. E.
  | author5-link = Donald Knuth
  | title = On the Lambert W function
  | journal = Advances in Computational Mathematics
  | volume = 5
  | page = 333
  | year = 1996
  | url = http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/LambertW.ps <!-- or http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf -->
  | format = [[PostScript]] <!-- or PDF -->
  | doi = 10.1007/BF02124750
| arxiv = 1809.07369
  }}</ref>

:<math>\mathrm{ssrt}(x)=e^{W(\ln x)}=\frac{\ln x}{W(\ln x)}</math>

The function also illustrates the reflective nature of the root and logarithm functions as the equation below only holds true when <math>y = \mathrm{ssrt}(x)</math>:

: <math>\sqrt[y]{x} = \log_y x</math>

Like [[square root]]s, the square super-root of {{mvar|x}} may not have a single solution. Unlike square roots, determining the number of square super-roots of {{mvar|x}} may be difficult. In general, if <math>e^{-1/e}<x<1</math>, then {{mvar|x}} has two positive square super-roots between 0 and 1; and if <math>x > 1</math>, then {{mvar|x}} has one positive square super-root greater than 1. If {{mvar|x}} is positive and less than <math>e^{-1/e}</math> it doesn't have any [[real number|real]] square super-roots, but the formula given above yields countably infinitely many [[complex number|complex]] ones for any finite {{mvar|x}} not equal to 1.<ref name = "Corless" /> The function has been used to determine the size of [[data cluster]]s.<ref>Krishnam R. (2004), "[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.8594 Efficient Self-Organization Of Large Wireless Sensor Networks]" - Dissertation, BOSTON UNIVERSITY, COLLEGE OF ENGINEERING. pp. 37–40</ref>

At <math> x = 1 </math> : 

<center><math> \mathrm{ssqrt}(x) = 1 + (x-1) -(x-1)^2 + \frac{3}{2} (x-1)^3 - \frac{17}{6} (x-1)^4 + \frac{37}{6}(x-1)^5 - \frac{1759}{120}(x-1)^6 + \frac{13279}{360} (x-1)^7 + \Omicron \left((x-1)^8 \right) </math></center>

==== Other super-roots ====
[[File:Cube super root.png|thumb|The graph <math>y=\sqrt[3]{x}_s</math>.]]
For each integer {{math|''n'' > 2}}, the function {{math|''<sup>n</sup>x''}} is defined and increasing for {{math|''x'' ≥ 1}}, and {{math|1=<sup>''n''</sup>1 = 1}}, so that the {{mvar|n}}th super-root of {{mvar|x}}, <math>\sqrt[n]{x}_s</math>, exists for {{math|''x'' ≥ 1}}.

However, if the [[Tetration#Linear approximation for the extension to real heights|linear approximation above]] is used, then <math> ^y x = y + 1</math> if {{math|−1 < ''y'' ≤ 0}}, so <math> ^y \sqrt{y + 1}_s </math> cannot exist.

In the same way as the square super-root, terminology for other super roots can be based on the [[nth root|normal roots]]: "cube super-roots" can be expressed as <math>\sqrt[3]{x}_s</math>; the "4th super-root" can be expressed as <math>\sqrt[4]{x}_s</math>; and the "{{mvar|n}}th super-root" is <math>\sqrt[n]{x}_s</math>. Note that <math>\sqrt[n]{x}_s</math> may not be uniquely defined, because there may be more than one {{mvar|n}}<sup>th</sup> root. For example, {{mvar|x}} has a single (real) super-root if {{mvar|n}} is ''odd'', and up to two if {{mvar|n}} is ''even''.{{Citation needed|date=October 2009}}

Just as with the extension of tetration to infinite heights, the super-root can be extended to {{math|1=n = ∞}}, being well-defined if {{math|1/''e'' ≤ ''x'' ≤ ''e''}}. Note that <math> x = {^\infty y} = y^{\left[^\infty y\right]} = y^x,</math> and thus that <math> y = x^{1/x} </math>. Therefore, when it is well defined, <math> \sqrt[\infty]{x}_s = x^{1/x} </math> and, unlike normal tetration, is an [[elementary function]]. For example, <math>\sqrt[\infty]{2}_s = 2^{1/2} = \sqrt{2}</math>.

It follows from the [[Gelfond–Schneider theorem]] that super-root <math>\sqrt{n}_s</math> for any positive integer {{mvar|n}} is either integer or [[Transcendental number|transcendental]], and <math>\sqrt[3]{n}_s</math> is either integer or irrational.<ref name="condor.depaul.edu"/> It is still an open question whether irrational super-roots are transcendental in the latter case.

===Super-logarithm===
{{main|Super-logarithm}}
Once a continuous increasing (in {{mvar|x}}) definition of tetration, {{math|<sup>''x''</sup>''a''}}, is selected, the corresponding super-logarithm <math>\operatorname{slog}_ax</math> or <math>\log^4_ax</math> is defined for all real numbers {{mvar|x}}, and {{math|''a'' > 1}}.

The function {{math|slog<sub>''a''</sub> ''x''}} satisfies:
:<math>\begin{array}{lcl}
\operatorname{slog}_a {^x a} &=& x \\
\operatorname{slog}_a a^x &=& 1 + \operatorname{slog}_a x \\
\operatorname{slog}_a x &=& 1 + \operatorname{slog}_a \log_a x \\
\operatorname{slog}_a x &>& -2
\end{array}
</math>

==Open questions==
Other than the problems with the extensions of tetration, there are several open questions concerning tetration, particularly when concerning the relations between number systems such as [[integer]]s and [[irrational number]]s:
*It is not known whether there is a positive integer {{mvar|n}} for which {{math|<sup>''n''</sup>[[pi|&pi;]]}} or {{math|<sup>''n''</sup>[[e (mathematical constant)|''e'']]}} is an integer. In particular, it is not known whether either of {{math|<sup>4</sup>&pi;}} or {{math|<sup>5</sup>e}} is an integer.
*It is not known whether {{math|<sup>''n''</sup>''q''}} is an integer for any positive integer {{mvar|n}} and positive non-integer rational {{mvar|q}}.<ref name="condor.depaul.edu">[http://condor.depaul.edu/mash/atotheamg.pdf Marshall, Ash J., and Tan, Yiren, "A rational number of the form {{math|''a''<sup>''a''</sup>}} with {{mvar|a}} irrational", Mathematical Gazette 96, March 2012, pp. 106–109.]</ref> Particularly, it is not known whether the positive root of the equation {{math|1=<sup>4</sup>''x'' = 2}} is a rational number.

== See also ==
*[[Ackermann function]]
*[[Big O notation]]
*[[Double exponential function]]
*[[Hyperoperation]]
*[[Iterated logarithm]]
*[[Symmetric level-index arithmetic]]

{{Commons category|tetration}}
== References ==
{{reflist|2}}

* Daniel Geisler, ''[http://www.iteratedfunctions.com/ iteratedfunctions.com]''
* Ioannis Galidakis, ''[https://web.archive.org/web/20090520164620/http://ioannis.virtualcomposer2000.com/math/exponents4.html On extending hyper4 to nonintegers]'' (undated, 2006 or earlier) ''(A simpler, easier to read review of the next reference)''
* Ioannis Galidakis, ''[https://web.archive.org/web/20060525195301/http://ioannis.virtualcomposer2000.com/math/papers/Extensions.pdf On Extending hyper4 and Knuth's Up-arrow Notation to the Reals]'' (undated, 2006 or earlier).
* Robert Munafo, ''[http://mrob.com/pub/math/hyper4.html#real-hyper4 Extension of the hyper4 function to reals]'' ''(An informal discussion about extending tetration to the real numbers.)''
* Lode Vandevenne, ''[http://groups.google.com/group/sci.math/browse_frm/thread/39a7019f9051c5d7/8c1c4facb7e4bd6d#8c1c4facb7e4bd6d Tetration of the Square Root of Two]'', (2004). ''(Attempt to extend tetration to real numbers.)''
* Ioannis Galidakis, ''[https://web.archive.org/web/20090420132250/http://ioannis.virtualcomposer2000.com/math/index.html Mathematics]'', ''(Definitive list of references to tetration research. Lots of information on the Lambert W function, Riemann surfaces, and analytic continuation.)''
* Joseph MacDonell, ''[http://www.faculty.fairfield.edu/jmac/ther/tower.htm Some Critical Points of the Hyperpower Function]''.
* Dave L. Renfro, ''[http://mathforum.org/discuss/sci.math/t/350321 Web pages for infinitely iterated exponentials]''
* {{cite journal | author = Knobel R | year = 1981 | title = Exponentials Reiterated | url = | journal = [[American Mathematical Monthly]] | volume = 88 | issue = | pages = 235–252 }}
* Hans Maurer, "Über die Funktion <math>y=x^{[x^{[x(\cdots)]}]}</math> für ganzzahliges Argument (Abundanzen)." ''Mittheilungen der Mathematische Gesellschaft in Hamburg'' '''4''', (1901), p.&nbsp;33–50. ''(Reference to usage of <math>\ {^{n} a}</math> from Knobel's paper.)''
* ''[http://ariwatch.com/VS/Algorithms/TheFourthOperation.htm The Fourth Operation]''
*Luca Moroni, [https://arxiv.org/pdf/1908.05559.pdf ''The strange properties of the infinite power tower''] (https://arxiv.org/abs/1908.05559)

==Further reading==
* {{MathWorld|id=PowerTower|title=Power Tower|author=Galidakis, Ioannis and Weisstein, Eric W.|access-date=5 July 2019}}

{{Hyperoperations}}
{{Large numbers}}

[[Category:Exponentials]]
[[Category:Binary operations]]
[[Category:Large numbers]]