# Main Page

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## Overview

My name is Daniel Geisler, daniellgeisler@gmail.com, welcome to my web site dedicated to promoting research into the question of what lies beyond exponentiation. I support the tetration community as a librarian and a historian. The original website is at tetration.org/original.

### Development

As a person who has networked with many others interested in tetration, I've seen a great number of people reinvent the same types of extension of tetration, thus the name of Knoebel's humorous article, Exponentials Reiterated.

I am frequently contacted by young mathematicians regarding their work on extending tetration. The topic of tetration runs profoundly deep. Even brilliant well educated mathematicians find it difficult to extend tetration. I most strongly encourage young mathematicians to obtain the best formal education they can get and work with a mentor who can help them advance their work. As an educator I have noticed stages of development in the quest to extend tetration. Each of the following foundations has a deep impact on the rate at which one can assimilate what is known about tetration. The important figures in the research of tetration are strong in multiple items. The foundations are not a destination but rather an endless path of developmental growth with specific phases.

1. A background in complex dynamics.
2. Knowing the history of the development of complex dynamics and tetration.
3. Understanding the analytic geometry of tetration fractals and its development. Can you view the tetration Mandelbrot set and the associated Julia fractals and see the story they tell? Can you look at a point on the tetration Mandelbrot and get a sense of the associated Julia set looks like? Can you reconcile what you see visually with your mathematical understanding? See The Quests to Decode the Mandelbrot set ... where important mathematicians in current complex dynamics use formal mathematics in lockstep with the rigorous exploration of the fractal geometry of the Mandelbrot set.
4. Mastery of computer algebra. Can you reconcile the computer algebra results with your mathematical understanding?

### Foundational Works

The purpose of this website is to provide resources for studying tetration. See Paulsen's work below for what is widely considered to be the most important current work on extending tetration and his tetration calculator.

## Idea

Tetration is defined as iterated exponentiation but while exponentiation is essential to a large body of mathematics, little is known about tetration due to its chaotic properties. The standard notation for tetration where ${\displaystyle a\in \mathbb {C} }$ is ${\displaystyle ^{1}a=a,\;^{2}a=a^{a},\;^{3}a=a^{(a^{a})}}$ and so on. Although as a rule all domains will be specified as used, the foundation of this website is complex analysis as ${\displaystyle a\in \mathbb {C} ,n\in \mathbb {N} }$ gives the universally accepted definition of ${\displaystyle ^{n}a}$. The proofs on this site are founded on complex analysis which are then used to develop complex dynamics.

Mathematicians have been researching tetration since at least the time of Euler but it is only at the end of the twentieth century that the combination of advances in a dynamical system and access to powerful computers is making real progress possible. 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. In fact, the Tetration Forum is a website dedicated to the subject.

Expressions like ${\displaystyle {}^{3}2=2^{2^{2}}=2^{4}=16}$ and ${\displaystyle {}^{2}i=i^{i}=e^{\log(i)i}=e^{-\pi /2}}$ are easy to compute in complex analysis, but expressions like ${\displaystyle {}^{1/2}2}$ and ${\displaystyle {}^{i}e}$ becomes very difficult, but not impossible. The expression ${\displaystyle ^{\frac {1}{2}}e}$ is a good case in point. Doesn't it seem like it should be a real number? The simplest approach depends on the fixed point where ${\displaystyle \exp(c)=c}$, or ${\displaystyle e=c^{(1/c)}}$, but using Lambert's W function we get ${\displaystyle c=-{\overline {W(-1)}}\approx .31813150520+1.3372357014I}$ which is complex. This results in solutions in terms of complex numbers. The expression ${\displaystyle ^{\frac {1}{2}}e}$ is a simplification of the question asked by Kneser, what is the solution of ${\displaystyle f(x)}$ for ${\displaystyle f(f(x))=e^{x}}$ where ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ as opposed to ${\displaystyle f:\mathbb {R} \to \mathbb {C} }$?

### Three Problems

A central theme of this article is that extending tetration and extending the Ackermann function can be achieved by extending iterated functions. So results will focus on extending complex dynamics, the iterated function ${\displaystyle f^{n}(z)}$ from ${\displaystyle z\in \mathbb {C} ,n\in \mathbb {N} }$ and ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$ to more general domains like ${\displaystyle n\in \mathbb {C} }$. This immediately allows tetration and the Ackermann function to be extended to the complex numbers. This process of extending ${\displaystyle n}$ is referred to as fractional iteration.

A final requirement in this article is the insistence that extensions must be consistent with what they are extending. Thus when ${\displaystyle n\in \mathbb {C} }$ is constrained to ${\displaystyle n\in \mathbb {N} }$, the resulting mathematics must be consistent with what is known about complex dynamics.

#### Extension Approaches

The following is a initial attempt to classify the approaches to extending tetration. While people are interested in an overview of the different extension methods for tetration, at present the best approach is to study the different mathematicians who work with tetration.

• Classical fixed point tetration
• Combinatorial
• Geisler
• Matrix methods
• Infinite composition and contour integrals

## Formal definitions

Define ${\displaystyle z,z_{0}\in \mathbb {C} }$ with ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$.

Given ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$, when the function ${\displaystyle f}$ is holomorphic and also analytic or locally equal to its own Taylor series. If functions ${\displaystyle g,h}$ are holomorphic, the function ${\displaystyle f=g/h}$ is a meromorphic function.

The fixed point of ${\displaystyle f}$ where ${\displaystyle f(z)=z}$ is commonly denoted as ${\displaystyle z_{0}}$, so that ${\displaystyle f(z_{0})=z_{0}}$.

${\displaystyle f^{0}=\operatorname {id} _{X}}$

and

${\displaystyle f^{n+1}=f\circ f^{n},}$

so that ${\displaystyle f^{n}(z_{0})=z_{0}}$.

When discrete iteration is extended from from ${\displaystyle n\in \mathbb {N} }$ to continuous iteration ${\displaystyle n\in \mathbb {R} }$ it is referred to as a flow.

An involution is defined as a function that is its own inverse, so ${\displaystyle f=f^{-1}}$ and ${\displaystyle f^{2}={\textrm {id}}_{X}}$.

### Arithmetic

Ackermann's original three-argument function ${\displaystyle \varphi :\mathbb {N} \times \mathbb {N} \times \mathbb {N} \to \mathbb {N} }$, ${\displaystyle \varphi (m,n,p)}$ is defined recursively as follows:

{\displaystyle {\begin{aligned}\varphi (m,n,0)&=m+n\\\varphi (m,0,1)&=0\\\varphi (m,0,2)&=1\\\varphi (m,0,p)&=m&&{\text{for }}p>2\\\varphi (m,n,p)&=\varphi (m,\varphi (m,n-1,p),p-1)&&{\text{for }}n,p>0\end{aligned}}}

##### Flows

The archetype of all flows is the additive flows ${\displaystyle {\mathsf {a}}_{t}}$ or additive family at given by ${\displaystyle {\mathsf {a}}_{t}(x)=t+x}$; the flow equation is just the associativity law ${\displaystyle (s+t)+x=s+(t+x)}$.

##### Functional Equations

The Abel equation, is a type of functional equation of the form ${\displaystyle f(h(x))=h(x+1)}$ or ${\displaystyle \alpha (f(x))=\alpha (x)+1}$. The forms are equivalent when ${\displaystyle \alpha }$ is invertible.

The Abel equation provides a way to transform iterated functions into the additive family, which has a flow, and then transform back to the original iterated function now iterated by a flow.

#### Multiplication

##### Symmetry
• Reflectional symmetry - the map ${\displaystyle z\to {\overline {z}}}$ or complex conjugation is an example of reflectional symmetry.
• Rotational symmetry - the map ${\displaystyle z\to i\cdot z}$ is an example of four fold rotational symmetry.
• Scaling symmetry - typically scaling symmetries are non-linear. The Sierpiński triangle is an example of scaling symmetry.

Hexagonal lattices display translational, reflectional and six fold rotational symmetry.

Logarithmic spirals display a combination of scaling and rotational symmetry.

##### Flows

The multiplicative flows ${\displaystyle {\mathsf {m}}_{t}}$ are given by ${\displaystyle {\mathsf {m}}_{t}(x)=c^{t}\cdot x}$ for some constant ${\displaystyle c\neq 1}$,say ${\displaystyle c=e}$ for example. Multiplicative flows can generally be considered as of coequal importance with the additive flow; the two are generally related by an equation ${\displaystyle {\mathsf {m}}_{t}=\exp \circ \;{\mathsf {a}}_{t}\circ \log }$, at least where ${\displaystyle \log }$ makes sense.

##### Functional Equations

The Schröder's equation is a functional equation of the form ${\displaystyle \Psi {\big (}h(x){\big )}=s\cdot \Psi (x)}$.

The Schröder's equation provides a way to transform iterated functions into the multiplicative family, which has a flow, and then transform back to the original iterated function now iterated by a flow.

#### Tetration

##### Iterated exponential notation

For ${\displaystyle a,z\in \mathbb {C} }$ and ${\displaystyle b\in \mathbb {N} }$ with ${\displaystyle m,n\in \mathbb {Z} }$,

${\displaystyle \exp _{a}^{b}(z)=a^{a^{\cdot ^{\cdot ^{a^{z}}}}}}$ with ${\displaystyle n}$ ${\displaystyle a}$ s.
${\displaystyle {\exp }_{a}^{0}(z)=z}$, ${\displaystyle {\exp }_{a}^{1}(z)=a^{z}}$
${\displaystyle {\exp }_{a}^{-b}(z)={\log }_{a}^{b}(z)}$
${\displaystyle {\exp }_{a}^{m}({\exp }_{a}^{n}(z))={\exp }_{a}^{m+n}(z)}$
##### Tetration of natural numbers

For ${\displaystyle a,b\in \mathbb {N} }$, tetration ${\displaystyle ^{b}a}$ is defined as ${\displaystyle \varphi (a,b,3)}$ or ${\displaystyle \exp _{a}^{b}(1)}$ in iterated exponential notation.

##### Tetration with a complex base

Complex analysis extends the exponential function, allowing ${\displaystyle x^{y}}$ to be defined where ${\displaystyle \varphi :\mathbb {C} \times \mathbb {C} \times 2\to \mathbb {C} }$ giving ${\displaystyle \varphi :\mathbb {C} \times \mathbb {N} \times 3\to \mathbb {C} }$. Let ${\displaystyle a\in \mathbb {C} }$ then the function, then

{\displaystyle {\begin{aligned}\varphi (0,0,2)&=0^{0}=1\\\varphi (a,0,3)&={}^{0}a=1\\\varphi (a,n,3)&=\varphi (a,\varphi (a,n-1,3),2)=\;a^{^{n-1}a}=\;^{n}a\end{aligned}}}

The iterative application of ${\displaystyle \varphi }$ gives us power towers of ${\displaystyle a}$ to height ${\displaystyle n}$, or ${\displaystyle ^{n}a=a^{{\Bigl (}{a^{\cdot ^{\cdot ^{a}}}}{\Bigr )}}}$ as well as fractals like the tetration Mandelbrot in Figures 1 and 2 as well as the tetration Julia fractals in Figure 3. A famous problem is whether ${\displaystyle ^{4}\pi }$ is an integer.

 Figure 1: Tetration Mandelbrot escape fractal Figure 2: Tetration Mandelbrot period fractal Figure 3: Tetration Julia fractal at a = -1

The tetration Mandelbrot escape fractal in Figure 1 is the escape value of ${\displaystyle n}$ where ${\displaystyle |^{n}a|\geq 10^{6}}$. When ${\displaystyle ^{n}a}$ is periodic it doesn't escape. This leads to a complementary fractal, the tetration Mandelbrot period fractal in Figure 2. This fractal displays the period of ${\displaystyle ^{n}a}$; moving through the colors of the rainbow, red is period 1, yellow is period 2, green is period 3 and so on.

When the values of while the tetration Julia fractal displays the dynamics of the map ${\displaystyle z\to (-1)^{z}}$.

The red area of Figure 2 displays the values for which ${\displaystyle ^{n}a}$ is period 1, in other words convergent. This is called the Shell-Thron region.

### Dynamical Systems

In the most general sense, a dynamical system is a tuple (T, X, Φ) where T is a monoid, written additively, X is a non-empty set and Φ is a function.

The function Φ(t,x) is called the evolution function of the dynamical system: it associates to every point x in the set X a unique image, depending on the variable t, called the evolution parameter. X is called phase space or state space, while the variable x represents an initial state of the system.

For any x in X and ${\displaystyle t_{1},t_{2}\in T}$

${\displaystyle \Phi (0,x)=x}$
${\displaystyle \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),}$

We often write

${\displaystyle \Phi _{x}(t)\equiv \Phi (t,x)}$
${\displaystyle \Phi ^{t}(x)\equiv \Phi (t,x)}$

if we take one of the variables as constant. The function

${\displaystyle \Phi _{x}:I(x)\to X}$

is called the flow through x and its graph is called the trajectory through x. The set

${\displaystyle \gamma _{x}\equiv \{\Phi (t,x):t\in I(x)\}}$

is called the orbit through x.

## Ultra Fractal

Exploring tetration fractals is a valuable aid in understanding tetration. Originally the best fractal software was fractint which was developed on CompuServe in the late Eighties. Ultra Fractal is fractint's successor both in general and specifically for its support of fractint formulas. Its combination of a perturbation algorithm for quick computing of pans and zooms and its complete CPU utilization makes it hard to beat.

 Zoom into Kneser's fixed point c of the exponential function Tetration Mandelbrot fractal Zoom into Tetration Mandelbrot fractal at z = -1.0