What Lies Beyond Exponentiation?
periods of tetration around real axis


Tetration by Escape Tetration by Period
Tetration by Escape Tetration by Period

My name is Daniel Geisler , welcome to my web site dedicated to promoting research into the question of what lies beyond exponentiation. 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 is \(^{1}a=a,  ^{2}a=a^a,  ^{3}a=a^{a^a},\) and so on. 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 dynamical systems and access to powerful computers is making real progress possible.

The big question in tetration research is how can tetration be extended to complex numbers. How do you compute numbers like \(^{.5}2\), and \(^{\pi i}e\) ? This web site will show how to compute these and other problems. See the Tetration page for a one page overview of extending tetration to the complex numbers.

I.N. Galidakis https://ingalidakis.com/ has a new website with significant work on tetration not available elsewhere!

Tetration isn't tetration anymore

The social abstract concept of tetration as a producer of fantastically large numbers has eclipsed it's original mathematical usage. As a researcher and mathematical community organizer I am quite displeased at how hard it is to find useful information about tetration research on search engines. Some tetration history. I am the senior researcher in tetration with fifty years of research and work to build a community for researches. This site is now twenty years old and before that I was on sci.math.research for a decade discussing tetration. 

Note: Google "tetration -cisco" to ignore Cisco Tetration and focus on tetration research. 

Book of Numbers

Joshua Cohen wrote the highly rated Book of Numbers where the largest company in the world in is named Tetration.

Cisco Tetration

Here's some insight into Cisco Tetration - we can assume they think the idea of tetration is cool, but they have refused to reply to my friendly attempt at contact. So tetration now refers to two disjoint groups - mathematical researchers who live on meager means, but that history will likely remember with kindness. Then you have engineers making from $150 K to $200 K a year. Just note that the folks that allegedly thought of everything didn't think to connect or make the slightest alliance with the owner of Tetration.org. I will share more thoughts later.   

So far I have been unable to reach either Joshua Cohen or Cisco Tetration management. :(

New at Tetration.org

Tetration.org is now twenty years old and in need of a complete rewrite which I am doing using MediaWiki with the Math extension. See http://iteratedfunctions.com/

Schroeder's Fourth Problem - OEIS A000311

The Schroeder tree graph of {{{1, 5}, 3}, {{2, 4}, 6}}

f[0] = 0;
s[1] = D[f[g[x]], {x, 1}] /. g[x] -> 0;
s[n_] := D[f[g[x]], {x, n}] /. g[x] -> 0 /. Derivative[1][f][0]*Derivative[n][g][x] -> 0 /. Derivative[1][g][x] -> l^(t - 1)
dyn[1] = l^t; dyn[n_] := dyn[n] = Simplify[Sum[s[n], t] /. D[g[x], {x, m_}] :> dyn[m] /. l -> Derivative[1][f][0]]
Hier[n_] := Block[{l = 1}, Sum[s[n], t] /. D[g[x], {x, m_}] :> dyn[m] /. t -> 1 /. f -> Exp]
Table[Hier[i], {i, 2, 15}]
Out[]= {1,4,26,236,2752,39208,660032,12818912,282137824,6939897856,188666182784,5617349020544,181790703209728,6353726042486272}

Tetration research: 1986 - 1991

In 1986 I had several conversations on extending tetration to the complex numbers with Stephen Wolfram. He suggested that I write my research up and he would edit and publish it in his journal. Unfortunately for me Wolfram had moved on to establishing Mathematica. Algebraic Exponential Dynamics is the article I submitted to Wolfram in 1990.

All Maps Have Flows  & All Hyperoperators Operate on Matrices

In 1986 Stephen Wolfram introduced me to the question of whether all maps are flows. Given the fifteen-year-old mathematics on Tetration.org, I have a simple proof that all maps are flows, that they are two different views of the same thing. Consider the Taylor series of an arbitrary smooth iterated function and it's representation as the combinatorial structure total partitions, the recursive version of set partitions. Each enumerated combinatorial structure has a symmetry associated with it. Let's say we want to consider \(C_2\), just remove all combinatorial structures inconsistent with \(C_2\). Because I can define \(GL(n)\) as the domain and the iterant, through representation theory, that if I can compute with matrices, I can compute within any symmetry.

Extending the Hyperoperators

Just as the exponential function of invertible matrices can be computed, all hyperoperations can be defined with invertible matrices.

Contact: daniellgeisler@gmail.com

Copyright  ©  2001-2020 Daniel Geisler