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. A one page overview of my research can be found at Math Equation.

New at Tetration.org


Changes in Tetration.org

Tetration.org was originally designed and written in a six week period eleven years ago. In that time a revolution in tetration research has occurred. Ioannis Galidakis has published a number of papers on tetration and a growing community of talented mathematicians collaborates on tetration research through sci.math.research the Tetration Forum, and more recently math.stackexchange.com and mathoverflow.net . Still, the central goal of publishing a universally accepted and understood extension of tetration to the real and complex numbers hasn’t been achieved yet. A number of algorithms have been proposed for extending tetration, but they have have not been proven to converge. The most obvious change to Tetration.org is that it is now using the MathJax JavaScript package to display \(\LaTeX\).


A theory of tetration is far more than a series of algorithms, it should expose a number of principles and provide surprising lessons.

Tetration is Dynamics

The connection between tetration and iterated functions is vital to understand. In mathematics the branch that studies iterated functions is called dynamics. Tetration of complex numbers must always be consistant with complex dynamics. The departure of any theory of tetration from complex dynamics serves to invalidate the theory. For example a theory that is not consistent with fixed points and limit cycles is fatally flawed.

Always Generalize

In conversation, Stephen Wolfram suggested that a theory of complex tetration could be generalized into a theory of fractional or continuous dynamics where \(f^n(z), n \in \mathbb{C}\) is defined. Not only does reducing tetration to complex dynamics provide a solid mathematical foundation for tetration, but it also makes it trivial to extend complex dynamics to pentation and the higher hyperoperators beyond tetration.

Contact: [email protected]
Copyright  ©  2001-2015 Daniel Geisler