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 1a=a, 2a=aa, 3a=aaa, 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. It is my hope that this site will have something of interest for everyone from high school students to mathematicians researching tetration. I particularly recommend taking a look at the section on fractals and maybe even getting a copy of Fractint and exploring tetration on your own.


Tetration sites with peer reviewed information

Check out Ioannis Galidakis' web site for his recent papers including On Extending hyper4 and Knuth's Up-arrow Notation to the Reals

Galidakis, Ioannis and Eric W. Weisstein,
Power Tower , From MathWorld--A Wolfram Web Resource
Nice website on background information relevant to tetration, with new content
constantly being added.

MacDonnell, J. F.
Some Critical Points of the Hyperpower Function,
International Journal of Mathematical Education, 1989, Vol. 20, #2, p.297

Informally peer reviewed

sci.math.research Number of good discussions on tetration research.

Tetration Forum Lively site on tetration research.

Other tetration sites

Andrew Robbins' Tetration web site.
Robbins has proposed a way to extend tetration to the real numbers.

Dmitrii Kouznetsov's Tetration and superlog web site.
Kouznetsov has also proposed a way of extending tetration.

Wikipedia: Tetration Worth checking out.

Site Map


This site greatly benefits from extensive links to the two following sites:

The following sites have been invaluable in the resources that they provide:
I would like to express my deep gratitude to the following people:
  • Ed Pegg Jr. at (check it out!) and the Mathematica Information Center. I am greatly indebted to him for his review and feedback of my Mathematica notebooks and website as well as his patience.
  • Dean Gooch at SRJC's math department for teaching me how to be a real mathematician, as well as many other things, and single handedly improving my view of academia.
  • Michael Somos for feedback on my web site and insightful correspondences.
  • Stephen Wolfram at Wolfram Research for conversations on the connection between tetration and dynamics, for his work in general, and for Mathematica. His vision of a unified dynamics has been the inspiration for my own work in dynamics.
  • Leif Smith at Pattern Research for numerous conversations and support over the years.
  • Gregory Henschel at the Department of Education's Institute of Educational Sciences. Although his contributions to this project of three decades are different from my own, I consider them of equal importance.