Tetration

Tetration is also referred to as iterated exponentials, hyperpowers and the dynamics of the exponential map. As R. A. Knoebel notes, many mathematicians have independently rediscovered tetration since the time of Euler. In Hilbert's famous speech, "Mathematical Problems", he states that a good mathematical problem should be clear, easy to comprehend, and "Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock our efforts." While questions regarding tetration are easy to state, it is seemingly impossible to make substantive progress in the field. At least this used to be true; the last decade has seen a series of articles on continuous iteration, first the closed form solution for the continuous iteration of the logistics equation, and then a general proof for the continuous iteration of real functions. The significance is that it now becomes meaningful to talk about expressions like ^πe and ^1/22.

Consider the impact of addition, multiplication and exponentiation on mathematics. Each succeeding operator is less pervasive throughout mathematics yet of greater elegance in connecting together different areas of mathematics in new interesting ways; just consider Lie groups as an example of the utility of exponentiation.

Tetration for Natural Numbers

${\ ^{n}a = \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop n}\,$

Alternate notations for tetration

${}^{n} a \equiv a \uparrow \uparrow n \equiv a \rightarrow n \rightarrow 2 \,$

Examples

${}^0 a \equiv 1 \,$

${}^1 a \equiv a \,$

${}^2 a = a^a \,$

The following three articles discuss the nuts and bolts of extending tetration to the complex numbers.

This page will focus on providing a bird's-eye-view of the history and possible future of tetration research. It will attempt to give a top-down overview of tetration to be supported by other entries providing the bottom-up axiomatic nuts and bolts treatment of the subject. Because of the speculative nature of this entry and its intent of discussing the significance of different lines and approaches to tetration research, I would heartily encourage other knowledgeable people to makes their own comments and edits so that this entry can achieve a more neutral point of view. My fascination with tetration has lead to a fascination with people fascinated with tetration, particularly people involved with tetration research. I’m very interested in how our collective understanding of tetration evolves in a seemingly organic manner through the interplay of people. I want to be an embedded journalist in the war on tetration, but that requires referring to the actual people working to advance our knowledge of tetration. Even though I intend to represent people’s positive contributions, please let me know or just edit out any inappropriate information.

Usually tetration is considered part of mathematics, but here we will consider tetration in the broader context of natural science. The approach is strongly influenced by the work of Stephen Wolfram as well as a couple of discussions with him in the eighties. People familiar with A New Kind of Science will have a definite advantage in understanding this material. Because tetration is based on exponentiation and exponentiation is so fundamental to mathematical physics the question of tetration's relevance to physics with be explored.

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