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Introduction

This document has several purposes; foremost, is an attempt to come out of the closet and to move forward as a whole person; it provides a base for my research into the psychodynamics of mathematical thought, uniting my interest in Yoga, psychology, and mathematics, information which may be of interest to others. Regardless of the scientific veracity of my work, as a human, I am on a journey that many other people are on. I wish them to know that they are not as alone as they may think.
 
Understand that this document is in large part my own internal mythology; memories of memories that I have used to reinforce my sense of self. It is my hope that the process of writing will serve as a form of spiritual archeology, allowing me to categorize the outer layers of my self and to excavate deeper, bringing the light into the dark.
 
For most of my life I have lead a secret second life as a "mad scientist" reveling in the megalomaniacal contemplation of mathematics and science. Beginning in 3rd grade, I began realizing that intellectually, I was profoundly different from other people. My vocabulary was growing at an incredible rate, but as other children commented on it, I adjusted my speech to sound like the other children. I had suffered a horrible humiliation the previous year when I had to take second grade over a second time. By third grade I remember having the desire to be the greatest scientist that had ever lived. During my freshman year of high school I was repeatedly shaken by the magnitude of my grasp of mathematics; many times I would read about some new piece or area of mathematics for the first time, only to find that I had given substantial though to the subject and already had a degree of familiarity with it. I was reading a vast amount of mathematics, but I was figuring out things even faster than I could read them. Frequently I would begin reading about a new subject, only to be struck with a vision of the relevant mathematical landscape in considerable detail, which continued reading then confirmed. Having deep insights is wonderful; I was doing things mathematically that I thought were beyond the capability of humans but it really bothered me, it made me feel like I wasn't quite human. The positive side was that I was willing to plunge into the deepest, most freighting mathematical abyss I could conceive of.
 

Exotic Arithmetic

I have a deep love of numbers; more than specific numbers, I am interested in the nature of their most universal and abstract forms — the types of numbers that must exist and the types of numbers that may exist. Arithmetic serves as the basic machinery for creating numbers.; Questions regarding the possible types of numbers quickly lead to question regarding the possible types of arithmetic. Table 1 displays a list of arithmetic operators; the first three rows contain the classical arithmetic operators, the rows beyond list the operators that display chaotic properties.
 
The development of quantum mechanics provides a good analogy to the development of arithmetic. Quantum mechanics draws a veil over the nature of the world at distances smaller than the atom. Piercing the veil required an understanding of new mathematical techniques as well as the construction of a series of increasingly powerful atom smashers.
 
Likewise, chaos draws a veil over arithmetic when going beyond exponentiation and piecing this veil also requires new mathematical techniques. Computer hardware and software serve the role of mathematical atom smashers, exposing the workings of tetration, pentation and beyond.
 

1965 - First Awakening

Consider the series of squares; 1, 4, 9, 16, and 25, as shown below. To calculate the next term begin by looking at the differences between neighboring terms of the series and write the difference down on the line below. The process is repeated until a row of zeros is generated. The differences are then added back in by moving from the lower right to the upper right.� This process is referred to as difference calculus and is a discreet form of differential calculus. Below is an example based on squares, below that cubes. I made these discoveries in 3rd and 4th grade respectively. The squares have two no-zero rows because squares are second-degree polynomials with only the first and second derivatives as non-zero; this is why the cubes have three non-zero rows. I discovered this technique in third grade while taking an IQ test. There were many problems where a series of numbers were presented and the answer consisted of determining the next number in the series. It seemed that since the number series problems were of the same form that there must be a unified way to solve all of them.

1 4 9 16 25 36
3 5 7 9 11
2 2 2 2
0 0 0



1 8 27 64 125 216
7 19 37 61 91
12 18 24 30
6 6 6
0 0

 
My father kept the relics of his tour of the Korean War in a large ornate box that our Catholic bible had come in. The box contained a combination padlock that I opened by the age of eight, by dint of hyperactivity; this gives some indication of the amount of mental energy that I was capable of bringing to bear on a single simple item. Also inside the box was the Book, not the Bible, but an Army manual on applied mathematics. The manual contained tables for the areas and volumes of different shapes, trigonometry tables, and most importantly a table of the squares, square roots, cubes, and cube roots of numbers. I used to gaze at the table of powers as if it held all the mystical secrets of the world. I find it amusing that my research into tetration had its roots in my research into squares in the 3rd grade. During this time I made a second discovery about squares; I noticed that 4*6 was one less than 5*5, just as 5*7 is one less that 6*6; I understood that this was a universal property of numbers. I ask many teachers why this was so, but they were unable to answer me. The answer is that (x-1)(x+1) = x2-1. I was discovering algebra on my own in 3rd grade, I could have easily learned it with a little help; this one incident has made me an advocate of school reform.
 
Three incidences helped with my scientific education. Mrs. Barnholt, a neighbor, gave me her son's old high school chemistry book and a college biochemistry book. These books gave me an acquaintance with mathematics without having to immediately tackle its abstractions. I spent a good deal of time searching, trying to find pictures of the different electron energy states of the elements, as are in current chemistry books. The incredible response that I obtained from a letter that I wrote to the Atomic Energy Commission was mind-boggling. I don't remember why I wrote the letter but I received a package of half a dozen booklets on particle physics and a thick white paper summarizing the physics of the previous year. These were well written booklets presenting the latest information in physics. There is no way to describe how much that meant to me. The lesson is that a small act of kindness to a child can have a profound effect on the child's development and direction in life; I like that. My father actually bought me several mathematics workbooks, It was the only time in my life that he had done anything like this; it may have given me a bit more of an emotional boost towards mathematics.
 
In 4th grade I obtained the first quadrant of the unit circle by plotting the Lorentz-FitzGerald contraction from special relativity on graph paper and correctly concluded that a velocity (boost) was a rotation through the complex plane. I also felt that the mathematical basis of quantum mechanics needed to be fixed when I read about the problems of infinities associated with the practice of renormalization. This is interesting since this had been an important area of my mathematical research and part of the reason I have invested so much time in it. Over a two-year period of time I gained a solid layman's understanding of science and mathematics but by the end of 4th grade I ended my intense study of science; I realized that I desperately needed to understand calculus to continue on.
 
During 3rd grade, my teacher Mrs. Vogel had an ongoing arithmetic competition in speed and accuracy with an owl broach worn by the winner; the owl went back and forth between my friend Julie and me. It can be seen in my case that nurture was a very large factor in my mathematical development.
 

1970 - Calculus

The summer after 7th grade, I taught myself algebra, matrix theory, and basic differential and integral calculus. I learned algebra in two weeks, sans trigonometry. I had to memorize the formulas for the solution to quadratic equations and the sum of geometric progression; everything else I could derive on my own. Although I didn't have a good grasp of the calculus, I spent 8th grade unsuccessfully trying to determine the formula for the volume of a sphere of arbitrary dimensions.
 

1971 - The Great Awaking

Pascal's triangle

I was introduced to the Revolution in 1971, as a freshman in High school. We fought against bourgeois dress codes, and in particular, male student's right to wear long hair. My nemesis was a gym teacher by the name of Squid. Well, Squid probably wasn't his legal name, but it was well earned. Squid looked like a teddy bear with a crew cut. He taught swimming, although he couldn't swim himself. Squid felt compelled to verbally attack me in the classroom for my long hair and wrong thinking. This was somewhat problematic for him as he was quite mentally slow for a gym teacher. I was able to verbally dance him around anywhere I wanted him to go — on stage in front of a large number of my peers, repeatedly. It was almost perfect. Almost. Squid also happened to teach a second activity, wrestling. It just so happened that the star quarterback of the football team was in my gym class. I guess Squid finally had a perceptive thought and wondered how I would do against the quarterback who had twenty-five pounds on me. I did damn well for a while, going 2-3 against the quarterback. An ill-considered maneuver on my part ended with a broken right arm.

Having a broken arm, my good arm, changed the direction of my life. The prime impact was that it redirected my hyperactivity into the mental realm. I began to spend a great deal of time reading and thinking about mathematics.

Difficult mathematical problems would constantly occur to me and many times I was able to come up with an answer, only to read for the first time, several months later, about the very problem I had worked on and discover that I had obtained the correct results. I determined that 1/2 ! equaled the square root of pi by summing an infinite binomial expansion, replicated Ramanujan’s summation of the positive integers equaling –1/12, and seriously pondered how to extend geometry to fractional dimensions.

After rediscovering the Euler identity from playing with the power series of the exponential function, I questioned whether higher analogs of i, the square root of negative one, and the complex number plane were necessary in advanced arithmetic. Searching through mathematics books, I found that the complex numbers were closed under addition, multiplication, and exponentiation. In other words, no mixture of complex numbers — addition, multiplication, and exponentiation — led to anything beyond complex numbers. An example of the lack of closure will help show what closure is. The real number line is closed under addition and multiplication; it's a happy little universe unto itself. The inclusion of exponentiation opens a crack in this universe; 1 squared is 1, -1 squared is also 1. Only a number with a magnitude of one can be squared to produce a number with a magnitude of one; yet 1 and -1 are not candidates. This necessitated going beyond the real number line and discovering the complex number plane, which contains numbers with a magnitude of one besides 1 and -1.

I recalled reading a small except about the Ackerman function, a function that mapped the positive integers to arithmetic operators. One property of positive integers is that adding one to them creates a new larger positive integer. This is mirrored by the recursive or repeated use of an arithmetic operator to create a new "bigger" arithmetic operator, an operator that generates larger numbers than its predecessor. Recursive addition is multiplication and recursive multiplication is exponentiation; but this process of recursive recursion can continue, leading to operators such as tetration, pentation, hexation, and beyond. This infinite collection of arithmetic operators is referred to as First Order Arithmetic. The Ackerman function defines how to calculate any First Order Arithmetic expression involving integers; it can be implemented as a very small program in many computer languages.

The Ackerman function leads to the possibility that something beyond complex numbers might be necessitated; but that it would have to be necessitated by one of the arithmetic operators beyond exponentiation. The problem was that the Ackerman function was defined only for integers, where things are boring and safe; I expected closure to break down in more unusual places. The more I tried to find any reference to the realm beyond exponentiation, the more I "felt" that there just wasn't anything out there. Ancient maps denote the ends of the world with annotations indicating, "There be monsters!"; that is what the Ackerman function is, a sign proclaiming, "There be monsters!" at the edge of the mathematical world. While contemplating the Ackermann function I began having visions of Ackermann molecules where the atoms were instances of the Ackermann function. Each atom was a beautiful small sphere radiating a different sweet soft pastel color. The Ackermann molecules were the expressions created from nested Ackermann functions. They appeared to float through the air in long interacting chains of different colored lights. I was totally enchanted. I felt that it would the loveliest thing imaginable to understand the deep mathematical properties of these charming constellations. Questions about possibilities of always being able to represent large molecules with equivalent smaller molecules lead me to reflect on algorithmic information theory.

This is the point at which I began my research into tetration, recursive exponentiation, with the primary goal of being able to define tetration between two complex numbers. I spent vast amounts of time through the 1970's researching tetration to no avail. The more I worked, the more my totem image for my work became digging through a vast, beautiful, extremely thick wall of diamond. Tetration was like some exotic alien metal out of Star Trek, absolutely perfectly unaffected by any known force applied to it. There was no rational reason for me to work on tetration; it was a case of unrequited love. I had reason to walk away from my research in tetration a dozen times, and I did. But I was completely compelled by a vision of transcendental beauty; it was like seeing the face of God in the distance.

1973 - Fame

Reviewing my SAT scores the summer after my sophomore year in high school, my counselor recommended that I seek entrance to Bradley University, which I did. Although I majored in physics, I was "discovered" by Dr. McGaughy, the chairman of the Mathematics department. Dr. McGaughy was someone I respected as both a person and a mathematician; he listened to my ideas about tetration with interest and respect. Entering college early had minimal impact on my mathematical studies but had a profound impact on my mathematical self-respect.
 

Late Seventies - The Computer Revolution

I enlisted into the Air Force in 1978, which profoundly impacted my mathematical life in three ways. In early June 1978 I could afford my first computer, a TI 58 Programmable Calculator, for about $105. I bought it with my first paycheck. The TI 58 supported the addition, multiplication, and exponentiation of complex numbers; I probably ran tens of thousand if not hundreds of thousands of calculations on the TI 58 and manually plotted the results on hundreds of sheets of graph paper. The second impact was training in computer hardware and programming, which have been a foundation of my mathematics work. The final impact was the experience I gained working in an environment where world-class science and research was being performed.
 
The five-year period after obtaining my first computer was when I did the foundational research in tetration. Countless plots showed that tetration's behavior was quite erratic; I could see why nothing was written on the subject. Chaos theory was emerging in this time frame; at first I saw similarities between the odd behavior of tetration and chaos. As time progressed I slowly began to realize that tetration was an actual example of chaos. This explained why three hundred years after the development of logarithms, the next arithmetic operator, tetration, had never been developed; significant research into chaotic systems without using computers is nightmarish, if not impossible. Chaotic systems contain an infinite amount of detail. Only computers can store and manipulate the great amounts of information needed to even crudely model chaotic systems.
 
Independent of the numerical explorations, I began gaining mathematical traction in deriving the equations underlying tetration. Exponentiation is non-commutative and has two type of expressions zn and az; tetration follows suit and supports two types of equations, nz and za.

 



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