What Lies Beyond Exponentiation?
periods of tetration around real axis

Math Equation

Modest Mouse

The universe works on a math equation
That never even ever really even is any end.
Infinity spirals out creation
We're on the tip of it's tongue, and it is saying
We ain't sure where you stand
You ain't machines and you ain't land
And the plants and the animals, they are linked

Does the universe works on a math equation? Is there really any end? And what about spirals?

Albert Einstein spent the later part of his life looking for a Unified Field Theory, a theory of everything in physics that would unify and supersede the separate theories of physics. Stephen Wolfram, hailed as Einstein's successor in the Eighties, pointed out that scientist might have a problem unifying the different branches of physics because the different branches mathematics underlying physics themselves were not unified. All of physics is a subset of the mathematical discipline of dynamics, yet the branches of dynamics are not unified.

Cellular automata, differential equations, recurrence equations, iterated functions are all different types of dynamical systems. A important difference it their ability to model and explain chaos. Almost all physical systems embody a degree of chaos. The central problem is that classical mathematics "breaks down" in the presence of chaos, yet everything in the universe is chaotic to a degree. The importance of this fact canít be overstated. Differential equations are the most important branch of dynamics because most laws in physics are represented using differential equations. Differential equations are also vital to the development of large and important parts of pure mathematics, but they are horrible in modeling chaotic systems. On the other hand, cellular automata are great for modeling chaotic systems, even universes, but there is no significant bridge between classical mathematics and cellular automata.

Cellular Automata

The following video is about a specific cellular automata named Life.

What is needed is a mathematical Rosetta stone capable of unifying and translating between the different mathematical dynamical systems. A universal dynamical system that is deeply integrated with classical mathematics, yet excels at modeling chaotic systems. It turns our that there are two formally accepted ways of defining physics in terms of mathematics. A physical system can be defined as either a partial differential equation or as a continuously iterated function. This second option is a surprise to virtually all mathematicians and physicist.

Let the following symbols be defied as:
x - space
n - discrete time
t - continuous time
f(x) - the “laws of physics” for an instant in time

Let f 0(x)=x, f 1(x)=f(x), f 2(x)=f(f(x)), f 3(x)=f(f(f(x)))
the expression f n(x) represents an iterated function. If time is continuous then the expression is a continuously iterated function f t(x).

Iterated Functions - f n(x)

The following video shows a deep zoom into the Mandelbrot set where \(f(z)=z^2+c\). Despite being referred to as the most complicated object in mathematics, the Mandelbrot set is based on the iteration of one of the simplest functions in existance, the quadratic equation.

Who hasn't looked at the Mandelbrot set and felt that they were seeing something fundemental about reality? Well, in fact you are seeing something fundemental about physics and reality! The Universe is a fractal - it's not just a cool idea, it is how reality works!

Song of the Universe - f t(x)

Any possible universe where space and time are continuous can be represented by the equation f t(x). If someone asked me what I study, I could rightfully tell them that I have devoted much of my life to studying the equation f t(x). Two questions come up, the first is how can you study something as abstract as f t(x)? The answer is that I employ a standard mathematical technique of finding the Taylor series of f t(x). The result is an infinitely long math equations that describes the Universe! I wonder if this is what Modest Mouse was talking about? The second question would come from people who know me. What about my interest in tetration? Well, often times in mathematics two seemingly different problems have the same answer. The mathematical problem of defining tetration resolved once it is understood how to continuously iterate functions. I consider defining tetration and continuously iterating functions to be two facets of the same problem.

Earlier I raised the issue of a mathematical Rosetta stone to unify the different branches of mathematical dynamics. I propose that continuously iterated functions can serve as the universal dynamical system. They are capable of displaying all the properties of complex systems, but are related to differential equations and recurrance equations.

Contact: daniel@tetration.org
Copyright  ©  2001-2015 Daniel Geisler