A Peculiar Class of Cellular Automata

Introduction

It has been suggested by several authors (like for instance Konrad Zuse in his work "Der rechnender Raum") that the mechanism, underlying physical phenomena is in fact a cellular automaton. In other words the universe can be regarded as one big cellular automaton and the physical phenomena observed are due to the emergent behavior of this cellular automaton. Whether this is true remains to be seen. However, it opens up a plethora of possibilities with the main goal of creating a cellular automaton or a class of cellular automata, exhibiting emergent behavior that best mimics the physical phenomena observed in reality. That in itself is an exciting playground filled with challenges like:

·        How to develop a cellular automaton emerging and growing out of a single cell (big bang singularity).

·        How to mimic the conversion of energy into mass and vice versa. (E = mc²)

·        How to mimic the phenomenon of two types of electrical charges.

·        How to mimic attractive forces over distance. Etc., etc.

Such challenges inherently lead to quite some metaphysical brainstorming in trying to implement them into the desired cellular automaton.
As such the author of this blog is developing an E-book.
This E-book describes the efforts undertaken by the author to develop such a cellular automaton. It does not have the pretension to be a work of scientific thoroughness in the area. It’s more a description of cheerful playing and experimenting with the concept. Just for the sheer fun of it. It should be emphasized that the development and visualization of cellular automata only became practicable with the introduction of fast, easy programmable computers. Computer technology and cellular automata are strongly intertwined. As such it is inevitable to avoid discussions of computer program build up in order to understand the development of the cellular automaton under investigation.   In the book example programs are given entirely written in Python. The reader is encouraged to use these programs as a starting point for his/her own experiments. Needless to say that the author's own experiments are still ongoing. So every now and then the book will be updated with new findings. (Time and opportunity given).
This of course still leaves open the question whether all this playing around will prove that the universe actually is one big cellular automaton. For instance, is there any evidence that space and time are in essence discrete opposed to being treated as continuous in most predominant areas of physics? As a matter of fact there are 2 peculiarities in physics pointing into this direction: Planck length and Planck time. Planck length is comprised from 3 different fundamental physical constants: the speed of light in vacuum, Planck's constant and the gravitational constant:

Planck time then is defined as the amount of time it would take a light beam to cross one Planck length. I.e. approximately 5.3875 x 10^-44 seconds. It should be emphasized that at the time of writing of this E-book, the physical significance of the Planck length (if any) is not yet known.
The Planck length is extremely small. For illustrative purposes, the (classic) diameter of a proton is estimated at 1 x 10^-15 m therefore the Planck length is about 1 x 10²⁰ smaller. This can be compared as the diameter of a hydrogen atom to the diameter of the sun. This directly leads to the feasibility of developing cellular automata based on such small discrete entities. I.e. within the limits of the available computational systems is it possible to observe emergent behavior requiring such an enormous amount of discrete entities? The answer is that it most likely is not possible. However, the solution might lie in taking large ensembles of entities and consider distribution and collision functions. A bit in the manner of the development of Lattice Boltzmann Methods from Lattice Gas Automata.
However, this is all very speculative. Therefore the final answer to the question whether the universe is one big cellular automaton might at the end of the day lead to another question: is not that a matter of belief?