Roger Eugene Hill
The life, career, scientific and spiritual insights of a physicist plus a few excursions into Complexity Science and Art.
REH
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With Louise. Tent Rocks,
New Mexico. Dec 2016.




















Create patterns using cellular automata



A little history

In the 1940’s, at what is now the Los Alamos National Laboratory (LANL), the mathematician John von Neumann was working on the problem of sef-replicating systems, such as those found in biology or cybernetics. At the same time his friend and colleague at LANL, Stanislaw Ulam, was studying a related problem of crystal growth. Ulam was using what today would be called a computer simulation technique. He represented the crystal surfaces as nodes in a lattice network and using one of the early computers built for the Manhattan Project, probably the ENIAC, he observed the changes to the surfaces as the nodes obeyed rules that he imposed on them. (This is supposition on my part: I have not actually seen any of Ulam’s publications related to this work.) Ulam suggested to von Neumann that he try a similar computational approach to the study of self-replicating systems. Von Neumann adopted this approach and gave the concept the name Cellular Automata (CA). He did considerable work with CA and in 1966 published a book entitled The Theory of Self-Reproducing Automata. [A.W. Burks (ed), Univ. of Illinois Press]. In the 1970’s CA surfaced in the popular culture in the form of the mathematician John Conway’s Game of Life largely as a result of Marvin Gardner's article in Scientific American: The Fantastic Combinations of John Conway’s New Solitaire Game ‘Life’ [Sci. Am., 223:120-123,1970]. In the 1980’s considerable work was done on CA by Stephen Wolfram. The concepts of CA are a central theme of his book A New Kind of Science [Wolfram Media, Inc., 2002]. This book contains a complete history of CA as well as a full exposition of the science and mathematics associated with CA. The rules used to identify CA that are defined in Wolfram’s book are the rules I have used in my program CA Pattern Maker. In fact, Wolfram’s book was in large part the inspiration for the development of this program.

What are Cellur Automata?

The analytical (pencil and paper) way to solve for the state of physical systems undergoing change is to solve differential equations that assume space and time are continuous quantities. Computer simulations of these systems, on the other hand, divide space and time into discrete elements. The elements of space are called “cells” and the elements of time are called “steps”. CA are arrays of cells whose state at a particular time step depends automatically and only on its own state and the state of its neighbor cells at the previous step. If the CA array is confined to a line where the neighbors are just the ones to the Left and Right, the CA is said to be one dimensional. Two dimensional CA arrays can involve just 4 neighbors (North,West,South and East) or all 8 neighbors including the diagonals (N,NW,W,SW,S,SE,E,NE). In principle the cells within the CA can have any number of states. Von Neumann’s self-reproducing robots, for example, had 29 states. But the CA cells we are considering here have only 2 states: dead or alive, empty or filled, white or colored, or, in the usual computer parlance, 0 or 1. The rules that the CA obey as they evolve are coded as a set of integer numbers according to the schemes defined in Wolfram's book. I have included a detailed explanation of how the integer codes relate to the rules imposed on 1D and 2D CA on a separate page.

You can view a collection of CA patterns here. The captions for 1D CA patterns give the rule number; e.g., CA161. For 2D patterns, the caption gives the rule number and number of steps; e.g., CA491_66 which means 4 neighbor rule 491 run for 66 steps. Patterns using 8 neighbor rules have captions like CA8_174826_200, meaning 8 neighbor rule 174286 run for 200 steps. Captions where the steps are given like 46x6 means the pattern was run for 46 steps 6 times in a row (very different from 276 steps all at once.)

A cross-platform (browser based) version of CA Pattern Maker has been developed as a Silverlight application. This version includes printing capability but I have not yet tried to implement image saving in Silverlight. The program will operate in any browser equiped with a Silverlight plugin (available for free, about 7MB).

Operating the Program

You can adjust the settings for the three stages of pattern setup:
1)Coloring:
You can create your own pallette of colors and/or change the background color of the pattern using a color-picker;
You can define how the colors in the pallette are sequenced through the time steps required to generate the pattern: randomly, uniformly with a specified number of steps per color, using the Fibonacci sequence where the steps per color progress through a specified portion of the Fibonacci sequence, or custom, where you choose the number of steps for each color in the pallette.
2)Pattern Type:
You can choose to generate 1D or 2D patterns or to play Conway's Game of Life.
You can specify the size of the cells (in pixels) and the rule number (code) for the 1D and 2D CA.
3)Initial cell positioning:
For the 1D and 2D CA you can choose to have the starting cell positioned at the center, you can manually position any number of starting cells, or, in 1D, you can randomly position the initial cells.
For the Game of Life you can select among a set of established starting patterns or create your own custom pattern.

You then push "Start" to generate: the entire pattern for 1D, the pattern after a specified number of steps for 2D, or, to begin the Game of Life.



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