Example of Chaos in Rule 23-3

The evolution property of Cellular Automata Rule 23-3 known as Conway's Life have been known to be chaotic. This means a small change in initial condition, will have a big effect on the evolution of said world.

This Morning (August 10th 2008,1 AM (GMT+8)) I had been lucky enough to observe this particular property of Rule 23-3 while doing some experiments involving 21 Conway's Worlds.

The experiments involved 7 worlds with worldsize of 30 x 30 pixels, 7 worlds with worldsize of 60 x 60 pixels and 7 worlds with worldsize of 120 x 120 pixels. The main objective of the experiments was to determine the average longevity of Conway's world in relation to the worldsize of said Conway's world.

What interest me the most was an unusually high longevity observed in one world with worldsize of 120 x 120 pixels. This particular initial condition below, was shown to have longevity of 9150 cycles.

Initial Condition	Death Condition

This unusually high longevity (about 2.135 Standard Deviation from the mean value) had made me wonder about the rarity or abundance of Conway's world with such long lifetime. So I alter the initial condition by killing one live cell from the original initial condition above. This had decreased the lifetime of the world down to 3059 cycles.

Initial Condition	Death Condition

Since killing one live cell in the initial condition made the longevity of the world being near to the mean longevity value, this might means that initial conditions that produce world with high longevity value to be rare. But more experiments must be done before taking any further conclusions.

If you are interested to help calculating the average lifetime of a Conway's world with certain worldsize, just download the program and do some experiments by yourself. Submit your experiment results to this particular maling list Yahoo!Groups : Cellular Automata Research.

Update on Cellular Automata Programs

I had updated my cellular automata programs. The previous version seems to have problem with the width of the screen in computers without some Japanese fonts installed.

The new links is here :

• Cellular Automata v1


• Cellular Automata v2

The Cellular Automata v1 is a program to simulate single state cellular automata. Thus the color of each cell could only be black or red (or other color, if you choose so.

The Cellular Automata v2 is a program to simulate multiple state cellular automata. The rule required are a bit more complex than the one for Cellular Automata v1. But once you find a good rule, you could have more fun. The Cellular Automata v2 package have three rules inside in (*.rul) files.

Unfortunately I am really busy with my web programming work, that I have no time left to do any deeper research on it. Anyway I am going to have some good surprise for programmers in the near future. Have fun!

Mathematical Artwork Thumbnails

The images below are thumbnails of mathematical artworks I had made. Click the image for greater resolution.

Cellular Automata

Several days ago, I read an entry about Cellular Automation in Wikipedia, which links to a page titled Conway's Game of Life. In that page I learned that there is a simple rule in Cellular Automation, capable of generating complex pattern. Since I had been interested in the concept for a long time, I made the program yesterday (April 1st 2008).

There are many rules, which can be used in a Cellular Automation program. A Cellular Automation's rule have to mention what neighborhood condition should be fulfilled, to make a cell born, unchanged, or dies. The image above for example, is generated using rule 1/1, which means that for each cell, having one neighbor is going to come to life, but will die of loneliness if it have no neighbor, or die of overcrowd if it have more than one neighbor.

You can see the lexicon of rules, and the possible outcome in this Index of Rule.

Gosper's Glider Gun (Rule 23/3)

One of the pattern in Cellular Automation that I'm fond of is Gosper's Glider Gun which use Rule 23/3. The Gun, continuously produce a glider, which is a pattern capable of traversing the map. Since the glider is going to leave away, as soon as its pattern emerged from the reaction in a Gosper's Glider Gun, the Gun appear to shoot, with a glider as its ammunition.

I used Gosper's Glider Gun, to make a pattern consisting of 16 Gosper's Glider Guns, to make a Collider above. The collision between glider are unpredictable. Sometimes they explosively annihilate, sometimes they calmly annihilate, depend on how they collide.

The Collider is not stable, it will slowly degenerate itself to become the pattern above. The pattern above consist of only p1-oscillator and still lives. ( For definition of p1-oscillator and still lives, see this
Lexicon Of Life ).

I had made the X-Collider above as well, using 16 Gosper's Glider Guns in 4 different phase. The X-Collider, will continue to make a lot of glider collisions, and exhibit chaotic behavior. Whether these will result in degeneration or oscillation, is yet unknown. I haven't tried to play it long enough.

By merging different pattern I found in this Lexicon Of Life, I made the Space station above from components such as gliders, glider duplicators, octagon IIs,
octagon IVs, non-monotonic spaceships, and sparky spaceships. ( See the definition of each component in the Lexicon Of Life )

Download Cellular Automata Installer here.

Credits