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Cellular Automata October 1, 2008

Posted by eric22222 in Math.
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(Part 1 of session 48 is up)

We start with a line of cells that can either be “on” or “off,” represented by 1 or 0. In this case, ‘#’ will be on and ‘_’ will be off (for the sake of visual clarity):

[___#__###_]

We update each cell according to the sum of the cell’s value and the two values on either side. If this sum is 1 or 2, the cell’s value becomes 1. If it is 0 or 3, the cell’s value is set to 0. The above string of cells is updated like so:

[___#__###_]
[__#####_##]

In addition, we assume that the values wrap around from left to right. That is, we say that the leftmost cell neighbors the rightmost. This is seen in the second update:

[___#__###_]
[__#####_##]
[###___####]

This seems to be completely chaotic until we look at a larger example. Starting with one cell on, we begin to see an interesting pattern:

[________________#________________]
[_______________###_______________]
[______________##_##______________]
[_____________#######_____________]
[____________##_____##____________]
[___________####___####___________]
[__________##__##_##__##__________]
[_________###############_________]
[________##_____________##________]
[_______####___________####_______]
[______##__##_________##__##______]
[_____########_______########_____]
[____##______##_____##______##____]
[___####____####___####____####___]
[__##__##__##__##_##__##__##__##__]
[_###############################_]

This is actually a pattern known as the Sierpinski Triangle:

Even after all that, I’ve only scratched the surface, but I’ll hopefully get to the cool stuff some other time.

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Comments»

1. Cellular Automata, continued « Eric Online - November 5, 2008

[…] Cellular Automata, continued November 5, 2008 Posted by eric22222 in General, Math. trackback (You may want to refresh yourself on the basics of cellular automata before reading this post by reading my previous one) […]


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