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conway 3d

by Talaya White (Submitted: 10/16/2009)

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Idiot's Guide to the Neighbor Adaptation
With 8 neighbors,
1 = 1/8
2 = 2/8
and so on.

But in a three dimensional world rather than a 2d one, you have neighbors above and below you, not just to the sides and in front of you. So how do we figure out how many neighbors we have in a 3d world, using what we know about our neighbors in a 2d world?
We have eight neighbors, and we're not counting ourselves. That means we're sitting in the middle of a 3x3 2d square. That's 3^2, the two standing for the second dimension. Make that 3^3, the second 3 standing for the third dimension, and we're sitting in the middle of 27 cubes. Subtract the one we're occupying because we can't be our own neighbor, and we have 26 neighbors.

2/8 = x/26 -> cross multiply -> 2*26 = x*8 -> x = 6.5

We'll round up*, and change the first rule of Conway's game to:
1. Any live cell with fewer than 7 living neighbors dies from underpopulation.

Do this with the rest of the numbers and adjust the rules accordingly:
2. Any live cell with more than 10 neighbors dies of overpopulation.
3. Any dead cell with between 7 and 10 (including 10) living neighbors becomes a live cell.*

*This is a rough adaptation done when I was bored at work. Change the inclusions as you see fit. Actually, you should change them even if you think I got it right, in the spirit of experimental exploration.