The face centered cubic (fcc) lattice has two cellular automata rules that produce enough complexity to do anything useful. One of these tends to create exponential growth that expands to fill the entire universe. The 3333 rule, however, does not. It lends itself well to interesting structures, and also has great symmetry. I’m fascinated by its elegant simplicity – the only rule is that each cell is alive in the next generation if it has three live neighbors in the previous generation. You may recall that in the fcc lattice each cell has 12 neighbors.
I’ve found a lot of fascinating patterns in this automata, and will write about some of these another time. But, today being Christmas, I am delighted to present to you (ahem) this 126-cycle oscillator that looks somewhat like a Christmas tree (shown here in green, of course). This oscillator is one of the most exciting I’ve seen so far. Over 126 generations, it goes from the tree shape (seen above pointing up and to the left), through somewhat chaotic configurations, to a similar tree pointing in the opposite direction, through more chaos, and finally back to the original shape.
The easiest way to start one of these is with a planar triangular arrangement of 27 cells on each side. After 78 iterations, this configuration turns into the “tree” shape shown earlier.
After 63 more generations, it turns into a tree facing the opposite direction. After another 63 generations, we are back to the original tree shape, facing left (from this viewpoint). The complete oscillator cycle is shown here in a looping GIF animation, starting at the generation after the first tree:
And, finally, if you want to see the planar triangular shape turn into the tree, here is an animation of that process: