Hi, I implemented a cellular automaton, and am experimenting with allowing the cell states to have continuous values. I color the states red for positive values, and green for negative values.
This gif shows "tangent bubbles" forming from the update rule
Cnew = (Cold+tan(avg))/2.0, where avg is the average value of the 4 neighbors of a cell.
I also wrote an idea for why this behavior is happening.
Idea
The tangent function is chaotic under repeated application to a starting number, but not in the same way as the z2+c map.
This is a sample of an orbit starting from x0 = 1.
1, 1.56, 74.69, -0.86, -1.17, -2.36, 0.99, 1.54, 30.62
-1.01, -1.61, 29.21, 1.37, 4.92, -4.82, 8.94, -0.53
x above 0 and below pi/2 will increase under tan(x), and x near pi/2 will have very large positive or negative tan(x). The result of this is that values will accelerate to near +-pi/2 and then can "explode" to a large value, and this value can drop down to a small value or explode to a different large value.
The update rule (Cold+avg)/2.0 is actually a form of the heat equation.
Starting from a single cell with value 1, an expanding bell function emerges
with a sum of values = 1. This process brings values closer to equilibrium.
The behavior of (Cold+tan(avg))/2.0 is a combination of the diffusive,
calming heat equation and the chaotic, hyper tangent function. A small region of cells with small values will be more affected by edge diffusion than the small increases the tangent provides, leading to this region dissipating toward 0. But there is a critical threshold where the value of a region will grow more than it dissipates, and this region forms a "bubble".
When the values get close enough to pi/2, the bubble pops, and a value explodes, making the value of neighbor cells change a lot, which causes the original value to change, leading to interesting dynamics in the region. The diffusion prevents values from forever cycling in a tangent orbit by themselves, and the tangent introduces a force against diffusion.
