I’ve seen some research online about using complex numbers in automata, and a while back in Stockholm, I noticed that you can form a closed set over , by simply multiplying any element of the set by any other, and I haven’t seen this used in automata anywhere else. I’ve obviously been a bit busy working on my A.I. software, Black Tree AutoML, but today I took some time out to put together a simple script that generates automata using this set. There’s only one rule, which is to simply take the product over a subset of , which means the behavior is determined entirely by the initial conditions, and the boundary conditions (i.e., the top and bottom and therefore immutable rows). This produces possible outcomes using uniform values for the initial conditions and the boundary conditions (i.e., the initial conditions and boundary conditions are independent of each other, and are each elements of ). Here are the 16 possible outputs, with 250 rows, and 500 iterations (i.e., columns):

Each output, again, has two variables: the initial row, set to a single value taken from , and the value of the top and bottom row, also taken from , which produces possible outputs. Unfortunately, the file names, which had the initial and boundary values in them, got screwed up during upload, and I’m not redoing it, but you can produce all of these using the code attached below, as there’s only 16 of them. You’ll note CA (1,1) and CA (-1,-1) are opposites (i.e., initial row = 1, boundary rows = 1; and initial row = -1, boundary rows = -1), in that one is totally white, and one is totally black. This follows from basic arithmetic, since if all initial cells are 1 (white), then any product among them will be 1. Similarly, if all initial cells are -1 (black), then any three cells will produce a product of -1 (the algorithm takes the product over exactly 3 cells). I also noticed what seem to be other algebraic relationships among them, where cells flip from green to blue, and black to white.

The shapes are simply beautiful, achieved using nothing other than multiplication, but beyond aesthetics, the notion more closely tracks an intuitive model of physics, in that there should be only one rule, that is then applied to some set of conditions, which is exactly what you get using this model. In contrast, typical cellular automata have many rules (e.g., for a three bit rule, there are eight possible combinations, producing possible rules). As a consequence, there’s a lot of interplay between the initial conditions, and the rule, when using typical automata. In contrast, in this model, there’s exactly one rule, and so the behavior is determined entirely by the initial conditions, which is exactly what you would expect in the real world, where I think it’s fair to say, we assume the existence of one, comprehensive rule of physics (see, e.g., Section 1.4 of my paper, A Computational Model of Time-Dilation). I’m not suggesting this set of automata define the fundamental rule of physics. The point is instead, if you want to model physics, using automata, and you’re being intellectually honest, you should probably use something like this, unless of course, you’re just trying to be practical, which is a different goal.