Iconography for Elementary Cellular Automata Based on Radial Convergence Diagrams

Consider a complete set of initial conditions for a finite elementary cellular automaton (ECA). This set can be indexed by integers using a Gray code or a binary to decimal conversion. During the ECA evolution, the set of such indices is mapped to itself, which can be illustrated with the help of radial convergence diagrams (RCD).

To construct an RCD, rescale indices to a unit interval and wrap it along the circumference of a circle. Each time an index is mapped to another index, connect them by a translucent spline line. Repeat the procedure for several steps of the ECA evolution (five in this implementation).

For most of the rules, as the ECA evolves, the set of possible states thins, ultimately converging to a limiting Cantor set attractor [1–5]. This is reflected in the RCD; splines for the most frequent mappings naturally have greater density. Being a fractal, the Cantor set is scale-invariant, so in turn RCDs look similar for all different ECA sizes and length of evolution.

Because RCDs are independent of the ECA's spatial and temporal sizes, they can be used as unique invariant icons. They contain a lot of information about the general dynamics of ECAs or their Wolfram class.