Planar Trinet Dynamics with Two Rewrite Rules

A deterministic network mobile automaton is proposed for the creation of planar trivalent networks (trinets) based on the application of only two simple rewrite rules. The possible Brownian dynamics of the control point are enumerated and explored. A useful behavioral complexity indicator is introduced, called the revisit indicator, exposing a variety of emergent features, involving periodic, nested, and randomlike dynamics. Regular structures obtained include one-dimensional graphs, oscillating rings, and the two-dimensional hexagonal grid. In two cases only, out of over a thousand that were inspected, a remarkably fair, random-like revisit indicator is found, with trinets that exhibit a slow, square-root growth rate. Some properties of these surprising computations are investigated. Finally, one two-dimensional case is found that seems to be unique in the way regularity and randomness are mixed.