Planar trinet dynamics with two rewrite rules

We propose a deterministic network mobile automaton for the creation of planar trivalent networks (trinets) based on the application of only two simple rewrite rules, and we enumerate and explore the possible brownian dynamics of the control point. A useful behavioral complexity indicator is introduced, called revisit indicator, exposing a variety of emergent features, involving periodic, nested and random like dynamics. Regular structures obtained include 1-D graphs, oscillating rings, and the 2-D, hexagonal grid. In two cases only, out of over a thousand we have inspected, a remarkably fair, random-like revisit indicator is found, whose trinets exhibit a slow, square-root growth rate; some properties of these surprising computations are investigated. Finally, one 2-D case is found that seems to be unique in the way regularity and randomness are mixed.