|Hard evidence having got me over a long obsession with a notional boundary between order and chaos with the realisation in 2009 that they are better seen as synergistic partners, and with chaos well defined mathematically as extreme sensitivity to initial conditions, I've finally started to think about an equivalent definition of order. This journey has been helped by Brendan Hill's insightful interest in just what it is that allows us to recognise emergent order in cellular automata. Is it our perception or is it intrinsic to the CA pattern?|
Wolfram's Class 1 (trivially, stable) and Class 2 CA can be taken to represent order. They key to Class 2 is that it shows repetition, allowing a highly compressed representation and greatly accelerated computation of far future states. (Class 1 can be seen as degenerate repetition.) Our perception is drawn to emergent order, presumably because the only way to win the Darwinian game is to deal effectively with anything that is reasonably predictable and trust that the world contains enough emergent order for enough to survive long enough before encountering fatal chaos. (This must also involve resilience in the face of most common chaos, but that is probably another story.)
So how to define order? An initial pattern might be judged to show order if it recurs repetitively. In a 2D CA this generally requires that a copy (or more) of the initial pattern appears again, potentially reflected and/or rotated and/or translated, with the added condition that any extraneous debris generated in the process is not going to interfere with the move being repeated. There would also appear to be some need for at least something in the intermediate patterns between repetitions to also repeat, so we should consider an individual (space)ship to show order, but not the streams of ships I've become too familiar with radiating from chaotic core with no pattern to their spacing and timing, that exclusion in turn excluding common enough rakes of ships which form typically diagonal lines.
There is also another discussion to be had on what those geometric rules for proximate order say about a larger scale view, but for now let us stick with a close correspondence between order and repetition. Then we can reflect how our attention is drawn to emergent local symmetries or to period doubling where repetition is progressively emerging in "delta" patterns outside the more slowly growing edge of chaotic core.