Registered: Nov 2008
Rule 30 Generalized Beyond CA.
I've taken a look at Rule 30 and agree it's an intriguing discovery. So I'm given to speculate about what makes it tick. It has been observed that there is order on both the left and right sides at the edges.
The question is, what happens further inward? I ask whether Rule 30 can be likened to the Three-Body Problem in celestial mechanics or something even more general.
The way I look at it is this: When there are two bodies their interaction is determined by their masses and distance. That is well known. When a third body is introduced their interaction suddenly becomes rather unpredictable. It has taken many years to solve the problem. Its behavior is illustrated here:
Intuitively one may speculate that when a third body interacts with two others, it is interacting with two changing bodies. This leads to its reacting unpredictably to the two bodies not only changing in relation to each other but to itself as well. Perhaps this behavior can be generalized to any three changing entities. Take, for example people. To the outside observer two unknown people's behavior are unpredictable. On knowing them, assuming they relate to each other, their behavior settles down to something predictable. Introduce a third party to them, and typically their immediate behavior becomes unpredictable. I think this has something to do with the multiplying of the variables involved. (If you will bear with me, I will get to Rule 30 momentarily.)
Suppose instead of two people encountering a third person, we substitute a NEW KIND of interaction between the two people. In general, we no longer know how the two people will react to each other. This is because the NEW situation draws out different and new things in different people. There are three variables: two people plus one of many arbitrary relations.
Back to Rule 30. The pattern behavior on the left edge of the generated pyramid is independent of the behavior on the right edge (except at the first cell). This continues to a modest extent inward from the edge. The CA rules may be the same for each edge, but the generated patterns are different due the asymmetry of the eight rules for generation. If I have it right, the left edge repeats patterns while the right edge repeats in a doubling fashion. Each pattern is independent of the other, the single exception being at the start where they coincide. Notice that generated patterns expand in WIDTH as further generation of the CA pyramid develops. Sooner or later, generation on the right overtakes and interferes with the pattern on the left and vice-versa. After the start each pattern takes off rapidly enough and remains independent of what is happening on the other side until its width becomes large enough it meets the other side's generation.
Here is my point: The two patterns generated on either side are independent until they meet and "clash." The points of clashing though I believe can be roughly mathematically calculated, is random in its results and has nothing to do with the patterns themselves. What we have are two different emergent patterns clashing in an arbitrary fashion. Is there any wonder that just as in the three-body problem or any three variable situation for that matter, one would expect chaotic, arbitrary if not random behavior on meeting?
Another way of thinking about this, is although the foundation of Rule 30 is the eight rules of CA generation, the patterns generated on the left and right edges in some sense, must be emergent and unpredictable. Therefore their interaction with each other would have characteristics brought about by their own nature and not directly by the eight rules at the foundation.
I have not examined Wolfram's other rules as yet, but I would expect it interesting to speculate on them as well.
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