Friday, December 01, 2006

Sierpinski, phone home

Several other posts on 'intelligent design' are on this blog

Randomness and form are acutely discussed in Chaos and Fractals: new frontiers in science by Peitgen, Jurgens and Saupe, Springer-Verlag, 1992. This book, a well-rounded introductory survey, makes the point that simple rules can yield complex results, anticipating Wolfram's chief conclusion in New Kind of Science by about a decade. In fact, the PJS book incorporates some of Wolfram's early findings on cellular automata.

A very interesting result of chaos theory is that rules governing a random walk (the "chaos game"), yield a highly probable overall order. This order very strongly tends to become more precise as the n-step iterative algorithm goes to infinity.

In particular, the chaos game has an attractor called the Sierpinski gasket, a form that shows up in deterministic processes as well, including some Wolfram diagrams.

This result is highly reminiscent of the wave-particle duality feature of a quantum mechanical double slit experiment conducted photon by photon. The constraints (the two slits and whatever those two slits might imply) influence the overall probabilities of where photons land, so that the pattern becomes a diffraction image typical of a wave.

Also, such a result tends to demonstrate that simple constraints on random behavior can yield sets associated with "order," a point of interest to network theorists and those interested in the emergence of orderly (low entropy) systems.

After a sufficient number of iterations, we might apply a runs test to a chaos game's Sierpinski gasket, by reading triangles via some "address" system, that identifies a specific triangle by a route taken from the apex to get there. I think we might linearize this by having an algorithm for formulating addresses. In general, sequences with very large or very small periods are considered very suggestive of nonrandomness. That is ABABABABAB has a low probability of being randomly generated, as does AAAAABBBBB.

By this, we would find that the triangles of a Sierpinski gasket have a periodicity consistent with low probability and low entropy; that is, consistent with order. But, using perhaps a box-count method, the order breaks down at the point level and we obtain a high entropy distribution. On the other hand, a careful assessment would give probable point densities per box and hence the consequent macro-structure. The order emerges as a consequence of the constraints that influence the probability distributions.

[Note that Wolfram, rather than emphasizing relatively high periodicity to focus on order or complexity, puts the stress on lowness of periodicity.]

In the chaos game, the attractor appears to be a consequence of conditional probabilities, but I need to do a bit more study to see exactly how that works.

So suppose a SETI television receiver picked up a transmission that could be broken down as a Sierpinski gasket. That is, the receiver picks up a signal from a source and translates the data onto a digitized screen. Suppose that at time t, the screen contained randomly lit pixels but at time t+k, a Sierpinski gasket began to emerge, which was even more pronounced at time t+(k+j). Now suppose the SETI observer didn't get around to looking at the monitor until t+(k+j). He or she might easily think that a signal had been sent by an intelligent being. Yet, we know that the image could have resulted from random processes constrained by simple natural limits.

Well, I'd like to know whether constraints exist for a random walk that has a Sierpinski carpet as an attractor. That's because an infinitely extended Sierpienski carpet contains every possible linear form (in the sense of topological equivalence). Hence, one might conceivably "discover" any symbolic message in a Sierpinski carpet. However, the fact that every message is implicit in this graph doesn't mean it is obvious. One must still use a winnowing process to find each message.

ADDENDUM: If a chaos game initial point falls on a vertex of the initial triangle, subsequent points always fall on the attractor, the Sierpinski gasket. If an initial point falls outside the initial triangle, each subsequent point draws nearer to the attractor, so that points converge to the attractor. As long as probabilities are equal, the attractor, over time, tends to fill out uniformly; otherwise not.
In the case of a double-slit experiment, the single quantum of energy is never recorded in the area of "destructive interference." So the area of constructive interference behaves exactly like a chaos attractor with the added idea that the initial condition of the double-slit and single quantum of energy is equivalent to starting out on the attractor.

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