In any case, the simulations as here presented seem to indicate a trend in which it becomes progressively easier for complex structure to evolve as some kind of surface and internal geometry setting becomes available. Secondarily, there is (automatically) a trend toward the production of systems organized at higher levels of equilibrium. In the first of the random numbers simulations, the mean r value for the correlation matrices associated with the input-to-double standardization matrices that sustained a three-dimensional representation was .188--higher even than the value for those that did not. The corresponding average r for the simulations in which four classes of points were randomly established on the surface of an earth-like spheroid was about .025. Similarly, the average r's for the analyses imposing latitudinally-oriented zones on the surface of the spheroid ran from about .036 to .052 (depending on spatial autocorrelation measure employed). For the randomly generated internal concentric zones analyses, the mean r values ran between about .065 and .102. For those concentric zones approximating the earth's actual zonation pattern the range was about .002 to .040.
There are other statistics of the entropy maximization process detailed here that are bound to be meaningful if the whole approach is fully validated; for example, the number of iterations it takes for a given matrix of values to double-standardize to stable z scores, and the degree of correlation existing between the original input scores and the double-standardized ones. At this point, however, it is entirely premature to deal with these.
Actually, however, and assuming the simulations to have been carried out competently enough to convey any kind of results, it can be said that they provide considerable support for the original thesis: that three-dimensional space may represent the results of an all-extending entropy maximization process--specifically, one that plays out as the special pattern of information flow occurring across its n = 4 subcomponents.
Although any number of other simulations can be imagined--on different metrics, using different distance and spatial autocorrelation measures and pre-arranged patterns--there is also no pressing reason not to bolt ahead and start looking at some real world systems. This will not necessarily be a straightforward matter, as most of these are more complexly arranged in space than across simple spheres or surfaces, and will present real challenges in terms of measurement. However, some actually are fairly straightforward...
Copyright 2006-2012 by Charles H. Smith.
All rights reserved.