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In this analysis natural systems are posed to subsystemize in a manner facilitating both structured information/energy sharing and an entropy maximization process projecting a three-dimensional, spatial, outcome. Numerical simulations were first carried out to determine whether

Historically, treatments of the three-dimensionality of mesoscale nature have remained largely on a philosophical/mathematical plane [

Suppose instead that space has properties implicit in each individual thing that makes it up. Space itself could be the result of some organizational principle whose results are emergent at all levels of the scale hierarchy. The principle could be probabilistic in its enaction, yet, by virtue of its nature, confining in such a way as to make many kinds of results impossible

This kind of thinking might be taken in many directions; one is toward the doctrine of final causes. This, one of the four classical Aristotelian causes, is most familiar in its application to design—for example, to the process of realization of artistic intent, as when a sculptor plans out his work before actually setting upon it. It is also relatable to teleology, and usually is. But there is no reason why teleology—especially as dictated by first causes (

Here is a structure whose evolution has taken place not only at its own level of organization, but simultaneously in response to a world of events beyond it. It can hardly be understood otherwise. The purpose of DNA is to build, from scratch as it were, a supra-DNA entity that fits into a pre-existing supra-individual environment. It is in essence an evolving program, one that is observing a final cause: the creation and maintenance of a living organism whose existence contributes to the operation of the ecological space supporting it.

In the spirit of exploration many years ago the senior author began to consider what kind of organizational principle might act as a wholly “natural” final cause. I soon came across writings of Benedict de Spinoza that could be related to modern ideas on subsystemization. A paper followed [

This is not the place to attempt to go through this model in detail [

The real world system investigated has been chosen for its relation to the authors’ actual backgrounds, because it provides a clear example of the approach, and because data sets on other potentially exploitable subjects could not be obtained. It is not a biological example (though one could argue it is an ecological one, as topography is the net result of biogeochemical processes), but biological applications are not difficult to imagine, as will be discussed later.

Geographers have been making use of entropy maximization analysis techniques for over four decades [

In the simulations, numerous matrices of varying dimension (3 × 3, 4 × 4, 5 × 5, and 6 × 6) filled with random numbers (simulating the universe of theoretically possible information flow/similarities data) were entropy-maximized, then examined for whether any of the resulting configurations corresponded to a Euclidean space. This can be determined through metric multidimensional scaling. “Successful” configurations not only had to correspond to an unambiguous three-dimensional geometric representation—as any zero-stress solution in three dimensions would—but also consist of element scores that as a group were equally weighted on the three cardinal orthogonal axes (that is, each point in the final spatial projection would be both the same distance from the origin, and the same set of distances from one another).

Thus the simulations examined a random sample of the universe of all possible combinations of

Summary of spatial projection simulations for matrix configurations of dimension 3 × 3 through 6 × 6. Circled numbers refer to data at the right margin giving the number of simulations in each test and the standard deviations accompanying the mean values plotted. The plotted numbers are the two

Only matrices of dimensionality

In the simulation analyses to be described, the results across thousands of initial matrices were next averaged to produce the final summary statistics (means) for each set of results (1) and (2) above; these are hereafter referred to as

For each matrix dimensionality of three through six in

Given the small but non-negligible portion of the simulations that do pass the spatial projection test (and their varying levels of internal redundancy), it becomes more reasonable to consider whether system patterns in a real world—or first, perhaps, simulated real world—context might do the same (as it is apparent that persistent flows of material/information in real space are often manifest as standing structural patterns). Some years ago, the senior author carried out additional simulations involving two- or three-dimensional spaces populated by randomly-grouped point patterns. These were executed in the same fashion as described above, but in these instances the initial matrix elements consisted of spatial autocorrelation statistics (

It is fair to ask what the

The senior author has performed pilot analyses on the patterns exhibited by several kinds of real world systems in an effort to determine whether they too pass the spatial projection test. The early results seem to suggest so, but most recently a more rigorous study on a set of stream basins in Kentucky was undertaken. Thirty-one basins were selected; to avoid possible complications these were chosen for their relative uniformity of size (ranging in area from 2.8 to 6.6 square miles), outflow into streams of markedly larger size, and absence of complicating surface conditions such as strip mining and karst topography. Basin limits were established manually from USGS topographic maps; elevations across each basin were then grid-sampled using ArcGIS platform data. Three sampling densities were applied for comparative purposes: for the 1× sampling the real-world distance from each sample point on the surface to each of its six nearest neighbors was 860 feet, for the 4× sampling, 430 feet, and for the 16× sampling, 215 feet.

It is a given that every point on the terrestrial surface of the earth constitutes a physical outcome devolving from co-mingling forces of elevation and reduction. These outcomes produce a range of gravitational potential energies that may be expected to interact with one another, especially with respect to their baseline, as conditions evolve. In the context of the present study, it was reasoned that these forces might internally organize,

How this happens at the microscale is bound to be tied to all sorts of processes invoking both larger- and smaller-scale systems responding to a myriad of influences. Adrian Bejan’s constructal theory [

The earlier pilot studies suggest that within some natural systems, the “imposed current” (in Bejan’s terminology) is so consistent and/or stable that physically observable boundaries between the subsystems emerge and persist. In more varying and/or multi-causal systems, however, the underlying organization may only be evident through the pattern of secondary indicators of its operation such as temperature or pressure. In the present instance, the topography itself reflects the impact of systemic interactions of gravitational forces. The central matter of interest here is whether evidence can be found that the system is self-organizing as a function of group relations, instead of a myriad of independently operating singularities. For each basin regular (triangular grid) samples of elevations were first classified into three, four, five, and six class structures using a nonhierarchical, information statistic-based, clustering algorithm [

Summary of variation explained statistics obtained for the clustering of the stream basins data (vectors) into two through six classes. The plotted points are the mean (n = 31) variations-explained for each classification, at three fineness levels of sampling. Colored line coding connecting points is for readability purposes only.

The summary statistics displayed in

A simple sum-of-squares-based (

Summary of spatial subsystemization properties of topography in 31 Kentucky stream basins, based on three fineness levels of sampling. The plotted values in the top three sets of four points are the

For the four-class solutions, 28 of the 31 16× spatial autocorrelation matrices do double-standardize to a symmetric state passing the spatial projection test described earlier. 28 of 31 also pass at the 4× sampling density, but only 18 of 31 at the 1× (again, likely reflecting the poorer resolution connected to the coarser samples). Parallel analyses employing two other metric spatial autocorrelation measures produced similar results. Remembering that the two-dimensional simulations only yielded about five percent success rates, this is remarkable. In theory, however, all 31 should have passed, so as a check the three nonconforming basins were subjected to a special 64× sampling (capturing even more detail in the topography), whereupon two of the three then produced symmetric

It possibly will be objected that the main tests merely represent a reconstruction of three-dimensionality that was already built into the setting, but this seems not to be so. First, the

As long as physical expression in three dimensions is viewed as an

The model discussed here considers the nature of spatial extension itself, through the patterns that secondarily characterize it. If it can be proved valid, it will apply equally to all existing natural systems’ internal characteristics, and how these change over time. Concerning stream basins in particular, a process of ongoing internal adjustments can be imagined wherein the feedbacks required to maintain the overall system’s integrity are ordered

Ultimately, however, the most important tests—and hopefully applications—of this model will fall within the biological sciences. Among the pilot studies mentioned earlier were analyses performed on color patterns on butterfly wings [

We thank members of the Center for Cave and Karst Studies at Western Kentucky University for their assistance; also J. Yan and K. Cary of the Department of Geography and Geology, Western Kentucky University, for their programming contributions and other help with the GIS data.

Bistochastization has been used in a variety of complex science contexts, including human migration, commodity economics, evolutionary genetics, and the analysis of marker genes, mutation rates and microarrays.

Plural “values,” because several variations on the constitution of the “surface zone” were investigated: Solid crust (oceanic plus continental) only, solid crust plus hydrosphere, and solid crust plus hydrosphere plus atmosphere (with the last, for three or four vertical widths).

To say this another way, practically any other imaginable combination of relative widths of the core–outer core–mantle–surface system produces higher mean correlation values, suggesting the earth as it is constituted is a very special place.