_{1}

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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 n × n input-output matrices could, once entropy-maximized, project a three-dimensional Euclidean metric. Only 4 × 4 matrices could; a small proportion passed the test. Larger proportions passed when grouped random patterns on and within two- and three-dimensional forms were tested. The pattern of structural zonation within the earth was then tested in analogous fashion using spatial autocorrelation measures, and for three time periods: current, 95 million years b.p. and 200 million years b.p. All expected results were obtained; not only do the geometries of zonation project a three-dimensional structure as anticipated, but also do secondary statistical measures reveal levels of equilibrium among the zones in all three cases that are nearly total, distinguishing them from simulations that do not incorporate a varying-surface zone-width element.

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 n × n input-output matrices could, once entropy-maximized, project a three-dimensional Euclidean metric. Only 4 × 4 matrices could; a small proportion passed the test. Larger proportions passed when grouped random patterns on and within twoand three-dimensional forms were tested. The pattern of structural zonation within the earth was then tested in analogous fashion using spatial autocorrelation measures, and for three time periods: current, 95 million years b.p. and 200 million years b.p. All expected results were obtained; not only do the geometries of zonation project a three-dimensional structure as anticipated, but also do secondary statistical measures reveal levels of equilibrium among the zones in all three cases that are nearly total, distinguishing them from simulations that do not incorporate a varying-surface zone-width element.

Keywords: Entropy Maximization, Earth’s Interior, Spatial Autocorrelation, Benedict de Spinoza, Spatial Extension, Space

In 1894, the British naturalist Alfred Russel Wallace (1823-1913) made the following surprising reply to an ongoing public discussion in the magazine Light as to whether there might be a fourth dimension:

Sir, —The discussion on this subject seems to me to be wholly founded upon fallacy and verbal quibbles. I hold, not only that the alleged fourth dimension of space cannot be proved to exist, but that it cannot exist. The whole fallacy is based upon the assumption that we do know space of one, two, and three dimensions. This I deny. The alleged space of one dimension—lines—is not space at all, but merely directions in space. So the alleged space of two dimensions—surfaces—is not space, but only the limits between two portions of space, or the surfaces of bodies in space. There is thus only one Space—that which contains everything, both actual, possible, and conceivable. This Space has no definite number of dimensions, since it is necessarily infinite, and infinite in an infinite number of directions. Because mathematicians make use of what they term “three dimensions” in order to measure certain portions of space, or to define certain positions, lines, or surfaces in it, that does not in any way affect the nature of Space itself, still less can it limit space, which it must do if any other kind of space is possible which is yet not contained in infinite Space. The whole conception of space of different dimensions is thus a pure verbal fantasy, founded on the terms and symbols of mathematicians, who have no more power to limit or modify the conception of Space itself than has the most ignorant schoolboy [

While this perspective may not quite satisfy today’s mathematicians or physicists, it is yet true that the question of why spatial extension is three-dimensional (at least, in a mesoscale sense) has never been resolved [

There are several reasons for not dismissing this foreign-sounding notion out of hand. First, it appears not to seriously conflict with more conventional ways of studying natural structure—that is, positivist/empiricist/mathematical approaches dwelling more on topology and/or causal sequences occurring between systems [

If the systems that comprise extended space are in fact organized in the manner I postulate, it seems a priori likely that they must exhibit some kinds of evidence of structure related to energy transfer and dissipation in the way suggested. These can be expected to stand as nonrandom patterns of differentiation in observable space, and therefore should be amenable to study through forms of pattern analysis. In particular, the patterns involved should reflect an entropy maximization process that sustains three-dimensionality itself; this is a difficult thing to envision, but reduces to the idea of a balance, as expressed in three dimensions, among its subsystems. One possible first step toward describing this organization is to conceive of a simple n by n flows/similarities matrix representing the summary characteristics of internal information flow (or sharing) such a system might exhibit: that is, as an i=1,n, j=1,n input-output matrix. Given such a numerical representation, the central question becomes whether the relations within any structures described in this fashion can be shown to project unambiguously as a three-dimensional space. Otherwise stated, will group-level interrelations (i.e., among a system’s primary subsystems, whether these be discretely separated from one another, or more complexly interlocated) reveal a cryptic Euclidean dimensionality to their fundamental organization? This question can be answered in a technical sense by subjecting related i=1,n, j=1,n data to a metric multidimensional scaling (MDS) operation: where an unambiguous solution in three dimensions is produced such that x_{1} + x_{2} + _{n} = 0, y_{1} + y_{2} + _{n} = 0, and z_{1} + z_{2} + _{n} = 0, each i = 1, n vector of relations contributing equally, ipso facto, to the group (spatial) structure. Ordinarily MDS is used to probe for internal structure within a set of similarities (or analogous) data. In such efforts the dimensional representation serves a similar purpose to the number of factors arriving in factor analysis. Here, however, we anticipate something more than just unambiguous representation as a three-dimensionally-set group of relations: in particular, each projected coordinate set in the output configuration will be indistinguishable from the other three (in the absence of rotations of the solution connected to particular axial scalings).

Investigation of these concerns proceeded through a series of simulations and empirical pattern analyses [

In this write-up I discuss only the results connected to matrix dimensions of n = 4, but in the actual tests I carried out parallel studies on n = 3, 5, and 6 dimensions as well—but none of these latter investigations produced any results interpretable as three-dimensional space, so are not detailed here [

In the first set of simulations, designed to sample the universe of all numerically conceivable combinations of values (and thus of potential flow rates or other kinds of relations), randomly-generated six-digit figures were used to fill the sixteen elements of the initial matrix. Two variations on this basic theme were investigated: 1) four by four matrices consisting of sixteen such values, entirely randomly assigned; 2) the same, but with all i, j values equaling all j, i values. Most recently (in 2008) I ran a total of 40,000 operations under the first set of conditions, and 25,000 under the second. As reported in the other paper [

These results are both intuitively and logically satisfactory. In each instance the proportion of input matrices that pass the test is small, but not so impossibly limiting as to suggest preclusion of diversity of result altogether (and even more importantly, within-system changes such as would be expected in any non-static natural entity— or for that matter, evolutionary process). Further, that the matrix dimension n = 4 turns out to be the only dimensionality that leads to results passing the test is in keeping with two important facts: 1) that it takes x + 1 coordinate points to specify an x-dimensional geometry; and 2) that equations with five or more variables are almost

never solvable (and here there is exactly only one n = 3 solution that sustains the kind of symmetry required).

The random numbers-based simulations were followed by a series of analyses carried out on contrived twoand three-dimensional test surfaces/figures. The object was to shed light on whether: 1) sets of regularly sampled coordinate points taken off these surfaces/volumes,^{1} when grouped into four subsets of points and then measured for their summary spatial autocorrelation properties,^{2} could yield 4 × 4 element matrices that when double-standardized passed the spatial extension test. The geometric surfaces tested included: 1) variously shaped two-dimensional enclosures, each at various “densities” of point sampling (with the sets of coordinates retrieved then randomly grouped into four subsets), 2) same, but sampling coordinate locations from the surface of a sphere, 3) same as #2, but instead of randomly generating subset compilations after the fact, sampling arbitrarily pre-arranged, latitudinally-parallel, hemispherically symmetric zones (with each analysis treating different combinations of widths of the zones), and 4) like #3, but involving samples of concentrically-arranged “zones” within the interior of the sphere (that is, as concentrically arranged about the x, y, z origin within it). These tests yielded a variety of results, though usually with around five percent of the two-dimensional simulations passing the three-dimensionality test, and as many as fifty percent of the three-dimensional simulations doing likewise [

Again, it can be argued that these results appear both intuitively and logically consistent with a priori expectations. In particular, one of the most advanced and complex systems—our biosphere—is organized upon a near-spherical planetary body, and through this kind of modeling an understanding of this might emerge: the pre-existing surface geometry may be such as to facilitate the kinds of stochastically-generated relationships leading to lasting complex structures.

Although the model entertained here must by its definition pertain to all naturally-developing systems (and at one level or another, all systems), this is not to expect that all of these will exhibit easily measurable characteristics exposing the supposed cryptic causal influence. Most complex natural systems (especially of smaller scale) are not emergent as, and/or on, simple geometries which are the outcome of neatly reducible causalities: instead, most have convoluted histories that represent reactions to different forces at different times, or to several different forces at once. For systems as complex and advanced as organismal bodies, for example, means of assessing actual information flow rates among subsystems may have to be found before analyses of the type discussed here can be attempted. (On the other hand, secondary characteristics of structures may be amenable to investigation: for example, brain physiology and health as evidenced—spatially—by internally varying temperatures, blood flow, or electrical activity). However, there are some natural systems that do exhibit fairly straightforward characteristics of pattern that can be linked to uniform causal influences, two of these being planetary surfaces and interiors.

Generally speaking, the past century’s increasing knowledge of the interior of the Earth has proceeded through incremental steps, leaning heavily on concurrent improvements both in physical theory and in means of instrumentation-based measurement. This knowledge can be used to present a test of the current model, one which anticipates the presence of structural underpinnings whose discovery would be unlikely through other philosophical starting points.

Before summarizing this research, one further aspect of measurement in the present context needs to be discussed. This concerns the level of redundancy of system structure (or information flow) conveyed through the values in the initial summary matrix, prior to its double-standardization. It is a well-known characteristic of developing natural systems that during their maturation period they tend to develop more and more internal feedback mechanisms that, generally, improve efficiency of operation through the elimination of redundant causal paths (see in particular the writings of the late Eugene and Howard Odum on the maturation of ecological systems). The random numbers simulations discussed earlier make it clear that an infinite variety of configurations of values in the initial summary matrix lead to double-standardized results that pass the three-dimensionality test. Some of these initial configurations may be comprised of individual vectors of values that are on average more highly—or lowly—correlated with the other three, thus exposing varying levels of nonrandomness within any implied corresponding system’s internal differentiation. In this way the mean value of the correlation matrix associated with an initial summary matrix may be considered at the least a measure of the extent of its system’s internal redundancy, and quite possibly, in real world terms, a measure of its net degree of disequilibrium as well (as it works its way toward dynamic equilibrium). Two such “mean correlation” statistics contribute to

It is therefore to be anticipated that real world systems measured and sampled as described here might exhibit a variety of nonequilibrial states that yet pass the present “spatial projection” test. Importantly, independent measures of a system’s equilibrium status at various points in its evolution can be expected to correspond to variations over time in its four by four matrix representations, a potentially important means of validation of the overall model. For example, in the studies on the hypsimetric properties of the stream basins [

In the present example, an attempt was made to contrast the spatial autocorrelation properties of a simulated but simplified zonal earth—a spheroid/geoid consisting of four concentric zones, including a surface zone of uniform width—with conditions obtaining in the actual earth, whose outermost (surface) zone, regardless of how defined (crustal lithosphere, same plus hydrosphere, same plus hydrosphere and atmosphere), varies greatly in thickness below various surface locations. Further, I looked not only at the latter configurations for the present day, but at two periods in the distant past: 95 and 200 million years ago [

The three-dimensional sample grid applied assigned about 16,000 to 21,000 within-geoid point locations, depending on the exact initial assumptions. These assignments were then partitioned into the four zonal groups through a combination of programming (based on data from the standard earth model) and estimation: assignment to the top of the mantle or the crust had to be assessed manually, based on the best information I had available.^{3} The “zone-level” spatial autocorrelation statistics were then calculated (i.e., between and within each i=1,4, j=1,4 pairing of sampled point subsets). Further, the problem was examined under the three different assumptions regarding what constitutes the “surface zone” of the earth (as distinct from the inner core, outer core, and mantle): crustal lithosphere (oceanic plus continental) alone, crustal lithosphere plus hydrosphere, and crustal lithosphere plus hydrosphere plus thirty kilometers of atmosphere. This process was repeated for the data from each of the time periods noted above.

The best overall results were obtained for a surface zone defined in terms of crustal lithosphere alone, but all three configurations for all three time periods both passed the three dimensionality test and yielded mean correlation coefficients ranging in value from about 0.002 to 0.003. An earlier pilot study, using a different set of sampled points at about the same sampling resolution, produced an even lower range of values, from about 0.001 to 0.003 [

There are two things to be said about these results. First, the range 0.001 to 0.003 represents an extremely small mean correlation value in the present context. In the fully random numbered simulations, only about one matrix in 367 produced mean values as low as 0.001, and one in 81 at 0.003. This, as compared with one in 31 for 0.006. I am unaware of a statistical test for these differences, but clearly the likelihood decreases at a nonlinear rate as zero is approached, so the difference between the actual and simulated data is far greater than it at first appears.

Second, though the degree of precision of sampling here seems considerable (at 16,000 to 21,000 points), it admittedly is still not sufficient to absolutely resolve the matter. In particular, at such low levels of measured spatial disequilibrium even a small amount of sampling bias may work against achieving a fully reliable final assessment. Nevertheless, in the internal differentiation of the earth we have a system of large scale and very long standing whose surface components, at least, provide evidence of having very nearly reached a state of dynamic equilibrium with respect to their participation in primary biogeochemical cycling processes. Atmospheric oxygen, for example, shows no overall trend of increase or decrease over the past three hundred million years (though of course it has experienced several lengthy up and down trends within that period [

The uniformity of results across the three time periods is thus both intriguing and consistent with the initial forecast. It can be safely predicted that a more concerted effort, involving perhaps ten or a hundred times as many real world-sampled points, will produce a spatial autocorrelation coefficient matrix whose associated correlation matrix yields a mean value even closer to zero—but that the parallel simulations not taking into account varying thickness will be less affected.

These findings, based on the same fundamental conceptual model as the earlier-discussed stream basins analysis [

For over three hundred years, the presumption among scientific investigators has been that space—actually, spatial extension—represents something “within which” matter changes form according to progressive local influences. The concept “within which”, however, is a rather fuzzy one (as the logical positivist Rudolf Carnap discovered when the system described in his Der LogischeAufbau der Welt ultimately collapsed as a result of his inability to provide a satisfactory definition for the elemental notion “located at” [

However skeptical one may be about the underlying philosophical perspective advanced here, the simulations and empirical analyses I have summarized—none of which require great sophistication of analytical technique— seem on their own merits worthy of further development. At the very least they lead one to question some basic assumptions about the role of chance in nature: especially as related to notions of final causes. (Final causes, that is—not first causes. The understanding presented here, while suggesting that the number of physical structures that can possibly evolve is considerably smaller than that is currently assumed, nevertheless still argues for a naturalistic appreciation of that evolution.) Proponents of gaian theory and the anthropic principle should take note.

In view of the simulations and empiricism presented here, the Wallace quote given earlier takes on greater interest. Wallace’s concept of “space” reduces to a statement about the full universe of causal events—whether these are describable in spatially-projected, “three-dimensional”, terms, or not. Here I have presented a model that suggests a way to account for the three-dimensionality of observable nature, but in the same context it becomes interesting to consider whether stable intra-system configurations of subsystemization may exist that are not “spatially projectable”, yet still have causal influence in the overall universe of events.

Wallace was fond of quoting the following words from A Preliminary Discourse on the Study of Natural Philosophy, by Sir John Herschel: “The perfect observer in any department of science, will have his eyes as it were opened, that they may be struck at once with any occurrence which according to received theories ought not to happen, for these are the facts which serve as clews to new discoveries” [