Abstract
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.
Highlights
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: How to cite this paper: Smith, C.H. (2014) “In Space” or “As Space”?: Spatial Autocorrelation Properties of the Earth’s Interior
The approach has the effect of canceling out the complication of absolute magnitudes of flow or similarity when the matter of interest is relative magnitudes across all elements—as would be the case, for example, when trying to compare rates of commuting among a set of places of greatly differing populations [18]. These results were contextualized with the aid of multidimensional scaling (MDS): it was determined, as might be expected, that only those matrices that converge to a matrix of z scores that is symmetric about the i = j diagonal can be projected as an unambiguous three-dimensional space via MDS, so attention was turned to looking for such solutions among simulated input data
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 zone, regardless of how defined, varies greatly in thickness below various surface locations
Summary
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 measurable characteristics exposing the supposed cryptic causal influence. 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. 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, exposing varying levels of nonrandomness within any implied corresponding system’s internal differentiation. In an earlier pilot study on another set of stream basins [23], it was possible to link this variation to independent measures of the basins’ surface energy characteristics (especially, as related to non-normality of elevational profile and other measures indicative of deviation from a system-wide balance between depositional and erosional forces)
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