Noninteger Dimensional Spaces and the Inverse Square Law

Abstract

Noninteger dimensionality, which shows up in many fields of physics and engineering, is generally viewed from the perspective of fractals and measured by the Hausdorff dimension. Motivated by information theoretic considerations and by extending the principle of complementarity, we propose a new interpretation in which it engenders a potential. The inverse square law may then be seen to originate from the properties associated with the noninteger dimensional space, and it further indicates a relationship between the dynamics and the actual value of the dimensions that may be used for experimental test of the interpretation.

Fractals with optimal information dimension

“…The selfsimilarity may be approximate, exact, or be defined statistically; it can also be associated with time series [2]. Recent papers show how the optimal dimensionality of e characterizes physical space [3][4][5], which opens up new applications to engineered systems. The application of fractals to physical systems, astronomy, complex and social networks, medicine and biology has been investigated for many years [6][7][8][9].…”

Asymptotic freedom and noninteger dimensionality

“…1.7-dimensional space sits within the 2-dimensional space). It may be argued that a fundamental characteristic of a noninteger space is that of a continual shrinking of the metrical relationships between objects 1,6 . This is seen most clearly when we visualize a 2-dimensional space obtained from a 3-dimensional space which will cause all the points in the third dimension to collapse to the extant two dimensions.…”

“…From the perspective of information theory, the optimum number of dimensions is the noninteger (and irrational) number e and since Nature chooses optimality, this should be the dimensionality of physical space 1,5 and the inverse square law itself may be seen as a consequence of noninteger dimensionality 6 . The information efficiency per dimension is E(d) = ln d…”