The article Preprint Size, scaling, and invariant ratios
compares the scaling and dimensional standpoints in relation to parts of a system such as an animal change in aspect when they become larger. Is there an advantage in using one standpoint over the other? A conceptual advantage? A modeling advantage? Is dimension more fundamental? Is one standpoint easier to understand? For the dimensional standpoint, does the difficulty of imaging weight proportional to 3 dim residing in the two dimensions of an animal bone’s weight-bearing cross-sectional area reveal that some aspect of dimension is overlooked in physics? Does the dimensional standpoint reveal a gap in understanding or is it just another way of looking at the same thing?
This question was prompted by an answer to a preceding question: Why scaling and not dimension, Galileo? Dr. Nicolis replied in part: There's no difference, since the scaling argument is nothing more and nothing less than dimensional analysis in the case at hand.
That seems right. But is it right? Which raises the question: Does the dimensional standpoint confer any advantages over the scaling standpoint? Hence the question. The problem: are there arguments that the dimensional standpoint has advantages over the scaling standpoint? In a sense, this problem has been implicit in my articles on metabolic scaling since 2008, and emerges somewhat in those articles beginning around the end of 2010 and beginning of 2011. In 2015, I concluded, in effect though not explicitly, that metabolic scaling and DE are best viewed from a dimensional standpoint. Using the dimensional standpoint in 2015 resolved issues that resisted resolution using the scaling standpoint.The implicit issue is thus now raised explicitly, in this question:
Does the dimensional standpoint confer any advantages over the scaling standpoint?
Scaling formulas can be easily and quickly obtained with the help of dimensional analysis (standpoint). Example, we mention the power law spectrum of fluid turbulence, and so forth.
How many times must it be pointed out that there's no difference between the two?
The scaling allows the application of models developed in one scale to be applied to other scales. However, the assumption of scaling and similarity must apply. For example, noting waves in water allowed the application to the diffraction of Young's experiment to develop a wave model easily. Also, the similarity of the Stern-Garlach experiment allowed the quantum characteristic of "spin" to mathematically develop. Dimensional analysis of these experiments become very complex very fast.
My suggestions are as the following:
(a) Adimensionalize to get your analytic model in a simplified scale-independent form,
(b) It is, in principle, now more convenient to execute your calculation schemes, and
(b) Dimensionalize to recover your analytic model back into the scale-dependent form for the final phase of your analyses.
The dimensional standpoint (perhaps) provides a simple and direct avenue to accounting for 3/4 metabolic scaling, Brownian motion's 4/3 fractal envelope and dark energy (so called). The strategy is to adopt a dimensional standpoint instead of (for a paradigm example) the scaling standpoint used by Galileo in 1638 in his weight bearing bone example and also used by Sarrus and Rameaux in 1838 in their analysis of why larger animals breathe more slowly. In that context, here is a quote of Sir William Lawrence Bragg that appears at the front of a book by Jacob Schwichtenberg called No-Nonsense Classical Mechanics : "The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them."
Dear Robert Shour,
Thanks for your informative answer. I agree with you in full.
In addition, there exist many scaling laws in Fluid Dynamics derived by applying the method of dimensional analysis.
With regards,
PK Karmakar
Physics, Tezpur University, India
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