WHAT DOES BENFORD’S LAW TELL US?
Currently I am exploring Benford’s famous law of first digits. The law seems to indicate that nature counts and builds things in geometric sequences rather than arithmetic sequences. Apparently nearly all geometric series converge to Benford’s law, which is scale invariant and base invariant.
http://people.math.gatech.edu/~hill/publications/PAPER%20PDFS/TheFirstDigitPhenomenonAmericanScientist1996.pdf
Here is a partial list of the natural distributions that obey the Newcomb/Benford Law.
surface areas of rivers
molecular weights
atomic element/isotope masses
E1 atomic transition lines in plasmas
universal physical constants
populations of 3,000 countries
surface areas of countries
full widths (lifetimes) of mesons and baryons
fibonacci sequence
half-lives of radioactive nuclei
exoplanet masses, radii, volumes, orbital periods,…
physical characteristics of pulsars
distances to galaxies
distances to stars in our Galaxy
death rates
blackbody radiation
prime numbers
river lengths
sizes of stored computer files
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One could argue on the basis of the universality of Benford’s law that nature’s fundamental geometry highly favors power laws, which implies scale invariance, which is a hallmark of self-similarity, which is a hallmark of fractal geometry. So maybe the fact that Benford’s law is nearly universal constitutes empirical proof that nature’s most fundamental geometry is fractal geometry.
This seems reasonable, especially for those who are fractalogists, but does it hold up mathematically? How strong is the case that Benford’s law is closely linked to fractals?
Thinking some more about the significance of Benford’s Law, which was actually discovered by Newcomb in 1881 but ignored for 57 years, I think we can indeed make a deductive/empirical argument that we definitely live in a fractal world.
Many of those who have studied the Newcomb/Benford Law have been left with the feeling that the law holds some universal and very fundamental revelation about nature that has not been fully understood. For example, R. A. Raimi stated: “What remains tantalizing is the notion that there is still some unexplained measure in the universe ..”.
Here is a simple argument that offers the explanation that Newcomb, Benford, Hill, Raimi, and many others have sought.
If nature’s geometry is dominated by fractal geometry, i.e., if we live in a fractal cosmos, then we would expect to find our world dominated by the logarithmic distributions that follow the Newcomb/Benford Law. Since that is exactly the surprising discovery that Newcomb stumbled upon, and Benford verified empirically, and mathematicians like Hill and Raimi have refined, the conclusion seems straightforward and unavoidable. We do in fact live in a fractal universe.
Of course the next question would be: Why is it a fractal world? My answer to that next question would be that the natural extension of Einstein’s relativity, which demonstrated the relativity of space, time, orientation, and states of motion, would be the extension to relativity of scale, which would require that the world must be fractal.
It is exceedingly easier to list the things that do not obey the Newcomb/Benford Law than those that do. Examples of the former are square root tables, specific heat tables (restricted distribution), values of 1/n, digits of pi (I think). The law is not obeyed in cases of true randomness and narrowly restricted distributions.
Nature has been trying to tell us something fundamental about the cosmos. Are we ready to listen yet?
Robert L. Oldershaw
http://www3.amherst.edu/~rloldershaw
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