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Long ago some guy (Benford?) discovered that in log tables, the
degree
to which pages were "worn" was not at all equal. Pages for numbers
beginning with "1" were far more beat up than pages for numbers
staring with "7", "'8" or "9".
A fascinating article on this phenomenon, known now as "Benford's
Law"
can be found in R.A. Raimi, Scientific American, Vol. 221, 109-120,
December 1969. Everyone interested in nature should be familiar with
this topic, wherein it is abundantly clear that contrary to common
assumptions the distributions of leading digits in measures of
natural
phemomena are NOT uniform.
The "anomalous" distribution holds for almost all geometric
sequences,
surface areas of rivers, specific heats of chemicals, E1 transition
lines in atomic spectra, universal physical constants, population
sizes, surface areas of countries, the fibonnaci sequence, nuclear
decay half-lives, death rates, blackbody radiation, prime numbers,
and
on and on and ....
The distribution is scale invariant, AND BASE invariant too!
There is only one plausible explanation I know of, and it can be
expressed in several ways: (1) Nature counts logarithmically rather
than arithmetically, (2) it is a totally fractal world, or (3) power
laws (inimately related to fractal phenomena) are EVERYWHERE IN
NATURE.
A few mathematicians and physicists have explored this topic in depth
and there is a body of literature that explores the issue from all
perspectives. The papers in The American Mathematical Monthly (see
Hill, or Raimi) are readable and authoritative.
No one has quite known exactly what to do with this undeniable fact
of
nature, and while everyone acknowledges that it is an empirical fact,
it has not really been incorporated into conventional science. It has
been like Continental Drift: "if you cannot explain it, or figure out
a mechanism for it, it must be wrong or is a red herring, or
whatever,
so long as it is not taken seriously, and definitely ignored in a
concerted manner.
When the fractal revolution comes, people will eventually say: "Well,
of course, Benford's remarkable discovery back in 1938 showed that
nature was self-similar/fractal/scale invariant decades ago. We knew
all about that!" Sigh, right and just so.
If you want to see endless power laws in nature, I highly recommend
the 2 articles by Raimi mentioned above as a starting point. The
Scientific American article is the one to start with.
And now we know how to incorporate this crucial message from nature
into our science. If we could just rouse the theoretical physicists
from their drunken string theory and loop-de-doop reveries, we could
explain the message to them.
Yours in science,
Rob
www.amherst.edu/~rloldershaw |
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