Fine structure constant and numerology
Andrew Jenner
(http://www.mathsoft.com/asolve/constant/fgnbaum/salamin.html) which
hypothesizes that the fine structure constant might be related to one
(or more) of the Feignbaum constants
(http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.html). Idly I
tried a few permutations on a pocket calculator to see if I could come
up with a formula relating them, but failed.
Then the other day I came across Robert Munafo's program RIES
(http://home.earthlink.net/~mrob/pub/ries/index.html), which attempts
to find just this sort of formula automatically (by brute force). I
modified it to take into account the Feignbaum alpha and delta
constants:
alpha = -2.50290787509
delta = 4.66920160910
(more digits can be found at
http://www.mathsoft.com/asolve/constant/fgnbaum/brdhrst.html)
and instructed it to search for the NIST value of the inverse
fine-structure constant (IFSC), 137.03599976(50)
http://www.geocities.com/Area51/Nebula/3735/fine.html).
It came up with the following amusing little formula:
2 2 pi
alpha pi (pi -1)
------------------
16
Or, for those of you viewing in a proportional font:
(alpha^2)(pi^2)(pi^pi-1)/16.
This gives a value for the IFSC of 137.03599968, which is well within
the range of error of the NIST value. Of course, I don't have any
justification of physical grounds as to why this should be the correct
value and expect that it is just a coincidence. However, I thought I
should post it here just in case the grand unified theory of physics
turns out to predict that the IFSC should be exactly this!
RIES came up with a few other formulae, but that was the prettiest one
which was within the margin of error.
I don't know if there is a more recent (and therefore more accurate)
experimental value for the IFSC, but if there is it would be
interesting to compare it to this and see if it still holds.
Andrew
--
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Uncle Al
>
> I recently read an article by Gene Salamin
> (http://www.mathsoft.com/asolve/constant/fgnbaum/salamin.html) which
> hypothesizes that the fine structure constant might be related to one
> (or more) of the Feignbaum constants
> (http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.html). Idly I
> tried a few permutations on a pocket calculator to see if I could come
> up with a formula relating them, but failed.
Given a starting number and a a target, there are an infinite number
of expressions that relate the two - especially if you only have to
match to a fairly small number of significant digits. Consider the
number of published generating functions for pi.
[snip]
> RIES came up with a few other formulae, but that was the prettiest one
> which was within the margin of error.
What does that mean? It means the relationship between the two is
arbitrary.
You can model a sine wave with a polynomial to arbitrary interpolative
precision. Economics, weather forecasting, and doomsayers do that
sort of thing all the time. Extrapolation outside the modeled
interval is meaningless, which accurately characterizes economics,
weather forecasting, and doomsayers.
--
Uncle Al
http://www.mazepath.com/uncleal/
http://www.ultra.net.au/~wisby/uncleal/
(Toxic URLs! Unsafe for children and most mammals)
"Quis custodiet ipsos custodes?" The Net!
Andrew Jenner
wrote:
>Given a starting number and a a target, there are an infinite number
>of expressions that relate the two - especially if you only have to
>match to a fairly small number of significant digits.
Quite, which is why I don't claim that my discovering is anything more
than silly numerology. I just thought that some people here might find
it interesting.
>> RIES came up with a few other formulae, but that was the prettiest one
>> which was within the margin of error.
>
>What does that mean?
One can define a complexity metric on expressions such as these, which
is exactly what RIES does. "Simpler" expressions (those containing
fewer symbols, to a first approximation) are considered to be prettier
than more complicated ones. The purpose of RIES is to find the
simplest one which fits the experimentally observed data. Of course,
"simplest" is a somewhat subjective concept, and one could probably
obtain a different expression by changing the weightings RIES uses to
calculate the complexity of an expression.
>It means the relationship between the two is arbitrary.
Yes, it is arbitrary - I never claimed otherwise. However, the fact
that it is such a simple expression makes it a bit less arbitrary than
a more complicated one with a similar value would be.
>You can model a sine wave with a polynomial to arbitrary interpolative
>precision. Economics, weather forecasting, and doomsayers do that
>sort of thing all the time. Extrapolation outside the modeled
>interval is meaningless, which accurately characterizes economics,
>weather forecasting, and doomsayers.
I fail to see what anything in that paragraph has to do with my post.