# Approximate formulas linking the fine structure constant and the first
Feigenbaum constant

##
The - somehow mysterious - fine structure constant α (see for example Wikipedia),
usually referred by its reciprocal 1/α ~ 137.035999679(94),
is thought to be related with I(0,x), the modified Bessel function of the first kind of order
0,
as you may easily see by a quick overlook on the net; see for example formula (1) at
https://accelconf.web.cern.ch/accelconf/LINAC08/papers/tup080.pdf

I have found (using the Inverse Calculator on the web) an approximate formula linking them even
with the - somehow mysterious - (first) Feigenbaum's constant
δ=4.6692016091... (see for example Wolfram MathWorld or Wikipedia)
by this relation

## δ ~ (1/α )^1/(I(0,1)sin(1) + Γ (5/12))[=4.669201650...]

##
which is exact up a relative error less than 1E-8 (or, 1 over 100 millions); and it's inverse, more interesting,

## 1/α ~ δ^(I(0,1)sin(1) + Γ (5/12))
[=137.0359957...]

##

which is exact up a relative error less than 3E-8.

That seems good enough, for a physical "constant" suspected of slight variations in time.

Bessel functions are notoriously related with goniometric functions and function Γ.

(Pay attention that here above α denotes - as used - the fine structure constant, not
the second Feigenbaum constant 2.502907875... usually denoted by the same
Greek letter.)

Alessandro Soranzo (2010)