"The various formulae look like they were inspired by the hallucinations of a mathematical poet. Among the simplest is a peculiar polynomial in π: 4π^{3} + π^{2} + π. Another takes the algebraic form (9/16π^{4})(π^{5}/5!)^{1/4}, and a third involves the 10th and 33rd prime numbers, 29 and 137, and takes the shape (29/π)cos(π/137)tan(π/29·137). A further formula in the collection is the strikingly simple (π^{2} + 137^{2})^{1/2}.

What draws these formulae together is that they are all remarkably close approximations to a number that physicists know very well: the so-called fine structure constant, *α* = *e*^{2}/*?c*, which plays a fundamental role in the quantum theory of electrodynamics. Arnold Sommerfeld first introduced *α* into physics in 1916 when proposing relativistic corrections to Bohr's model of the atom — corrections that gave 'fine structure' to its energy levels and to atomic spectra, hence the name. Later, during the development of quantum field theory, *α* emerged as a far more profound dimensionless number linking quantum theory with electrodynamics and relativity."

Nature Physics http://www.nature.com/nphys/journal/v6/n11/full/nphys1839.html