The Alpha Constant of Universe: a 20th century mystery

Famous Physicist Richard Feynman said about the alpha constant:

There is a most profound and beautiful question associated with the observed coupling constant, e the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to -0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!

electromagnetic force: one of the four forces of nature. Electromagnetic interactions hold electrons in atoms, hold atoms in molecules, and are used in all electronic devices.

fine structure constant: usually denoted as α, the lower case Greek alpha, is the dimensionless ratio e2/(hbar*c) = 1/137.03599976... [in cgs units, or e2/(4*pi*epsilono*hbar*c) in MKS units], which gives the strength of the electromagnetic interaction. Here e is the electron charge, hbar is Planck's constant divided by 2*pi, and c is the speed of light.