After a brief review of the golden ratio in history and our previous exposition of the fine-structure constant and equations with the exponential function, the fine-structure constant is studied …
The possible variation of the fine-structure constant, a, has inspired many people to work on modifications and/or generalizations of the current "standard" theories in which the electromagnetic field is involved. Here we first point out the amazing similarity between Bekenstein's model, describing the variation of α by a varying charge, and the Hojman-Rosenbaum-Ryan-Shepley torsion potential model. This observation invites us to consider a geometric theory of gravity in which a varying α originates from another kind of dynamic quantity of spacetime, i.e., vector torsion. Since the vector torsion field is weak and also not strongly coupled with fermions it is difficult to detect it directly. The detection of a time-varying α could thus provide some promising evidence for the existence of torsion.
The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The Kepler triangle, with its golden ratio proportions, is also a Pythagorean triangle. Combining classical harmonic proportions derived from Kepler’s triangle with quartic equations determine an approximate value for the fine-structure constant that is the same as that found in our previous work with the golden ratio geometry of the hydrogen atom. These results make further progress toward an understanding of the golden ratio as the basis for the fine-structure constant.