Physical Mathematics and The Fine-Structure Constant

Abstract

Research into ancient physical structures, some having been known as the seven wonders of the ancient world, inspired new developments in the early history of mathematics. At the other end of this spectrum of inquiry the research is concerned with the minimum of observations from physical data as exemplified by Eddington's Principle. Current discussions of the interplay between physics and mathematics revive some of this early history of mathematics and offer insight into the fine-structure constant. Arthur Eddington's work leads to a new calculation of the inverse fine-structure constant giving the same approximate value as ancient geometry combined with the golden ratio structure of the hydrogen atom. The hyperbolic function suggested by Alfred Landé leads to another result, involving the Laplace limit of Kepler's equation, with the same approximate value and related to the aforementioned results. The accuracy of these results are consistent with the standard reference. Relationships between the four fundamental coupling constants are also found.

Fine-structure constant from Sommerfeld to Feynman

“…His work with advanced algebras was suggestive toward the following quartic equation from our previous work. [17]. This calculation of the inverse fine-structure constant gives the same approximate value as ancient geometry combined with the extension of Raji Heyrovska's work on the golden ratio structure of the hydrogen atom [18].…”

“…Another approximation from our previous work [17] gives the same approximate value for the inverse finestructure constant. − ≃…”

“…≃ Royal Cubit and 440 Royal Cubits equal 432 long cubits of 21 inches. ≃ ( ), see the reference to Alfred Landé in our previous work [17].…”

“…Along with = − − + , = − + ( × ) and = + + . Also, ≃ × , reference our previous extension of Eddington's work [17].…”

Golden Ratio Geometry and the Fine-Structure Constant

“…The inverse fine-structure constant calculated with quartic polynomials and the main harmonic parameters of the Great Pyramid have suggested the golden ratio as the mathematical basis for the fine-structure constant. From previous work "... several more formulations for the finestructure constant with the same approximate value have connections with prime number theory, the real fixed point of the hyperbolic cotangent, anomalous magnetic moment of the electron, Laplace limit of Kepler's equation and harmonic proportions of the Cosmological Circle" [3,30,37].…”