Golden Ratio Geometry and the Fine-Structure Constant

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Abstract:

The golden ratio is found to be related to the fine-structure constant, which determines the strength of the electromagnetic interaction. The golden ratio and classical harmonic proportions with quartic equations give an approximate value for the inverse fine-structure constant the same as that discovered previously in the geometry of the hydrogen atom. With the former golden ratio results, relationships are also shown between the four fundamental forces of nature: electromagnetism, the weak force, the strong force, and the force of gravitation.

Sherbon, M.A. (2019). Golden Ratio Geometry and the Fine-Structure Constant. Journal of Advances in Physics, 16(1), 362 -368. https://doi.org/10.24297/jap.v16i1.8469

 

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Fine-structure constant from Sommerfeld to Feynman

Abstract: 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.Sherbon, M. (2019). Fine-structure constant from Sommerfeld to Feynman. Journal of Advances in Physics, 16(1), 335-343. https://doi.org/10.24297/jap.v16i1.8402

https://cirworld.com/index.php/jap/article/view/8402?fbclid=IwAR0r_WkjVFccAa-UeUOboR_kwL1hfao0YL6ubZaPF6awH3n0HH-iqDNA6HM

Physical Mathematics and the Fine-Structure Constant

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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.

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Inverse fine-structure constant is a root of: x⁴ – 136x³ – 136x² – 818x + 1 = 0. This gives a value for x = 137.035 999 168 ….

From the CODATA UPDATE (2016) Inverse fine-structure constant = 137.035 999 160 (33). The other root of the equation is approximately 1/818 and 818 = (136+1/3)6 = (4×136)+(2×137).

Sherbon, M. “Physical Mathematics and the Fine-Structure Constant.” Journal of Advances in Physics. 14, 3 (Oct. 2018), 5758-5764.  DOI: https://doi.org/10.24297/jap.v14i3.7760.

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Fine-Structure Constant from Golden Ratio Geometry

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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 in the context of other research calculating the fine-structure constant from the golden ratio geometry of the hydrogen atom. This research is extended and the fine-structure constant is then calculated in powers of the golden ratio to an accuracy consistent with the most recent publications. The mathematical constants associated with the golden ratio are also involved in both the calculation of the fine-structure constant and the proton-electron mass ratio. These constants are included in symbolic geometry of historical relevance in the science of the ancients.

α is the fine-structure constant, φ is the Golden Ratio, A is the Golden Apex of the Great Pyramid and K is the polygon circumscribing constant. 2016 CODATA: 137.035 999 160 (33).

Sherbon, M.A. “Fine-Structure Constant from Golden Ratio Geometry,” International Journal of Mathematics and Physical Sciences Research, 5, 2, 89-100 (2018). RG:322797654

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