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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From the exponential function of Euler’s equation to the geometry of a fundamental form, a calculation of the fine-structure constant and its relationship to the proton-electron mass ratio is given. Equations are found for the fundamental constants of the four forces of nature: electromagnetism, the weak force, the strong force and the force of gravitation. Symmetry principles are then associated with traditional physical measures.

International Journal of Physical Research, 5, 2, 46-48 (2017) hal-01312695

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Global Journal of Science Frontier Research: A Physics and Space Science, 15, 4, 23-26 (2015). hal-01174786v1ORCID:0000-0001-9720-8008

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What a surprise! This work might be a milestone achievement in the history of science. Edward G. Nightingale may not fully realize the breakthrough discovery that he has made. While it seems as though it aims to support the theories of Robert Bauval et al, it has a much larger application. He begins with Plato’s Lambda and calculates several different types of cubits. And this shows that the Lambda did not originate with Plato or the Pythagoreans and was probably part of the initiate knowledge they brought back with them to Greece from their study in Egypt.

Jocelyn Godwin, in *Harmonies of Heaven and Earth*, states “Albert von Thimus (1806-1878), a polymathic researcher into ancient harmonic theory, …” called Plato’s Lambda (from Iamblichus) the Lambdoma. “Thimus developed the Lambdoma into a square diagram, which he called the ‘Pythagorean Table’ … Both von Thimus and his spiritual heir, Hans Kayser, believed that this was the fundamental diagram of the lost ancient science of Harmonics, hinted at by Plato … as the culmination of all learning, but never revealed publicly. Others say that von Thimus was mistakenly projecting on the the ancient Pythagoreans his own discovery of a scheme typical of early nineteenth-century mathematical theory.” (Rene Guenon, *The Symbolism of the Cross*)

From the Pythagorean Table or Lambdoma described by Thomas Hightower in *The Musical Octave* … “Mathematicians and scientists have studied the Lambdoma since its discovery. It is said to hold the many esoteric secrets of the relationship between matter and spirit, including being a numerical representation of the World Soul…. In the 1920s Hans Kayser, a German scientist, developed a theory of world harmonics based upon the Lambdoma. He found that the principles of harmonious structure in nature and the fundamentals of harmonics were essentially the same….

Kayser believed that this knowledge of harmonics had become lost and had created a major schism between science and the spirit. He hoped that a true understanding of this relationship would create a bridge between the matter and soul…. According to Kayser, the whole number ratios of musical harmonics corresponds to an underlying framework existing in chemistry, physics, crystallography, astronomy, architecture, spectroanalysis, botany and the study of other natural sciences. The relationship expressed in the periodic table of elements, an understanding of the formation of matter, resembles the overtone structure in music….”

What Edward has done is to demonstrate von Thimus to be correct, and has opened the way to further discovery on the Great Pyramid itself. Laird Scranton reminds us of the Hermetic axiom, “As above, so below. As within, so without.” The Great Pyramid is as Manly P. Hall stated a scale model of both the microcosm and the macrocosm. The precession of the equinoxes for earth is an analog for the precession of the electron in hydrogen. Herodotus reported that the Great Pyramid was a “wonder of physics.”

Bruce Cathie rediscovered some of this harmonic theory from the ancient world grid artifacts. William Conner, in *Harmonic Mathematics*, developed a revised Pythagorean Table and showed some of the basic dimensions of the Great Pyramid encoded these harmonics. Robert K.G. Temple, in *Egyptian Dawn*, has also made a significant contribution to the plan on the Giza Plateau, describing the importance of the Pythagorean Comma and the Golden Angle of Resurrection. Goodreads

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“All integral laws of spectral lines and of atomic theory spring originally from the quantum theory. It is the mysterious organon on which Nature plays her music of the spectra, and according to the rhythm of which she regulates the structure of the atoms and nuclei.”

― Arnold Sommerfeld, * Atombau Und Spektrallinien *

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Wolfgang Pauli was influenced by Carl Jung and the Platonism of Arnold Sommerfeld, who introduced the fine-structure constant. Pauli’s vision of a World Clock is related to the symbolic form of the Emerald Tablet of Hermes and Plato’s geometric allegory otherwise known as the Cosmological Circle attributed to ancient tradition. With this vision Pauli revealed geometric clues to the mystery of the fine-structure constant that determines the strength of the electromagnetic interaction. A Platonic interpretation of the World Clock and the Cosmological Circle provides an explanation that includes the geometric structure of the pineal gland described by the golden ratio. In his experience of archetypal images Pauli encounters the synchronicity of events that contribute to his quest for physical symmetry relevant to the development of quantum electrodynamics.

Journal of Science, Vol. 2, No. 3, pp.148-154 (2012). SSRN: abstract=2147980 . ~ .

]]>Abstract: The history of science is polarized by debates over Plato and Aristotle’s holism versus the atomism of Democritus and others. This includes the complementarity of continuous and discrete, one and the many, waves and particles, and analog or digital views of reality. The three-fold method of the Pythagorean paradigm of unity, duality, and harmony enables the calculation of fundamental physical constants required by the forces of nature in the formation of matter; thereby demonstrating Plato’s archetypal viewpoint. papers.ssrn.com/abstract=1766049 *SSRN Classics: Journal of Philosophical & Scientific Texts*

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Abstract: Plato’s theory of everything is an introduction to a Pythagorean natural philosophy that includes Egyptian sources. The Pythagorean Table and Pythagorean harmonics from the ancient geometry of the Cosmological Circle are related to symbolic associations of basic mathematical constants with the five elements of Plato’s allegorical cosmology: Archimedes constant, Euler’s number, the polygon circumscribing limit, the golden ratio, and Aristotle’s quintessence. Quintessence is representative of the whole, or the one in four, extraneously considered a separate element or fifth force. This relationship with four fundamental interactions or forces also involves the correlation of constants with the five Platonic solids: tetrahedron, hexahedron, octahedron, icosahedron, and dodecahedron. The values of several fundamental physical constants are also calculated, and a basic equation is given for a unified physical theory in the geometric universe of Plato’s natural philosophy.

*SSRN Classics: Journal of Philosophical & Scientific Texts (July 21, 2010) SSRN: 1646568
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