The Golden Mean Number System: Its Platonic Roots, Paradigmatic Symmetry and Quantum Parameters


  • Scott Olsen Humanities Department, College of Central Florida
  • Mohamed S. El Naschie Department of Physics, Faculty of Science, University of Alexandria



golden section, One and Indefinite Dyad, the Greater and the Lesser, continuous geometric proportion, geometric mean, arithmetic mean, harmonic mean, pre-quantum particle, pre-quantum wave, Einstein spacetime, Unruh temperature, Hardy entanglement, Barbero-Immirzi parameter, Plato’s Sun, Divided Line and Cave


The golden mean number system may well be the most powerful tool in the physicist/mathematician’s arsenal. It emerges naturally out of Plato’s two principles, the One and the Indefinite Dyad of the Greater (Φ) and the Lesser (ϕ or 1/Φ). Herein we unpack its structural backbone in the golden series of exponential powers, …, ϕ7, ϕ6, ϕ5, ϕ4, ϕ3, ϕ2, ϕ, 1, Φ, Φ2, Φ3, Φ4, Φ5, Φ6, Φ7, … , along with its perfect combinatorial properties of addition and subtraction in growth and diminution, as well as, through multiplication and division via application of the modular Φ. We unravel the underlying paradigmatic symmetry of any given golden power serving simultaneously as geometric, arithmetic and harmonic means. And in the process, we reveal how the quantum parameters, including the pre-quantum particle, pre-quantum wave, Einstein spacetime, Unruh temperature, Hardy entanglement and the Barbero-Immirzi parameter, emerge naturally within Plato’s famous Republic similes of the One, Divided Line and Cave. The golden mean number system has now reemerged most completely and successfully in the E-Infinity theory.

Author Biographies

Scott Olsen, Humanities Department, College of Central Florida

Professor Emeritus of Philosophy and Religion
Humanities Department, College of Central Florida
Ocala, Florida

Mohamed S. El Naschie, Department of Physics, Faculty of Science, University of Alexandria

Distinguished Professor of Physics and Engineering
Department of Physics, Faculty of Science, University of Alexandria
Alexandria, Egypt


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How to Cite

Olsen, S., & El Naschie, M. S. (2022). The Golden Mean Number System: Its Platonic Roots, Paradigmatic Symmetry and Quantum Parameters . European Journal of Applied Sciences, 10(2), 319–327.