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European Journal of Applied Sciences – Vol. 10, No. 2
Publication Date: April 25, 2022
DOI:10.14738/aivp.102.12138. Olsen, S., & Naschie, M. S. E. (2022). The Golden Mean Number System: Its Platonic Roots, Paradigmatic Symmetry and Quantum
Parameters. European Journal of Applied Sciences, 10(2). 319-327.
Services for Science and Education – United Kingdom
The Golden Mean Number System: Its Platonic Roots,
Paradigmatic Symmetry and Quantum Parameters
Scott Olsen
Professor Emeritus of Philosophy and Religion
College of Central Florida, Ocala, Florida
M. S. El Naschie
Distinguished Professor of Physics and Engineering
Department of Physics, Faculty of Science
University of Alexandria, Alexandria, Egypt
ABSTRACT
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 Sun, Divided Line and Cave. The golden mean number system has now
reemerged most completely and successfully in the E-Infinity theory.
Keywords: 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.
INTRODUCTION
“The golden mean number system is the only number system which includes all integers, rational
and irrational numbers in addition to zero and countable as well as uncountable infinities in a
natural way and does not break down anywhere. It is the mathematics of the constructor theory
and the lingua franca of nature and consequently it is the foundation and unification of rational
existence.” [1,24]
The starting and the ending points of this paper are the mathematical and philosophical golden
mean number system of Plato, that great Pythagorean philosopher who was an initiate of the
ancient Egyptian (Saitic) and Greek (Eleusinian) mystery traditions. Under an oath of strict
silence (regarding the mysteries’ central revelation of continuous “golden” proportional
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symmetry) with the penalty of death, Plato brilliantly concealed (and yet revealed for the astute
student/reader) the profound truth in his Dialogues when viewed in conjunction with his
principles of the One and Indefinite Dyad. [2,3,4] The teachings were carried forward very
subtly through an oral tradition (consistent with Plato’s agrapha dogmata – unwritten lectures,
including his On the Good) only to later emerge quite forcefully in the famous eclectic Greco- Egyptian school, the Museum of Alexandria. There Egyptian, Persian, Indian, Chinese, Greek,
Roman and Arabic civilizations were melded together to form an international shrine of science,
art and wisdom. The school’s library was even known at one point to have in excess of 400,000
papyrus scrolls, the largest repository of recorded information, knowledge and wisdom in the
ancient world. Unfortunately, it suffered successive decimation at the hands of various political
and religious extremists, the scrolls eventually serving as fuel for the public baths. [5] Forced
underground once again, aspects of the doctrine reemerged later amongst various
Pythagoreans, Platonists, number theorists, the Tubingen platonic school in Germany, and
finally most fully and pragmatically in the 20th and 21st centuries’ through the brilliant
development of E-Infinity theory. [1-4,6-15,18,24-29]
And now, the present authors, along with their E-Infinity collaborators, [24-29] are focused on
achieving a wider recognition of the doctrine. One of us (Olsen) established quite clearly that
the golden mean number system is derived from Plato. [4,2,3] And the other (El Naschie) has
been using the resulting system to literally crack open the deepest paradoxes of modern-day
physics, certainly from at least 1994 when the seminal paper was published, Is Quantum Space
a Random Cantor Set with a Golden Mean Dimension at the Core? [6] [7-15]
PLATONIC BASIS FOR THE GOLDEN MEAN NUMBER SYSTEM, AND THE EMERGENCE OF
QUANTUM PARAMETERS IN PLATO’S THREE REPUBLIC SIMILES
“It is in this way, when they preserve the standard of the mean that all their works are good
and beautiful.... The Greater [Φ] and the Lesser [1/Φ or φ] are to be measured in relation, not
only to one another [Φ:φ or φ:Φ], but also to the establishment of the standard of the mean
[Φ:1:φ or φ:1:Φ].... [T]his other comprises that which measures them in relation to the
moderate, the fitting, the opportune, the needful, and all the other standards that are situated
in the mean between the extremes.” [Plato, Statesman 284a1-e8; bracketed material inserted
by author]
In an earlier work [2] it was established that Plato did indeed have (as espoused by Aristotle
[16]) two principles, the One (or Good) and the Indefinite Dyad (the Greater and Lesser, Excess
and Deficiency, or the More and the Less). And most significantly, that the Greater referred to
the major form of the golden ratio (i.e. Φ≈1.618033...) and the Lesser referred to the minor
form of the golden ratio (i.e. φ≈0.6180339...). And from these two principles, the One and the
Indefinite Dyad, Plato derived his ontology and epistemology. In addition, when the One is
placed in relation to the Greater and Lesser, it becomes apparent that the One is itself the
“golden” geometric mean between the Lesser and Greater φ:1:Φ = φ:1 :: 1:Φ. And this of course
comports with Plato’s assertion at Timaeus 31b-32a that “continuous geometric proportion” is
how separate, even disparate items are bonded or united together (e.g. the immaterial and
material, being and becoming, mind and body). And as is clearly demonstrated [4], Plato’s
ontology and epistemology of the Intelligible and Sensible realms, as reflected in his three
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Olsen, S., & Naschie, M. S. E. (2022). The Golden Mean Number System: Its Platonic Roots, Paradigmatic Symmetry and Quantum Parameters.
European Journal of Applied Sciences, 10(2). 319-327.
URL: http://dx.doi.org/10.14738/aivp.102.12138
Republic similes of the Sun (502d-509c), Divided Line (509d-511e) and Cave (514a-521b), are
rooted in continuous geometric proportion (see figures 1 and 2)
.
Figure 1. Plato’s Divided Line showing his ontology and epistemology [4]
Figure 2. Plato’s Republic similes in “golden” geometric proportion [4]
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In Figure 3 below, by beginning with the golden sectioning of the One, we show how the
quantum parameters naturally emerge as the fractal golden powers.
Figure 3. Republic similes with quantum parameters
THE THREE MEANS OF TWO GIVEN NUMBERS
Let us denote the three well-known means of any two numbers (a and c) using the chart
below:
AM is the arithmetic mean of a and c: AM = !"#
$ or 2AM = a+c.
GM is the geometric mean of a and c: GM = √�� or GM2=ac.
HM is the harmonic mean of a and c: HM = $!#
!"#
=
$%&!
$'& = %&!
'& .
There is a well-known relation [17] between the means that can be found by substitution:
HM = (GM)2/ AM .
2 AM HM (GM) Ä =
Key
1 – the One = the Good
– the golden section
P – pre-quantum particle (φ1
)
W – pre-quantum wave (φ2
)
E – Einstein spacetime (φ3)
U – Unruh temperature (φ4)
H – Hardy entanglement (φ5)
B – Barbero-Immirzi parameter (φ6)
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Olsen, S., & Naschie, M. S. E. (2022). The Golden Mean Number System: Its Platonic Roots, Paradigmatic Symmetry and Quantum Parameters.
European Journal of Applied Sciences, 10(2). 319-327.
URL: http://dx.doi.org/10.14738/aivp.102.12138
Using this substitution relation above, one of us [El Naschie, 18] derived a “golden” harmonic
mean via the Barbero-Immirzi parameter [24-29] that connects the fundamental work of David
Gross and Edward Witten to that of Lee Smolin [19] and Carlo Rovelli [20,9]. By establishing
that D=10 superstrings and loop quantum gravity are simply two sides of a “harmonic” golden
coin, he was able show that their respective pioneers are talking about the same idea using
different mental pictures.
UNCOVERING THE PARADIGMATIC SYMMETRY OF ANY GIVEN GOLDEN POWER, ACTING
SIMULTANEOUSLY AS A GEOMETRIC, ARITHMETIC AND HARMONIC MEAN
“...the more deeply we probe Nature’s secrets the more profoundly we are driven into Plato’s
world of mathematical ideals as we seek our understanding.” – Sir Roger Penrose [30]
It was during the writing of our new book, A Grand
Unification of Science, Arts and Consciousness:
Rediscovering the Pythagorean Plato’s Golden Mean
Number System, [4] that one of us (Olsen) uncovered the
underlying paradigmatic symmetry arising out of Plato’s
One and Indefinite Dyad. What led to this discovery was
that several years earlier, Lance Harding had been
studying the geometric relations involved in Le Corbusier’s
The Modulor [21] and found a variety of significant
geometric (GM), arithmetic (AM) and harmonic (HM) mean
relationships surrounding the One and Indefinite Dyad and
their exponential powers in the golden series. His findings
[22] included, amongst others, the following highly
meaningful relationship: see figure 4, Harding diagram.
Then more recently, Adam Tetlow,
unaware of Harding’s discovery,
pointed out his own finding regarding
the three means along a portion of the
golden series of exponential powers:
see
figure 5, Tetlow diagram.
Reflecting on the observations of both Harding and Tetlow,
and perceiving the significance of the fact that these mean
relationships maintain their internal structure and can
slide back and forth along the golden series, Olsen realized
that the One is simultaneously the “golden” geometric,
arithmetic and harmonic means of the Indefinite Dyad
when the Greater squared and Lesser squared are included in the golden powers under
consideration: see figure 6, Olsen diagram of paradigmatic symmetry.
Figure 4. Harding diagram
Figure 5. Tetlow diagram
Figure 6. Olsen diagram of
paradigmatic symmetry
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Following Plato’s assertion regarding continuous geometric proportion at Timaeus 31b-32a,
the most important mean here is the fact that One is the geometric mean between (and
therefore, bonding together) the Greater and the Lesser through continuous geometric
proportion. But this also “means” that the One actually plays the role of the Golden Mean, i.e.
One is the Golden Mean between the Greater and the Lesser. We maintain that the Golden Mean
is the geometric mean par excellence of all geometric means. Plato understood very well that It
is the archetypal Form or Standard for all geometric means. Furthermore, notice the interesting
bilateral symmetry of the geometric mean combined with “mirrored” asymmetry of the
arithmetic and harmonic means that arises in the following sequence of means associated with
the paradigmatic symmetry, see figure 7.
Figure 7: Unpacked paradigmatic symmetry displaying mirrored asymmetries of AM and HM
This is of course in addition to the fact that the golden section which gives rise to the golden
ratios is itself an asymmetric cut, thus in another sense marrying together symmetry and
asymmetry.
Also notice below (in figure 8) that the paradigmatic symmetry can be centered on any golden
power, as done here with the Greater (Φ), Hardy entanglement (φ5), and the Barbero-Immirzi
parameter (φ6), respectively.
Figure 8. All three means centered on Φ, φ5 or φ6, respectively, each example displaying an
instantiation of paradigmatic symmetry
Geometric mean (in black):
if GM = term n, then it is
between terms n-1 and n+1
Arithmetic mean (in green):
if AM = term n, then it is
between terms n-2 and n+1
Harmonic mean (in red):
if HM = term n, then it is
between terms n-1 and n+2
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Olsen, S., & Naschie, M. S. E. (2022). The Golden Mean Number System: Its Platonic Roots, Paradigmatic Symmetry and Quantum Parameters.
European Journal of Applied Sciences, 10(2). 319-327.
URL: http://dx.doi.org/10.14738/aivp.102.12138
Finally, let us note here that MIT physicists, led by Pablo Jarillo-Herroro discovered that bi- layered graphene twisted by “about 1.1 degrees” allows electrons to become easily entangled
and superconductive [31,32,33]. We (the authors) predict they will eventually discover this
angle is precisely φ12 x 360o = 1.118023205...degrees. [4] This should give rise to the discovery
of further quantum parameters as illustrated below in our Olsen-El Naschie entanglement (φ12)
centered paradigmatic symmetry.
Figure 9. Olsen-El Naschie entanglement (φ12) centered paradigmatic symmetry.
CONCLUSION
“The essential features of [quantum interconnectedness] are that the whole universe is in some
way enfolded in everything, and that each thing is enfolded in the whole.” - David Bohm [23]
At the root of all the deep questions within physics lies the number system one employs in the
process. Together with our E-Infinity theory collaborators, we have found that the golden mean
number system is indeed the lingua franca of nature. Its backbone emerges naturally as the
golden series of exponential powers out of Plato’s principles of the One and Indefinite Dyad of
the Greater (Φ) and Lesser (φ). And it all begins with the golden section, or as the ancient
Egyptians called it, the primordial scission. It is highlighted by its naturally recursive nature,
similar to but more profound, than its derivative Fibonacci series of numbers. The latter being
perfectly additive and approximately geometric, whereas the former is both perfectly additive
and perfectly geometric. Furthermore, it harbors the most stunning internal structure of
paradigmatic symmetry linking all aspects of the golden powers together in an incredible
symphony of interdependence. And finally, we have shown how the quantum mechanical
parameters of the pre-quantum particle (φ), pre-quantum wave (φ2), Einstein spacetime (φ3),
Unruh temperature (φ4), Hardy entanglement (φ5) and Barbero-Immirzi parameter(φ6) can be
naturally and effectively aligned within Plato’s three similes of the Sun, Divided Line and Cave.
This is the process of how the whole universe is fractally and holographically enfolded into each
and every part. The golden mean number system is in fact the lingua franca of nature and holds
the key to unpacking the fractal nature of the universe – penetrating into its outer fabric and
inner mysteries. It has reemerged most completely and satisfactorily in the modern era of high
energy physics and cosmology through the stunning simplicity of the many computational
successes of E-Infinity theory. In the end, what we have on display is literally the answer to the
greatest philosophical question of all: “How does the One become the Many?”
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References
[1] Mohamed S. El Naschie, Complementing the Deep Ideas of Deutsch-Marletto Constructor Theory with the
Computational Power of the Pythagorean-Plato Number System. European Journal of Applied Sciences, 10(1)
(2022). https://doi.org/10.14738/aivp.101.11792
[2] Scott Olsen, The Indefinite Dyad and the Golden Section: Uncovering Plato’s Second Principle, Nexus Network
Journal: Architecture and Mathematics, 2002a, 4, No. 1 (2002) p. 97-110.
[3] Scott Olsen, Golden proportional symmetry and the divided line – solving the platonic puzzles in one fell
swoop, Symmetry: Culture and Science, Volume 32, Number 2, (2021) pp. 161-164.
https://doi.org/10.26830/symmetry_2021_2_161
[4] Scott Olsen, Leila Marek-Crnjac, Ji-Huan He and Mohamed S. El Naschie. A Grand Unification of Science, Arts
and Consciousness: Rediscovering the Pythagorean Plato’s Golden Mean Number System, Kindle Press (2021) 184
pages.
[5] David Fideler, Introduction: Cosmopolis, or the New Alexandria, Alexandria: The Journal of the Western
Cosmological Traditions, Vol. 2, ed. David Fideler, Phanes Press (1993).
[6] Mohamed S. El Naschie, Is Quantum Space a Random Cantor Set with a Golden Mean Dimension at the Core?
Chaos, Solitons & Fractals, 4(2) (1994) pp. 177-179.
[7] Mohamed S. El Naschie, L. Da Vinci’s Turbulence, Multi-dimensional golden spinors, Prandtl’s Boundary layer
and quantum mechanical spin are all implied by the golden mean number system. European Journal of Applied
Science. Vol. 9, No 6 (2021) pp. 134-136.
[8] Mohamed S. El Naschie, A review of E-Infinity theory and the mass spectrum of high energy particle physics,
Chaos, Solitons & Fractals, Vol. 19, Issue 1 (2004) pp. 209-236.
[9] Mohamed S. El Naschie, Proving superstring theory using loop quantum mechanics (gravity), Chaos, Solitons
& Fractals, Volume 26 (2005) pp. 43-45.
[10] Mohamed S. El Naschie, Elements of a New Set Theory Based Quantum Mechanics with Applications in High
Energy Quantum Physics and Cosmology, International Journal of High Energy Physics, Volume 4, Issue 6 (2019)
pp. 65-74.
[11] Mohamed S. El Naschie and Leila Marek-Crnjac. Set Theoretical Foundation of Quantum Mechanics, NASA’s
EM Drive Technology and Minimal Surface Interpretation of the State Vector Reduction of the Quantum Wave
Collapse, Chaos and Complexity Letters, Volume 12, No. 2 (2018) pp. 85-100.
[12] Mohamed S. El Naschie, From Nikolay Umov via Albert Einstein’s
to the Dark Energy Density of the Cosmos , World Journal of Mechanics, Volume 8, No. 4
(2018) pp. 73-81.
[13] Mohamed S. El Naschie, Symmetria Massima of the Fractal M-Theory Via the Golden Mean Number System -
A New Language for A Deep Dialogue between Man and Nature, International Journal of Artificial Intelligence and
Mechatronics, Volume 7, Issue 3 (2018) ISSN 2329-5121. pp. 11-14.
[14] Mohamed S. El Naschie, The quantum gravity Immirzi parameter - A general physical and topological
interpretation. Gravitation and Cosmology, Vol. 19, Issue 3 (2013) pp 151-155. Note: it is more accurate to refer to
this parameter as Barbero-Immirzi parameter.
[15] Mohamed S. El Naschie, On a Fractal Version of Witten’s M-Theory. International Journal of Astronomy and
Astrophysics, Vol. 6, No. 2 (2016) pp 135-144.
[16] H.J. Kramer, Plato and the Foundations of Metaphysics, ed. and trans., J.R. Caton, New York: State University
of New York Press (1990).
[17] Ilya Nickolaevich Bronstein and Konstantin Adolfovic Semendjaew, Handbuch der Mathematik (pocket book
of mathematics), Harri Deutsch Publishing, Zurich und Frankfurt (1965) (in German).
[18] Mohamed S. El Naschie, Superstrings and Loop Quantum Gravity are Two Sides of the Same Golden Coin,
European Journal of Applied Sciences, 10(1) (2022) 387–392.
2 E = kmc 2 E = γmc 2 E mc = (21/ 22)
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Olsen, S., & Naschie, M. S. E. (2022). The Golden Mean Number System: Its Platonic Roots, Paradigmatic Symmetry and Quantum Parameters.
European Journal of Applied Sciences, 10(2). 319-327.
URL: http://dx.doi.org/10.14738/aivp.102.12138
https://doi.org/10.14738/aivp.101.11579.
[19] Lee Smolin. Three Roads to Quantum Gravity. Weidenfeld and Nicholson, London (2009).
[20] David Gross and Carlo Rovelli, String Theory or Loop Gravity?
https://YouTube.com/watch?V=AUYYIRSRPZW (2021) (See also Mohamed El Naschie facebook.
https=//www.Facebook.com/profile.php?id=100007609216640).
[21] Le Corbusier, The Modulor: A Harmonious Measure to the Human Scale Universally Applicable to Architecture
and Mechanics, London, Faber (1961).
[22] Scott Olsen, Divine Proportion: The Mathematical Perfection of the Universe, unpublished MS.
[23] David Bohm and Basil Hiley, The Undivided Universe: An Ontological Interpretation of Quantum Theory,
Routledge (1968).
[24] Mohamed S. El Naschie, High Energy Physics and Cosmology as Computation, American Journal of
Computational Mathematics, 6 (2016) pp. 185-189.
[25] Garnet Ord, Transfinite physics - Treading the path of Cantor and Einstein, Chaos, Solitons & Fractals Volume
25, Issue 4 (2005).
[26] Alexey Stakhov, assisted by Scott Olsen, The Mathematics of Harmony: From Euclid to Contemporary
Mathematics and Computer Science, World Scientific, Singapore (2009).
[27] Leila Marek-Crnjac, Cantorian Space-Time Theory: The Physics of Empty Sets in Connection with Quantum
Entanglement and Dark Energy, Lambert Academic Printing (2013).
[28] Peter Weibel, Garnet Ord and Otto E. Rossler, eds., Spacetime Physics and Fractality: Festschrift in honor of
Mohamed El Naschie on the occasion of his 60th birthday, Springer Verlag, Vienna and New York (2005).
[29] Mohamed S. El Naschie and Scott Olsen, When zero is equal to one: A set theoretical solution of quantum
paradoxes, Fractal Spacetime and Noncommutative Geometry in Quantum and High Energy Physics, 1(1) (2011)
pp. 11-24.
[30] Sir Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Jonathan Cape, Random
House Group Ltd. (2004) p. 1029.
[31] Catherine Zandonella, “Experiments explore the mysteries of ‘magic’ angle superconductors,” ScienceDaily,
Princeton University (2019). www.sciencedaily.com/releases/2019/07/190731131122.htm.
[32] Mohamed S. El Naschie and Tomasz Kapitaniak, “Soliton chaos models for mechanical and biological elastic
chains,” Physics Letters A, Vol. 147, no. 5,6 (1990).
http://kapitaniak.kdm.p.lodz.pl/papers/1990/125_PhysicsLettersA1990-147-5-6.pdf.
[33] Scott Olsen, The Golden Section: Nature’s Greatest Secret, Wooden Books, Glastonbury (1986).