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Exact formula for the fine-structure constant α in terms of the golden ratio φ

Stergios Pellis

sterpellis@gmail.com

ORCID iD: 0000-0002-7363-8254

Greece

13 October 2021

Abstract

The purpose of many sciences is to find the most accurate mathematical formula that can be found in the

experimental value of fine-structure constant. Attempts to find a mathematical basis for this dimensionless constant

have continued up to the present time. However,no numerological explanation has ever been accepted by the physics

community. In this paper we will present the exact expression for the fine-structure constant. A exact expression for

the fine-structure constant in terms of the golden angle,the relativity factor and the fifth power of the golden mean:

α-1=360·φ-2-2·φ-3+(3∙φ)-5

Finally we will present the continued fractions for the fine-structure constant.

Keywords

Fine-structure constant , Dimensionless physical constants ,Golden ratio , Golden angle , Relativity factor , Fifth

power of the golden mean

1. Introduction

Golden ratio φ is an omnipresent number in nature,found in the architecture of living creatures as well as human

buildings,music,finance,medicine,philosophy,and of course in physics and mathematics including quantum

computation. It is the most irrational number known and a number-theoretical chameleon with a self-similarity

property. The golden ratio can be found in nearly all domains of Science,appearing when self-organization processes

are at play and expressing minimum energy configurations. Several non-exhaustive examples are given in biology

(natural and artificial phyllotaxis, genetic code and DNA) physics (hydrogen bonds,chaos, superconductivity),

astrophysics (pulsating stars,black holes), chemistry (quasicrystals,protein AB models),and technology (tribology,

resistors, quantum computing, quantum phase transitions, photonics). The fifth power of the golden mean appears in

Phase transition of the hard hexagon lattice gas model, Phase transition of the hard square lattice gas

model,One-dimensional hard-boson model, Baryonic matter relation according to the E-infinity theory,Maximum

quantum probability of two particles,Maximum of matter energy density,Reciprocity relation between matter and

dark matter,Superconductivity phase transition, etc. Among the numbers in the Fibonacci range,the numbers 5and 13

seem to be the most important. Whereas number 5 is involved in the definition of the golden mean,number 13 is

found as a helix repetition number for instance in tubulin protein,thought to be the location from where our thinking

and consciousness originates.

The fine-structure constant is one of the most fundamental constants of physics. It describes the strength of the force

of electromagnetism between elementary particles in what is known as the standard model of particle physics. In

particular, the fine-structure constant sets the strength of electromagnetic interaction between light (photons) and

charged elementary particles such as electrons and muons. The quantity α was introduced into physics by A.

Sommerfeld in 1916 and in the past has often been referred to as the Sommerfeld fine-structure constant. In order to

explain the observed splitting or fine structure of the energy levels of the hydrogen atom, Sommerfeld extended the

Bohr theory to include elliptical orbits and the relativistic dependence of mass on velocity. One of the most important

numbers in physics is the fine-structure constant α which defines the strength of the electro-magnetic field. It is a

dimensionless number independent of how we define our units of mass,length,time or electric charge. A change in

these units of measurement leaves the dimensionless constant unchanged. The number can be seen as the chance that

an electron emits or absorbs a photon. It’s a pure number that shapes the universe to an astonishing degree. Paul Dirac

considered the origin of the number «the most fundamental unsolved problem of physics». The constant is

everywhere because it characterizes the strength of the electromagnetic force affecting charged particles such as

electrons and protons. Many eminent physicists and philosophers of science have pondered why α itself has the value

that it does,because the value shows up in so many important scenarios and aspects of physics. Nobody has come up

with any ideas that are even remotely convincing. A similar situation occurs with the proton-electron mass ratio μ,not

because of its ubiquity,but rather how chemistry can be based on two key electrically charged particles of opposite

electric charge that are opposite but of seemingly identical magnitude while their masses have a ratio that is quite

large yet finite. These two questions have a huge bearing on why physics and chemistry behave the way they do. The

product of the two quantities appears,at least at first glance,not to be so important. There is a dream,which,albeit more

often not confessed,occupies the most secret aspirations of theoreticians and is that of reducing the various constants

of Physics to simple formula involving integers and transcendent numbers. The fine-structure constant plays an

important role in modern physics. Yet it continues to be a mystery as to exactly what it represents and why it has the

mystical value it has.

2. The search for mathematical expression for the fine-structure constant

The mystery about the fine-structure constant is actually a double mystery. The first mystery – the origin of its

numerical value – has been recognized and discussed for decades. The second mystery – the range of its domain – is

generally unrecognized.

— M. H. MacGregor (2007). The Power of Alpha.

When I die my first question to the Devil will be: What is the meaning of the fine structure constant?

— Wolfgang Pauli

“God is a pure mathematician!' declared British astronomer Sir James Jeans. The physical Universe does seem to be

organized around elegant mathematical relationships. And one number above all others has exercised an enduring

fascination for physicists: 137,0359991.... It is known as the fine-structure constant and is denoted by the Greek

letter alpha (α).”

― Paul Davies

“While twentieth-century physicists were not able to identify any convincing mathematical constants underlying the

fine structure, partly because such thinking has normally not been encouraged, a revolutionary suggestion was

recently made by the Czech physicist Raji Heyrovska, who deduced that the fine structure constant, ...really is

defined by the [golden] ratio ....”

― Carl Johan Calleman, The Purposeful Universe: How Quantum Theory and Mayan Cosmology Explain the Origin

and Evolution of Life

The fine-structure constant plays an important role in modern physics. Yet it continues to be a mystery as to exactly

what it represents and why it has the mystical value it has. The elementary charge of electron e was proposed by

Stoney in 1894 and discovered by Thomson in 1896,then Planck introduced the energy quanta h∙ν in 1901 and

explained it as photon E=h∙ν by Einstein in 1905. Planck first noticed in 1905 that e2/c and h have the same

dimension. In 1909,Einstein found that there are two fundamental velocities in physics: c and e2/h requiring

explanation. He said, “It seems to me that we can conclude from h=e2/c that the same modification of theory that

contains the elementary quantum eas a consequence,will also contain as a consequence the quantum structure of

radiation.” Albert Einstein was the first to use a mathematical formula for the fine-structure constant α in 1909. This

expression is:

with numerical value α=0,00733038286 with an error accuracy of 0,45%. Later many scientists used other

mathematical formulas for fine-structure constant but they are not at all accurate. These are Jeans 1913,Lewis Adams

1914,Lunn in 1922,Peirles in 1928 and others. Arthur Eddington was the first to focus on its inverse value and

suggested that it should be an integer,that the theoretical value is α-1=136. In his original document 1929 he applied

the value:

α-1=16+1/2×16×(16-1)=136

However,the experiments themselves consistently showed that α-1≃137. This forced him to look for an error in his

original theory. He soon came to the conclusion that:

α-1=137

He thus argued that the extra unit was a consequence of the initial exclusion of every elementary particle pair in the

universe. In the document of 1929,Eddington considered that fine-structure constant relates in a simple way to the

cosmological constants,as given by the expression:

where Nthe cosmic number,the number of electrons and protons in the closed universe. Eddington always kept the

name and the symbol α:

The first to find an exact formula for the fine-structure constant αwas the Swiss mathematician Armand Wyler in

1969. Based on the arguments concerning the congruent group,the group consists of simple Lorentz transformations

such as the space-time dimensions that leave the Maxwell equations unchanged. The first form of the Wyler constant

type is:

With numerical value αw=0,00729735252... At the time it was proposed,they agreed with the experiment to be

within 1.5 ppm for the value α-1.

3. Measurement of the fine-structure constant

Based on the precise measurement of the hydrogen atom spectrum by Michelson and Morley in 1887,Arnold

Sommerfeld extended the Bohr model to include elliptical orbits and relativistic dependence of mass on velocity. He

introduced a term for the fine-structure constant in 1916. The first physical interpretation of the fine-structure

constant was as the ratio of the velocity of the electron in the first circular orbit of the relativistic Bohr atom to the

speed of light in the vacuum. The 2018 CODATA recommended value of αis:

α=0.0072973525693(11) (1)

with standard uncertainty 0,0000000011×10-3 and relative standard uncertainty 1,5×10-10.For reasons of

convenience,historically the value of the reciprocal of the fine-structure constant is often specified. The 2018

CODATA recommended value is given by:

α-1=137,035999084(21) (2)

with standard uncertainty 0,000000021×10-3 and relative standard uncertainty 1,5×10-10.There is general

agreement for the value of α,as measured by these different methods. The preferred methods in 2019 are

measurements of electron anomalous magnetic moments and of photon recoil in atom interferometry. The most

precise value of αobtained experimentally (as of 2012) is based on a measurement of g using a one-electron

so-called "quantum cyclotron" apparatus,together with a calculation via the theory of QED that involved 12672

tenth-order Feynman diagrams:

α-1=137,035999174(35) (3)

This measurement of αhas a relative standard uncertainty of 2,5×10-10.This value and uncertainty are about the

same as the latest experimental results. Further refinement of this work were published by the end of 2020,giving the

value:

α-1=137,035999206(11) (4)

with a relative accuracy of 81 parts per trillion.

4. Fine-structure Constant,Anomalous Magnetic Moment,Relativity Factor and the Golden Ratio that

Divides the Bohr Radius

Dr. Rajalakshmi Heyrovska in [8] has found that the golden ratio φ provides a quantitative link between various

known quantities in atomic physics,research in this book chapter entitled "The golden ratio in the creations of Nature

arises in the architecture of atoms and ions". While searching for the exact values of ionic radii and for the

significance of the ionization potential of hydrogen,Dr. Heyrovska has found that the Bohr radius can be divided into

two Golden sections pertaining to the electron and proton. More generally,it was found that φ is also the ratio of

anionic to cationic radii of any atom,their sum being the covalent bond length. After that she showed,among other

facts,that many bond lengths in organic and inorganic molecules behave additively,and are the sum of the covalent

and the ionic radii,whether partially or fully ionic or covalent. An interpretation and a value of the fine-structure

constant α-1 has been discovered in terms of the golden angle. Dr. Heyrovska proposed another interpretation of α

based on the observation that it is close to the golden angle. Fine-structure constant can also be formulated for the

first time exclusively in terms of the golden ratio as follows:

α-1=360·φ-2-2·φ-3

with numerical value:

α-1=137,03562809…

5. Exact formula for the fine-structure constant

There is a dream,which,albeit more often not confessed,occupies the most secret aspirations of theoreticians and is

that of reducing the various constants of Physics to simple formula involving integers and transcendent numbers. We

propose the exact formula for the fine-structure constant αin terms of in terms of the golden angle,the relativity

factor and the fifth power of the golden mean:

α-1=360·φ-2-2·φ-3+(3∙φ)-5 (5)

with numerical values:

α-1=137,03599916476564........ (6)

α=0,00729735256498292........ (7)

The numerical value (6) is the average of the measurements (2),(3) and (4). We believe that the formula (5) is the

exact formula for the fine-structure constant α.Another beautiful forms of the equations are:

(8)

Other equivalent expressions for the fine-structure constant are:

α-1=(362-3-4)·φ-2-(1-3-5)·φ-1 (9)

α-1=(362-3-4)+(3-4+2·3-5-364)·φ-1 (10)

α-1=1-2·φ-1+360·φ-2-φ-3+(3∙φ)-5 (11)

α-1=φ0-2·φ-1+360·φ-2-φ-3+(3∙φ)-5 (12)

α-1=φ0-2·φ-1+360·φ-2-φ-3-241·3-5·φ-4-(3∙φ)-5 (13)

α-1=(174.474·φ+86.995)∙(243∙φ5)-1 (14)

α-1=(87.480·φ3-486·φ2+1)∙(243∙φ5)-1 (15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

6. Continued fraction for the fine-structure constant

The pattern of the continued fraction for the fine-structure constant is:

The continued fraction for the fine-structure constant is:

In the notation of Carl Friedrich Gauss the fine-structure constant is:

8. Conclusions

We presented new exact expression for the fine-structure constant αin terms of the golden angle,the relativity factor

and the fifth power of the golden mean:

α-1=360·φ-2-2·φ-3+(3∙φ)-5

New interpretation and a very accurate value of the fine-structure constant has been discovered in terms of the

golden radio. These equations are simple,elegant and symmetrical in a great physical meaning. These exact

expressions should be studied further.

References

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[2] www.wolframalpha.com/

[3] www.math.stackexchange.com/

[4] www.mathworld.wolfram.com/

[5] www.numberempire.com/

[6] http://physics.nist.gov/cuu/Constants/

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