Golden Quartic Polynomial and Moebius-Ball Electron

A symmetrical quartic polynomial, named golden one, can be connected to coefficients of the icosahedron equation as well as to the gyromagnetic correction of the electron and to number 137. This number is not a mystic one, but is connected with the inverse of Sommerfeld’s fine-structure constant and this way again connected with the electron. From number-theoretical realities, including the reciprocity relation of the golden ratio as effective pre-calculator of nature’s creativeness, a proposed closeness to the icosahedron may point towards the structure of the electron, thought off as a single-strand compacted helically self-confined charged elemantary particle of less spheri-cal but assumed blunted icosahedral shape generated from a high energy double-helix photon. We constructed a chiral Moebius “ball” from a 13 times 180˚ twisted double helix strand, where the turning points of 12 generated slings were arranged towards the vertices of a regular icosahedron, belonging to the non-centrosymmetric rotation group I532. Mathematically put, we convert the helical motion of an energy quantum into a stationary motion on a Moebius stripe structure. The radius of the ball is about the Compton radius. This chiral closed circuit Moebius ball motion profile can be tentatively thought off as the dominant quantum vortex structure of the electron, and the model may be named CEWMB (Charged Electromagnetic Wave Moebius Ball). Also the gyromagnetic factor of the electron (g e = 2.002319) can be traced back to this special structure. However, nature’s energy infinity principle would suggest a superposition with additional less dominant (secondary) structures, governed also by the golden mean. A suggestion about the possible structure of delocalized hole carriers in the superconducting state is given.


Introduction
In this contribution, we dealt with the question, what indeed could be the relationship between a helical photon and a compacted elementary entity like the electron, composed of a burst helical photon, and why is an electron as stable as it is? We here apply dominant principles of nature such as symmetry, golden ratio sophistication, reciprocity, calculus, eternal repetition and entanglement. In this way, helical twisting is a frequently observed tool in nature's quiver, applied to objects from microscopic to cosmic scale. When we were children, we played with twisted twine that, after being folded at half the length, became stable double helically curled cords of chosen chirality. As a crystallographer, I explained chirality using a coil spring that, when rotated about 180˚, maintained the sense of curling, but when looked in front of a mirror, the image showed an opposite sense of chirality. This may help to understand the new exciting theories about the structure of the photon and the electron as helically curled wavy entities or self-confined elementary particles derived from compacted photons, quoting only the most impressive references among many others [1] [2] [3] [4] [5].
Another extended publication by Markoulakis and Antonidakis [6], published at the beginning of this year, is highly recommended. What indeed happens, if a high-energy photon of sufficient energy hits matter, forming a pair of particles with opposed chirality (pair creation of the electron besides the positron)? Either the energy-rich helically twisted photon already consists of strands of opposite chirality respectively charge or different chirality is immediately generated when it hits matter to form the positron besides the electron. These fundamental particles may be formed from single strand "circular ribbons" of the photon maintaining twisting that were glued together like an odd multiply screwed Moebius strip. The relationship between geometry and physical properties of a simple Moebius strip can be followed by a contribution of Starostin and Van der Heijden [7]. However, the proposed Moebius ball is very more complicated to be tackled mathematically, but may nevertheless represent an energetically most favorable state.
Noticeably, helical left-handedness and its opposite could be realized, when a single cellulosic fiber helix or cucumber tendrils were supported at both ends to minimize the twisting energy [8] [9].
However, what can we say about an elliptically twisted helix structure of the photon in contrast to the postulated circular helical structure?
In all such considerations, the golden ratio should be involved as nature's effective-evolutionary pre-calculator, dominating all areas of science, life and cosmos. So we begin in Chapter 2 with a short essay of number theory and the Last but not least and more practical, the golden quartic may also be important in the field of life coding [13] and in quantum information science [14].
This contribution can be understood as a supplement and continuation of an already published work of the present author [13] reviewing also some earlier published work. However, he well knows that objective truth is a continually evolving matter of science. Human beings are only able to approximate the truth, and cherished truths have to be thrown overboard from time to time. The article of Nobel laureate Franc Wilczek about the enigmatic electron may bring this verity to the point [15].

Reciprocity, Calculus, and the Golden Ratio Sophistication
The golden ratio is the most irrational number with the simplest infinite continued fraction representation at all and a very adaptable number-theoretical chameleon. Special attention is paid to the reciprocity property of the golden ratio as effective pre-calculator of natures creativness. We use the definition However, the golden ratio is frequently used by others as the reciprocal of this value Another important number, the fifth power of the golden ratio, should be considered in more detail, because it is connected to phase transitions from microscopic [16] [17] up to cosmic scale [18] There is another nice continued fraction approach to represent giving the quadratic equation 2 1 0 r r − − = with the solution 1 1 1.6180339887 The infinite series expansion of the functions its simplest geometrical sequences of powers of the variable x delivers further insight into the secret of this number and its fractal nature Remembering, the time transformation according to the new IRT theory [18] resulted in this simple formula A special application is given for x ϕ = , where Noticeably, the series expansion of the obverse function 1 1 x + has been first described by none other than Isaak Newton already in 1671 [20] as special case of the general binomial series. Replacing in Equation (2) x by −x or specialize the binomial series for arbitrary rational values of n one gets for n = −1 a series with terms of alternating signs Again one can replace x by φ and get the result, albeit a trivial one (see also Equation (4)) It seems as if a snake is biting its tail. The series expansion allows an insight into the fractal nature of these relations and its close proximity to Newton's "calculus as an algebraic analogue of arithmetic with infinite decimals", quoting the reflection of Stillwell [21]. On can also state a scientific contiguity to El Naschie's ε-Infinity Theory [19] as well as to the Feynman diagrams in quantum electrodynamics, following Feynman's honestly excellent Nobel lecture [22] [23].
Application of such polynomial approximations is utilized by a new relativistic nuclear fusion approach of the present author based on the Information Relativity Theory [18]. Furthermore, the relationship to double-well structures and its connection to quantum information research is considered as a fascinating scientific field [13] [14].

Beyond Chiral Platonic Solids Showing Golden Ratio Geometry
Two simple dual platonic solids continue to fascinate mathematicians, crystallograsphers, physicists as well as biologist, because nature likes obviously a combination of sixfold with fivefold symmetry. In times of an ongoing viral pandemic, we are confronted with the highly effective packaging and transportation of genetic information within the envelope (capside) of an icosahedron and its unwrapping to produce fatal effects in the human body. The dense packing of equal spheres (atom clustering), for instance with Fibonacci numbers 13 and 55, has been studied by Mackay, where a central sphere is repeatedly surrounded by n layers of icosahedral shells [24]. In the following we want to deals with the regular icosahedron besides its dual solid, the regular pentagonal dodecahedron, belonging to the five platonic solids. We discard centrosymmetry and focus on chiral I532 rotation symmetry of order 60 ( Coordinates of vertices see Appendix A.1. Remarkably, the ratio of the in-sphere volume V sph to the polyhedron volume V p of the icosahedron shows Fibonacci number 13 as an approximation in the relation [25] (see also Appendix A.1) The icosahedron can be represented by algebraic equations that map for instance the positions of its vertices respectively face centers using projective geometry. A stereographic projection of the vertices of an icosahedron with unit Journal of Applied Mathematics and Physics circumradius from the south pole of its circumsphere onto the plane z = 0 delivers the simplest equation [26] [27] [28] [29] ( ) ( ) 10 5 , The projection of the face centers of this solid with unit in-radius onto a complex plane resulted in the equation ( ) 20 15 10 Based on the preliminary work by Gordon [30], Klein showed the connection between the regular icosahedron, one of the five Platonic bodies, and the solution of the quintic polynomial. Instead of following Klein The four root of this polynomial have been calculated giving In addition, it yields The reader is frequently confronted with Fibonacci number 13, which obviously plays an important role besides ϕ and 5 ϕ when assessing bio-coding and related storage and processing of information. In the following we will work with another quartic polynomial and with coefficients like number 228 just considered.

Golden Quartic Polynomial Approach
The minimal polynomial of the golden ratio respectively its uneven powers (proof was reported by [31]) is given by Now we introduce a simple approach to decompose important numbers such as number 137 using the following relation, which has been recently applied by the present author [13] ( ) This approach can be recast in the depressed quartic polynomial equation exhibiting a symmetrical double-well structure (see Figure 2) The roots for the quartic can easily be calculated by the relation indicating that Full quartic polynomials can be fortunately solved by applying the procedure given recently by Tehrani [33].
Exemplarily, for n = 5, a = 1 we are confronted with the golden mean as roots of the quartic Because of the intimate connection with the golden ratio the name golden quartic polynomial was introduced by the present author.
Besides this number series, as a trivial result, numbers that are well approximated to an integer can be generated for  Table 1.
Therefore, this simple quartic is a very interesting one when we are dealing with the prerequisites of life and related matter governed by the golden mean or when we discuss possibilities of information processing. In the next chapter we apply the golden quartic polynomial to analyze fundamental numbers that are connected with the electron.
Very recently, the present author became aware about contributions by McMullin [34] as well as by Totland dealing with quartic polynomials in context with the golden ratio [35]. Also Penn demonstrated by an interesting YouTube contribution that every quartic polynomial is golden [36].

Beyond Number 137
Number 137 is not a mystic one, but is connected with the reciprocal of Sommerfeld's fine-structure constant α and this way connected with the electron [37]. The fine-structure constant α describes the coupling respectively measure of the strength of the electromagnetic force that determines the interaction between Journal of Applied Mathematics and Physics The solution 1 137.035999168 x = would approximate the value of 1 α − very precisely. Again we recommend the new approach of Tehrani to solve such a quartic [33].
However, the actual sequence of matching decimals may not be requested due to the ongoing correction of the physical theory of the electron. Instead, we use a less precise golden mean corrected approximation of the form [ Interestingly, for number n = 137 and a = 1, the roots of the golden quartic polynomial (Equation (21)) resulted in 1,2 11.61863 x = ± near 11 ϕ + and 136.9861 137 11 However, if one tentatively chooses 137.0360 n = (inverse fine-structure constant), the result for the roots is Interestingly, adding to the value of 11 ϕ + the value of the gyromagnetic correction factor of the electron, 0.00231878 e g ∆ = (see Chapter 6), we get almost the inverse fine-structure constant In a previously given contribution [13] we have examined the coefficients of the icosahedron equation in a similar way as given above. Using the coefficient 228 we found ( ) One obtains a slightly larger value using the integer number 13 ϕ ϕ The other coefficient of the icosahedron equation is number 494. When working with this number, we obtain again In this way, coefficients of the icosahedron equation or numbers connected with the Kummer surface can traced back to the number 137 respectively the inverse fine-structure constant. All numerical findings point towards an intimate connection of electron's quantum vortex structure near an icosahedron shape.
A puzzle points to the importance of geometrical relations. Bowen and Mulkern [4] reported about a relation between the Compton radius of the electron r c , the Bohr radius of the first electron orbit in the hydrogen atom r B , and the fine-structure constant α, possibly caused by electron's photon absorption ability

The Gyromagnetic Correction Factor of the Electron
Now we turn to the gyromagnetic correction factor of the electron. The g e factor of the electron, conceived as a classical charged particle, is determined by the relation However, the spin as half-integer quantum number of the electron was introduced without any physical justification [46]. Recently, a first attempt has been undertaken by He et al. [47] to connect the golden mean with the ad hoc spin-1/2 construct. Such golden mean approach may be the result of dark halo movement around the stretched electron in the sense of the Information Relativity theory.
Remembering that the "anomalous part" of the gyromagnetic factor ∆g e was recently given by a simple golden mean representation with sufficient accuracy [48] 6 ln 1 0.002319312 24 while a series expansion yields a value more accurate up to the tenth decimal place In this previous publication the present author did not connect the result with any icosahedral structure of the electron, but number 24 in the denominator was a strong indicator of what is being deduced in the following chapters.
The result may be compared to the high accuracy of the best known experimental value for g e determined as one-electron cyclotron transition for an elec- The calculation with The number 266.6 is very interesting. The division by integers frequently delivers numbers with repeating decimals, exemplified by 266. 24 6 11. 1 = [42]. In the following chapter, we develop an idea about the vortex structure of the electron and consequences for its anomalous g-factor as indicator for such structure.

Proposed Quantum Vortex Structure of the Electron
The photon may be considered as being composed of two partial waves, where a normal part is accompanied by a "dark" pilot wave of opposite chirality. The photon model of Gauthier [2] supposes that such dipole entity intrinsically formed from two helical strands of opposite chirality and charge is stable. When the oppositely charged strands are separated, both simultaneously formed particles move away from each other quickly due to their magnetic repulsion, even excessing their enormous electrostatic attraction. The magnetic force and electrostatic force would balance each other out at a distance of r 0 (see Appendix The motion profile can better be understood, if the icosahedron is unfolded onto a plane like a paper model, following the "spiraling" pattern on the formed hexagonal net of triangles.

A.2)
The mirrored structures of the sister particles will certainly influence each other at the very beginning of formation, inscribing the opposed chiral revolving.
We can learn more about a possible double helix unzipping mechanism from our knowledge about the DNA replication process. Furthermore, when combining Sommerfeld's fine-structure constant α with the golden mean φ, one can approximate number 13 once more Interestingly, in the proton structure the down quark has a charge of −13e.
The length l 0 of a single turn of a helix having height z 0 and radius r 0 can be calculated by unrolling the helix to a plane and then applying the Pythagorean according to [2], we get for l 0 The term 5 indicates for the first time that the golden ratio is involved. Now, the entire length of 0 13 l ⋅ must be compacted into a 12-sling Moebius ball to find its radius. The length of a single sling yields Remarkably, by using the in-sphere radius of an icosahedron (see also Appendix A.1), the edge length of the icosahedral cage then gives In this way, a self-confined chiral ball structure with about the Compton radius 13 3.86 10 m c r − ≈ × is generated that did not forgot where it was originating from. It remains to be shown, whether all the properties of the fundamental particle named electron can be confirmed as a self-stabilizing dynamic system that does not radiate. Once again it may be noticed that the icosahedron picture couples the structure with the golden ratio in harmony with what nature would expect.
However, for the electron instead of an icosahedron also the regular dodecahedron as its dual polyhedron can be chosen as confining cage. Then, these slings would hit the 12 face midpoints. If one operates with the next higher uneven Fibonacci number 21, the procedure given above would deliver 20 slings, which would be directed towards the face midpoints of the regular icosahedron, or to the vertices of the dodecahedron. We notice that the vortex structure of the electron and the parametric equations of a Horn torus given by Gauthier are also quite similar to these of a Moebius curve (see Appendix A.4) [2]. Nevertheless, his elegantly worked out physical interpretation can be transferred with few changes to our vortex construct.
With respect to the gyromagnetic correction factor of the electron ∆g e , also an icosahedron based numerical interpretation can be found, combining the Equation (44) with terms from the icosahedron Equation (14) respectively the golden quartic polynomial (see last line of Table 1), and the circumsphere radius r circ of an icosahedron with unit triangle edge length [13] [25] Alternatively, turning back to Schwinger's classical and simple QED approximation for the anomalous part of the g-factor as α/π [52], we can apply a modified π > π  as result of the Moebius stripe charge calculation according to Equation (108) (see Appendix A.5), but modified for the more complex Moebius ball structure. We can possibly verify a value of 3.146644586 ). Thereby we would be able to confirm the experimental ∆g e that has been reduced by IRT mass correction giving [18] [42] 0.00231909 e g α π It would provide a plausible physical understanding of such purely geometric icosahedron-based relationships, beyond Feynman's approach [53], connecting the anomalous g-factor little larger than 2 to the chaotic vortex motion of the electron. If the charge would reside on the surface of the ball, a g-factor of only 5/3 results [5]. However, if the electron charge is spread along the striped inner structure of the chiral Moebius ball, we should be able to prove by fortune the true anomalous g-factor in full glory from its chiral asymmetry at one fell swoop.
The next task is to find an analytical expression for the chaotic motion along multiple Moebius stripes with forced icosahedral symmetry and proceed according to the total charge approach given in Appendix A.5. The exorbitantly accurate determined experimental value ∆g e will make it easier to choose a very reliable structural electron model. Then we can unravel structurally the spin-orbital separation of the electron in quasi 1D Mott-insulators, too [54]. When the time is ripe, we will have learnt whether the structure of the electron is a fundamental one and that the "anomalous" g-factor may be more simply as thought and unequivocally a universal constant. Also the electron spin construct [46]

Superconducting Strands
There are indications that both conventional as well as unconventional super-conductivity is basically governed by hole-carriers, which are guided without resistance in a reversible way through the lattice [56] [57] [58]. When asking, what the structure of delocalized hole-carriers in the superconducting state would be, as all-convincing test case of our approach, the assumed chiral Moebius stripe governed property of the single electron could be relevant.
A delocalized electron hole may also be portrayed by a helical strand able to transport positive charge. During the unfolding of involved delocalizing Moebius electron balls a nested double-helical wavy entity of equal strand chirality (DNA case) could be formed, which can easily be unzipped just above the superconducting transition temperature T c and compacted again into two separated "particles". In this way the equi-chiral wavy entity is different to the photon, composed of two half-photons of opposed chirality and charge. The mathematical and experimental verification should be a worthwhile task for future cooperation.
In addition, the reader may study a seminal contribution by Schiller about tangled strands, elementary particles and the fine-structure constant [59].
Remembering, the present author suggested linking the optimum hole doping Both relations document the fractal nature of the electronic response in superconductors. It was suggested recently that the same is true for conventional superconductors [56]. Also Prester had reported before about evidence of a fractal dissipative regime in high-T c superconductors [60].

Challenge for Modeling of Inorganic or Organic Moebius Ball Structures
We pose the question whether it could be possible to synthesize an inorganic or

Conclusion
This contribution may once more promote understanding of the relationship between number theory, topology and physical properties with the focus on the ever-present golden ratio to decipher the fractal nature of such relations. Also Fibonacci number 13 is a constant companion in our discussion. Starting from a golden quartic polynomial, first ever we deduce the possible structure of the elementary particle electron as a vortex construct in form of an icosahedralshaped chiral Moebius ball with a radius little more than the Compton radius r c .
It is a beautiful geometric structure composed of 12 single-strand slings in chaotic motion, even when it would not be connected to an elementary particle. The Moebius ball vortex structure is a self-confined non-radiating charged compaction of helical structures proposed for the photon, and may be in this way a variation of the photon-electron theory of Gauthier with the changed supposition that the starting photon consists of two oppositely charged double helix strands in contrast to his single helix half-photon approach. Electron's icosahedron go- In a mathematical as well as philosophical sense, the geometry of folded Moebius strands including the Moebius ball mediates between eternity and symmetry respectively chiral asymmetry. Even in Escher's fine art work, combining mathematics with art, such Moebius icosahedron drawing could not be found. However, it is one of the most beautiful chiral motion profiles connecting icosahedral symmetry with analytical realities given by Moebius stripes and its quantitative solution a demanding task of differential geometry.
The inner integral could be exactly solved, followed by an appropriate numerical integration of the outer integral. Using linked terms ( ) 1 3 cos a u = + (108) Consequences and the connection with the golden mean will be discussed in a following comprehensive contribution. Remarkable is also the approximate relation between number 13 and the circle constant π by solving the quadric equa-