In this contribution results from different disciplines of science were compared to show their intimate interweaving with each other having in common the golden ratio φ respectively its fifth power φ ^{5} . The research fields cover model calculations of statistical physics associated with phase transitions, the quantum probability of two particles, new physics of everything suggested by the information relativity theory ( IRT ) including explanations of cosmological relevance, the ε - infinity theory, superconductivity, and the Tammes problem of the largest diameter of N non-overlapping circles on the surface of a sphere with its connection to viral morphology and crystallography. Finally, Fibonacci anyons proposed for topological quantum computation (TQC) were briefly described in comparison to the recently formulated reverse Fibonacci approach using the Jani čko number sequence. An architecture applicable for a quantum computer is proposed consisting of 13-step twisted microtubules similar to tubulin microtubules of living matter. Most topics point to the omnipresence of the golden mean as the numerical dominator of our world.
The golden mean or golden ratio 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 [
However, recently the unification of physics was advanced by Suleiman’s information relativity theory (IRT) correcting an overlooked flaw in Newton’s theory with respect to time displacements between observer and moving bodies [
Whereas the unification of physics is intimately connected with the golden number system, in this contribution the main focus is simply on phase transformation governed by the fifth power of the golden mean and insights beyond this topic.
Herein the golden mean is denoted by φ. Its inverse number is just φ − 1 = 1 + φ . One yields numerically
φ = 5 − 1 2 = 0.6180339887 ⋯ (1)
φ − 1 = 5 + 1 2 = 1 + φ = 1.6180339887 ⋯ (2)
It is important to notice that in numerous papers, for instance such about
topological quantum computation (TQC), the golden mean is taken as 5 + 1 2 .
However, in emerging matrix representations for Fibonacci anyon models actually the inverse of this number is always found in the matrix elements. So the choice of the golden ratio according to Equation (1) is well founded and should be maintained.
While in nature this number emerges as result of chaotic-statistical evolution, on the other hand, the fifth power of φ is frequently linked to phase transitions from microscopic to cosmic scale. The secrete behind both number may be revealed by considering their infinitely continued fraction representations [
φ = 5 − 1 2 = 1 1 + 1 1 + 1 1 + ⋯ (3)
φ 5 = 1 11 + 1 11 + 1 11 + ⋯ (4)
Rogers-Ramanujan presented a continued fraction of the form [
r ( τ ) = q 1 / 5 1 + q 1 + q 2 1 + q 3 1 + ⋯ (5)
where q = e 2 π i τ . For τ = 0 one gets the golden mean φ, and for τ = i = − 1 the interesting result is [
r ( i ) = 5 + 5 2 − 1 + 5 2 = 2 + φ − 1 − φ − 1 ≈ π φ 5 + ( π φ 5 10 ) 2 (6)
which can be approximated by terms containing φ 5 . About a nested infinitely continued fraction calculating φ ˜ = 0.619071 ⋯ see [
However, if one deals especially with the behavior of matter suffering from phase transitions, then the ideas of Baxter caused the deepest impact on physics [
It remains to be noted that it was a long way before Baxters ideas were fully accepted by down-to-earth physicists, and it is hoped that the new ideas formulated by Suleiman respectively El Naschie and scientific fellows will be soon taken over by the scientific community.
Simplistic models of statistical mechanics covering a large system of particles are important in linking microscopic and macroscopic average properties and behavior of matter undergoing a transition from disordered states to ordered ones by density increase or temperature reduction. The Ising model of a ferromagnet was the first exactly solved two-dimensional model defined on a square lattice [
In the following we want to pay tribute to Baxter’s important research in the field of statistical mechanics [
Z ( z ) = ∑ n = 0 N 3 z n g ( n , N ) (7)
where the positive real variable z denotes activity or fugacity, and g ( n , N ) defines the number of allowed ways of placing n particles (molecules) on the lattice. One can write the partition function per site of an infinite lattice as
κ = lim n → ∞ Z 1 / N (8)
Low activity is equivalent to homogeneity, whereas high activity complies with heterogeneity. In this way, a phase transition is expected. Baxter discovered two distinct regions of positive fugacity with criticality at the point z = z_{c}_{ }
0 ≤ z < z c and z c < z < ∞ (9)
Quite surprisingly, the result for the critical z_{c} is numerically equal to the reciprocal fifth power of the golden ratio
z c = φ − 5 = 11 + φ 5 = 11.0901699 ⋯ (10)
A second (non-physical) singularity was found for z n p = − φ 5 = − 0.090169943 ⋯
The two singularities are the roots of the simple quadratic equation
z c 2 − a z c − 1 = 0 , (11)
where a = 11. For a = −1 one simply gets the golden mean φ = 5 − 1 2 as one of
the solutions. So we are once again faced with the importance of the golden mean in statistical mechanics.
The behavior of the mean density ρ(z) and the order parameter R(z) at the
critical point has been calculated, too. By using the normalized value z r = z z c one confirms
ρ ( z ) ~ ρ c − 5 − 3 2 ( 1 − z r ) 2 3 = ρ c − 1 φ 5 + φ − 5 ( 1 − z r ) 2 3 , z → z c − (low density branch) (12)
where
ρ c = 5 − 5 10 = φ 5 = 0.276393202 ⋯ (13)
R ( z ) = 3 5 [ 1 5 5 ( z r − 1 ) ] 1 9 = 3 5 [ 1 φ 5 + φ − 5 ( z r − 1 ) ] 1 9 , z → z c + (high density branch) (14)
It is worth to note that the critical exponents 2/3 and 1/9 are conjectured to be universal. Further one should notice that the pre-factors given in the original publication are simply the reciprocal of the sum of the fifth power of the golden mean and its inverse
5 − 3 2 = 1 5 5 = 1 φ 5 + φ − 5 = 11 + 2 φ 5 . (15)
For a more precise study of the hard hexagon partition function in the complex fugacity plane see the valuable contribution published by Assis et al. [
Following Tracy et al. [
z ( τ ) = { − ς 5 ( τ ) disordered regime ς − 5 ( τ ) ordered regime (16)
which is compatible with Baxters results for the hard hexagon approach [
ω 1 = φ 2 + i 2 1 + φ − 2 , | ω 1 | = 1 , (17)
and applying the h_{5} order-5 transformation
ς → h 5 ( ς ) = ω 1 − φ ς ω 1 φ + ς (18)
then due to this transformation the Hauptmodul H(ς) is invariant by the involution ς → − ς − 1 and as well by the order-5 transformation ς → ω 1 ⋅ ς
H ( ς ) = H ( h 5 ( ς ) ) = H ( − ς − 1 ) = H ( ω 1 ⋅ ς ) (19)
It was also Baxter [
Baxter showed that this model is integrable resulting in two critical points at fugacities z_{c}_{ }
z c = ± φ 5 (20)
Again the fifth power of the golden mean is involved in the result. Baxter’s famous work was seemingly overlooked and therefore not quoted in the Science article about quantum criticality, which was experimentally verified in the golden ratio bearing spin dynamics next a phase critical point of the quasi-one-dimensional Ising ferromagnet CoNb_{2}O_{8} published by Coldea et al. [
Beyond it should be mentioned that the quantum Hamiltonian of a one-dimensional hard-boson model was found to map the transfer matrix of Baxter’s hard-square model with a quantum critical point at [
V w = φ 5 (21)
where V represents the nearest-neighbor (two sites apart) interaction, U the chemical potential of the bosons, and w 2 = V 2 + U V . The Hilbert space of the chosen one-dimensional quantum approach is shown to be identical of the space of states along a line of the hard-square model [
El Naschie’s E-infinity (ε^{∞}) theory [
e M = φ 5 2 = 0.04508497 (22)
e E D = 1 − e M = 5 2 φ 2 = 0.9549150 (23)
e D M = 3 2 φ 4 = 0.218847 (24)
e P D = 2 φ − 1 2 = 0.736068 (25)
e M + e D M + e P D = 1 (26)
The readers are faced in the resulting formulas with the golden mean, especially with its fifth power in the beautiful relation
e M + e E D = φ 5 2 + 5 2 φ 2 = 1 (27)
where matter is completed by its dark wavy surrounding.
In a previously published contribution the present author recast both the matter amount e_{M} and dark matter one e_{DM} into a suitable form by combining φ 5 and the inverse number to suggest a reciprocity relation between both matter entities according to [
e M = 1 10 5 φ 5 = 0.04508 ⋯ (28)
e D M = 1 10 ( 5 φ 5 ) − 1 = 0.22180 ⋯ (29)
Then a persuasive equation for the pure dark energy e_{PD} can be written down [
e P D = 1 − 1 10 ( 5 φ 5 + ( 5 φ 5 ) − 1 ) = 0.7331 ( 73.31 % ) (30)
Such quantum entanglement based coincidence means that the constituents of the cosmos should not be considered independent of each other, which was impressively confirmed later by the IRT theory of Suleiman [
Hardy’s maximum quantum probability of two quantum particles [
P = p τ 2 1 − p τ 1 + p τ (31)
The maximum of P yielded
P max = 1 − φ 1 + φ φ 2 = φ 5 = 0.090169943 ⋯ (32)
The Hardy function, displayed in
Now we want to study this function in more detail, replacing the variable p_{τ} by x and write the function P as
f ( x ) = x 2 1 − x 1 + x (33)
Integration of f ( x ) leads to
1 2 ∫ 0 1 f ( x ) d x = ln ( 2 ) − 2 3 = 0.026480514 ⋯ ≈ 1 12 π (34)
The first derivative of f ( x ) resulted in
f ′ ( x ) = d f ( x ) d x = 2 x ( 1 − x − x 2 ) ( x + 1 ) 2 (35)
By comparing the nominator term − ( x 2 + x − 1 ) with Equation (11) the reference to the golden ratio becomes evident. The remaining multiplicative term caused a discontinuity at x = − 1 .
Higher derivatives of f ( x ) lead to the following results
f ″ ( x ) = 2 ( 1 − x 3 − 3 x 2 − 3 x ) ( x + 1 ) 3 (36)
f ‴ ( x ) = − 12 ( x + 1 ) 4 (37)
Special values for f ( x ) , f ′ ( x ) and higher derivatives were summarized in the following
If one connects the original function with energy, then its third derivative should be somewhat like a pressure. More surprisingly, the third derivative comes along as a very simple function with an origin value of −12 at x = 0 and a value of −12φ^{4} at x = φ, underscoring the magic importance of number 12, found for instance as the number of vertices of the Platonic solid icosahedron.
x | f ( x ) | f ′ ( x ) | f ″ ( x ) | − f ‴ ( x ) / 12 |
---|---|---|---|---|
0 | 0 | 0 | 2 | 1 |
1/3 | 5/90 = 1/18 | 5/24 = 0.208333 | −5/16 = −0.31250 | (3/4)^{4 } |
1/2 | 1/12 | 1/9 | −22/27 = −0.8148148 | (2/3)^{4 } |
φ | φ 5 | 0 | − ( 1 + φ 6 ) | φ 4 |
1 | 0 | −1/2 | −3/2 | (1/2)^{4} |
However, nature presents much more relationships to keep in mind, where the golden mean or its fifth power is involved, and superconductivity is no exception. So we must reassess the theory considering the dark matter surrounding the moving electrons, which dive into the dark after marriage, or in other words, become superconducting under special conditions. Recently, the present author suggested linking the optimum hole doping σ 0 of high-T_{c} superconductors with the golden mean again in the form of its fifth power [
σ 0 ≈ 8 π φ 5 = 0.2293 (38)
It was suggested recently that the same is true for conventional superconductors [
v F v K ≈ 2 π φ 5 = 0.0571 (39)
Both relations document the fractal nature of the electronic response in superconductors. Also Prester had reported before about evidence of a fractal dissipative regime in high-T_{c} superconductors [
In
The prospect of developing a resilient theory of superconductivity is promised. Quantum entanglement of two moving electrons is influenced by local interaction of their interwoven dark matter surroundings, quoting the cogwheel picture of Suleiman [
In 1930 the Dutch biologist Tammes [
distance between two points x , y ∈ S 2 . For α = 0 the potential V = ln ( 1 | x − y | )
is used. Given a fixed configuration of N points ω N = ( x 1 , x 2 , ⋯ , x N ) , the α-energy sums up as [
E ( α , ω N ) : = { ∑ 1 ≤ i < j ≤ N ln 1 | x i − x j | , if α = 0 ∑ 1 ≤ i < j ≤ N | x i − x j | α , if α ≠ 0 (40)
Of importance is the determination of extremal energies for N points on the sphere [
ε ( α , N ) : = { inf ω N ∈ S 2 E ( α , ω N ) , if α ≤ 0 sup ω N ∈ S 2 E ( α , ω N ) , if α > 0 (41)
Our concern should now be the special case ε ( 0 , N ) for which a conjecture already exists [
ε ( 0 , N ) ≥ − 1 4 ln ( 4 e ) N 2 − 1 4 N ln N + O ( N ) (42)
lim N → ∞ E ( 0 , N ) − ( − 1 4 ln ( 4 e ) N 2 − 1 4 N ln N ) N = μ (43)
with an estimate of μ = − 0.026422 ⋯ [
μ = 2 3 – ln ( 2 ) = − 0.0264805 ⋯ (44)
Quoting Saff et al. [
E ( 0 , ω ^ N ) − ε ( 0 , N ) ≤ 114 ⋅ ln N ≈ 3 | μ | ln N (45)
The interesting connection between minimal discrete energy on the sphere and matter energy density of the cosmos according to the information relativity theory [
Nowadays we are faced with the worldwide pandemic spread of a very mobile corona virus. Therefore, the viral self-assembly and reproduction as a thermodynamic process comes ones again into focus. Viral self-assembly is intimately related to the before mentioned Tammes problem [
Models for the viral spread are quite complex, but may be likewise considered as noise dynamical phase transition with a critical point [
Heretically put, the deadly virus acts as a corrective to human stupidity by clipping human dominance and restlessness for the good of the environment. Optimistically, the mankind, if it survives at all, could be able in the distant future to control respectively destroy any small pathogenic bodies via teleported information, influencing the dark matter halo of the particles.
In anticipation of the explanations in the next chapter, we may ask whether quantum computation with (stable) viral structures could be possible in future or by use of artificial microtubules build from helically rolled up hexagon lattice entities similar to tubulin proteins that then may operate under ambient conditions. The author recommends reading the seminal contributions of Penrose and Hameroff [
The concept of a special arrangement of microtubules into a “heavy” hexagonal lattice with numbers that follow the Fibonacci number sequence and with vertices that minimize frequency collisions to protect the system from de-coherence, has been effectively refined again and again in nature by evolution. Not surprisingly, it resembles somewhat the qubit architecture of IBM’s last quantum computer (QC) [
ballistic speed are entangled through their pronounced dark matter halo with the environment and can by that pass on and store information. This is an important result of the IRT of Suleiman [
The golden mean φ as the most irrational number found in nature with the simplest continued fraction representation at all is approximated by the division of two consecutive numbers of the Fibonacci number sequence [
J n + 2 = 8 ( J n + 1 − J n ) (46)
Comparable to the quadratic Equation (7), where solutions lead to the golden mean, we can formulate a quadratic equation for the reverse sequence, too. It yields
J 2 − 8 J + 8 = 0 (47)
giving fundamental number j_{1,2} as its two solutions [
j 1 = 4 + 8 = 2 ( 2 + 2 ) = 6.82842712475 ⋯ (48)
j 2 = 4 − 8 = 2 ( 2 − 2 ) = 1.171572875 ⋯ (49)
As in the case of the golden mean also the reverse number j 1 − 1 seems to be important
j 1 − 1 = 1 − j 2 − 1 = j 2 8 = 0.1464466094067 ⋯ (50)
Again, both numbers can be approximated by the ratio of consecutive Janičko numbers [
This number sequence may be important in calculations with quantum bits. The particle-wave duality is essential in the quantum information theory, where the unit of information is given by the quantum bit (qubit) coined by Schumacher [
| ψ 〉 = p r | 0 〉 + e i ϕ 1 − p r | 1 〉 (51)
where 0 ≤ p r ≤ 1 is the probability of the bit being in the 0 state, and 0 ≤ ϕ ≤ 2 π is the phase of the quantum. All possible values of pr and the phase ϕ are reachable by application of quantum gates, realized by reliable and repeatable Josephson junctions. For instance, operating the quantum logical T-gate as quantum circuit with a phase of π/4 that is mapped by the unitary matrix
T = ( 1 0 0 e i π / 4 ) (52)
one gets for p r ( 0 ) = j 2 − 1 = 0.8535533 ⋯ respectively for p r ( 1 ) = j 1 − 1 = 0.1464466 ⋯ . In this way the connection of a single qubit with Janičko numbers is demonstrated as before suggested by Janičko itself [
In contrast, topological quantum computation using Fibonacci anyons as qubits [
For a multi-anyon system the number of fusion paths resulted exactly in the Fibonacci number, and the move accompanied by the consecutive fuse of the anyons from left to right can be represented by a F matrix. For more than three anyons consistency equations for the fusion trees in form of pentagon relations deliver the entries of the F-matrices [
F τ τ τ τ = ( F 11 F 1 τ F τ 1 F τ τ ) = ( φ φ φ − φ ) (53)
where again φ is the golden number (
The braiding operator B can be derived by matrix multiplication of F-matrices with R-matrices. When the R-matrix is given by diagonal spin factors
R = d i a g ( e 4 π i 5 , e − 3 π i 5 , e − 3 π i 5 , e 4 π i 5 , e − 3 π i 5 ) (54)
and the extended 5 × 5 F-matrix by
F = ( 1 1 1 φ φ φ − φ ) (55)
then one obtains finally the B matrix as [
B = F − 1 R F = ( e 4 π i / 5 e − 3 π i / 5 e − 3 π i / 5 φ e − 4 π i / 5 − φ e − 2 π i / 5 − φ e − 2 π i / 5 − φ ) (56)
“Pentagon” moves respectively “hexagon” ones are third order polynomial equations in many variables with much more equations than variables, but not always existing solutions.
Freedman, Larsen and Wang [
p r = | V ( L , t ) ( t 1 2 + t − 1 2 ) g − 1 | 2 (57)
where V ( L , t ) is the Jones polynomial of a link L [
of unity, expressed in the parlance of a crystallographer, with r ∈ { 1 , 2 , 3 , 4 , 6 } are excluded. Field and Simula [
Fibonacci anyons and the Jones polynomial at the fifth root of unity t = exp ( 2 π i 5 ) .
The magnitude of the Jones polynomial [
V H o p f ( e 2 π i 5 ) = p r ( | 0 〉 5 ) ⋅ φ − 1 = φ (58)
with only two elementary physical braiding operations used before measuring the annihilation probability of the anyons upon fusion [
This short survey may serve to demonstrate the importance of the golden number for the development of quantum computation that should stimulate research in a direction along which way nature has always evolutionary and subtly worked. In this sense the seminal ideas of Penrose and Hameroff [
Besides the compounds currently applied such as GaAs hetero-structures or graphene, which show the fractional quantum Hall effect (FQHE), the investigation of other inorganic (semiconducting) compounds is recommended, for instance ferroelastic compounds or compounds that show Fibonacci-like unit-cell decoration [
by an angle of α = atan ( 5 3 3 ) − 30 ∘ = 13.898 ∘ to generate the blue outlined
unit-cell with lattice parameter a that has 13 times the volume of the small cell. One may repeat this procedure to obtain a super-cell and so on. Furthermore, it should be possible to roll up this 2D net into a microtubule (
d 1 ≈ 13 a s u b ⋅ cos a s u b ⋅ cos ( α ) / π = 3.5 ⋅ a / π . Interesting is d 6 = 21 ⋅ a π showing a further Fibonacci number. By a full turn one gains a height in filament direction of h 1 = 39 2 a s u b ≈ π ⋅ a s u b . If you look through the microtubule perpendicular to the filament direction, you will see the mirror image on the back somewhat offset.
In practice it may be a sophisticated task to produce such a tubule, but nowadays it should be possible. The future will show whether such a composite is applicable for QC, after the right compound with optimal physical properties is found. One could start with doped graphene rolled up in the explained Fibonacci twisting. The Kitaev honeycomb lattice approach may be applied in an extended form to this special lattice variant [
Interestingly, the helically twisted tubulin microtubules show strongest reinforcement of the ordered pattern when the protofilament number n equals 13 [
A very stimulating question is what happens if the microtubule is compressed or dilated due to vibration in filament direction finally going through an incommensurability state represented by 13 + φ 5 = ( 2 + φ − 1 ) 2 = 13.0901699 ⋯ instead of 13 twist steps (see Appendix). One may tentatively characterize such a behavior also by a phase transition. A highly effective quantum computer could possibly be realized based on such a principle. Besides the explained longitudinal adaption of the twisted net, its sense of rotation, optimal diameter, and the freedom given by the decoration of the net itself, there are many more degrees of freedom, if one constructs a hexagonal lattice of such 13-step microtubules with non-hexagonal internal symmetry. Neighboring filament can be rotated one again the other violating the hexagonal symmetry. In addition, the border of the assembled filaments may be important. This is what nature has already optimized to equip its creatures with perception, memory and intelligence.
This contribution reviewed and extended results of different scientific disciplines that involve the golden mean φ as very important number of nature. It covered especially phase transitions governed by the fifth power of φ, found in Baxter’s hard lattice approaches of statistical mechanics as well as in energy density relations of the cosmos, comparing results of the ε-infinity theory of El Naschie with the conclusive findings of Suleiman’s scale-free information relativity theory (IRT) besides Hardy’s quantum entanglement probability. Also superconductivity is described as an inherently fractal property connected with the interaction of two moving electrons entangled and finally paired by way of their dark wavy surroundings. Since Klein’s research on the icosahedron many others solved and combined fundamental scientific questions beyond icosahedral structures. From Tammes problem of decorating a unit sphere with hard circles one may be guided to viral morphology as well as phase transition, self-assembly and reproduction of viruses. Are there connections between the energy density of the cosmos with Tammes problem?
The golden mean is involved in the development of modern topological quantum computation, too. The performance of topological quantum computation with Fibonacci anyons can give a vague idea of how the human brain actually processes and stores information. It may be highly interesting to develop artificial self-assembling structures based for instance on the tubulin protein or viruses that can form helically wounded microtubules with the ability to store and to process information. A proposal was made for a twisted microtubule tailors from a rolled up Fibonacci net. Quantum computation is easier understandable, if one accepts the suggestions of Suleiman that quantum entanglement is a local realistic phenomenon “where each moving particle will be permeated by the dark matter halo of the other one making the two physically entangled” [
The author appreciates valuable suggestions given by a competent and cooperative reviewer.
The author declares no conflicts of interest regarding the publication of this paper.
Otto, H.H. (2020) Phase Transitions Governed by the Fifth Power of the Golden Mean and Beyond. World Journal of Condensed Matter Physics, 10, 135-158. https://doi.org/10.4236/wjcmp.2020.103009
Roots of unity: By solving the quartic polynomial ϕ 5 ( x ) = ∑ n = 0 4 x n one yields the four primitive roots of unity ω i (i = 1 to 4) here notably expressed by terms of the golden mean
ω 1 , 2 = φ 2 ± i 2 1 + φ − 2 = φ 2 ± i 2 2 + φ − 1 (59)
ω 3 , 4 = − φ − 1 2 ± i 2 1 + φ 2 = − φ − 1 2 ± i 2 2 − φ (60)
About the Fibonacci number 13: This number is very special because its square root, coming along as the unit-cell—sub-cell parameter ratio of the Fibonacci net approach, can be approximated by 2 + φ − 1
13 = 3.60555 ⋯ ≈ 2 + φ − 1 = 3.6180339887 ⋯ (61)
Furthermore, the exact adjustment connects the number 13 with the fifth power of φ, which number was the main concern of this contribution
2 + φ − 1 = 5 φ − 1 = 13 + φ 5 = 13.0901699 ⋯ = 3.6180339887 ⋯ (62)
When adding to the Hardy function [
f ( x ) = 3 + ( 1 − x ) x 2 ( 1 + x ) (63)
An exceptional result is obtained for the x value at zero of this function
x = 5 3 ( 10 2 3 + 10 1 3 + 10 0 3 ) = 5 × 2.59867450788 ⋯ = 12.993372 ⋯ ≈ 13 (64)
Crystallography of the Fibonacci 2D-Lattice: The reciprocal sub-cell is put up according to [
( 31 4 ¯ 0 ) : h 2 + k 2 + h k = 13 (65a)
( 52 7 ¯ 0 ) : h 2 + k 2 + h k = 39 = 3 × 13 (65b)
Also remarkable, the sine of the twist angle α = 13.898 ∘ can be approximated well by the ratio of π and number 13 + φ 5
sin ( 13.898 ) = 0.240192 ⋯ ≈ π 13 + φ 5 = 0.239996 ⋯ = sin ( 13.886 ) (66)
The Fibonacci net can be generated by application of symmetry operations of the two-dimensional symmetry group p6. One needs three positions to get the 13 lattice points. Two general six-fold positions with coordinates in fractions of again
13, besides the origin, yield the lattice points:
Cabinet of curiosities: With this title Fang et al. [
β = a r c c o s ( 3 φ − 1 − 1 4 ) = 15.5224 ⋯ ∘ (67)
However, one can approximate this angle very well by a relation that uses the fifth power of φ
β ≈ 3 ⋅ a r c s i n ( φ 5 ) = 15.52015 ⋯ ∘ (68)
Geometric interpretation of φ^{5}: The angle of β / 3 = 5.173 ⋯ ∘ is also near the difference of the half inner angles between pentagon and hexagon being 36 ∘ − 30 ∘ = 6 ∘ . The difference in the triangle areas between both regular polygons yielded a value near φ 5 = 0.090169943 ⋯
A 6 6 − A 5 5 = a 2 4 ( 3 − 1 + 1 φ + 0.5 ) = 0.088917221 ⋯ a 2 (69)
where A 6 and A 5 are the full areas of the hexagon respectively pentagon with edge length a . When the pentagon is only slightly rotated about the edge against the hexagon plane, and then the projected area is subtracted, one can well adapt the exact value of φ 5 .