Reciprocity Relation between Alternative Gravity Formulas

We compare Newton’s force law of universal gravitation with a corrected simple approach based on Bhandari’s recently presented work, where the gravitation constant G is maintained. A reciprocity relation exists between both alternative gravity formulas with respect to the distances between mass centers. We conclude a one-to-one mapping of the two gravitational formulas. We don’t need Einstein’s construct of spacetime bending by matter.

A Different Approach to High-Tc Superconductivity: Indication of Filamentary-Chaotic Conductance and Possible Routes to Room Temperature Superconductivity

The empirical relation of between the transition temperature of optimum doped superconductors Tco and the mean cationic charge , a physical paradox, can be recast to strongly support fractal theories of high-Tc superconductors, thereby applying the finding that the optimum hole concentration of σo = 0.229 can be linked with the universal fractal constant δ1 = 8.72109… of the renormalized quadratic Hénon map. The transition temperature obviously increases steeply with a domain structure of ever narrower size, characterized by Fibonacci numbers. However, also conventional BCS superconductors can be scaled with δ1, exemplified through the energy gap relation kBTc ≈ 5Δ0/δ1, suggesting a revision of the entire theory of superconductivity. A low mean cationic charge allows the development of a frustrated nano-sized fractal structure of possibly ferroelastic nature delivering nano-channels for very fast charge transport, in common for both high-Tc superconductor and organic-inorganic halide perovskite solar materials. With this backing superconductivity above room temperature can be conceived for synthetic sandwich structures of less than 2+. For instance, composites of tenorite and cuprite respectively tenorite and CuI (CuBr, CuCl) onto AuCu alloys are proposed. This specification is suggested by previously described filamentary superconductivity of “bulk” CuO1﹣x samples. In addition, cesium substitution in the Tl-1223 compound is an option.

Ratio of In-Sphere Volume to Polyhedron Volume of the Great Pyramid Compared to Selected Convex Polyhedral Solids

The architecture of the Great Pyramid at Giza is based on fascinating golden mean geometry. Recently the ratio of the in-sphere volume to the pyramid volume was calculated. One yields as result RV = π ⋅ φ5, where is the golden mean. It is important that the number φ5 is a fundamental constant of nature describing phase transition from microscopic to cosmic scale. In this contribution the relatively small volume ratio of the Great Pyramid was compared to that of selected convex polyhedral solids such as the Platonic solids respectively the face-rich truncated icosahedron (bucky ball) as one of Archimedes’ solids leading to effective filling of the polyhedron by its in-sphere and therefore the highest volume ratio of the selected examples. The smallest ratio was found for the Great Pyramid. A regression analysis delivers the highly reliable volume ratio relation , where nF represents the number of polyhedron faces and b approximates the silver mean. For less-symmetrical solids with a unique axis (tetragonal pyramids) the in-sphere can be replaced by a biaxial ellipsoid of maximum volume to adjust the RV relation more reliably.

Phase Transitions Governed by the Fifth Power of the Golden Mean and Beyond

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 φ5. 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 Janičko 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.

Reciprocity as an Ever-Present Dual Property of Everything

Reciprocity may be understood as relation of action and reaction in the sense of Hegel’s philosophical definition. Quoting Kant, freedom and ethical necessities are reciprocally limited. In this contribution, a more mathematical than philosophical reflection about reciprocity as an ever-present dual property of everything was given. As a crystallographer, the author is familiar with the action of Fourier transforms and the relation between a crystal lattice and its reciprocal lattice, already pointing to the duality between particles and waves. A generalization of the reciprocity term was stimulated by results of the famous Information Relativity (IR) theory of Suleiman with its proven physical manifestation of matter-wave duality, compared to the set-theoretical E-Infinity theory developed by El Naschie, where the zero set represents the pre-quantum particle, and the pre-quantum wave is assigned to the empty set boundary surrounding the pre-particle. Expectedly, the most irrational number of the golden mean is involved in these thoughts, because this number is intimately connected with its inverse. An important role plays further Hardy’s maximum quantum entanglement probability as the fifth power of φand its connection to the dark matter. Remembering, the eleven dimensions in Witten’s M-theory may be decomposed into the Lucas number L5 = 11 = φ−5 – φ5. Reciprocity is indeed omnipresent in our world as piloting waves that accompany all observable earthen and cosmic matter. As a side effect of the IR theory some fundamental constants such as the gyromagnetic factor of the electron, Sommerfeld’s fine-structure constant as well as the charge of the electron must be marginally changed caused by altered relativistic corrections. Consequences also arise for our vision about the evolution of life and consciousness.

Super-Hydrides of Lanthanum and Yttrium: On Optimal Conditions for Achieving near Room Temperature Superconductivity

Recently, many seminal papers deal with the syntheses, stability and superconducting properties of super-hydrides like LaH10 or YH10 under high pressure, reporting critical temperatures near room temperature. In the first run one will assume that the involved metal atoms contribute a number of 3 electrons to the pairing pool corresponding to their valence. However, another possibility may be that the cationic valence is somewhat smaller, for instance only 2.29, resulting in a nominal electron number per cation of σ0 = 0.229 ≈ 3/13 instead of 0.3. Then, we will have a numerical equality to the optimum hole number in the cuprate high-Tc superconductors, a number that reflects the fractal nature of electronic response in superconductors. However, if one still keeps up the oxidation state of +3 of lanthanum, one will need 13 hydrogen atoms to match the optimum σ0. Such composition may be found at the phase boundary between the observed LaH10 and LaH16 phases. Partial ionic replacement is suggested to shift the super-hydride composition into the σ0 optimum. Micro-structural phenomena such as multiple twinning and ferroelastic behavior as observed with cuprates may also influence the superconductivity of super-hydrides. Finally, epitaxial growth of super-hydrides onto a specially cut diamond substrate is proposed.

Magic Numbers of the Great Pyramid: A Surprising Result

Recently attention has been drawn to the frequently observed fifth power of the golden mean in many disciplines of science and technology. Whereas in a forthcoming contribution the focus will be directed towards Fibonacci number-based helical structures of living as well as inorganic matter, in this short letter the geometry of the Great Pyramid of the ancient Egyptians was investigated once more. The surprising main result is that the ratio of the in-sphere volume of the pyramid and the pyramid volume itself is given by π⋅φ5, where φ = 0.618033987⋅⋅⋅ is nature’s most important number, the golden mean. In this way not only phase transitions from microscopic to cosmic scale are connected with φ5, also ingenious ancient builders have intuitively guessed its magic before.

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 spherical 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 (ge = 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.

Beyond a Quartic Polynomial Modeling of the DNA Double-Helix Genetic Code

By combination of finite number theory and quantum information, the complete quantum information in the DNA genetic code has been made likely by Planat et al. (2020). In the present contribution a varied quartic polynomial contrasting the polynomial used by Planat et al. is proposed that considered apart from the golden mean also the fifth power of this dominant number of nature to adapt the code information. The suggested polynomial is denoted as g(x) = x4 - x3 - (4 - ϕ2 )x2 + (4 – ϕ2)x + 1, where is the golden mean. Its roots are changed to more golden mean based ones in comparison to the Planat polynomial. The new coefficients 4 – ϕ2 instead of 4 would implement the fifth power of the golden mean indirectly applying . As an outlook, it should be emphesized that the connection between genetic code and resonance code of the DNA may lead us to a full understanding of how nature stores and processes compacted information and what indeed is consciousness linking everything with each other suggestedly mediated by all-pervasive dark constituents of matter respectively energy. The number-theoretical approach to DNA coding leads to the question about the helical structure of the electron.

Nuclear Fusion Research and Development Need New Relativistic Mass and Energy Corrections Given by the Information Relativity Theory

Hundred years after the conjecture of the British astronomer Eddington that the sun is powered by nuclear fusion of hydrogen, new physics theory may help make energy harvesting by nuclear fusion soon a reality. Researchers as well as investors funding fusion megaprojects are asked to deal with new relativistic corrections for mass and energy proposed by Suleiman in his Information Relativity Theory (IRT). These corrections were calculated in this contribution. It will help to decide whether a venture will be successful and to save big investments when in doubt. The assumed optimal kinetic energy for controlled nuclear fusion must be corrected to a somewhat higher level. At very high kinetic energy in the upper GeV range, it remains not enough baryonic mass to be transformed in energy. The fusion probability faded out to zero already at the golden limit of the recession speed of between target nucleon and projectile nucleon. Cold nuclear fusion, if ever possible, is recommended for protons rather than deuterons at highest experimental possible temperatures around 1000 (K) and needs fine-tuned kinetic nucleon energy. It would be also of interest whether a golden ratio based nuclear fuel confinement chamber could be beneficial. In this connection, also cold nuclear fusion setups should be discussed. Nature is governed by the golden ratio and criticality of physical systems influenced by it, and nuclear physics is not an exception. Computer simulations of the underlying controlled nuclear fusion processes should gain profit from IRT corrected starting information and may tackle anew possible low energy nuclear transmutations considering the wave-like dark components of matter and energy.

Mass Constituents of a Flat Lattice Multiverse: Conclusion from Similarity between Two Universal Numbers, the Rocksalt-Type 2<i>D</i>Madelung Constant and the Golden Mean

In fairly good agreement with the consensus range of dark energy to matter this ratio of the critical density is suggested to be connected with the golden mean &phi;=0.6180339887, yielding for dark energy to matter mass fractions .?Assuming the baryonic matter to be only 4.432%, the ratio of matter to baryonic matter would be , and further the ratio of dark matter to baryonic one . If one subtracts from the dark matter a contribution of antimatter with the same mass of baryonic matter, according to the antigravity theories of Villata respectively Hajdukovic, the remaining mass ratio would yield . Replacing the “Madelung” constant α of Villata’s “lattice universe” by &phi;, one reaches again 1 + &phi;as the ratio of the repulsive mass contribution to the attractive one. Assuming instead of a 3D lattice a flat 2D one of rocksalt type, the numerical similarity between the Madelung constant and φ&minus;1 could not be just coincidence. The proposed scaling of the cosmological mass fractions with the square of the most irrational universal number &phi;may indicate that the chaotic cosmological processes have reached a quite stable equilibrium. This may be confirmed by another, but similar representation of the mass constituents by the Archimedes’ constant &pi;, giving for respectively for the dark components . However, the intimate connection of φ with its reciprocal may ignite the discussion whether our universe is intertwined with another universe or even part of a multiverse with the dark constituents contributed from there.

Pyroelectric Bi_5-x(Bi₂S₃)₃₉I₁₂S: Fibonacci Superstructure, Synthesis Options and Solar Cell Potential

Previously, synthetic hexagonal bismuth sulfide iodide (polar space group P63, a = 15.629(3) ?, c = 4.018(1) ?, Z = 2) has been described by the rather unsatisfactory fractional formula Bi19/3IS9 [1] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400310037003600350038003400370039000000 -[3] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400310037003600350038003400370036000000 . A redetermination of the structure using old but reliable photographic intensity data indicated the presence of additional split positions and reduced atomic occupancies. From the observed pattern of this “averaged” structure a consistent model of a superstructure with lattice parameters of a' = √13·a = 56.35(1) ?, c' = c, and a formula Bi5-x(Bi2S3)39I12S emerged, with 2 formula units in a cell of likewise P63 space group. Structural modulation may be provoked by the space the lone electron pair of Bi requires. When Bi on the 0, 0, z position of the “averaged” cell is transferred to two general six-fold sites and one unoccupied twofold one of the super-cell, more structural stability is guaranteed due to compensation of its basal plane dipole momentum. Owing to the limited intensity data available, more details of the superstructure are not accessible yet. Some physical properties and solar cell application are discussed together with suggestions of ambient temperature synthesis routes of c-axis oriented nano-rod sheets.

Reciprocity Relation between the Mass Constituents of the Universe and Hardy’s Quantum Entanglement Probability

In this short contribution, a reciprocity relation between mass constituents of the universe was explained governed by Hardy’s maximum entanglement probability of φ5 = 0.09017. While well explainable through a set-theoretical argumentation, the relation may also be a consequence of a coupling factor attributed to the normed dimensions of the universe. Also, very simple expressions for the mass amounts were obtained, when replacing the Golden Mean φ by the Archimedes’ constant π. A brief statement was devoted to the similarity between the E-Infinity Theory of El Naschie and the Information Relativity Theory of Suleiman. In addition, superconductivity was also linked with Hardy’s entanglement probability.

Modeling of a Cubic Antiferromagnetic Cuprate Super-Cage

Convex polyhedral cuprate clusters are being formed through lateral frustration when the a and c lattice parameters of the tetragonal ACuO2 infinite layer structure will become identical by substitution of a large cation (A = Ba2+). However, the corner-shared CuO2 plaquettes of the infinite network suffer a topotactic rearrangement forming edge-connected units, for instance Cu18O24 cages (polyhedron notation [4641238]) with 2 compound (space group P4/ nmm) will be discussed. The possibility to construct a cuprate super-cage with m3m symmetry (polyhedron notation [4641242438]) is being reported. This super-cage still consists of edge-connected CuO2 plaquettes when fully decorated with copper ions, but with different curvatures, arranged in circles of 9.39 ? of diameter with 139.2° Cu-O-Cu antiferromagnetic super-exchange interaction. On the one hand, the realization of such a quite stable cuprate super-cage as a candidate for high-Tc superconductivity depends on whether a template of suitable size such as the cation or C(CH3)4 enables its formation, and on the other hand the cage can further be stabilized by highly charged cations located along the [111] direction. Synthesis options will be proposed based on suggested cage formation pathways. An X-ray powder pattern was calculated for a less dense cluster structure of Im3m space group with a lattice parameter of a = 14.938 ? and two formula units of Cu46O51 to facilitate future identification. Characteristic X-ray scattering features as identification tool were obtained when the electron distribution of the hollow polyhedron was approximated with electron density in a spherical shell.

Comment to Guynn’s Fine-Structure Constant Approach

Sommerfeld’s fundamental fine-structure constant α once more gives reason to be amazed. This comment is a Chapter of a publication in preparation dealing mainly with golden ratio signature behind Preston Guynn’s famous matter/space approach. As a result we present a relation of α to the galactic velocity , mediated by the circle constant π, which points to an omnipresent importance of this constant and its intrinsic reciprocity pecularity: α ≈ π2|βg| respectively . The designation fine-structure constant should be replaced simply by Sommerfeld’s constant. We present golden mean-based approximations for α as well as for electron’s charge and mass and connect the word average value of interaction coupling constant αs(mz) with |βg|.

Comparison of the Vibrational Spectra of Copper Polysilicate, CuSiO₃, with Those of the Prototypic Copper Polygermanate, CuGeO₃

Orthorhombic copper polysilicate, CuSiO3, is isotypic to the spin-Peierls compound CuGeO3 and represents a further example of a quasi-one-dimensional spin = 1/2 antiferromagnetic Heisenberg chain system. This is a representation of the first Raman and IR/FIR spectra for CuSiO3, measured at room temperature on polycrystalline samples. A comparison of the optical phonons, predicted by a factor group analysis, with those observed for the CuGeO3 prototype, is presented. A mode assignment for the silicate is given. Surface effects due to a very small crystallite size may cause additional broad bands observed in the Raman spectrum of CuSiO3. From the analysis of the Davydov doublet an intralayer-to-interlayer bond strength of about 40 is derived for the silicate, which is about 20% lower than for the isotypic germanate, allowing for different magnetic responses at low temperature.

Crystal Growth of Cu6(Ge,Si)6O18&#183;6H2O and Assignment of UV-VIS Spectra in Comparison to Dehydrated Dioptase and Selected Cu(II) Oxo-Compounds Including Cuprates

Low-dimensional quantum spin systems with the Cu2+ central ion are still in the focus of experimental and theoretical research. Here is reported on growth of mm-sized single-crystals of the low-dimensional S = 1/2 spin compound Cu6(Ge,Si)6O18&#183;6H2O by a diffusion technique in aqueous solution. A route to form Si-rich crystals down to possible dioptase, the pure silicate, is discussed. Motivated by previously reported incorrect assignments of UV-VIS spectra, the assignment of dd excitations from such spectra of the hexahydrate and the fully dehydrated compound is proposed in comparison to dioptase and selected Cu(II) oxo-compounds using bond strength considerations. Non-doped cuprates as layer compounds show higher excitation energies than the title compound. However, when the antiferromagnetic interaction energy as Jz&#183;ln(2) is taken into account for cuprates, a single linear relationship between the Dqe excitation energy and equatorial Cu(II)-O bond strength is confirmed for all compounds. A linear representation is also confirmed between 2A1g energies and a function of axial and equatorial Cu-O bond distances if auxiliary axial bonds are used for four-coordinated compounds. The quotient Dt/Ds of experimental orbital energies deviating from the general trend to smaller values indicates the existence of H2O respectively Cl&minus;axial ligands in comparison to oxo-ligands, whereas larger Dt/Dqe values indicate missing axial bonds. The quotient of the excitation energy 2A1g by 2&#183;2Eg-2B2g allows checking for correctness of the assignment and to distinguish between axial oxo-ligands and others like H2O or Cl&minus;.

X-Ray Powder Diffraction: Why Not Use CuKβ Radiation?

CuKβ radiation with a wavelength of λ = 1.3923 ? is recommended for crystal structure determination from X-ray powder diffraction using the Rietfeld method. A highly sensitive image plate detector is able to collect enough intensity to record a brilliant X-ray powder pattern in a reasonable time, compared to CuKα1 radiation used today. Especially atomic displacement coefficients could be determined more precisely with the much greater number of reflections recorded. A double-radius Guinier camera attached to a micro-focus rotating anode tube ensures increased brilliance besides high resolution. A simple construction specification is presented to make smart cylindrically bent Ge(111) or Si(111) X-ray monochromators that deliver focused CuKβ radiation. The highly linear response of image plate detectors allows removing of fluorescence radiation simply as background of the pattern. The proposed equipment is a cost-efficient alternative to a liquid gallium-metal-jet X-ray source with maximum power load and a similar wavelength of λ(GaKα1) = 1.34013 ?.

Galactic Route to the Strong Coupling Constant αs(mz) and Its Implication on the Mass Constituents of the Universe

Some fundamental physical quantities need an alternative description. We derive the word average value of interaction coupling constant αs(mz) from the observed maximum galactic rotation velocity by the simple relation , where is the velocity, at which the difference between galactic rotation velocity and Thomas precession is equal, and α is Sommerfeld’s constant. The result is in excellent agreement with the value of αs = 0.1170 ± 0.0019, recently measured and verified via QCE analysis by CERN researchers. One can formulate a reciprocity relation, connecting αs with the circle constant: . It is the merit of Preston Guynn to derive the Milky Way maximum value of the galactic rotation velocity βg, pointing to its “extremely important role in all physics”. The mass (energy) constituents of the Universe follow a golden mean hierarchy and can simply be related to the maximum of Guynn’s difference velocity respectively to αs(mz), therewith excellently confirming Bouchet’s WMAP data analysis. We conclude once more that the golden mean concept is the leading one of nature.

New Superionic Memory Devices Can Provide Clues to the Human Memory Structure and to Consciousness

Since the work of Penrose and Hameroff the possibility is discussed that the location of human memory and consciousness could be connected with tubulin microtubules. If one would use superionic nano-materials rolled up to microtubules with an electrolyte inside the formed channels mediating fast ionic exchange of protons respectively lithium ions, it seems to be possible to write into such materials whole image arrays (pictures) under the action of the complex electromagnetic spectrum that composes these images. The same material and architecture may be recommended for super-computers. Especially microtubules with a protofilament number of 13 are the most important to note. We connected such microtubules before with Fibonacci nets composed of 13 sub-cells that were helically rolled up to deliver suitable channels. Our recent Fibonacci analysis of Wadsley-Roth shear phases such as niobium tungsten oxide , exhibiting channels for ultra-fast lithium-ion diffusion, suggests to use these materials, besides super-battery main application, in form of nanorods or microtubules as effectively working superionic memory devices for computers that work ultra-fast with the complex effectiveness of human brains. Finally, we pose the question, whether dark matter, ever connected with ultrafast movement of ordinary matter, may be responsible for synchronization between interactions of human brains and consciousness.