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Golden ratio and Klein bottle Logophysics: the Keys of the Codes of Life and Cognition (Spurred by Liu & Sumpter, “Is the golden ratio a universal constant for self-replication?”) -comment: final revised version in press in Quantum Biosystems
Abstract and Figures
Abstract : We present 1) a novel unified conception of science, cognition and
phenomenology in terms of the Klein Bottle logophysics, 2) as a supradual creative
agency based on self and hetero-reference and multistate logic associated to the
non-orientable topologies of the Möbius strip and Klein Bottle surfaces, 3) related
to the Golden ratio in several areas of biology (particularly genomics), cognition,
perception, physics and music, and the multiple biochemical codes of life , 4)
semiosis and topological folding in the genesis of life , 5) the torsion geometries
and non-orientable topologies, their relation to active time and chronomes,
standing waves and cyclical process, providing an ontology for “chance” and apply
them to 6) human-bodyplan, neurosciences, music cognition, structure and
processes of thinking, particularly Quantum Mechanics, creativity and the logics
of the psyche; 7) a universal principle of self-organization and the genesis of life,
the π-related visual cortex and holography; 8) as an harmonic principle in the
brain’s pattern formation, pattern recognition and morphogenesis, and the
topological paradigm to neuroscience proposing an explanation for the higher
dimensional organization of brain connectomes based on the Klein Bottle as the
metaform for patterns; 9) higher-order cybernetics, ontopoiesis and autopoiesis
in Systems Biology,the psyche’s bi-logic; 10) the supradual nature of
phenomenology, its relation to cosmological cycles, and an examination of the
forceful omission of supraduality in academic philosophy vis-à-vis the foundations
of science and philosophy in ancient Greece; 11) a rebuttal of Dr. Liu et all’s PLOS
article claiming the appearance of Phi in genomes as accidental, in terms of the
supradual ontopoiesis hereby presented and by reviewing several codes of life
discovered by Pérez, which elicit their unity already starting at the level of the periodic table of elements and Life compounds atomic mass; and 12) the Golden
mean in the rituals of whales and the supradual logophysics of social organization
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Gravitomagnetic equations result from applying quaternionic differential operators to the energy–momentum tensor. These equations are similar to the Maxwell’s EM equations. Both sets of the equations are isomorphic after changing orientation of either the gravitomagnetic orbital force or the magnetic induction. The gravitomagnetic equations turn out to be parent equations generating the following set of equations: (a) the vorticity equation giving solutions of vortices with nonzero vortex cores and with infinite lifetime; (b) the Hamilton–Jacobi equation loaded by the quantum potential. This equation in pair with the continuity equation leads to getting the Schrödinger equation describing a state of the superfluid quantum medium (a modern version of the old ether); (c) gravitomagnetic wave equations loaded by forces acting on the outer space. These waves obey to the Planck’s law of radiation.
The golden ratio, ϕ = 1.61803…, has often been found in connection with biological phenomena, ranging from spirals in sunflowers to gene frequency. One example where the golden ratio often arises is in self-replication, having its mathematical origins in Fibonacci’s sequence for “rabbit reproduction”. Recently, it has been claimed that ϕ determines the ratio between the number of different nucleobases in human genome. Such empirical examples continue to give credence to the idea that the golden ratio is a universal constant, not only in mathematics but also for biology. In this paper, we employ a general framework for chemically realistic self-replicating reaction systems and investigate whether the ratio of chemical species population follows “universal constants”. We find that many self-replicating systems can be characterised by an algebraic number, which, in some cases, is the golden ratio. However, many other algebraic numbers arise from these systems, and some of them—such as and 1.22074… which is also known as the 3rd lower golden ratio—arise more frequently in self-replicating systems than the golden ratio. The “universal constants” in these systems arise as roots of a limited number of distinct characteristic equations. In addition, these “universal constants” are transient behaviours of self-replicating systems, corresponding to the scenario that the resource inside the system is infinite, which is not always the case in practice. Therefore, we argue that the golden ratio should not be considered as a special universal constant in self-replicating systems, and that the ratios between different chemical species only go to certain numbers under some idealised scenarios.
In 2009, the Commission on Isotopic Abundances and Atomic Weights (CIAAW) of the International Union of Pure and Applied Chemistry (IUPAC) introduced the interval notation to express the standard atomic weights of elements whose isotopic composition varies significantly in nature. However, it has become apparent that additional guidance would be helpful on how representative values should be derived from these intervals, and on how the associated uncertainty should be characterized and propagated to cognate quantities, such as relative molecular masses. The assignment of suitable probability distributions to the atomic weight intervals is consistent with the CIAAW’s goal of emphasizing the variability of the atomic weight values in nature. These distributions, however, are not intended to reflect the natural variability of the abundances of the different isotopes in the earth’s crust or in any other environment. Rather, they convey states of knowledge about the elemental composition of “normal” materials generally, or about specific classes of such materials. In the absence of detailed knowledge about the isotopic composition of a material, or when such details may safely be ignored, the probability distribution assigned to the standard atomic weight intervals may be taken as rectangular (or, uniform). This modeling choice is a reasonable and convenient default choice when a representative value of the atomic weight, and associated uncertainty, are needed in calculations involving atomic and relative molecular masses. When information about the provenance of the material, or other information about the isotopic composition needs to be taken into account, then this distribution may be non-uniform. We present several examples of how the probability distribution of an atomic weight or relative molecular mass may be characterized, and also how it may be used to evaluate the associated uncertainty.