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Number theory and the unity of science
... A decagon is formed by a two pentagons, with one rotated by 36 degree from the other. The crystallographic structure of DNA, stress patterns in nanomaterials, the stability of atomic nuclides and the periodicity of atomic matter depend on the Golden ratio [9]. The golden ratio, or the mathematical ratio of the Phi, has been discovered to be the only mathematical arrangement that can reproduce itself indefinitely without variation. ...
... Black holes warp space in their vicinity so much that in classical General Relativity, nothing, not even light, can escape. However, when quantum effects are included, black holes can lose energy via a process known as Hawking radiation [9]. The golden ratio is the precise point at which the modified heat of a black hole goes from positive to negative, and it is part of the equation for the lower constraint on black hole entropy. ...
... Regarding the golden ratio again, Boeyens and Thackeray [86] are quoted by Mamombe [61], "We suggest that there is a strong case that the so-called, 'Golden Ratio' (1.61803 ...) can be related not only to aspects of mathematics but also to physics, chemistry, biology and the topology of space-time." [86]. ...
... Regarding the golden ratio again, Boeyens and Thackeray [86] are quoted by Mamombe [61], "We suggest that there is a strong case that the so-called, 'Golden Ratio' (1.61803 ...) can be related not only to aspects of mathematics but also to physics, chemistry, biology and the topology of space-time." [86]. Xu and Zhong state, "... we would like to draw attention to a general theory dealing with the noncommutativity and the fine structure of spacetime which comes to similar conclusions and sweeping generalizations about the important role which the golden mean must play in quantum and high energy physics." ...
After a brief review of the golden ratio in history and our previous exposition of the fine-structure constant and equations with the exponential function, the fine-structure constant is studied in the context of other research calculating the fine-structure constant from the golden ratio geometry of the hydrogen atom. This research is extended and the fine-structure constant is then calculated in powers of the golden ratio to an accuracy consistent with the most recent publications. The mathematical constants associated with the golden ratio are also involved in both the calculation of the fine-structure constant and the proton-electron mass ratio. These constants are included in symbolic geometry of historical relevance in the science of the ancients. Published version: International Journal of Mathematics and Physical Sciences Research, 5, 2, 89-100 (2018).
... Regarding the golden ratio again, Boeyens and Thackeray [86] are quoted by Mamombe [61], "We suggest that there is a strong case that the so-called, 'Golden Ratio' (1.61803 ...) can be related not only to aspects of mathematics but also to physics, chemistry, biology and the topology of space-time." [86]. ...
... Regarding the golden ratio again, Boeyens and Thackeray [86] are quoted by Mamombe [61], "We suggest that there is a strong case that the so-called, 'Golden Ratio' (1.61803 ...) can be related not only to aspects of mathematics but also to physics, chemistry, biology and the topology of space-time." [86]. Xu and Zhong state, "... we would like to draw attention to a general theory dealing with the noncommutativity and the fine structure of spacetime which comes to similar conclusions and sweeping generalizations about the important role which the golden mean must play in quantum and high energy physics." ...
After a brief review of the golden ratio in history and our previous exposition of the fine-structure constant and equations with the exponential function, the fine-structure constant is studied in the context of other research calculating the fine-structure constant from the golden ratio geometry of the hydrogen atom. This research is extended and the fine-structure constant is then calculated in powers of the golden ratio to an accuracy consistent with the most recent publications. The mathematical constants associated with the golden ratio are also involved in both the calculation of the fine-structure constant and the proton-electron mass ratio. These constants are included in symbolic geometry of historical relevance in the science of the ancients. International Journal of Mathematics and Physical Sciences Research, 5, 2, 89-100 (2018).
... In the thirteenth article, The Golden Ratio in Nature: A Tour across Length Scales, the authors carry out a mathematical review of the golden ratio, which is common in very diverse fields and in phenomena of different length scales, from the galactic to the atomic. Its presence in various natural phenomena is then reviewed, the tendency of which suggests that it is a fundamental constant of nature [47,48]. ...
Scientific research, in the era of new technologies and globalization, is becoming a science industry, meaning it increasingly requires interdisciplinarity and transdisciplinarity to achieve its goals of scientific progress, with this also being a requirement for its application to the well-being of society [...]
... Inversely, if the golden ratio were existed at the birth phase of the universe, then our universe should follow the "golden ratio" that might be continued and be observed at present time (that we see everywhere now) and in future also. As the "golden ratio" is observed presently in all living and non-living objects everywhere in nature from micro to macro scales [62] one can say-"golden ratio" had present at the birth phase of our universe and played an important role in creation and evolution of the universe. ...
... The examples discussed above show that there is a tendency for φ to appear at all length scales, sometimes in surprising places. This tendency for the Golden ratio to appear in such a wide variety of phenomena has even led to suggestions that the Golden ratio is a fundamental constant of nature [78,79]. ...
The Golden ratio is an irrational number that has a tendency to appear in many different scientific and artistic fields. It may be found in natural phenomena across a vast range of length scales; from galactic to atomic. In this review, the mathematical properties of the Golden ratio are discussed before exploring where in nature it is claimed to appear; beginning at astronomical scales and progressing to smaller lengths, until reaching those of atomic and quantum physics. For each phenomenon discussed, the evidence for the presence of the Golden ratio is assessed. In making such a tour across length scales, it is illustrated just how prevalent this single number is within the natural universe.
... The examples discussed above show that there is a tendency for φ to appear at all length scales, sometimes in surprising places. This tendency for the Golden ratio to appear in such a wide variety of phenomena has even led to suggestions that the Golden ratio is a fundamental constant of nature [78,79]. ...
The Golden ratio is an irrational number that has a tendency to appear in many different scientific and artistic fields. It may be found in natural phenomena across a vast range of length scales; from galactic to atomic. In this review, the mathematical properties of the Golden ratio are discussed before exploring where in nature it has been found; beginning at astronomical scales and progressing to smaller lengths, until reaching those of atomic and quantum physics. In making such a tour across length scales, it is illustrated just how prevalent this single number is within the natural universe.
... These quarter circles ultimately form the Fibonacci spiral (Jun-Sheng, 2019). These concepts take shape in naturally occurring places, such as in the crystallographic structure of DNA, in botanical phyllotaxis, in the curvature of elephant tusks, etc. (Boeyens & Thackeray, 2014). This paper, however, focuses on its incorporation in paintings, such as how it had been used in Leonardo da Vinci's, The Mona Lisa (Kemp, 2004). ...
The goal of this study was to examine paintings from three different art movements--the Renaissance, Baroque, and Romantic eras--and determine if there were differences in the observed usage of the Fibonacci sequence during each period. By doing this, I hoped to further the understanding of the techniques and characteristics of paintings made during these art movements. Prior research shows that the Fibonacci sequence has been noticed in some major Renaissance pieces, such as the Mona Lisa and the Vitruvian Man; however, there is no apparent research done specifically on how this mathematical principle has appeared in other art movements, nor are there any existing comparisons made between art movements in regard to utilization of this sequence. Data was collected through content analysis based on random samples of paintings that I formulated. For each art movement, through content analysis, it was then determined whether or not each painting incorporated the Fibonacci sequence. The results had shown that the Renaissance era had utilized the sequence in 60% of its paintings, the Baroque in 40%, and the Romantic in 30%. Therefore, it can be concluded that the Fibonacci sequence had differed in how often it had been incorporated when comparing the Baroque and Romantic eras to the Renaissance; however, this is limited due to various factors during the analyzation process. Further research should try to examine various other art movements, and if modern era art is more prone to utilize the Fibonacci sequence as knowledge of this mathematical concept is more universally known.
This work investigates the integration of Fibonacci patterns and Golden Ratio principles into proteinoid-based systems, connecting fundamental mathematical concepts with contemporary biomimetic approaches. Proteinoids are thermal proteins that can self-assemble and have enzyme-like capabilities. They provide a distinct platform for biomimetic information processing. Our study examines the impact of integrating Fibonacci sequences and the Golden Ratio (ϕ = 1.618) into the design and synthesis of proteinoids on their structural organization and response characteristics. We developed two categories of stimuli: auditory signals generated using frequencies derived from the Fibonacci sequence, and electrical patterns that correspond to the proportions of the Golden Ratio. The proteinoid microsphere assemblies were subjected to these stimuli, and their electrical and structural responses were recorded and analyzed. The results indicate that proteinoid systems reveal unique reactivity to acoustic stimuli based on the Fibonacci sequence, exhibiting heightened sensitivity to particular combinations of frequencies and demonstrating nonlinear amplification effects. The proteinoid assemblies exhibited distinctive temporal dynamics and emergent oscillatory behaviors when exposed to voltage patterns inspired by the Golden Ratio, which were not detected with ordinary input signals. These findings provide opportunities for developing advanced bioinspired information transfer and security systems and might improve our understanding of information processing in early chemical systems.
Different studies have examined impression formation regarding shapes. However, few studies have examined whether shapes directly evoke the emotions. An experiment was conducted to investigate characteristics of forms that evoke negative emotions among adolescents and young children. The results indicated that both adolescents and young children had negative emotions regarding circle-aggregate shapes. It might be possible that characteristics of circle-aggregate shapes include signals that developmentally and evolutionally facilitate approach-avoidance behaviors.
It is demonstrated that all stable (non-radioactive) isotopes are formally interrelated as the products of systematically adding alpha particles to four elementary units. The region of stability against radioactive decay is shown to obey a general trend based on number theory and contains the periodic law of the elements as a special case. This general law restricts the number of what may be considered as natural elements to 100 and is based on a proton:neutron ratio that matches the golden ratio, characteristic of biological and crystal growth structures. Different forms of the periodic table inferred at other proton:neutron ratios indicate that the electronic configuration of atoms is variable and may be a function of environmental pressure. Cosmic consequences of this postulate are examined.
Questions of alpha taxonomy are best addressed by comparing unknown specimens to samples of the taxa to which they might belong. However, analysis of the hominin fossil record is riddled with methods that claim to evaluate whether pairs of individual fossils belong to the same species. Two such methods, log sem and the related STET method, have been introduced and used in studies of fossil hominins. Both methods attempt to quantify morphological dissimilarity for a pair of fossils and then evaluate a null hypothesis of conspecificity using the assumption that pairs of fossils that fall beneath a predefined dissimilarity threshold are likely to belong to the same species, whereas pairs of fossils above that threshold are likely to belong to different species. In this contribution, we address (1) whether these particular methods do what they claim to do, and (2) whether such approaches can ever reliably address the question of conspecificity. We show that log sem and STET do not reliably measure deviations from shape similarity, and that values of these measures for any pair of fossils are highly dependent upon the number of variables compared. To address these issues we develop a measure of shape dissimilarity, the Standard Deviation of Logged Ratios (sLR). We suggest that while pairwise dissimilarity metrics that accurately measure deviations from isometry (e.g., sLR) may be useful for addressing some questions that relate to morphological variation, no pairwise method can reliably answer the question of whether two fossils are conspecific.
New relations which concern the existence and stability of atomic nuclei are presented, together with a discussion of evidence for the relations given earlier as exhibited by the newer data on the existence of isotopes. It is shown that the four series, the helium, uranium, lithium and beryllium series, exhibit a considerable amount of regularity and are now almost continuous. The more abundant species of odd atomic number keep in general to a constant isotopic number as the atomic number increases, or else the isotopic number increases by the same amount as the atomic number. There is a general tendency as the atomic number increases for the isotopic number of the most abundant isotope of elements of both even and odd atomic number (a) to remain constant, (b) to increase at the same rate as the atomic number or (c) to decrease along a line of constant electronic number.
In this paper we show that the formalism of O. Klein's version of the five-dimensional relativity can be interpreted as a four-dimensional theory based on projective instead of affine geometry. The most natural field equations for the empty space-time case are a combination into a single invariant set of the gravitational and electromagnetic field equations of the classical relativity without modification. This seems to be the simplest possible solution of the unification problem. When we drop a restriction on the fundamental projective tensor which was imposed in order to reduce our theory to that of Klein a new set of field equations is obtained which includes a wave equation of the type already studied by various authors. The use of projective tensors and projective geometry in relativity theory therefore seems to make it possible to bring wave mechanics into the relativity scheme.
G. Suwa et al. and E. Delson's News and Views address one of the most contentious issues in palaeoanthropology, that of the boundaries (if any) between hominid species, in the context of an exciting new fossil from Konso in Ethiopia. Suwa et al. attribute this specimen to Australopithecus boisei, but note that it has some similarities to the South African robust australopithecines first described by Robert Broom more than 50 years ago.
Morphometric analysis of early Pleistocene African hominin crania in the context of a statistical (probabilistic) definition of a species
- JF Thackeray
- E Odes