Golden Ratio Research Articles

There Is no Universal Standard of Beauty.

This article discusses the various factors that contribute to facial beauty and the different contexts in which they are significant. The authors examine the roles of the golden ratio, ideal proportions, symmetry, and specific facial features in what constitutes an attractive face. They also investigate the influence of cultural and evolutionary perspectives on facial beauty and the impact of social media on our perception of beauty. The article provides a comprehensive overview of the topic and emphasises the significance of facial beauty and its impact on our lives. The authors acknowledge that the concept of beauty is difficult to define with absolute precision, and that the objective-subjective argument regarding facial beauty remains unsolved. However, they argue that physical attributes undoubtedly influence perceptions of attractiveness, but beauty extends beyond surface-level features. They also suggest that embracing diversity in beauty standards fosters a more inclusive society that celebrates the unique qualities of every person.Level of Evidence V This journal requires that authors assign a level of evidence to each article. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors www.springer.com/00266 .

Evaluation of the Golden Proportion, Golden Percentage, and Recurring Esthetic Dental Proportion in Kenyans of African Descent.

To evaluate the validity of the Golden Proportion, Golden Percentage, and Recurring Esthetic Dental (RED) Proportion among Kenyans of African descent with naturally well-aligned teeth. Standardized frontal photographic images of the smiles of 175 participants aged 18-35 years were obtained, and Adobe Photoshop was used to analyze and measure the frontal widths of the maxillary central and lateral incisorsand canines in triplicate. The average teeth widths were calculated to determine the existence of the Golden Proportion, Golden Percentage, and RED Proportion, and their validity using independent sample t-tests to compare the differences in the mean teeth widths at α < 0.05. The number of male and female participants was 107 (61.1%) and 68 (38.9%), respectively. The Golden Proportion between the maxillary central and lateral incisors was found in 4.0% on the right and 2.8% on the left of all the participants, but between the maxillary lateral incisors and canines was found in only 0.6% on the right of male participants (p < 0.0001). The RED Proportion between the maxillary lateral and central incisors was in the range of 67%-70%, and between the canines and lateral incisors was 82%-84% (p < 0.0001). The proportion of RED was not constant, and gradually increased distally. The Golden Percentage of 15% was observed in the lateral incisors bilaterally; however, in the central incisors and the canines, the Golden Percentage was 22% and 12%, respectively. The Golden and RED Proportions were invalid determinants of anterior teeth proportions. The Golden Percentage existed only in the lateral incisors. The Golden Proportion, RED Proportion, and Golden Percentage theories may not be applicable to all populations when designing smiles. Racial and ethnic backgrounds are important considerations to establish objective quantifiable values of anterior tooth proportions that are beneficial for esthetic restorations.

Predicting effects of tunnel throttling of annular freeway vehicular flow by a continuum model

Fluid flow throttling is common in industrial and building services engineering. Similar tunnel throttling of vehicular flow is caused by the abrupt number reduction of roadway lane, as the tunnel has a lower lane number than in the roadway normal segment. To predict the effects of tunnel throttling of annular freeway vehicular flow, a three-lane continuum model is developed. Lane III of the tunnel is completely blocked due to the need of tunnel rehabilitation, etc. There exists mandatory net lane-changing rate from lane III to lane II just upstream of the tunnel entrance, which is described by a model of random number generated through a golden section analysis. The net-changing rate between adjacent lanes is modeled using a lane-changing time expressed explicitly in algebraic form. This paper assumes that the annular freeway has a total length of 100 km, a two-lane tunnel of length 2 km with a speed limit of 80 km/h. The free flow speeds on lanes I, II and III are assumed to be 110, 100 and 90 km/h respectively. Based on the three-lane continuum model, numerical simulations of vehicular flows on the annular freeway with such a tunnel are conducted with a reliable numerical method of 3rd-order accuracy. Numerical results reveal that the vehicular flow has a smaller threshold of traffic jam formation in comparison with the case without tunnel throttling. Vehicle fuel consumption can be estimated by interpolation with time averaged grid traffic speed and an assumed curve of vehicle performance. The vehicle fuel consumption is lane number dependent, distributes with initial density concavely, ranging from 5.56 to 8.00 L. Tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling.

On Absolute Continuity of the Erdös Measure for the Golden Ratio, Tribonacci Numbers, and Second-Order Markov Chains

On Absolute Continuity of the Erdös Measure for the Golden Ratio, Tribonacci Numbers, and Second-Order Markov Chains

AI-enhanced "Two-thirds Guidelines" for Lipolifting: Addressing Multiple Hallmarks of Facial Aging.

Facial aging involves complex changes such as volume loss, ligament weakening, and skin quality alterations. The "two-thirds guidelines" emerge as a novel strategy to combat these aging signs, drawing from an extensive analysis of 2800 facial fat grafting procedures conducted over two decades. Guided by facial lipolifting data, including patient age, fat type (microfat and nanofat), and injection depth, this study devises a systematic framework for multilayer fat rejuvenation and ligament restoration. The two-thirds guidelines advocate injecting two-thirds of the patient's age for microfat and one-third for nanofat, with specific injection codes for lower, middle, and upper facial regions. A prospective study involving 400 patients confirms the efficacy of the two-thirds guidelines. However, applicability may vary for patients outside SD ranges, particularly concerning facial proportions and body mass index. Patients within the golden ratio range (1.4-1.9) report high satisfaction rates and a 50% fat graft uptake, with minimal complications. For patients outside this range, an artificial intelligence (AI) program was implemented. The two-thirds guidelines offer a comprehensive approach to facial rejuvenation, addressing volume loss, ligament weakening, and skin quality. They are applicable in early aging stages, promising enduring and natural outcomes while mitigating effects of weight fluctuations. These guidelines provide a safe, replicable, and adaptable approach to facial fat grafting, either standalone or in combination with facelift techniques, with minimized overfilling risks. A dataset obtained from 2800 patients serves as the foundation for developing an AI program tailored to aid doctors in diagnosing and treating similar cases.

ϕ−Multiplicative Calculus

In this paper, we present a novel mathematical framework termed “ϕ-multiplicative calculus”, which serves as a Golden Fibonacci calculus to fundamental concepts in multiplicative calculus. This innovative calculus introduces a parameter ϕ (Golden ratio), offering a nuanced extension of traditional calculus. Our work encompasses the establishment of ϕ-multiplicative calculus and the demonstration of essential theorems concerning derivatives, integrals, and their operation properties within this mathematical framework. The paper contributes to the academic discourse by providing a comprehensive exploration of the proposed ϕ-multiplicative calculus, presenting a robust foundation for further investigations in this specialized mathematical domain.

Rapid computation of Hopf bifurcation points of continuous and discrete systems through minimization

For an autonomous nonlinear system, the Hopf bifurcation point along the equilibrium path is a critical feature that indicates whether the values of the parameters change from exhibiting fixed-point behavior to having a periodic orbit. To solve these problems, we developed a method of transforming an eigenvalue problem based on the Jacobian matrix at equilibrium into a minimization problem, enabling the rapid identification of a solution. Specifically, this generalized eigenvalue problem is solved by identifying the vector variable after reducing the number of eigenequations by one in the nonhomogeneous linear system. This can be achieved by normalizing the value of a selected nonzero component of the eigenvector and then moving the column containing this component to the other side of the equation. An appropriate merit function was established in terms of the Euclidean norm of the eigenequation, and this merit function was minimized using the golden section search algorithm to determine the eigenparameters of the bifurcation point. The accuracy of the method for identifying the parameter values and the corresponding imaginary eigenvalues at the Hopf bifurcation points was evaluated for numerous examples for both the continuous and discrete systems. The method was both fast and accurate. Moreover, its stability in the presence of noise was investigated, and the method was robust.

5G Radiation Source Location Based on Passive Virtual Aperture Technology by Single-Satellite

With the development of 5th-Generation Mobile Communication (5G) technology and the deployment of low-Earth orbit satellites, using satellites to locate 5G radiation sources is of great significance in commerce and the military as an important task of integrated sensing and communication. Recently, passive virtual aperture technology has been introduced into passive location to improve accuracy, but the existing method, using matched filters to search the Doppler information to realize the location, has the disadvantages of high complexity and poor range resolution. In this paper, an improved 5G radiation source location based on a virtual aperture is proposed, which uses the improved Golden Section search-fractional Fourier algorithm (GSS-FRFT) to improve the existing passive virtual aperture location methods. First, the received signals are coherently accumulated to convert the time gain into spatial gain, and the subcarrier phase information is extracted by Fast Fourier Transform based on the 5G signal characteristics to obtain the azimuth signal. Then, an improved high-order GSS-FRFT algorithm is proposed to analyze the Doppler information, and signal focusing and satellite ephemeris data are used to estimate the effective velocity and solve the radiation source location. The simulation results show that the proposed method can improve the location accuracy compared with other single-satellite location methods and has high resolution, high accuracy and low complexity compared with the existing passive virtual aperture location method.

Weyl superconductivity and quasiperiodic Majorana arcs in quasicrystals

Weyl superconductivity is a topological phase in three-dimensional crystals in which the Weyl equation describes quasiparticle excitation near band-touching points in momentum space called Weyl nodes. For quasicrystals which lack translational symmetry, a theory of Weyl superconductivity has not been established, in spite of recent extensive studies on quasicrystalline topological phases. Here, we demonstrate the occurrence of quasicrystalline Weyl superconductivity by extending the definition of Weyl superconductivity to periodically stacked, two-dimensional superconducting quasicrystals. We identify quasicrystalline Weyl nodes—topologically protected point nodes in one-dimensional momentum space corresponding to the stacking direction—in terms of a topological invariant given by a change in the Bott index in quasicrystalline layers. We find that these Weyl nodes exist in pairs and that Majorana zero-energy modes protected by the nonzero Bott index between a pair of quasicrystalline Weyl nodes appear on surfaces. These Majorana zero modes form an infinite number of arcs in momentum space, densely and quasiperiodically distributed as a function of momentum in the direction of surfaces within each quasicrystalline layer. In Ammann-Beenker (Penrose) quasicrystals, the quasiperiodicity of Majorana arcs is governed by the silver (golden) ratio associated with the quasicrystalline structure. Published by the American Physical Society 2024

Comparison of the width-to-height ratio in maxillary anterior teeth in different populations: A systematic review and meta-analysis

The width-to-height (W/H) ratio values of maxillary anterior teeth are parameters used in esthetic dentistry. However, a meta-analysis of studies evaluating these parameters in individuals from different countries, taking sex into account, is lacking. The purpose of this systematic review and meta-analysis was to determine and compare the W/H ratio values seen in the maxillary anterior teeth of different populations. A systematic search was performed using the PubMed/Medline, Google Scholar, and Web of Science databases to identify English and non-English language articles reporting the golden ratio, golden percentage, tooth ratio, tooth size, W/H ratio, maxillary tooth width, maxillary tooth height, maxillary anterior tooth size, and dental esthetics. The titles, full text, and abstracts were scanned by 2 investigators independently to identify articles that fulfilled the inclusion criteria. Using a meta-analysis software program, data extracted from each selected study were statistically combined using the random-effects model, and the weighted mean differences were calculated. The search protocol resulted in a total of 987 articles, but only 15 articles fulfilling the inclusion criteria were included in the meta-analysis. While the sample size varied between 50 and 412 in the 15 analyzed studies, a total of 2304 individuals from 12 countries were evaluated. As a result of the evaluation of the general W/H means of the maxillary anterior teeth, the largest teeth were seen in Korea and Pakistan, and the general mean W/H ratios of the central, lateral, and canine teeth showed no statistically significant difference between sexes (P>.05). As a result of the meta-analyses of the W/H ratios of the maxillary central, lateral, and canine teeth, the differences between populations were found not to be statistically significant.

Златното сечение в педагогиката: приложението му в сугестопедичната методика на проф. д-р Георги Лозанов и проф. д-р Евелина Гатева

The article examines the application of the golden ratio to the suggestopedic methodology of Prof. Dr. Georgi Lozanov and the contribution of Prof. Dr. Evelina Gateva to the globalized artistic construction of the entire educational process. According to the research and application of the methodology, both during the existence of the Institute of Suggestology and Suggestopedia (1966–1991) and in its application today, the golden section, acting together with the other six laws of suggestopedia, supports the calm and full-fledged assimilation of a huge volume of material, which is stored in long-term memory and is used over time as full-fledged knowledge, and at the same time participates in revealing the potential reserves of the personality. The work of the teacher in the preparation for the learning process is fundamental—the coordination of all its constituent parts so as to offer the learners harmoniously arranged content, beauty, and ease in the learning process.

A Golden Ratio Algorithm With Backward Inertial Step For Variational Inequalities

In this paper we study the convergence analysis of a Golden Ratio Algorithm with a backward inertial step and a fully adaptive step size procedure for the purpose of approximating solutions of variational inequalities in Hilbert spaces. We present a weak convergence result when the operator is quasi-monotone and locally Lipschitz continuous, and a strong convergence result in the setting of strong pseudo-monotonicity. Our algorithm features one operator evaluation and one projection computation at the current iteration. We recover interesting algorithms in the literature, for instance, the Golden Ratio Algorithm. Numerical experiments were conducted to validate the theoretical convergence analysis.

A Method with Double Inertial Type and Golden Rule Line Search for Solving Variational Inequalities

In this work, we study a new line-search rule for solving the pseudomonotone variational inequality problem with non-Lipschitz mapping in real Hilbert spaces as well as provide a strong convergence analysis of the sequence generated by our suggested algorithm with double inertial extrapolation steps. In order to speed up the convergence of projection and contraction methods with inertial steps for solving variational inequalities, we propose a new approach that combines double inertial extrapolation steps, the modified Mann-type projection and contraction method, and the line-search rule, which is based on the golden ratio (5+1)/2. We demonstrate the efficiency, robustness, and stability of the suggested algorithm with numerical examples.

Serendipitous relationship between discrete distribution models representing random orientations of fillers in composite materials and the golden ratio

The purpose of this study is to establish simple discrete distribution models capable of expressing the two and three-dimensional random orientation states of fillers in composite materials. Initially, a new method is proposed to divide the elastic constants into isotropic and anisotropic parts. Employing this method, for the composite material containing ellipsoidal fillers with various orientations in the material, the macroscopic elastic constants and thermal expansion coefficients of the material are derived based on the Mori–Tanaka method. Subsequently, the analysis of the macroscopic properties is performed for fillers oriented two and three-dimensional discrete distributions. As a result, it is found that when both angular pitches of the azimuth angle and the rotational angle of the fillers in a plane are at most 72°, the macroscopic elastic constants and the thermal expansion coefficients of the material exhibit in-plane isotropy, leading to the realization of a two-dimensional random state. Furthermore, if the angular distribution of the zenith angle and azimuth angle corresponds to the vertices of the regular dodecahedron and icosahedron, and the angular pitch of the rotational angle of the fillers is at most 72°, the macroscopic properties of the material become isotropic, resulting in a three-dimensional random state. Notably, the angle of 72° and the orientation angles of the vertices of the regular dodecahedron and icosahedron are associated with the golden ratio. Therefore, this analysis suggests a serendipitous relationship between the golden ratio and the angular pitch that simplifies the random orientation distribution of fillers.

Golden ratio of the r+/r- for the structure-selectivity in the thermoelectric BaZn2-xCdxSb2 system

A total of five Zintl phase compounds in the solid-solution BaZn2-xCdxSb2 (0.10(2) ≤ x ≤ 2) system have been successfully synthesized by the molten Pb-flux method. Both powder X-ray and single-crystal X-ray diffraction analyses proved that the title compounds displayed the phase transition between the BaCu2S2-type phase (space group Pnma, Z = 4) and the CaAl2Si2-type phase (space group P3¯m1, Z = 1) depending on the Zn/Cd mixed ratio. Furthermore, this kind of structure selectivity was dictated by the radius ratio criterion between the cationic and the anionic elements (r+/r–): three Zn-rich compounds with the r+/r– > 1 (BaZn1.90(1)Cd0.10Sb2, BaZn1.75(2)Cd0.25Sb2, and BaZn1.62(2)Cd0.38Sb2) preferred to adopt the BaCu2S2-type phase, while two Cd-rich compounds with the r+/r– ≤ 1 (BaZn0.16(2)Cd1.84Sb2 and BaCd2Sb2) preferred to crystallize in the trigonal CaAl2Si2-type phase. A series of TB-LMTO calculations using the two hypothetical models: BaZn1.5Cd0.5Sb2 and BaZn0.5Cd1.5Sb2, proved that a resonance peak in the density of states of BaZn1.5Cd0.5Sb2 implied the enhanced Seebeck coefficient of the three Zn-rich compounds, and the reduction of band gap observed in a band structure of BaZn0.5Cd1.5Sb2 influenced the electrical conductivity of the Cd-rich BaZn0.16(2)Cd1.84Sb2. Temperature-dependent thermoelectric property measurements showed the maximum ZT of 0.54 for the quaternary BaZn1.62(2)Cd0.38Sb2, which should mainly be attributed to the reduced κtot due to the Zn and Cd disordering in the anionic frameworks.

Golden ratio-based leverage targeting and the ESG performance of US and European listed firms

Our paper aims to assess whether golden ratio-based leverage targeting is linked to better ESG performance. To answer this research question, we study how ESG performance affects the distance between the leverage targets and the levels defined by the golden ratio and examine the temporal dynamics of leverage in relation to the distance between leverage targets and the golden ratio levels. Our main findings show that when firms in the European manufacturing sector have better ESG performance, they choose to have leverage targets closer to the levels defined by the golden ratio. In this case, firms adjust their book leverage considering the distance between the leverage targets and the golden ratio levels. Our results highlight that prudential sustainability reporting regulation and higher ESG exposure can guide firms toward a more harmonized capital structure.

Throughput fairness in backscatter-assisted cognitive radio networks with short packets

This work studies the throughput fairness among multiple Internet of Things (IoT) nodes in a backscatter assisted cognitive radio network, where the primary transmitter conveys its long-packet information to its receiver while multiple IoT nodes alternate backscattering their short-packet information to the information receiver via backscatter communication (BackCom). Specifically, we devise a non-convex problem aimed at ensuring the throughput fairness among IoT nodes concerning their transmitted data bits, by jointly optimizing the short-packet blocklength, packet error rate, and power reflection coefficient of each IoT node. Employing the block-coordinated-decent (BCD), the original problem is decoupled into two subproblems, both of which are solved by the proposed golden section based iterative algorithm and the proposed successive convex approximation (SCA) based iterative algorithm, respectively. Then a BCD based iterative algorithm is developed to solve the original problem. Simulations demonstrate the rapid convergence and superiority of the proposed algorithm over several baseline schemes in achieving the fairness of transmission bits.

A machine learning decision criterion for reducing scan time for hyperspectral neutron computed tomography systems

We present the first machine learning-based autonomous hyperspectral neutron computed tomography experiment performed at the Spallation Neutron Source. Hyperspectral neutron computed tomography allows the characterization of samples by enabling the reconstruction of crystallographic information and elemental/isotopic composition of objects relevant to materials science. High quality reconstructions using traditional algorithms such as the filtered back projection require a high signal-to-noise ratio across a wide wavelength range combined with a large number of projections. This results in scan times of several days to acquire hundreds of hyperspectral projections, during which end users have minimal feedback. To address these challenges, a golden ratio scanning protocol combined with model-based image reconstruction algorithms have been proposed. This novel approach enables high quality real-time reconstructions from streaming experimental data, thus providing feedback to users, while requiring fewer yet a fixed number of projections compared to the filtered back projection method. In this paper, we propose a novel machine learning criterion that can terminate a streaming neutron tomography scan once sufficient information is obtained based on the current set of measurements. Our decision criterion uses a quality score which combines a reference-free image quality metric computed using a pre-trained deep neural network with a metric that measures differences between consecutive reconstructions. The results show that our method can reduce the measurement time by approximately a factor of five compared to a baseline method based on filtered back projection for the samples we studied while automatically terminating the scans.

Comparison of the Size of the Facade Element of Vernacular Architecture of Lamban Pesagi with the Golden Section as an Initial Variable Parametric Design Method

Comparison of the Size of the Facade Element of Vernacular Architecture of Lamban Pesagi with the Golden Section as an Initial Variable Parametric Design Method

Calculating toxic pressure for mixtures of endocrine disruptors

Incidence of autoimmune disorders, birth defects, and neurological diseases rose over the past 50 years due to increasing variety and quantity of pollutants. To date, there appear few methods capable to evaluate and predict mixture effects by endocrine disruptors (EDs). For the first time, we have developed calculus to determine mixture effects by all kinds of EDs.Our method uses the golden ratio ϕ and draws from bifurcation and chaos theory. Using also the concept of molecular mimicry, we developed the equation: effect=100%1+e5·∑KiCi−niϕ3. We successfully tested the equation using a range of cohort studies and biomarkers, and for different pollutants like heavy metals, thyroid hormone mimickants, chromate/chlorate, etc.The equation is simple enough to use with only minor prior knowledge and understanding of basic algebra. The method is universal and calculation is data 'light', requiring only pollutant concentrations [C], potencies K and an integer n for endocrinal involvement. This study offers a comprehensive framework to assess the health effects of pollutant exposure across diverse populations, envisioning far-reaching impact, and presenting practical examples and insights.