Golden Ratio Research Articles

The mathematics on classical ballet: connections between the beautiful and the exact

The article addresses the intrinsic relationship between classical ballet and mathematics, exploring the presence of geometric elements and symmetry in the techniques and movements of this form of dance. The research aims to explore the potential use of the golden ratio and mathematical modelling to analyze patterns, proportions, and understand the underlying physical principles of the dancers' leaps and turns. This qualitative and applied research, based on bibliographical analysis and a case study, provides a profound insight into the interaction between art and science, demonstrating how mathematics translates into beauty and harmony in classical ballet. These conclusions enrich the understanding of the interconnection between seemingly distinct disciplines, highlighting mathematics as a form of artistic expression and classical ballet as a means of mathematical expression.

Estimation of Tooth Dimensions and Golden Divine Ratio in Extracted Human Permanent Maxillary and Mandibular Canines in a Cohort of Tamil Ethnicity.

Background Teeth serve many functions, and aesthetics is one of the most important aspects served by teeth, perceived by the limbic system of the human brain. The golden divine ratio is the unique proportion often correlated with beauty. The present study was devised to estimate the dimension of human permanent canines and approximation to the golden divine ratio. Materials and methods The present study included 47 extracted human permanent canines retrieved from the tooth repository of our institute's Department of Oral Biology. Using digital vernier calipers (Themisto TH-M61 digital vernier caliper, 0-150mm/ 6 inch, JIPVI Ecommerce Pvt. Ltd, India, 2022), the following measurements were taken: Mesio-distal and labiolingual dimensions of the crown, crown length, root length, root to crown ratio (R/C) and the tooth to root ratio (T/R). The data were analyzed using Statistical Package for Social Sciences (SPSS) software version 26. Results All the dimensions' mean and standard deviations were calculated for both maxillary and mandibular canines. While the means of mesio-distal and labiolingual dimensions of the crown approximated the values reported in the literature, there was some variation in crown and root lengths. The mean crown lengths of the upper and lower canines were 10.34mm and 9.76mm, respectively, while the root lengths were 16.52 and 15.54mm, respectively. The R/C of both sets and the T/R of the upper canine only followed the golden ratio. T/R of the lower canine was slightly higher (1.64) Conclusion Although the number of included teeth was less, owing to the fact that canines are rarely extracted, our results provided new values of canines for updation in a unique population. More studies are required for comparative anthropological data updates.

An Eye-Tracking Study on Viewer Compliance with the Gestalt Closure Principle: Analyzing the Impact of Gap Size and the Golden Ratio

Abstract The Gestalt principle of closure refers to brain activity filling in missing information to perceive objects as complete, even when partially hidden. Designers leverage this principle to imply shapes without explicitly showing them. However, evaluating viewer responses poses challenges. The author conducted a comprehensive eye-tracking experiment to determine when viewers mentally completed a gap in a circle and explored the potential influence of the golden ratio on the closure principle. The golden ratio, approximately 1.618, is associated with visual harmony and is used in art and design to create pleasing proportions. Findings showed that viewers tend to mentally close gaps up to 135° but avoid closure beyond 150°. This suggests the golden ratio’s relevance in applying the closure principle to design.

An analysis of maxillary anterior teeth dimensions for the existence of golden proportion in the representative North Indian population.

Esthetic appearance of the face is a major concern in self-image among the majority of the population. The dimensions, anatomy, and arrangement of maxillary anterior are the key factors in the esthetic appearance of the face. The present study aimed to measure the mesiodistal dimensions of maxillary anterior teeth and analyzing the golden proportion in the representative North Indian population. Subjects were seated in the upright position in a dental clinical chair. A digital Vernier caliper was used to record the mesiodistal dimensions of central incisors (CIs), lateral incisor (LI), and canine (C) teeth in maximum smile position. A total of three sets of recordings were made by a single observer to avoid the measurement bias of both left and right sides of teeth. Digital images were also captured. Obtained measurements were used for the calculation of ratios and statistical analysis was done. The prevalence of golden ratio between right CI/LI was observed in 3.1% males and 3.2% females. The calculated median ratio was 1.2 for males and 1.3 for females which differs significantly (P > 0.05) from the golden ratio. The divine proportion between visible portion of right C/LI was seen in 39.5% males and 32.3% females with calculated median value for males being 0.6 and that for females was 0.7. The prevalence of golden proportion between the CI/LI in esthetically pleasing smiles was too small (mean-3.9 in males and 2.75 in females) while the golden ratio of 0.6 was found in the majority of the population irrespective of the gender.

Magnetic properties of the frustrated Ising chain

Using the Kramers–Wannier transfer matrix method, we studied several decorated Ising chains. The exact expressions for thermodynamic characteristics, including the ground state characteristics, were obtained. We considered several modeling chains with different signs and absolute values of exchange constants for the nearest- and the next-nearest neighbors. For these models, we calculated the magnetization curves. The critical values of magnetic fields and corresponding magnetization plateau parameters were obtained. Analytic expressions for the ground state entropy were obtained for the chains with different interaction constants. The dependencies of the number of states with minimum energy (the degeneration of the ground state) as the function of the number of particles were found. It was shown that these dependencies are expressed in terms of well-known numerical sequences, namely Lucas numbers and Pell numbers, which, in the limit of a large number of particles, are proportional to the powers of the golden and silver sections. Therefore, the ground state entropy (per particle) of the systems under consideration can be described in terms of these sections and, therefore, is nonzero.

Evaluation and Comparison of Facial Appearance Using the Golden Ratio: An Anthropometric Study in Preschool and School-Going Children of Santhal Tribe in West Bengal.

Introduction Golden ratio and beauty are two inseparable sides of the same coin and have been studied for centuries by the Greeks. This divine ratio is defined as an invincible parameter in aesthetic dentistry to measure looks, symmetry, and balance. Being beautiful and handsome also boosts confidence in today's children and therefore is a top priority for young growing kids. However, there is no study done to define facial measurements based on the golden ratio in preschool and school-going children that can succor aesthetics in formative years. The purpose of this research was to evaluate facial proportions in the vertical dimension, quantify them in reference to the golden ratio, and analyze the association with gender among preschool and school-going children of the Santhal tribe in the Bankura district. Materials and methods A total of 399 subjects, 198 children of 3-5 years with primary teeth and 201 children of 6-12 years with mixed dentition, were selected from schools in villages of Bankura district, West Bengal, India. The subjects were made to relax in a sitting position and a digital vernier caliper was used to record the following vertical facial parameters: total facial height (TFH), trichion-gnathion distance (Tr-Gn), and subnasale-gnathion distance (Sn-Gn). The total facial height was correlated with sexual dimorphism and dentition. The ratio of Tr-Sn/Sn-Gn was calculated and compared with the golden ratio. The ratio was classified as normal (if it was between 1.6 and 1.699, i.e., normal to the golden ratio value), long (if it was more than 1.699, i.e., more than the golden ratio value), or short (if it was below 1.6, i.e., less than the golden ratio value). This facial analysis based on the golden ratio was correlated to sexual dimorphism and dentition. The data were recorded, compared with gender, and statistically analyzed using the unpaired t-test and Chi-square test. Results The total facial height was larger in males than females in both primary and mixed dentition; however, the value was highly significant in primary dentition. Tr-Sn/Sn-Gn ratios were lower in the longfacial categoryin both males and females in both preschool and school-going children. The values were statistically significant in these ratios for both dentitions. Conclusion The majority of children in the Santhal tribe of Bankura in West Bengal did not conform to the golden ratio and showed long faces. There was a significant association of facial features with sexual dimorphism. Clinical significance The early prediction of facial features in children and its confirmation with the established golden ratio can be considered an imperative parameter to comprehend facial aesthetics and symmetry.

Automated Maximum Torque per Ampere Identification for Synchronous Reluctance Machines with Limited Flux Linkage Information

The synchronous reluctance machine is well-known for its highly nonlinear magnetic saturation and cross-saturation characteristics. For high performance and high-efficiency control, the flux-linkage maps and maximum torque per ampere table are of paramount importance. This study proposes a novel automated online searching method for obtaining accurate flux-linkage and maximum torque per ampere Identification. A limited 6 × 2 dq-axis flux-linkage look-up table is acquired by applying symmetric triangle pulses during the self-commissioning stage. Then, three three-dimensional modified linear cubic spline interpolation methods are applied to extend the flux-linkage map. The proposed golden section searching method can be easily implemented to realize higher maximum torque per ampere accuracy after 11 iterations with a standard drive, which is a proven faster solution with reduced memory sources occupied. The proposed algorithm is verified and tested on a 15-kW SynRM drive. Furthermore, the iterative and execution times are evaluated.

GOLDEN PROFILOGRAPHY: AESTHETICS OF FACIAL PROPORTIONS IN A GROUP OF STUDENTS FROM THE FACULTY OF DENTISTRY OF THE UNIVERSITY OF CUENCA

Introduction and objective. The approach to the harmony of a facial profile involves multiple studies, of which in this study the golden ratio and profiling are analysed to see the aesthetics and facial pattern of the patients. Golden Profilography is defined as the drawing of the vertical facial profile of aesthetic proportions obtained by means of the golden section from a photograph of the facial profile. Material and methods. The Golden Profilography is a photograph of the facial profile with aesthetic proportions obtained from measurements of the patient's facial profile enlarged to life size, from which we take the measurement corresponding to the vertical nasal axis divided by the golden section. Results. We carried out this study in students of the Seventh Cycle of the Faculty of Dentistry of the University of Cuenca from different regions of the country, who underwent a radiographic examination in order to determine their profile, clearly for academic purposes. A total of 27 students. All of them were profiled in the same way using the aforementioned software such as Nemocast. Conclusions. The facial profile obtained by the selected software verifies the golden aesthetics and it is evident that in a group of students a higher index of a mesofacial profile is reflected, which is historically known as a normal profile.

Balanced Nontransitive Dice: Existence and Probability

A triple $(A,B,C)$ of dice is called nontransitive if each of $P(A<B)$, $P(B<C)$, and $P(C<A)$ is greater than $\frac12$ and called balanced if $P(A<B)=P(B<C)=P(C<A)$. From the result of Trybuła, it is known that $P(A<B)$ is less than $\frac{-1+\sqrt{5}}{2}$, the golden ratio, for every balanced nontransitive triple $(A,B,C)$ of dice. Schaefer asked whether this upper bound is tight, and Hur and Kim conjectured that the upper bound can be reduced to $\frac12+\frac19$. In this paper, we characterize all possible probabilities $P(A<B)$ for balanced nontransitive triples $(A,B,C)$ of dice. Precisely, we prove that, for every rational $\frac12 <q<\frac{-1+\sqrt{5}}{2}$, there exists a balanced nontransitive triple $(A,B,C)$ of dice with $P(A<B)=q$, which disproves Hur and Kim's conjecture and answers Schaefer's question.
 We also characterize all triples $(m,n,\ell)$ of positive integers such that there exists a balanced nontransitive triple $(A,B,C)$ of dice, where $A$, $B$, and $C$ are $m$-sided, $n$-sided, and $\ell$-sided dice, respectively. This generalizes Schaefer and Schweig's result showing the existence of a balanced nontransitive triple of $n$-sided dice for every $n\ge 3$.

Lithium-Ion Supercapacitors and Batteries for Off-Grid PV Applications: Lifetime and Sizing

The intermittent nature of power generation from photovoltaics (PV) requires reliable energy storage solutions. Using the storage system outdoors exposes it to variable temperatures, affecting both its storage capacity and lifespan. Utilizing and optimizing energy storage considering climatic variations and new storage technologies is still a research gap. Therefore, this paper presents a modified sizing algorithm based on the Golden Section Search method, aimed at optimizing the number of cells in an energy storage unit, with a specific focus on the unique conditions of Denmark. The considered energy storage solutions are Lithium-ion capacitors (LiCs) and Lithium-ion batteries (LiBs), which are tested under different temperatures and C-rates rates. The algorithm aims to maximize the number of autonomy cycles—defined as periods during which the system operates independently of the grid, marked by intervals between two consecutive 0% State of Charge (SoC) occurrences. Testing scenarios include dynamic temperature and dynamic load, constant temperature at 25 °C, and constant load, considering irradiation and temperature effects and cell capacity fading over a decade. A comparative analysis reveals that, on average, the LiC storage is sized at 70–80% of the LiB storage across various scenarios. Notably, under a constant-temperature scenario, the degradation rate accelerates, particularly for LiBs. By leveraging the modified Golden Section Search algorithm, this study provides an efficient approach to the sizing problem, optimizing the number of cells and thus offering a potential solution for energy storage in off-grid PV systems.

Fractal penis and golden penis. The hidden dimensions of penile tissue

The penis is an organ with a unique and complex structure. It is formed by a fractal architecture, meaning its design is based on highly complex repetitive patterns. In addition, its dimensions follow what are known as "golden proportions", also known as "the divine proportion". These proportions are a mathematical relationship that is found in many natural structures and, in the case of the penis, provide stability and improve its performance. This design is not the result of chance, but rather the result of an underlying mathematical message. Fractal geometry, based on complex numbers, is a key tool in understanding the internal structure of the penis, which cannot be described using traditional Euclidean geometry. In addition to fractal geometry, the presence of golden proportions in the penis suggests that its design is influenced by mathematics. To fully understand this relationship, additional studies are required and the creation of innovative mathematical models that include tools such as artificial intelligence. In conclusion, the penis is an example of how mathematics and nature are closely related and how mathematical complexity can improve the functioning of natural structures.

ANALYSIS OF THE RELATIONSHIP BETWEEN THE GOLDEN RATIO AND THE FACIAL PARAMETER THROUGH CBCT IN SEVENTH CYCLE STUDENTS OF THE FACULTY OF DENTISTRY OF THE UNIVERSITY OF CUENCA, PERIOD SEPTEMBER 2023 - FEBRUARY 2024.

The clinical skill necessary to modify dentofacial structure through orthodontic procedures, maxillofacial surgery or plastic surgery involves having a deep understanding of facial harmony. This encompasses the evaluation of facial aesthetics, proportions and symmetry to achieve optimal results.To analyze the relationship between the elements of the golden ratio with the facial parameter using CBCT and lateral cephalic of the seventh cycle “A” students of the Faculty of Dentistry of the University of Cuenca. The group of participants in the study was made up of students of the seventh cycle Orthodontics course at the University of Cuenca, which includes 34 people between the ages of 20 and 30 years, belonging to both sexes, with and without orthodontic treatment. with class I malocclusion complete dentition. The facial type of each patient was analyzed in relation to the presence or absence of the divine proportion (1.618) between head height and width. bizygomatic. Additionally, a data collection matrix was developed to evaluate craniofacial anthropometric measurements. It was observed that 0% are brachyfacial, 0% are mesofacial and 100% are dolichofacial. Furthermore, the vertical relationship and harmonic profile component of the golden ratio was present in a higher percentage (66.66%) in dolichofacial individuals when compared to the other facial types.The morphological facial type that had the most predominance in our study was the dolichofacial with absence of brachyfacials and mesofacials, in the students there was more golden ratio in the vertical relationship and harmonic profile regardless of the sex of the students.

Two subgradient extragradient methods based on the golden ratio technique for solving variational inequality problems

Two subgradient extragradient methods based on the golden ratio technique for solving variational inequality problems

Multi-strategy synthetized equilibrium optimizer and application.

Improvement on the updating equation of an algorithm is among the most improving techniques. Due to the lack of search ability, high computational complexity and poor operability of equilibrium optimizer (EO) in solving complex optimization problems, an improved EO is proposed in this article, namely the multi-strategy on updating synthetized EO (MS-EO). Firstly, a simplified updating strategy is adopted in EO to improve operability and reduce computational complexity. Secondly, an information sharing strategy updates the concentrations in the early iterative stage using a dynamic tuning strategy in the simplified EO to form a simplified sharing EO (SS-EO) and enhance the exploration ability. Thirdly, a migration strategy and a golden section strategy are used for a golden particle updating to construct a Golden SS-EO (GS-EO) and improve the search ability. Finally, an elite learning strategy is implemented for the worst particle updating in the late stage to form MS-EO and strengthen the exploitation ability. The strategies are embedded into EO to balance between exploration and exploitation by giving full play to their respective advantages. Experimental results on the complex functions from CEC2013 and CEC2017 test sets demonstrate that MS-EO outperforms EO and quite a few state-of-the-art algorithms in search ability, running speed and operability. The experimental results of feature selection on several datasets show that MS-EO also provides more advantages.

The Impact of Golden Ratio Application on User Satisfaction: A Study on Horizontal Scrolling in User Interface (UI) Design

The utilization of the golden ratio scheme is well-established for enhancing aesthetic appeal and visual harmony in various design contexts. However, its application in user interface (UI) design, particularly concerning mobile screen elements, has been less explored. The potential benefits of incorporating the golden ratio into UI elements in mobile applications remain unclear, primarily due to a lack of comprehensive understanding. This study investigates the impact of implementing the golden ratio in UI design, including its influence on user satisfaction. A sample of 114 participants, comprising designers, participated in the study. Result: the simple linear regression analysis revealed a 7.5% coefficient, indicating a positive relationship between the integration of the golden ratio and heightened user satisfaction. These findings offer valuable insights for UI designers, especially for designing mobile applications requiring expanded space, such as catalog apps. Nevertheless, further research is necessary to explore the golden ratio’s application in UI design for devices with smaller screens, like smartwatches or digital cameras.

Unraveling the "golden ratio": a pilot study investigating acute-to-chronic workload ratio in breast cancer patients undergoing active treatment.

Training load monitoring is a common practice in sports medicine for supporting athletes' health and performance. Despite progress in exercise oncology research for breast cancer patients, training load monitoring is underutilized. This study retrospectively investigated the relationship between maintained training load within a defined range and physical and health outcomes of ten breast cancer patients during active anticancer treatment who underwent a 12-week exercise program. Intervention consisted of endurance and resistance training, three times a week, with each session lasting 30-45min. Assessments were conducted at baseline, 6 and 12 weeks after enrollment, evaluating physical function (6-min walk test-6MWT, and sit-to-stand), muscle strength, body composition, sleep quality (Pittsburgh Sleep Quality Index-Pittsburgh Sleep Quality Index), quality of life (EORTC-QLQ-C30), heart rate variability and physical activity levels (International Physical Activity Questionnaire-International Physical Activity Questionnaire). The Physiological Cost Index/Energy Expenditure Index (PCI/EEI) was estimated using the 6MWT and Heart rate. Training load monitoring was performed by session rating of perceived exertion (sRPE, relative intensity multiplying with session duration). Acute-to-Chronic Workload Ratio (ACWR) (7:28, rolling average) was calculated accordingly. Analyses were performed within-subjects across time points and between-subjects, comparing those who maintained from weeks 6-12 an ACWR of 0.8-1.3 with those who did not. Adherence rates were similar between groups. Physical function improved in the total sample with large effect sizes (Δ6MWT = 56.5m [95%CI: 6-100m], effect size [w] = 0.52, p = 0.006; ΔSit-to-Stand = 1.5 [95%CI: 1-5], effect size [w] = 0.681, p < 0.001), demonstrating greater changes in patients with higher ACWR. Sleep quality improvements were higher in the appropriate ACWR group (p = 0.016). A positive correlation was demonstrated between global health status and 6MWT change from baseline to 12 weeks (ρ = 0.689, p = 0.04). Despite a small sample size, patients maintaining sufficient relative training load presented greater physical fitness and sleep quality improvements. Thus, training load monitoring may enhance exercise program benefits in breast cancer patients under active treatment.

A performance analysis of a hybrid golden section search methodology and a nature-inspired algorithm for MPPT in a solar PV system

A performance analysis of a hybrid golden section search methodology and a nature-inspired algorithm for MPPT in a solar PV system

Золотое сечение в римском эпосе (на материале «Фиваиды» Стация)

Golden section poetic phenomena studies comprise a wide range of topics, starting from basic prosodic elements to the work’s entire composition structure. The primary purpose of this paper is to establish the connection between the golden section and the literary composition and form and to analyze it on the material of the episode of aristeia and katabasis of Amphiaraus in Statius’ “Thebaid” (VII, 690–823). The following objectives are included: to define and measure the segments of the golden section; to determine the marking points dividing the segments which, as a rule, contain salient features — literary devices, hero’s speech, poet’s apostrophe etc.; to identify the functions and role of the golden section in the episode’s composition. The methodology uses general scientific methods such as philological textual analysis and mathematical tools. The supportive research is grounded on parallel comparison of episodic structure, framed by visual techniques and segment dividing points. It is observed that proportional segmentation is achieved by typical epic techniques such as epitaph or catalogue of victims serving the closing part of the narrative. As for the simile’s structural functions, they mark the poetic period by opening and, more frequently, ending it and highlight the culmination moments functioning similar to the hero’s speeches. The overall examination concludes that the golden section in “Thebaid” manifests itself with striking precision in relation to separate parts and to the whole poem as well. Similarly, it applies to corresponding stages of emotional tension and moments of culmination and the correlating literary devices and techniques. The utilization of the golden section provides the symmetrical composition and the triadic structure of the episode, demonstrating the poetic skillfulness of Statius and mastery of attracting the reader’s attention to the most significant points.

ОБГРУНТУВАННЯ КОНЦЕПЦІЇ МОДЕЛЮВАННЯ УПРАВЛІННЯ ПІДПРИЄМСТВАМИ ТУРИСТИЧНО-РЕКРЕАЦІЙНОГО КОМПЛЕКСУ

The article substantiates the concept of management of enterprises of the tourist and recreational complex as an effective model, which is built on the basis of an integral indicator of assessing the status of enterprises of the TRC. Management of the enterprises of TMC is impossible without obtaining an assessment of the current status, which is characterized by a set of indicators and factors that are interdependent and contradictory, so it makes sense to build an integral indicator for assessing the status of TMC enterprises by: allocating indicators of the performance of TMC enterprises and building a binary tree, the vertices of which are the selected factors; weighting the vertices of the resulting binary tree, which is carried out on the basis of the principle of "golden section" and applying the built weighted tree to decision-making. In the concept of modeling the management of retail and entertainment complex enterprises, rating assessment is used to determine the rating of a retail and entertainment complex enterprise and is implemented in the form of a dynamic rating model. The proposed conceptual model should ensure the adoption of strategic management decisions, improvement of the final results of activity, rationalization of the structure of formation of resources and costs, achievement of the goals of strategic development of the enterprises of the shopping and entertainment complex through the mechanism of strategic development, which includes methodological support, which contains an algorithm for determining the level and vector of strategic development of the enterprises of the shopping and entertainment complex, assessment of the strategic position of enterprises and selection of strategic alternatives of the enterprises of the shopping and entertainment complex, as well as instrumental support. The algorithm for determining the level of strategic development of retail and entertainment complex enterprises consists of seven stages and begins with diagnostics of factors influencing the efficiency of functioning and strategic development of retail and entertainment complex enterprises and ends with the definition of a specific strategy for managing retail and entertainment complex enterprises based on the level of strategic development of retail and entertainment complex enterprises.

Center of mass of 2D and 3D objects and the golden ratio

This study delves into the problem of finding the centre of mass for excised polygons in two and three dimensions, revealing intriguing connections and occurrences. We begin with rectangle and rhombus with sides in the ratio of the golden ratio, showcasing certain excised polygons with their centre of mass on the edge. Furthermore, we extend this analysis to 3D shapes like cuboids, pyramids, and cylinders, discovering three ways of excision, resulting in balanced objects on the edge. Exploring physics related to the centre of mass uncovers a fascinating mathematical coincidence between the golden ratio and the Fibonacci series. Additionally, we demonstrate and verify the center of mass location on the edges through 3D-printed objects.