Golden Ratio Research Articles

Integrated Transcriptomic and Metabolomic Analysis Reveal the Dynamic Process of Bama Hemp Seed Development and the Accumulation Mechanism of α-Linolenic Acid and Linoleic Acid.

Bama County is a world-famous longevity county in the Guangxi Province, China. Bama hemp is a traditional seed used in hemp cultivation in the Bama County. The seeds contain abundant unsaturated fatty acids, particularly linoleic acid (LA) and linolenic acid in the golden ratio. These two substances have been proven to be related to human health and the prevention of various diseases. However, the seed development and seed oil accumulation mechanisms remain unclear. This study employed a combined analysis of physiological, transcriptomic, and metabolomic parameters to elucidate the fatty acid formation patterns in Bama hemp seeds throughout development. We found that seed oil accumulated at a late stage in embryo development, with seed oil accumulation following an "S″-shaped growth curve, and positively correlated with seed size, sugar content, protein content, and starch content. Transcriptome analysis identified genes related to the metabolism of LA, α-linolenic acid (ALA), and jasmonic acid (JA). We found that the FAD2 gene was upregulated 165.26 folds and the FAD3 gene was downregulated 6.15 folds at day 21. Metabolomic changes in LA, ALA, and JA compounds suggested a competitive relationship among these substances. Our findings indicate that the peak period of substance accumulation and nutrient accumulation in Bama hemp seeds occurs during the midstage of seed development (day 21) rather than in the late stage (day 40). The results of this research will provide a theoretical basis for local cultivation and deep processing of Bama hemp.

Go(Φ)d is Number: Plotting the Divided Line & the Problem of the Irrational

Go(Φ)d is Number: Plotting the Divided Line & the Problem of the Irrational

An Optimization Algorithm for Embedded Raspberry Pi Pico Controllers for Solar Tree Systems

Solar photovoltaic (PV) systems stand out as a promising solution for generating clean, carbon-free energy. However, traditional solar panel installations often require extensive land resources, which could become scarce as the population grows. To address this challenge, innovative approaches are needed to maximize solar power generation within limited spaces. One promising concept involves the development of biological tree-like structures housing solar panels. These “solar trees” mimic the arrangement of branches and leaves found in natural trees, following patterns akin to phyllotaxy, which correlates with the Fibonacci sequence and golden ratio. By adopting an alternative 1:3 phyllotaxy pattern, three solar panels can be efficiently arranged along the stem of the solar tree structure, each rotated at a 120-degree displacement. Optimizing the performance of solar trees requires effective maximum power point tracking (MPPT), a crucial process for extracting the maximum available power from solar panels to enhance the overall efficiency. In this study, a novel metaheuristic algorithm called horse herd optimization (HHO) is employed for MPPT in solar tree applications. Moreover, to efficiently manage the generated power, a cascaded buck–boost converter is utilized. This converter is capable of adjusting the DC voltage levels to match the system requirements within a single topology. The algorithm is implemented using MATLAB and embedded within a Raspberry Pi Pico controller, which facilitates the generation of pulse-width modulation (PWM) signals to control the cascaded buck–boost converter. Through extensive validation, this study confirms the effectiveness of the proposed HHO algorithm integrated into the Raspberry Pi Pico controller for optimizing solar trees under various shading conditions. In essence, this research highlights the potential of solar tree structures coupled with advanced MPPT algorithms and power management systems to maximize solar energy utilization, offering a sustainable solution for clean energy generation within limited land resources.

Optical trimer: A theoretical physics approach to waveguide couplers

Abstract We study electromagnetic field propagation through an ideal, passive, three-dimensional, triangular three-waveguide coupler using a symmetry-based approach that capitalizes on the underlying su(3) symmetry. The planar version of this platform has already demonstrated its utility in photonic circuit design, enabling optical sampling, filtering, modulating, multiplexing, and switching. We aim to provide a practical tutorial on using group theory for the analysis of photonic lattices for those less familiar with abstract algebra methods.
This approach serves as a powerful tool for optical designs. To illustrate this, we focus on the equilateral trimer, connected to the Discrete Fourier Transform, and the isosceles trimer, related to the golden ratio, providing stable single waveguide output. We also explore a scenario where the coupling in an equilateral coupler changes linearly with propagation distance. Going beyond the standard optical-quantum analogy, we show that coupled-mode equations for intensity and phase allows us to calculate envelopes for inputs within an intensity class, as well as individual input field amplitudes. This approach streamlines the design process by eliminating the need for point-to-point propagation calculations, highlighting the power of group theory in the field of photonic design.

Angle Trisection with Growth Rate of The Golden Ratio

In this paper, we explore the triangulation of angles using the golden ratio and geometric compass principles. By dividing angles into three equal parts based on the golden ratio, we demonstrate a method for solving this geometric problem. Furthermore, we extend our exploration into the third dimension, where a geometric compass can effectively divide a two-dimensional angle into three equal parts. Additionally, our investigation delves into the realm of six-dimensional space-time, where we observe the simultaneous movement of two arms of the compasses based on the golden ratio to achieve the division of every angle into three equal parts. We emphasize the significance of the growth rate associated with the golden ratio in resolving complex mathematical and physical problems. This study provides valuable insights into the application of geometric principles and the golden ratio in solving challenging problems within mathematics and physics.

Optimization of the Reaction Temperature During Phosphoric Acid Production Using Fibonacci Numbers and Golden Section Methods

Optimization of the Reaction Temperature During Phosphoric Acid Production Using Fibonacci Numbers and Golden Section Methods

Chaotic opposition Golden Sinus Algorithm for global optimization problems

Optimization techniques are required to find the best solutions to challenging problems in many engineering disciplines. Metaheuristic algorithms that effectively increase search performance using various evolutionary strategies have become increasingly popular in recent years. The Golden Sinus Algorithm (GoldSa) is a population-based optimization algorithm that uses the sine function and the golden ratio. In this study, a chaotically enhanced opposition-based Golden Sinus Algorithm (Co-GoldSa) has been proposed to improve the efficiency of the exploitation and exploration ability of the GoldSa method. While designing this approach, it is first necessary to analyze the chaotic behavior of the GoldSa parameters. For this purpose, three chaotic GoldSa methods have been developed using eight chaotic maps to determine the effect of the chaotic maps on the parameters with different behaviors. Secondly, the opposition-based learning strategy is adjusted to the cGoldSa to enhance the searching ability. To investigate the proficiency of the proposed Co-GoldSa method, it has been examined with well-known and newly introduced metaheuristic approaches on benchmark functions and classical engineering design problems. Besides, an efficient framework of the multilevel thresholding image segmentation has been presented based on the Co-GoldSa method since the efficient processing of pathological images is quite important in medical diagnostics. The experimental outcomes reveal the superiority of the proposed method in solving global optimization problems, image segmentation, and engineering problems. Thus, the outcomes of the benchmark functions, image segmentation, and classical engineering problems support that the proposed Co-GoldSa approach can be considered a promising method for resolving challenging optimization problems.

Golden ratio method for solving monotone variational inequality problems in Hadamard spaces

In this article, a generalized variational inequality problem in the setting of Hadamard spaces is introduced and analyzed. For approximating a solution of the problem when the underlined mapping is monotone, an adaptive algorithm that requires the computation of only one proximal operator at each iteration is proposed. The work gives a step towards investigating monotone variational inequality problems in geodesic settings.

Preconditioned golden ratio primal-dual algorithm with linesearch

Preconditioned golden ratio primal-dual algorithm with linesearch

Using the Golden Section to Approximate π

Summary We derive one specific example of a Machin-like formula for approximating π using the unit circle and the reciproval of the golden section. The efficiency of our derived formula is compared to other, similar formulas using Lehmer’s measure.

Optimizing bimetallic plasmonic structures for enhanced surface plasmon resonance sensing through golden ratio design

Optimizing bimetallic plasmonic structures for enhanced surface plasmon resonance sensing through golden ratio design

To Approach the Factors of Influencing Consumer Preferences for Corrective Eyewear Design

Correctional eyeglasses used for myopia, hyperopia, astigmatism, and presbyopia require adaptation and prolonged wearing, similar to assistive devices. Unlike decorative glasses, which can be freely changed or not worn as desired and coordinated with overall attire, wearing eyeglasses alters facial appearance. The aesthetics, comfort, and functional quality of eyeglasses directly affect the wearer's willingness, thus impacting visual health and quality of life. Therefore, corrective eyeglass frames need to balance between aesthetic appearance and visual health. This study employs methods such as the Kano Model, Important Performance Analysis (IPA), and Choquet integral to explore the key preference factors for the design of corrective eyeglass frames. Differing from traditional 5-point equidistant semantic questionnaires, this study employs a non-equal interval semantic questionnaire with a golden ratio scale to differentiate adjacent semantic importance ratios, highlighting weighted emphasis on aesthetics. The results provide reference for eyeglass designers and related industries. It is expected to design eyewear that better meets the needs of the Taiwanese population, encouraging people to wear corrective eyewear willingly, thereby further maintaining visual health, improving quality of life, enhancing visual health, and quality of life.

Fault distance measurement method for wind power transmission lines based on improved NSGA II

A fault distance measurement method that is suitable for long-distance wind power transmission lines is proposed by this paper. By combining the voltage and current at both ends of the line during normal operation, a dynamic line parameter correction method based on improved NSGA II is proposed to ensure measurement errors caused by parameter changes are avoided. Based on the construction of the fault distance measurement function, a global optimization method that is based on adaptive golden section search is proposed, where only the power frequency measurement data at both ends of the bus is required and synchronization is not required for the measurement of fault distance. A large number of experimental results demonstrate that the method this paper proposes is highly adaptable, with a relative distance measurement error of less than 0.2 km, and high accuracy is maintained in distance measurement even with certain degrees of noise interference.

Using the golden ratio in detecting the aesthetics of Mosque facades: The case of contemporary Mosques in Erbil city

From ancient Greece till the present, the golden ratio has been considered one of the most significant criteria of beauty in art and architecture. Architectural forms are part of the visual language that is bound by the rules of geometry and proportion. This study aims to identify the extent to which contemporary mosques in the city of Erbil utilize the golden ratio in the aesthetics of their facades. The research attempts to answer the question: was the golden ratio implemented in designing the facade of contemporary mosques in Erbil? The study adopted a mixed quantitative and qualitative analysis method. To achieve the study's aim and answer its questions, six mosques built in the last decade in Erbil were selected. Qualitatively, survey and documentation followed. Quantitatively, Phi matrix software, AutoCAD, and graphic software were applied as a mathematical approach. The study concluded that only one case used the golden ratio in the design of its façade by 100%, while it did not exceed 75% in most other mosques. The study draws the attention of designers and architects in the field of mosque architecture to the adoption of the golden ratio in the designs of mosque facades because of its role in achieving the aesthetic aspect.

Smile aesthetics in Pakistani population: dentist preferences and perceptions of anterior teeth proportion and harmony

BackgroundThis study aimed to evaluate dentist perceptions of attractive smiles in the Pakistani population, considering different dental proportions.MethodsMaxillary casts and digital images were used to create symmetrical representations of anterior teeth. dentists’ preferences for good and bad teeth proportions, width/height ratios, and various dental proportions (golden, recurring esthetic dental (RED), golden percentage, Preston, and local/observed) were assessed using one sample and paired t-test. The Chi-square test was used to determine the gender disparities and factors affecting smile attractiveness. A p-value of ≤ 0.05 was taken as significant.ResultsThe RED proportion emerged as the preferred choice for normal-sized teeth, with specialists and general dentists favoring it over the golden proportion. For tall teeth, the golden proportion was predominantly preferred. The golden percentage received limited preference for aesthetic smile construction.ConclusionsThe smiles created using the principles of RED proportion were opted as the most attractive by local dentists. Factors such as tooth arrangement, color, and midline were highlighted as essential considerations in aesthetic smile construction.

Generalized Freudenthal duality for rotating extremal black holes

Freudenthal duality (FD) is a non-linear symmetry of the Bekenstein-Hawking entropy of extremal dyonic black holes (BHs) in Maxwell-Einstein-scalar theories in four space-time dimensions realized as an anti-involutive map in the symplectic space of electric-magnetic BH charges. In this paper, we generalize FD to the class of rotating (stationary) extremal BHs, both in the under- and over-rotating regime, defining a (generalized) rotating FD (generally, non-anti-involutive) map (RFD), which also acts on the BH angular momentum. We prove that the RFD map is unique, and we compute the explicit expression of its non-linear action on the angular momentum itself. Interestingly, in the non-rotating limit, RFD bifurcates into the usual, non-rotating FD branch and into a spurious branch, named “golden” branch, mapping a non-rotating (static) extremal BH to an under-rotating (stationary) extremal BH, in which the ratio between the angular momentum and the non-rotating entropy is the square root of the golden ratio. Finally, we investigate the possibility of inducing transitions between the under- and over- rotating regimes by means of RFD, obtaining a no-go result.

Ассимиляционный потенциал природы и территорий: нетривиальные следствия управления устойчивым эколого-экономическим развитием

Ecological-economic systems, which are in a state of unstable dynamic equilibrium under the influence of natural and anthropogenic factors, strive for sustainability if human economic activity is able to maintain the assimilation potential at the level of reproduction of natural resources and environmental quality by devoting part of the income to the environmental agenda. Within the framework of the law of balanced environmental management T=1.618·t·a, sustainable environmental and economic development is achieved by allocating funds at the level (t) of the GDP growth rate (T) to the environmental agenda for the regions. The value of the ecological situation coefficient (ESC) in the law of balanced environmental management for the Russian Federation varies around the golden numbers 0.618, 1.618, 2.618, depending on the level of economic development in the biosphere, artificial maintenance of the quality of the environment in it, as well as in the future possibility of terraforming the solar planets systems. An indicator of sustainable, harmonious environmental and economic development is a linear relationship between the assimilation potential (a) and the EES at the level of golden numbers 0.618, 1.618, 2.618. In this case, sustainable environmental and economic development can be achieved in any environmental condition as long as balance is maintained between the rate of economic growth and allocations of funds for the restoration of the assimilation function of nature. But they must be directed to the environmental agenda in strict accordance with the golden proportion relative to the main directions of environmental activities, even in conditions of transformed (converted) nature, and in the near future, in conditions of terraforming. However, then all development costs will go exclusively to maintaining the habitat in multiples of the golden numbers of 1.618 and 2.618 in the coefficient of the law of balanced environmental management.

Angle Trisection with Growth Rate of The Golden Ratio

In this paper, we explore the triangulation of angles using the golden ratio and geometric compass principles. By dividing angles into three equal parts based on the golden ratio, we demonstrate a method for solving this geometric problem. Furthermore, we extend our exploration into the third dimension, where a geometric compass can effectively divide a two-dimensional angle into three equal parts. Additionally, our investigation delves into the realm of six-dimensional space-time, where we observe the simultaneous movement of two arms of the compasses based on the golden ratio to achieve the division of every angle into three equal parts. We emphasize the significance of the growth rate associated with the golden ratio in resolving complex mathematical and physical problems. This study provides valuable insights into the application of geometric principles and the golden ratio in solving challenging problems within mathematics and physics.

Blind frequency offset estimation method based on a minimum phase correction error for a full spectrum modulated NFDM system.

To improve the spectral efficiency of a full spectrum modulated nonlinear frequency division multiplexing (FS-NFDM) system, a blind frequency offset estimation (FOE) method has been proposed. The approach based on the minimum phase correction error can achieve high estimation accuracy of sub-MHz without need of any training symbols. Furthermore, in order to reduce the computational complexity, an eigenvalue-shift method is used to get a coarse search interval of FO, and then the one-dimensional optimization algorithm based on golden section search and parabolic interpolation is used to get the optimal FOE for the coarse search interval. The feasibility and reliability of the proposed blind FOE approach have been demonstrated in both BTB and fiber transmission scenarios. Compared with the grid search method, the proposed solving scheme can save hundreds of times of the searches. The experimental results reveal that the proposed method is robust to the amplified spontaneous emission noise and phase noise and has the capabilities of a wide FOE range and a high FOE accuracy.

Evaluating the validity of recurring esthetic dental proportion in young adults

Objectives: An important aspect of cosmetic dentistry is developing a geometric or numerical proportion to associate the consecutive width of maxillary anterior teeth. New hypotheses in this area include the Golden proportion, golden percentage, and the recurrent esthetic dental (RED) proportion. This study used digital photos and computer analysis to examine the existing RED proportion associated with the widths of maxillary anterior teeth in people with permanent dentition. Material and Methods: Sixty young adults (28 males and 32 females) were photographed from the front. Digital measurements were taken of all maxillary anterior teeth. Results: To determine whether the RED proportion exists in the permanent dentition, the mean width proportion of the maxillary lateral incisor to central incisor and maxillary canine to lateral incisor was established. The average lateral incisor to central incisor width proportion of the right and left side was determined to be 70% and 71%, respectively. The average canine to lateral incisor width proportion was determined to be 80% and 78%, respectively. Conclusion: The maxillary anterior teeth’s consecutive widths did not exhibit RED proportions in the data that were examined. People’s anterior tooth proportions differ based on their race and ethnicity. Thus, the RED ratio is not the only standard for rebuilding aesthetically acceptable smiles globally.