Golden Section Ratio Research Articles

The golden ratio in baseball: the influence of historical eras on winning percentages in major league baseball.

The golden section or golden ratio (61.8% or 0.618) is a mathematical phenomenon that appears in art, literature, music and nature with such ubiquity that it is thought to be a fundamental principle of aesthetic organisation. The golden ratio also manifests in sport, particularly as the proportion of wins to losses required to win a Major League Baseball championship. This study extends early work on the golden ratio in baseball by incorporating more than three decades of additional data. This study involved a historically contextualized examination of how winning percentages have changed across the seven historical eras of modern baseball, including analyses of the relative contribution of offensive and defensive statistics to championship winning teams. Data was extracted from Baseball Reference and included statistics for 398 championship winning teams from both the American and National Leagues between 1901 and 2019. Pearson correlation coefficients were computed for winning percentage with indicators of offensive and defensive performance during each era. Main and interaction effects of Era and League on winning percentage were examined using factorial ANOVA, with follow-up analyses examining whether the golden ratio was included in each factor's 95% confidence interval. Our findings suggest that winning percentages for championship teams were closest to the golden ratio during eras where the relative contribution of offense and defense was most closely balanced: the Integration Era (1942-1960) and the Expansion Era (1961-1976). Previous scholarship theorizes that the golden ratio represents an aesthetic ideal or a Gestalt archetype. If this aesthetic theory is applied to sporting competition, these results suggest that baseball may be most aesthetically appealing to fans when offense and defense is balanced in such a way as to ensure that championship teams win 61.8% of their games.

An Intelligent Cost-Reference Particle Filter with Resampling of Multi-Population Cooperation.

Although the cost-reference particle filter (CRPF) has a good advantage in solving the state estimation problem with unknown noise statistical characteristics, its estimation accuracy is still affected by the lack of particle diversity and sensitivity to the particles' initial value. In order to solve these problems of the CRPF, this paper proposed an intelligent cost-reference particle filter algorithm based on multi-population cooperation. A multi-population cooperative resampling strategy based on ring structure was designed. The particles were divided into multiple independent populations upon initialization, and each population generated particles with a different initial distribution. The particles in each population were divided into three different particle sets with high, medium and low weights by the golden section ratio according to the weight. The particle sets with high and medium weights were retained. Then, a cooperative strategy based on Gaussian mutation was designed to resample the low-weight particle set of each population. The high-weight particles of the previous population in the ring structure were randomly selected for Gaussian mutation to replace the low-weight particles in the current population. The low-weight particles of all populations were resampled in turn. The simulation results show that the intelligent CRPF based on multi-population cooperation proposed in this paper can reduce the sensitivity of the CRPF to the particles' initial value and improve the particle diversity in resampling. Compared with the general CRPF and intelligent CRPF with adaptive MH resampling (MH-CRPF), the RMSE and MAE of the proposed method are lower.

Post-Construction Analysis of Scale and Proportion: A Case Study of the Woodbury Towers, USA, in Reflecting Cogent Design Principles

Critically assessing your own design work after it has been constructed for several years is an exciting question of self-peer review that has to set aside the initial design amplitude while letting the theories perform in order to discover some unexpected results with respect to the original design. This paradoxical mechanism between practicing and teaching when career switching from a professional practitioner to a university instructor can harmoniously integrate theory and praxis to illuminate future paths and amplify innovation. Here, I use the Woodbury Towers in the Woodbury Irvine Community as a case study to explore planning design practice in context of the Southern California landscape as the subject of the self-peer review process. Creating community is building a place for people to live and share their experience. Community basically refers to sets of people who may identify themselves with a place in terms of notions of commonality, shared values or solidarity in particular contexts. Landscape architecture can contribute to a sense of community by contributing theories, experiences and design vision in the development process. In this paper, the synchronization of design scale and proportion considered essential, has been explored to reflect theoretical functions and aesthetic characteristics of good design since the Classical Era. The principle of The Golden Section ratio simulated with a digital superimposition technique is the main tool for this research method which led to discovery of the subconscious ideas in the post-construction phase.

Research on power density improvement for interior permanent magnet synchronous machine based on permanent magnet minimization

In order to improve the power density of the permanent-magnet synchronous machine and reduce the cost of effective materials, this paper investigated approaches of power-density improvement and discussed the influence of flux-weakening level and salient ratio parameters on the characteristics of torque and power for the interior permanent magnet synchronous machine (IPMSM) based on the permanent magnet (PM) minimisation. Firstly, the model of PM minimisation can be established based on the principle of equal area of stator and rotor magnetomotive force. According to the minimisation model, a formula is derived for calculation of structural parameters of the permanent magnets in the IPMSM. Besides, the relation among the interior multilayer permanent magnets is established. Then, comparing and analysing the extremum characteristics of the maximum torque output and maximum power output of the motor under different parameters, the regular influence on the IPMSM can be summarised between the parameter and the internal power factor angle. So, it can be proposed that the corresponding internal power factor angle with respect to the golden section ratio of the maximum torque output is around −35°. Finally, combined with the results of the finite element and analytical calculation, a 45 kW PM motor for electric vehicles was manufactured and tested in no-load, load and thermal experiment, respectively. The peak torque of the prototype lasted for 94 s and reached 2.3 times the rated torque. Compared with the conventional PM motor used for the electric vehicle, the power density was increased by more than 0.5 kW/kg which can verify the correctness of the theoretical analysis.

Evolution pathway of process risk of marine traffic with the STAMP model and a genetic algorithm: A simulation of LNG-fueled vessel in-and-out harbor

Risk evolution characteristics of an LNG-fueled vessel had been simulated by using the systems-theoretic accident modeling and processes (STAMP) model and a genetic algorithm (GA). First, the STAMP model was established to deconstruct the structure of the risk evolution which was affected by external environmental components. Second, given that the risk performance and maritime traffic develop in the time dimension, and the situation changes in the spatial dimension, a GA was used to provide a timeline for the STAMP model. Third, the coupled STAMP-GA model was applied to the LNG-fueled vessel system. Based on the golden section ratios for levels of risk performance, a Cloud model was introduced to provide input data for the GA model. Last, combined with the data for the LNG-fueled vessel and using a route scenario for a specific anchorage to a designated berth, the risk evolution performance was simulated to explore the risk characteristics of the LNG supply process. The results show that the risk evolution mode of the LNG-fueled vessel follows a bathtub-shaped distribution curve. The novel STAMP-GA model proves to be efficient in analyzing the mutation of risk components and the coupled process in the risk evolution.

Non-Cooperative Game Forwarding Leveraging User Trustworthiness in Mobile Edge Networks

Given frequent changes of network topology caused by limited computing power, strong mobility and weak reliability of most nodes in mobile edge computing (MEC) networks, a Non-Cooperative Game forwarding strategy based on user Trustworthiness (NCGT) is proposed to deal with low security and efficiency of data transmission. NCGT firstly considers device residual energy ratio, contact probability, service degree and link stability between devices to measure the reliability of nodes. Then, leverages Entropy Weight (EW) method and Golden Section Ratio (GSR) to develop a security optimal neighbors screening model based on multi-attribute decision-making, which ensures that high-performance security nodes are selected as forwarding game objects. Third, NCGT takes forwarding and non-forwarding as the policy set, designs the benefit function, and gets forwarding probability of nodes through Nash equilibrium, to reduce a large number of redundancy, competition and conflict in forwarding requests and improve its broadcasting efficiency. The simulation results show, NCGT is more effective against black hole and witch attacks than S-MODEST and AODV+FDG when there exists malicious nodes. Meanwhile, with the increasing of network load, NCGT with or without GSR always performs best in the terms of data delivery rate, delay, transmission energy consumption and system throughput in MEC environment.

Quality Prediction of Continuous Casting Slabs Based on Weighted Extreme Learning Machine

A slab quality prediction model based on machine learning plays an important role in improving final slab quality. However, the class imbalance of continuous casting datasets has a negative impact on the training of basic machine-learning models. In this study, weighted extreme learning machine (WELM) models are constructed to predict the slab quality of under different operation patterns by feeding millions of data. The results show that WELM models can achieve better prediction performance on the two types of continuous casting datasets than the basic algorithms. The superiority of WELM is demonstrated by the relatively high-precision identification of every kind of slab. The performance of WELM models with different weighting schemes is studied and the model with the golden section ratio weighting method is recommended for application as a quality prediction model. Meanwhile, WELM can still maintain a good predictive performance and generalization ability when training a large amount of data. This model can satisfy the demands for slab quality prediction and optimize the continuous casting process.

Золотое сечение и золотые прямоугольники при построении икосаэдра, додекаэдра и тел Архимеда, основанных на них

A brief history of the development of the regular polyhedrons theory is given. The work introduces the reader to modelling of the two most complex regular polyhedrons – Platonic solids: icosahedron and dodecahedron, in AutoCAD package. It is suggested to apply the method of the icosahedron and dodecahedron building using rectangles with their sides’ ratio like in the golden section, having taken the icosahedron’s golden rectangles as a basis. This method is well-known-of and is used for icosahedron, but is extremely rarely applied to dodecahedron, as in the available literature it is suggested to build the latter one as a figure dual to icosahedron. The work provides information on the first mentioning of this building method by an Italian mathematician L. Pacioli in his Divine Proportion book. In 1937, a Soviet mathematician D.I. Perepelkin published a paper On One Building Case of the Regular Icosahedron and Regular Dodecahedron, where he noted that this “method is not very well known of” and provided a building based “solely on dividing an intercept in the golden section ratio”. Taking into account the simplicity and good visualization of the building based on golden rectangles, a computer modeling of icosahedron and dodecahedron inscribed in a regular hexahedron is performed in the article. Given that, if we think in terms of the golden section concepts, the bigger side of the rectangle equals a whole intercept – side of the regular hexahedron, and the smaller sides of the icosahedron and dodecahedron rectangles are calculated as parts of the golden section ratio (of the bigger part and the smaller one, respectively). It is demonstrated how, using the scheme of a wireframe image of the dual connection of these polyhedrons as a basis, to calculate the sides of the rectangles in the golden section ratio in order to build an “infinite” cascade of these dual figures, as well as to build the icosahedron and dodecahedron circumscribed about the regular hexahedron. The method based on using the golden-section rectangles is also applied to building semiregular polyhedrons – Archimedean solids: a truncated icosahedron, truncated dodecahedron, icosidodecahedron, rhombicosidodecahedron, and rhombitruncated icosidodecahedron, which are based on icosahedron and dodecahedron.

Stochastic and empirical models of the absolute asymmetric synthesis by the Soai-autocatalysis.

Absolute asymmetric synthesis (AAS) is the preparation of pure (or excess of one) enantiomer of a chiral compound from achiral precursor(s) by a chemical reaction, without enantiopure chiral additive and/or without applied asymmetric physical field. Only one well-characterized example of AAS is known today: the Soai-autocatalysis. In an attempt at clarification of the mechanism of this particular reaction we have undertaken empirical and stochastic analysis of several parallel AAS experiments. Our results show that the initial steps of the reaction might be controlled by simple normal distribution ("coin tossing") formalism. Advanced stages of the reaction, however, appear to be of a more complicated nature. Symmetric beta distribution formalism could not be brought into correspondence with the experimental observations. A bimodal beta distribution algorithm provided suitable agreement with the experimental data. The parameters of this bimodal beta function were determined by a Pólya-urn experiment (simulated by computer). Interestingly, parameters of the resulting bimodal beta function give a golden section ratio. These results show, that in this highly interesting autocatalysis two or even perhaps three catalytic cycles are cooperating. An attempt at constructing a "designed" Soai-type reaction system has also been made.

Golden Section Heat Engines and Heat Pumps

In this study the efficiency of heat engines and the coefficient of performance of heat pumps are reconsidered following the idea of the golden section ratio. Two models are analyzed: the golden section Carnot cycle and the golden section endo-reversible cycle. The observed expressions for efficiency and coefficient of performance are cast in terms of the golden section ratio. Single (steam turbines[40Wilfried, U., OMMI, vol. 2, issue 3, pp. 1-9, 2003]) and double (combined cycles[Ishikawa, M., Terauchi, M., Komori, T., Yasuraoka, J., Mitsubishi Heavy Industries, Ltd., Technical Review Vol. 45 No. 1, pp. 15-17, 2008]) heat engine cycles are considered. Expressions for the working fluid temperatures at the hot and cold sides are derived. Calculations performed on steam parameters, with the supercritical temperature of 610<SUP>o</SUP>C, show that the relative differences of the golden section single heat engine, when compared to the endo-reversible results, fell within the 2- 4% range, with relative power reductions of 15% from the maximum power point. Finally, it was shown that system efficiency differs by approximately 3.6% from the efficiency derived based on the golden section.

Beyond the Golden Section and normative aesthetics: Why do individuals differ so much in their aesthetic preferences for rectangles?

Interest in the experimental aesthetics of rectangles originates in the studies of Fechner (1876), which investigated Zeising’s suggestion that Golden Section ratios determine the aesthetic appeal of great works of art. Although Fechner’s studies are often cited to support the centrality of the Golden Section, a century of subsequent experimental work suggests it has little normative role in rectangle preferences. However, rectangles are still of interest to experimental aesthetics, and McManus (1980) used a paired comparison method to show that although population preferences are weak, there are strong, stable, statistically robust and very varied individual preferences. The present study measured rectangle preferences in 79 participants, particularly assessing their relationship to a wide range of background measures of individual differences. Once again weak population preferences but strong and varied individual rectangle preferences were found, and computer presentation of stimuli, with detailed analyses of response times, confirmed the coherent nature of aesthetic preferences for rectangles. Q-mode factor analysis found two main factors, labeled “square” and “rectangle,” with participants showing different combinations of positive and negative loadings on these factors. However, the individual difference measures, including Big Five personality traits, Need for Cognition, Tolerance of Ambiguity, Schizotypy, Vocational Types, and Aesthetic Activities, showed no correlation at all with rectangle preferences. Individual differences in rectangle preferences are a robust phenomenon that clearly requires explanation, but at present their variability is entirely unexplained.

Data-Based Stochastic Approach to Absolute Asymmetric Synthesis by Autocatalysis

Abstract Statistical distribution of enantiomeric excesses obtained by two sets of parallel experiments of absolute asymmetric synthesis by asymmetric autocatalysis was analyzed. It has been found that experimental data give an excellent fit to a bimodal β distribution formula, where the components are in a golden section ratio. The parameters of this higher order β distribution were found by computer-simulated Pólya urn model experiments. The urn model experiments indicate that the Soai-autocatalysis might operate by three concerted cooperating catalytic cycles. These results may provide also a general model of asymmetric autocatalysis.

Construing stigmatized identities: A golden section study

In order to examine whether persons belonging to stigmatized groups are evaluated more negatively than others, this study employed the golden section hypothesis, which predicts that people organize their evaluations in a ratio of roughly 62% positive to 38% negative. Surveys were administered to 154 participants, with 143 surveys analysed. Participants rated nine different identities (business person, politician, lawyer, police officer, college professor, criminal, mental patient, homeless person, and self) along 12 different bipolar dimensions (generous-stingy, pleasant-unpleasant, true-false, fair-unfair, active-passive, energetic-lethargic, sharp-dull, excitable-calm, strong-weak, bold-timid, hard-soft, and rugged-delicate). These dimensions had well-established positive and negative poles. Non-stigmatized identities were evaluated slightly more positively than the golden section hypothesis predicts, while stigmatized identities were rated in a 'reverse' golden section pattern. A criminal identity was rated more negatively than the golden section hypothesis predicts and the self more positively. Follow-up analyses suggested that the percentages of positive adjectives assigned to the self and non-stigmatized others were consistent with a model that expands upon the golden section hypothesis. Stigmatized identities are rated using a reverse golden section ratio. This has implications for understanding how people construe stigmatized people. While the golden section hypothesis did not hold for non-stigmatized identities, this may be because the data are better explained by a more refined model that is slightly different from the golden section.

Multiscale multiresolution genetic algorithm with a golden sectioned population composition

AbstractA new genetic algorithm (GA) strategy called the multiscale multiresolution GA is proposed for expediting solution convergence by orders of magnitude. The motivation for this development was to apply GAs to a certain class of large optimization problems, which are otherwise nearly impossible to solve. For the algorithm, standard binary design variables are binary wavelet transformed to multiscale design variables. By working with the multiscale variables, evolution can proceed in multiresolution; converged solutions at a low resolution are reused as a part of individuals of the initial population for the next resolution evolution. It is shown that the best solution convergence can be achieved if three initial population groups having different fitness levels are mixed at the golden section ratio. An analogy between cell division and the proposed multiscale multiresolution strategy is made. The specific applications of the developed method are made in topology optimization problems. Copyright © 2007 John Wiley &amp; Sons, Ltd.

The biology of vernix caseosa

The biology and physical properties of the uniquely human skin cream 'vernix caseosa' are discussed. This material coats the foetal skin surface during the last trimester of gestation and provides multiple beneficial functions for the foetus and newborn infant. Vernix has a complex structure similar to stratum corneum but lacks lipid lamellae and is more plastic due to the absence of desmosomal constraints. In utero, vernix is made in part by foetal sebaceous glands, interacts with pulmonary surfactant, detaches into the amniotic fluid, and is swallowed by the foetus. At the time of birth, vernix has a remarkably constant water content approximating 80%. Postnatally, vernix is simultaneously a cleanser, a moisturizer, an anti-infective, and an anti-oxidant. Vernix facilitates acid mantle development and supports normal bacterial colonization. Its hydrated cellular structure and unusual lipid composition provide a 'best' solution for the needs of the foetus and newborn, not least of which is the attraction of caregivers. Vernix is an important natural biomaterial of potential interest to cosmetic scientists, and other disciplines involved in product development and therapies targeting the complex interface between the stratum corneum and a changing terrestrial environment.

The human stratum corneum as extended, covalently cross-linked biopolymer: Mathematics, molecules, and medicine

A novel mathematical and molecular hypothesis is proposed to account for the peculiar organization of human epidermis. Mathematically, the organization of the interfollicular epidermis is hypothesized to be a tetratomic identity manifesting a gravitational logic in the arrangement of its functional compartments. The squares of the natural numbers; i.e., 1, 4, 9, and 16 are taken, on empirical grounds, to correspond to the number of cell layers in the respective epidermal strata (germinativum, spinosum, granulosum, and corneum). The outer two strata, overlying the Langerhans cells, constitute the 'living' and 'dead' components of the traditional 'epidermal barrier'. Together, these two strata illustrate in their union of 9 + 16 = 25 cells, a way of conceiving the skin surface (the body-environment identity) as both closure and contact. The organization of human epidermis into functional units based on phi, the golden section ratio, builds upon this gravitational logic. Finally, the fact that the extensively cross-linked proteolipid envelope of the cornified epidermal cell is a single multi-gene molecule is deemed scientifically incontrovertible. The molecular hypothesis in need of validation and verification is whether the corneodesmosomal 'rivets' linking one corneocyte to another are covalently bonded structures. If so, the cornified scaffolding of the stratum corneum constitutes a highly organized, extended, multi-gene, polymer molecule strategically located precisely at the shared surface of the body and environment. This hypothesis places the differentiated structure of the epidermis, an ectodermal derivative like the brain, front and center in the translation of molecular biology to clinical bedside care.

Shape and Order in Organic Nature: The Nautilus Pompilius

S piral forms abound in organic nature, and the beautiful Nautilus pompilius is frequently cited as an exquisite example of mathematical laws governing growth (Fig. 1). Cook [2] speaks of the Curves of Life to epitomize the geometric diversity in nature. In contrast to Cook, Colman's [3] Nature's Harmonic Unity follows Zeising's [4] thesis that the Golden Section [5] is the central principle in art and nature. Colman draws a maze of traces regulateurs [6] over natural and human-made objects to support his argument that the complicated forces in life and art conform to the law of the extreme and mean proportional [7]. Colman's method is open to the criticism that there is no objective way of determining whether a proposed analysis is the one intended and not simply another way of explaining the shape and proportions of an artifact by an entirely different procedure. It is a well-known fact that absolute uniformity of growth forces leads to straight lines and circles; however, growth is not uniform-it is this feature that signals the production of curves. Stevens explains the process: lesson of curvature is quite general and has nothing to do with what tissue or material is involved. Forms curl so that the faster growing or longer surface lies outside and the slower growing or shorter surface lies inside, there being more room outside than inside [8]. Huxley is more precise: If the length-width growth-ratio remains constant, but the absolute magnitudes of both components are greatest at one margin [and] least at the opposite margin, and are intermediately graded around the two sides of the mantel, [the form] will be distorted, and, so long as the growth-ratios concerned remain constant, will grow into a true logarithmic spiral in a single plane [9]. Since all growth in organic nature that follows this rule yields a logarithmic spiral, there is nothing mysterious about spirals. Clearly, it is the growth-ratios that control the shape of a spiral, and, as is soon apparent from a study of shells, there is no one particular spiral that fits all shells-some have tighter curves than others [10]. Every planar logarithmic spiral has an acute angle P3, termed the angle of the spiral, which controls its shape. This angle, between a tangent to any point on the curve and its polar radius vector, is constant for a particular spiral. The idea that phenomena in organic nature and art exhibit unity in obedience to some quintessential force in Nature, particularly the Golden Section ratio (), is a mystic belief that pervades the literature on proportion [11,12]. A recent example is that of Fletcher, who states that the Golden Section describes the shape of the Nautilus pompilius [13]. Fletcher arrives at this conclusion by examining the proportions of a rectangle enclosing a cross-section of the shell, and careful measurement confirms that her rectangle ABCD has its sides AB and BC in a ratio 1:(/)4), where =( (5+1)1/2. However, an examination of an accurate reconstruction of Fletcher's diagram (Fig. 2) reveals that whereas AB, BC and AD (the left, bottom and top edges of the enclosing rectangle) are tangential to the shell's boundary curve, the position of the right edge is dictated by the arbitrary position of point X, the terminal point of the curve. In other words, the shell should be rotated about its

Unity and the Golden Section: Rules for Aesthetic Choice?

Simple geometric figures were divided internally into two parts; the resulting area-to-area divisions were used to evaluate the aesthetic appeal of the unity (1:1) and golden section (1.618:1) ratios, among others. Three experiments incorporated a variety of figures and each utilized a different method (a multiple-choice procedure, a production method, or paired comparisons) for testing preferences. In the third experiment, several paired comparisons were of special interest. Each paired comparison pitted a square against a rectangle whose dimensions were in the proportion of the golden section; in addition, each figure of a test pair was divided internally to yield either the unity or golden section ratio

Symmetry and the Organization of Form: A Review Article

Symmetry and the Organization of Form: A Review Article

The St. Cuthbert Gospel Binding and Insular Design

The late 7th-century Gospel of St. John preserved among the relics of St. Cuthbert retains its original binding, whose decorative designs embody in an exemplary way the principles of composition that informed early Insular art. The designs on the front and back covers are geometrical not only for showing the influence of compass and straight-edge. They are geometrical in conception as well, having proportions among their major dimensions that are all based on the two true measures of geometry, the golden section ratio and the square root of two. This essay analyses the forms of the designs and provides line-drawings to show how they could have been created.