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Objective:The purpose of this work was to establish a statistical relationship of the results obtained with two different techniques. The comparison was made between the ideal facial height proposed in the Silva cephalometric method (SCM) and the facial height obtained using Euclid's rectangle or golden rectangle. Materials and Methods:38 cephalometric radiographs of dentistry students of Universidad de Cuenca, of both sexes, were used. All of these radiographs had suitable characteristics for cephalometric tracing in the NemoStudio 2022 software. Once the cephalometric tracing was made using these software tools, SCM was performed on each one of them to collect the data, and then we proceeded to overlap Euclid’s rectangle on all the images. The correlation of these two methods was established with Pearson’s correlation coefficient. Results:It was concluded that the Pearson correlation coefficient was [r]: 0.9271 Conclusions:5% of the patients presented compatibility with the logarithmic spiral of the golden ratio.

Geometric properties of finite Blaschke products have been intensively studied by many different aspects. In this paper, our aim is to study geometric properties of finite Blaschke products related to the golden ratio $\alpha =\frac{1+\sqrt{5}}{2}$. Mainly, we focus on the relationships between the zeros of canonical finite Blaschke products of lower degree and the golden ratio. We show that the geometric notions such as "golden triangle, "golden ellipse" and "golden rectangle" are closely related to the geometry of finite Blaschke products.

Every reader knows about the Golden Rectangle (see [1, pp. 85, 119], [2, 3]), and that it can be subdivided into a square and a smaller copy of itself, and that this process can be continued indefinitely, converging towards the intersection point of diagonals of any two successive rectangles in the sequence. The circumscribed logarithmic spiral passing through the vertices and converging to the same point is also familiar (see [3, 4]), and is analogous to the circumcircle of a regular polygon or a triangle. The approximate logarithmic spiral obtained by drawing a quarter-circle inside each of the squares is equally well known [3, p. 64]. Perhaps slightly less familiar is the inscribed spiral, which is tangential to a side of every rectangle, like the incircle of a triangle or a regular polygon. It does not (quite) coincide with the spiral passing through the point of subdivision of each side, as discussed in [3, pp. 73-77]. The Golden Rectangle, its subdivisions, and the circumscribed and inscribed spirals are illustrated in Figure 1.

All of us want to promote pleasing architecture but is there a clear theory that explains what makes a design good. Why do some buildings look pleasing to us while the others not so much? Why do a few of them give us a sense of wellbeing whereas a few of them make us feel out of place. Architecture never has a single answer and is very subjective. It is evident that there could be a tangible relationship between the visual aspect of a building and the intangible that is the way we feel about it. This research tries to understand these tangible characteristics of a built form and tries to objectify aesthetics by analyzing it through the lens of scale and proportion. “Geometry existed before creation” - Johan Kepler “Nature seems to be written in the language of mathematics” - Galileo Galilei These statements form the basis of this research. It will start with the study of the relationship of aesthetics of façade and their proportions by overlaying the guiding principles attached to these attributes on several built forms. Once established as to how through history buildings have related to different simple and complex equations the research moves on to perceive a certain number of buildings in current scenario. These buildings may or may not have been built on some geometrical guiding principles. Later through survey the research will try and identify whether or not an aesthetically pleasing building has some relationship with scale. It will try and identify the exact equation that guided the form of the building. The relationship between proportion and aesthetics will be established by running experiments, overlaying and calculating proportion theories on to the façade of that particular built form.

The Roman baths in the city of Évora, discovered in 1987 and integrated into the current building of the Paços do Concelho, proved to be one of the most important examples of this typology at the national level. This structure was located close to the Forum and asserted itself as one of the most important public buildings in the city. Thanks to the archaeological intervention of 2019, it was possible to record and study new structures in the complex. Several researches suggest that the architecture of this period uses a vast set of geometric and spatial rules in order to make the space harmonious and sensorially pleasant to human experience. In this sense, the present study proposes to analyze this thermal complex by using geometry with the aim of better understanding its spatial and volumetric organization and also the way in which the aforementioned complex was related to the coeval urban context. Las termas romanas de la ciudad de Évora, descubiertas en 1987 e integradas en el edificio actual de los Paços do Concelho, resultaron ser uno de los ejemplos más importantes de esta tipología a nivel nacional. Esta estructura estaba ubicada cerca del Foro y se afirmó como uno de los edificios públicos más importantes de la ciudad. Con la intervención arqueológica de 2019, se lograron registrar y estudiar nuevas estructuras en el complejo. Varias investigaciones sugieren que la arquitectura de este periodo utiliza un vasto conjunto de reglas geométricas y espaciales para hacer que el espacio sea armonioso y sensorialmente agradable para la experiencia humana. En este sentido, el presente estudio se propone analizar este conjunto termal, utilizando la geometría para tratar de comprender mejor su organización espacial y volumétrica y también la forma en que dicho conjunto se relacionaba con el contexto urbano de la época.[pt] As termas romanas da cidade de Évora, descobertas em 1987 e integradas no actual edifício dos Paços do Concelho, revelaram-se um dos mais importantes exemplares desta tipologia a nível nacional. Esta estrutura localizava-se próximo do Forum e afirmava-se como um dos mais importantes edifícios públicos da cidade. Com a intervenção arqueológica de 2019, foi possível registar e estudar novas estruturas do complexo. Várias pesquisas sugerem que a arquitectura deste período utiliza um vasto conjunto de regras geométricas e espaciais de forma a tornar o espaço harmonioso e sensorialmente agradável à vivência humana. Nesse sentido, o presente estudo propõe-se fazer uma análise deste complexo termal, utilizando a geometria para tentar compreender melhor a sua organização espacial e volumétrica e também a forma como o referido complexo se relacionava com o contexto urbano da época.

The inherent rules of Geometry, as well as the forms described by them, have always inspired artistic creation and suggested canons to achieve perfection in the creating of many works of art. Let us think about the beauty of some geometric shapes used in painting, architecture, sculpture. The natural forms have almost always inspired those built, where the aesthetic aspect is obtained through a deep dialogue between shape, function, materials, and technique. Everything tends to achieve a common goal; and thanks to this communion of knowledge the forms develop themselves differently, in relations with the historical period in which they are used within the various design choices. The dialogue between different fields is fundamental to find those optimal choices to achieve the beauty in each construction in progress. The aim of this work is to understand how it is possible to obtain even very complex three-dimensional surfaces starting from a simple geometric figure—such as a flat polygon—through synthetic processes and simple transformations. These surfaces, often described by structures of extraordinary beauty, are recognizable in the configuration of many natural growth’s forms, in some admirable architectural constructions characterized by perfect and futuristic shapes and, at last, in the description of some contemporary and delicate beauty sculptures where the light is added to the shape to mark its beauty.

“Investigate, because of the golden rectangle” offers mathematics students and enthusiasts inspiration for mathematical play by way of a guided construction of the golden rectangle. The discussion is illustrated with numerous hand-drawn sketches. A golden rectangle is a rectangle whose side lengths are in the golden ratio, which is, where the Greek letter (pronounced “phi”) is approximately equal to. Readers learn that an indirect, even haphazard, approach in mathematical play may lead to unanticipated discoveries. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

This paper presents an interesting deduction of the Golden Spiral equation in a suitable polar coordinate system. For this purpose, the concepts of Golden Ratio and Golden Rectangle, and a significant result for the calculation of powers of the Golden Ratio ϕ using terms of the Fibonacci sequence are mentioned. Finally, various geometrical considerations that help us deduce the sought equation are presented.

In researches of basic theory of TCM, the Luoshu has direct relationship with the Jiugong bafeng theory. In this paper based on analysis of mathematical relations among the Fibonacci sequence, the Fibonacci rectangle and golden rectangle, according to geometric properties of the golden rectangle as well as numeric distribution rules implied in Luoshu, relationship between the Fibonacci sequence including its modular series by 10 and Luoshu is studied, and inherent and consistent mathematical relationship between them is found in spatial orientation, digital running sequence, numeric grouping and generating rules, etc. The study results of this paper can help to build up close mathematical association between Luoshu and the Fibonacci sequence, which can lay key theoretical foundation for the ultimate breakthrough for scientific mysteries containing in Luoshu and the Jiugong bafeng theory in TCM.

Background and Objectives: The maxillary central incisor is the dominant element of anterior dental composition and hence should be restored or replaced with proper proportion of width and length for better aesthetic results. However, the literature is not clear regarding verifiable guidelines for the determination of proportions of the teeth. The aim of this study was to investigate the existence and suitability of Golden Rectangle, Recurring Aesthetic Dental Proportion, and Golden percentage between the widths of maxillary anterior teeth in individuals with natural dentition, with the aid of digital photographs and computer analysis. Material and Methods: Frontal full-face digital photographs of the subjects (in smile) were made under standardized conditions using a digital camera and a tripod stand was used to place and orient the camera in the standardized position (camera was positioned 1 meter away from the patient; and the lens of the camera was adjusted at the patients' lip level). Imaging software (Adobe Photoshop CS5; Adobe Systems, Inc, San Jose, Calif.) was used to mark the anatomic landmarks and to digitally analyze the photograph. The entire process of proportion analysis was done by a single observer. Results: The RED proportion was not found to exist between the six maxillary anterior teeth. The values suggested in the golden percentage were not applicable on the subjects of this study. However, a slight modification of these percentages can be adopted taking into consideration the ethnicity differences of the subjects in this study. The values obtained were 24%, 15%, 11% in males and 23%, 15%, and 11% in females. Golden rectangle concept can be used for choosing dimensions of maxillary central incisors which are esthetically pleasing. Conclusion: RED proportion is an unsuitable methods to relate the successive widths of the maxillary anterior teeth. The golden percentage theory seems to be applicable to relate the successive widths of the maxillary anterior teeth if percentages are adjusted taking into consideration the ethnicity of the population. Golden rectangle concept is suitable for choosing dimensions of maxillary central incisors which are esthetically pleasing.

It can be found quite often in the literature that many well-known architects have employed either the golden rectangle or the Fibonacci rectangle in their works. On contrary, it is rare to find any specific reason for using them so often. Recently, Shekhawat (2015) proved that the golden rectangle and the Fibonacci rectangle are one of the best connected rectangular arrangements and this may be one of the reasons for their high presence in architectural designs. In this work we present an algorithm that generates n - 4 best connected rectangular arrangements so that the proposed solutions can be further used by architects for their designs.

The golden ratio has universal application and provides maximum performance in an artistic composition. This article discusses how knowledge about golden ratio can be used in jewelry design. The golden rectangle and golden pentagon were used as grids to generate jewelry with harmonic proportions.

The close relationship between art and science is a distinctive feature of the Renaissance, and, as we have observed again and again, Leonardo represents one of the highest points of this synthesis. The scientific knowledge that he obtained via his studies became a fundamental instrument also for his work as an artist and vice versa. As discussed in the previous chapter, it was thanks to his dear friend Luca Pacioli that Leonardo was able to fully appreciate the beauty of geometry and arithmetic. Luca Pacioli’s support was also critical, more generally, in enabling him to achieve a satisfactory knowledge of mathematics.

© Published under licence by IOP Publishing Ltd. Treemaps, a visualization method of representing hierarchical data sets, are becoming more and more popular for its efficient and compact displays. Several algorithms have been proposed to create more useful display by controlling the aspect ratios of the rectangles that make up a treemap. In this paper, we introduce a new treemap algorithm, generating layout in which the rectangles are easier to select and hierarchy information is easier to obtain. This algorithm generates rectangles which approximate golden rectangles. To prove the effectiveness of our algorithm, at the end of this paper several analyses on golden rectangle treemap have been done on disk file system.

The objectives of Physical Chemistry: Concepts and Theory are presented. The book is organized in accordance with the mathematics used: continuum for thermodynamics; discrete for quantum theory; statistics and probabilities for statistical mechanics; and the mathematics for potential-driven changes at short and long times for kinetics. The Reader is presented with material developed from the fundamental mathematical form of differential equations rather than integrated expressions to provide a better understanding of the interrelationships between subfields within the area of physical chemistry. This approach reveals how experimental conditions and a judicious choice of theoretical expressions can reduce the equations to a suitable form for the system at hand. The introduction of how mathematics describes the world we experience is at three levels: Nature is Patterns, Nature is Numbers, and Nature is Recursion Relationships. Many objects in Nature are related to the Golden Ratio, such as apple seeds, spiral galaxies, and mollusk shells. Numbers give a quantitative description of Nature and the power of prediction. The complexity of Nature is embraced in the relationships between the numbers as manifested in recursion relationships, in which a few numbers can be used to generate a more complete description of Nature. The study of prime numbers is an example of how a scientific study on a system evolves through the use of mathematical models. It is through mathematics that information is gained about a system, which leads to a simple understanding as reflected in movies and television series such as Star Trek.

As part of a larger research aimed at developing design aids for architects, this paper presents the “automated” generation of the “best connected” rectangular floor plans, satisfying given topological and dimensional constraints. It has been seen that architects, knowingly or unknowingly, have often used either the golden rectangle or the Fibonacci rectangle in their works throughout history. But it was hard to find any specific reason for such use, other than aesthetic. In 2015, Shekhawat showed that they are among the best connected rectangular arrangements (dimensionless rectangular floor plans) and that this may well be another reason for their frequent use in architectural design. In this work, an alternative algorithm is presented which generates n − 3 best connected rectangular arrangements, being n the number of rooms. Then, this concept is further extended for constructing the best connected dimensioned rectangular floor plans. The goal is to provide an optimal solution for the rectangular space allocation problem, while satisfying given topological and dimensional requirements.

A polyomino is called odd if it can tile a rectangle using an odd number of copies. We give a very general family of odd polyominoes. Specifically, consider an L-shaped polyomino, i.e., a rectangle that has a rectangular piece removed from one corner. For some of these polyominoes, two copies tile a rectangle, called a basic rectangle. We prove that such a polyomino is odd if its basic rectangle has relatively prime side lengths. This general family encompasses several previously known families of odd polyominoes, as well as many individual examples. We prove a stronger result for a narrower family of polyominoes. Let L n denote the polyomino formed by removing a 1 × (n−2) corner from a 2 × (n−1) rectangle. We show that when n is odd, L n tiles all rectangles both of whose sides are at least 8n 3, and whose area is a multiple of n. If we only allow L n to be rotated, but not reflected, then the same is true, provided that both sides of the rectangle are at least 16n 4. We also give several isolated examples of odd polyominoes.

Maxillary central incisors are of critical value in the dental esthetics. The golden rectangle ratio concept may play a role in selecting the optimum width and length of this tooth. The aim of this study is to investigate the existence of this ratio among individuals with natural dentition and routinely used acrylic denture teeth sets and to validate its role in dental esthetics. The clinical crown length of maxillary left and right central incisor (CI) is measured, mean is calculated and combined crown width of left and right central incisors is measured. Crown width to crown length ratio of natural dentition and acrylic denture teeth sets of different manufacturers was calculated and compared with the Golden Rectangle ratio and statistically analysed. Golden rectangle was found to have a significant relationship with esthetic appearance of maxillary central incisors. The results of this study will help in proper selection of teeth for all restorative procedures.

It is generally accepted that compositions in deconstructive architecture are irrational, fragmented, and do not follow proportional systems or principles of architecture, such as harmony, continuity, and unity. These compositions are understood as the result of compilations of random geometries that are often non-rectilinear, distorted, and displaced. In spite of this, deconstructive architecture is widely accepted and practiced in the last couple of decades. On the other hand, geometrical proportions have long been considered as a self-guided method of aesthetically proven designs. This paper examines the hypothesis that the golden rectangle as a proportional system is manifested, to a varying degree, in deconstructive architecture. Methodologically, the hypothesis was tested using two inter-related methods. First, Tension Points of three famous examples of deconstructivist architecture were identified using the Delphi method by a panel of experts. Second, a matrix of displaced golden rectangles was used to test the degree of correspondence between the tension points of the case studies and the golden rectangle. It was found that deconstructive architecture is not a type of “free-form” architecture; and that conventional proportional systems and aesthetics laws, such as the golden ratio, are partially manifested in its compositions and forms, thus confirming the hypothesis. This paper argues that since architects are trained to capture proportional systems and design according to certain organizational and proportional principles, this would inevitably be consciously or unconsciously reflected on their designs.

This paper analyses the presence of the Fibonacci numbers and the golden section within a characteristic project of one of the twentieth century’s leading architects: the Pastoor Van Ars church in The Hague designed by Aldo van Eyck. By means of a thorough analysis of the building based on field work and consultation of unpublished documents, it is possible to show that the compositional scheme of the plan of the church seems to have been designed according to Fibonacci relationships and the golden rectangle.

Objective To introduce diagram of Nine-mansions in which another diagram named Xian-Tu is nested together with the mathematical knowledge hidden in them. Methods Exploring with algebraic method the diagram of Nine-mansions and diagram of Xian-Tu nested in it and comparing the diagrams with the eight-pointed-star veins. It widely exists in the Neolithic Age's antiques found in China. Results The golden rectangle and golden spiral were found hidden in the diagrams and they show close relationship with eight-pointed-star veins. Conclusion The mathematical knowledge does exists in the diagram of Nine-mansions and Xian-Tu, which may mean something for Traditional Chinese Medicine (TCM) for its discourse on Nine-mansions and Eight-winds obviously employed the diagram of Nine-mansions for its medical purpose.

The drawing method of a regular polygon has been explained in a variety of different methods in the relevant literatures. Most of the students against the plotting described procedure to complete the drawing step by step, but often without knowing the principle. It's important to develop a reasonable drawing scheme with simple geometry applied in the engineering drawing technology. A new model for regular pentagon (regular polygon, n=5) drawing will be developed with Discovery teaching method (as experimental group). The performance is better than Demonstrating method (as reference group) that students contrast the drawing procedure to finish a regular pentagon step by step. We chose the students of vocational high school provided with engineering background as a sample space. The pass probability for the experimental group with Discovery teaching method is p=.692 which is higher than the reference group p=.544. The golden rectangle is also applied to map the relationship of regular pentagon and developed an interest for engineering drawing.

The two most striking characteristics of the sequence of primes are that there are many of them but that their density is rather slim.

The paper mainly conducts the study on the application of The Golden Ratio in landscape design through the planning of Liuqing Park. Exploring firstly the popularity of The Golden Ratio in plants, animals, and human, and as well as the exploitability in art, architecture, and formal garden design, it makes creative designs to the determination of a fountain and the path annexed to the minor roads. Under the guide of The Golden Ratio, detailed drawing methods to locate the position and the paths have been given, and to be a good try to combine mathematics and landscape design.

Gustav Fechner is widely respected as a founding father of experimental psychology and psychophysics but fewer know of his interests and work in empirical aesthetics. In the later 1800s, toward the end of his career, Fechner performed experiments to empirically evaluate the beauty of rectangles, hypothesizing that the preferred shape would closely match that of the so-called 'golden rectangle'. His findings confirmed his suspicions, but in the intervening decades there has been significant evidence pointing away from that finding. Regardless of the results of this one study, Fechner ushered in the notion of using a metric to evaluate beauty in a psychophysical way. In this paper, we recreate the experiment using more naturalistic stimuli. We evaluate subjects' preferences against models that use various types of object complexity as metrics. Our findings that subjects prefer either very simple or very complex objects runs contrary to the hypothesized results, but are systematic none the less. We conclude that there are likely to be useful measures of aesthetic preference but they are likely to be complicated by the difficulty in defining some of their constituent parts.

Suppose that a rectangle with a longitudinally symmetric slit is given. Then we want to find another rectangle with a square removed and a conformal mapping between them so that the ordered set of four vertices is mapped on the other ordered set of four vertices. We also determine the modulus of the image rectangle and the size of the square removed.

which is called the golden ratio [5] or its reciprocal [4], depending on the author. Almost any discussion oftthe golden rectangle will include the image in figure 1, which is constructed by starting with a golden rectangle, partitioning it into a square and a smaller golden rectangle, and then repeating the process indefinitely. We will refer to any such set as a golden set. Authors occasionally include the golden set in discussions of self-similarity [5, 6], because it is the union of a square and a smaller copy of itself, scaled by a factor of y. Some popularizations of the golden ratio [4, 7] also include examples of truly self-similar, planar fractals to illustrate properties of y. Now the golden set is not truly self-similar, because it is not a union of smaller copies of itself, but it suggests a problem: Can we construct a self-similar fractal made of golden sets?

Walken Farmhouse (Gutshaus Walken) was designed by Hugo Haring in 1922. The purpose of this paper is to derive the proportional relationship in the floor plan of the House. Many approximations of 1:1, 1:2, 1:√2 and 5:7 are extracted from the dimensions of the first floor plan. This paper clarifies that design of the first floor based on a composition of geometric shapes such as squares and rectangles whose horizontal to vertical ratio are 1:2, 5:7, and few 1:√2. Those shapes have the proportional relationship.

The golden quadratic x 2 − x − 1 = 0, when re-expressed as (x)(1) = 1/(x − 1), x = 1.618, can be interpreted as the algebraic expression of division in extreme and mean ratio (DEMR) of a line of length x = 1.618 into a longer section of length 1 and a smaller of length (x − 1). It can, however, also be interpreted as the formulation of the area of a golden rectangle of sides x = 1.618 and 1, and as the system of equations constituted by y = x, and y = 1/(x − 1). Based on the well-known connection existing between the first two of these interpretations, the authors address the problem of finding out the thread connecting the golden rectangle with the system of equations referred to above. The results obtained indicate first that this system, like the golden rectangle, also carries in its geometry the essential traits of DEMR; and, second, that it implicitly subsumes the simpler rectangular geometry of its alternative interpretation. The process of developing these connections brought forward a heretofore apparently unreported golden trapezoid of sides Φ, 1, ϕ and \(\sqrt{2}\).

To Renaissance mathematician Luca Pacioli, it was the Divine Proportion. To German astronomer Johannes Kepler, it was a precious jewel. The only proportion to increase simultaneously by geometric progression and by simple addition, the Golden Section achieves unity among diverse elements in remarkably efficient ways. We explore the Golden Ratio 1: ϕ , also known as the Golden Mean, and its appearance in the regular pentagon and other geometric constructions.

The present research envisages the aesthetic design optimization strategy employed to achieve an aesthetically pleasing sustainable design of a Human cum Electric Powered Vehicle. The perspective of golden ratio is implicated as a tool for predicting, analyzing and iterating the design for an aesthetic and sustainable design. The vehicle—Vajrayana, is a two seater, tricycle with convertible rooftop cum wind shield type aerodynamic body. This paper is concerned, first with the problem of designing a streamlined vehicle shape having a convertible rooftop cum wind shield type aerodynamic body, followed by a sustainable artistic approach, to approximate, various iteration with golden ratio-especially in the form of the golden rectangle. The final design is simulated to validate that golden ratio, with the concept of sustainability and aerodynamics can be tailored to give pleasant vehicle design and overall performance.

Geometer Mark Reynolds discusses four approximatons for finding the Golden Section of the circumference of a circle from the root-2 rectangle.

The processes of manipulation of the square that lead to the production of ad quadratum, the golden rectangle, and the sacred cut exploit the properties of the square, but only by virtue of the existence of a primal figure, a first and original square. That ‘first’ square has to be the result of a simple, clear, effective and efficient method of construction: a method that constituted its original geometry and established its perfection as the instrument for all subsequent geometrical and metrological manipulations. The paper deals directly with this puzzle of the first square, tracing its origins to the emergence of the western architectural tradition in ancient Egypt. The question that it will ask and answer is: “how did that original square get there in the first place?” It will do this by showing that there are recurring references to a very specific pair of measurements—the 20-digit remen and the 28-digit cubit—and that there are recurring discussions of techniques for their application.

This paper describes a student/faculty research project as one example in an area that appears to offer other opportunities for undergraduate research. Our project was to investigate whether people tend to prefer the rectangular shape known as the golden rectangle to other rectangular shapes. We examine methods used in previous experiments, identify some sources of bias in them, and offer the results of an experiment of our own design.

The above figure which has been famous since The Welfare Costs of Tariffs, Monopolies and Theft (Tullock, 1967) has led to a very large amount of research, and even a larger amount of references to the importance of the rent-seeking cost. Basically, what the article claimed was that the total cost of monopolies, tariffs and crimes was represented not only by the triangle at the right, usually called the Harberger triangle, but also by the rectangle. It argued that the development of monopoly, tariff, or other special privilege normally involved the investment of resources. There is no reason why these resources should receive a higher return than resources invested in other activities, hence the rectangle, which represented the return on them should more or less equal the amount of resources invested. In the words, the social waste from rent seeking was much greater than the previous studies had indicated.

This paper gives algorithms to dissect a rectangle sequentially into polygons that can be reassembled to a rectangle, which is similar to the initial rectangle, and [Formula: see text]-similar equilateral triangles or regular hexagons and also pentagons or heptagons or octagons with symmetry.

Associating geometric arrangements of 9 Loshu numbers modulo 5, investigating property of golden rectangles and characteristics of Fibonacci sequence modulo 10 as well as the two subsequences of its modular sequence by modulo 5, the Loshu-Fibonacci Diagram is created based on strict logical deduction in this paper, which can disclose inherent relationship among Taiji sign, Loshu and Fibonacci sequence modulo 10 perfectly and unite such key ideas of holism, symmetry, holographic thought and yin-yang balance pursuit from Chinese medicine as a whole. Based on further analysis and reasoning, the authors discover that taking the golden ratio and Loshu-Fibonacci Diagram as a link, there is profound and universal association existing between researches of Chinese medicine and modern biology.

Can we produce a ratio ψ to rival the golden ratio ϕ? In this article I shall construct a variant of the classical golden rectangle by choosing a different geometrical property. The rectangle described has more advanced mathematical properties than its golden relation, though it is not extravagant. In this sense it could be described as a supergolden rectangle.

We present a simple texture that predicts the cotangent of the solar neutrino mixing angle to be equal to the golden ratio. This prediction is 1.4{sigma} below the present best-fit value and final SNO and KamLAND data could discriminate it from tri-bimaximal mixing. The neutrino mass matrix is invariant under a Z{sub 2} x Z{sub 2}{sup '} symmetry: that geometrically is a reflection along the diagonal of the golden rectangle. Assuming an analogous structure in the quark sector suggests a golden prediction for the Cabibbo angle, {theta}{sub C}={pi}/4-{theta}{sub 12}{approx_equal}13.3 deg., up to the uncertainties comparable to V{sub ub}.