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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.

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.

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 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.

The Fibonacci sequence of numbers is known since the ancient past. It is considered as a proportion which can be frequently appreciated in nature. The associated ratio, called as the golden ratio is manifested in the works of art, nature, galaxies, monuments etc. Many famous mathematicians, philosophers, and artists have used this ratio in their work. Humans distinguish this proportion as pleasing, as it enlightens beauty when applied in living or non living entities. The ideal proportion is directly related to golden proportion. It ranges from 1 to 1.618. Moreover, the proportion can be applied in facial aesthetics also. Studies have showed that the beautiful faces have facial measurements close to golden ratio. Thus, the treatment following the standard will help in obtaining optimal facial aesthetics. The review is written with an aim to highlight the historical perspectives, its varied applications in the past and present; and further scope of research in future.

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.

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.

We investigated the golden rectangle and the related Fibonacci spiral and golden spiral. The coordinates of the shrinkage points of a golden rectangle were derived. Properties of shrinkage points were discussed. Based on these properties, we conduct a comparison study for the Fibonacci spiral and golden spiral. Their similarities and differences were looked into by examining their polar coordinate equations, polar radii, arm-radius angles, and curvatures. The golden spiral has constant arm-radius angle and continuous curvature, while the Fibonacci spiral has cyclic varying arm-radius angle and discontinuous curvature.

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.

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.

ResumenEn este trabajo se diseñó y estableció un jardín xerófito con fines de divulgación científica. Ésto se realizó en las áreas verdes del Campus I de la Facultad de Estudios Superiores Zaragoza, de la Universidad Nacional, al oriente de la Ciudad de México. El diseño del jardín consistió en trazar y dibujar, en la superficie del suelo, una espiral logarítmica, una estrella de Vitrubius, un pentágono fractal y un rectángulo áureo. Se utilizaron estacas de 10 cm de alto, pintadas de blanco, para delimitar las curvas y plantas crasas para remarcar las figuras geométricas. Se trasplantaron 29 especies vegetales, con 144 individuos en total, y se instaló un atrapanieblas como modelo de cosecha de agua atmosférica para riego vegetal. La divulgación científica de los conceptos geométricos se ha realizado mediante visitas guiadas. Se concluye que es factible establecer un jardín urbano de cero riego en el Oriente de la capital mexicana, sin embargo, es necesario reforzar las estructuras de las curvas geométricas. También, se requiere establecer páneles informativos sobre las figuras dibujadas sobre el suelo y cédulas con datos técnicos de las especies vegetales, para facilitar la divulgación de los conceptos científicos del diseño base del jardín y la explicación de la función del atrapanieblas.AbstractIn this work, a xerophytic garden was designed and established for science popularization purposes. This was made on a green area of the Facultad de Estudios Superiores Zaragoza of the National University, at the East of Mexico City. The garden design was outlined by drawing on the soil surface, a logarithmic spiral, a Vitrubius star, a fractal pentagon and a golden rectangle. For this, white painted 10 cm high stakes were buried to delimit the curves, and succulent plants were planted to remark the geometric figures. Twenty nine plant species were transplanted, with 144 total individuals, and a catch-fog mesh device was installed with a view to exhibit an atmospheric water harvesting model for plant irrigation. The science popularization of geometric concepts was made by means of guided visits. It was concluded that it is feasible to establish an urban garden without irrigation in the East of Mexico City. Nevertheless, it is necessary to strengthen the structure of geometric curves. Also, it is required to install information panels about the soil drawings and technical fact sheets on plant species in order to facilitate popularization of the underlying scientific concepts of garden design and the explanation of catch-fog device function.