Golden Rectangle Research Articles

INVESTIGATING THE HIDDEN GEOMETRY SYSTEM AT THE ENTRANCE OF THE HISTORICAL HOUSES OF KASHAN FROM THE QAJAR PERIOD TO THE BEGINNING OF THE PAHLAVI PERIOD

Geometry is one of the most important aspects hidden in the design of historical buildings in a way, it can be said that architecture without the presence of geometry is meaningless and lacks functional validity. The ratio and proportion that can be seen in the entrance and front facade of the historical houses of Kashan is a function of geometry, which is named hidden geometry in this research. The geometric system at the entrance of historical houses is one of the most important topics that help to better understand the historical building. The findings show that the entrance design of the houses is under the hidden geometry system, and five types of geometric systems are used to shape it, including symmetry, Fibonacci in golden triangles, Fibonacci in golden rectangles, dynamic rectangles, and regular polygons.

Golden Ratio Flap Designed Using the Golden Ratio Rectangle.

Rotation flaps are arcuate repairs that redistribute tension vectors and recruit adjacent and/or distant tissue laxity. The incision curve could be a logarithmic spiral curve to reduce the length of an incision. We propose a rotation flap-the golden ratio flap-designed using a golden rectangle. The flap incision is an arc though the major square which is beside the minor square of a golden rectangle. The defect is attached to the line of another minor square and diagonal to the major square. The bottom line runs from the incision end to the tangent point of the circle or the oval, and the perpendicular height line runs from the bottom line to the cross point of the flap incision. These parameters were analyzed retrospectively for four superficial sarcomas that were treated using a rotation flap with an incision approximating the logarithmic spiral curve. The ratio of height to bottom of the golden ratio flap design was highly similar to the preoperative flap design in the four cases assessed. With the new design, the ratio of bottom to the defect diameter (minor axis in the oval defect) was 1.3, and for the height, it was 1.4. The golden ratio flap, designed using the golden rectangle, is reproducible. The parameters of height and bottom approximate the flap shape, or the length and width, respectively. For clinical applications, step-by-step guidance for drawing the new flap are also proposed.

POWER LINES OF THE FORMAT AS A METHOD OF TEACHING COMPOSITION

The issue of this study isthe contradiction between the need to optimize the training of students of art and graphic faculties of pedagogicaluniversities in the field of composition and the insufficient development of methodsof teaching composition adapted for art and pedagogical education. The research methods used were the study of literature on a given topic, conversations with artists, teachers and art theorists, as well as the practice of teaching compositionat the Institute of Fine Arts of MoscowPedagogical State University for several years. In the course of the study, we studied the concept of «power linesof the format» from the point of view of the geometry of the picture and touched upon some philosophical aspects of this concept related to intuition and implicit knowledge, relying on the works by R. O. Jacobson and L. A. Mikeshina.The main functions of power lines as constructions underlying compositional schemes are indicated. On specific examples, such types of power linesas the square and the surplus element, the central axes, «thirds», «quarters» and«fifths» are considered. Special attention is paid to the concept of the «golden section» as one of the cornerstones of composition: its definition, method of construction, and role in the picture are given. The golden rectangle as well as increase and decrease of numbers in the golden proportion are considered altogether with the placement of the construction elements based onit. The Fibonacci spiral as a power line of the format, which is a special caseof the application of the golden section rule, is drawn. The article also studiesin detail the diagonals of the format and two different types of their classification associated with the names of such theorists as N. M. Tarabukin and A. A. Kovalev.The result of the study was the development of a compositional exercise called «Power lines of the format», the method of its implementation, as well as its introduction into the practice of teaching compositionat the Faculty of Art and Graphic Artsof Moscow Pedagogical State University.

Maxillary Anterior Teeth Dimensions and Relative Width Proportions: A Narrative Literature Review.

Predictable results in the aesthetic treatment of anterior teeth can be obtained by resorting to the concept of dental aesthetics and, in particular, defining the ideal tooth dimensions and proportions to obtain a harmonious smile. Considering the great variety of articles dealing with the topic, and the lack of updated reviews, this narrative literature review aims to evaluate current knowledge on anterior teeth dimensions and to verify the existence and the potential applications of the anterior teeth proportioning theories (Golden Proportion, Golden Percentage, RED Proportion, and Golden Rectangle). PubMed, Embase, Cochrane Library, and Google Scholar databases were comprehensively searched using different keywords and term combinations. The research includes articles published up to June 2023, no time limits were set, and only articles in English were included. Inclusion criteria comprehended reviews, clinical studies, and in-vitro studies. A total of 66 articles were selected. Two main topics were identified: "Anterior teeth dimensions", "Golden Proportions, Golden Percentage, RED Proportions, and Golden Rectangle". As far as tooth dimensions are concerned, different width ranges are recognized for men and women and for different ethnic groups. Perfectly symmetric contralateral elements are found in low percentages of subjects. The correlation between dental dimensions and facial parameters is not always present, and it strongly depends on the sample's ethnicity and gender. Ideal tooth proportions were only partially identified.

Limits of Golden Constructions

Summary A golden rectangle is characterized by the fact that if an inscribed square is removed from one end, then the remaining rectangle is similar to the original one. By iterating this process of removing a square, one obtains an infinite sequence of nested golden rectangles which converges to a point. One can construct other sequences of rectangles by starting from arbitrary, not necessarily golden, rectangles. The goal of this paper is to analyze the behavior of these sequences, primarily by modeling the process using linear algebra.

Comparation between golden rectangle and ideal facial height proposed by the Silva Cephalometric Method (MCS) through digital analysis (NemoStudio 2022) in a university population

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.

Golden Tiles and Arctangent Identities

Summary We use plane tilings with golden rectangles to derive arctangent identities.

Golden right triangles and the golden quadrilateral

A golden right triangle of first type is a right triangle such that the shortest side is the golden section of the hypotenuse as defined in [1], [2]. If we consider a golden rectangle and one of its diagonals, we obtain a right triangle with the shortest side being the golden section of the other side; this right triangle will be named golden right triangle of second type.

Використання властивості періодичності для генерування комбінаторних конфігурацій

Introduction. Sign combinatorial spaces that exist in two states: convolute (tranquility) and deployed (dynamics), are considered. Spaces, in particular biological, physical, informational and some others, for which the axioms of sign combinatorial spaces, are valid, have a combinatorial nature. When they are deployed, combinatorial numbers (Fibonacci numbers) are formed, through which logarithmic spirals appear in living nature. These spirals are formed due to the finite sequences that take place during the deployment of the agreed spaces and which are presented geometrically using polar coordinates. Formulation of the problem. The logarithmic spiral is geometrically represented through a “golden rectangle” in which one side is 1,618 times longer (“golden” number or golden section). The presence of the golden ratio in nature is manifested through Fibonacci numbers, which are formed from an arithmetic triangle from elements of finite sequences formed by the deployment of sign combinatorial spaces. But this spiral is transmitted through the “golden rectangle” indirectly. The problem is to trace its formation in nature through constructed sequences, the elements of which are represented by polar coordinates. The approach proposed. Using the finite sequences that are formed during the unfolding of sign combinatorial spaces and the representation of their elements in polar coordinates, we can trace the dynamics of the formation of logarithmic spirals in nature. Conclusion. Representation of biological space as a sign combinatorial space can explain various phenomena in nature. When unfolding these spaces from the convolute spaces finite sequences are formed, the sums of the members of which determine the number of combinatorial configurations in a subset of isomorphic combinatorial configurations and form an arithmetic triangle (Pascal’s triangle). Fibonacci numbers and, accordingly, a golden number are formed from an arithmetic triangle. The logarithmic spiral fits into a golden rectangle. The dynamics of the formation of the logarithmic spiral is traced due to the finite sequences formed as a result of the deployment of the sign combinatorial spaces, the elements of which are presented in polar coordinates.

Potential of technology to deepen understanding of the hidden complexity of the Golden ratio – From investigations with polygons to generalized golden rectangles

Potential of technology to deepen understanding of the hidden complexity of the Golden ratio – From investigations with polygons to generalized golden rectangles

Finite Blaschke products and 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.

Rectangles and spirals

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.

A parametric analysis of the “digitally-derived geometric design” of the façade of the Macau Holy house of Mercy

ABSTRACT The Macau Holy House of Mercy was built in 1569 and rebuilt in the 18th century. Located in the historic center of Macao, the Macau Holy House of Mercy is included on the World Heritage List of UNESCO.With the Macau Holy House of Mercy, especially the façade of its neo-classical arcade added in 1905, as the research subject, this study applied the methods of historical document research, field measurement and digital analysis, especially parametric analyses with the assistance of the modeling software of Rhinoceros 6.0 and the software plug-in of Grasshopper. And the following results were obtained: 1) its designers applied three-section partition in the horizontal and vertical directions and equal division in stratification in the overall dimension of the façade; 2) its designers also applied modularization (modular) design and combination in the columns and details of doorways; 3) this study found that the (successive) partition of the façade is closer to the “Golden Section“ than “Root Rectangle Partition”; further analyses discovered the Golden Rectangle and Golden Spiral (successive division) in the façade. The underlying aesthetics-mathematics pattern of ”digitally-derived geometric design” is of significant historical value.

Analysis of Geometric Proportions on Maxillary Anterior Teeth for Esthetic Smile Design: An In vivo Study.

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.

بررسی ساختار و سامان‌بندی‌های به‌کار رفته مخمل‌های عثمانی کاخ موزه توپکاپی

بررسی ساختار و سامان‌بندی‌های به‌کار رفته مخمل‌های عثمانی کاخ موزه توپکاپی

A Technology to Synthesize 360-Degree Video Based on Regular Dodecahedron in Virtual Environment Systems

The paper proposes a new technology of creating panoramic video with a 360-degree view based on virtual environment projection on regular dodecahedron. The key idea consists in constructing of inner dodecahedron surface (observed by the viewer) composed of virtual environment snapshots obtained by twelve identical virtual cameras. A method to calculate such cam-eras’ projection and orientation parameters based on “golden rectangles” geometry as well as a method to calculate snapshots position around the ob-server ensuring synthesis of continuous 360-panorama are developed. The technology and the methods were implemented in software complex and tested on the task of virtual observing the Earth from space. The research findings can be applied in virtual environment systems, video simulators, scientific visualization, virtual laboratories, etc.

Golden Wires and Rectangle: A Cost-Effective Treatment for Tuberculosis of the Thoracic Spine.

Study Design:This was a retrospective study.Purpose:The purpose of this study was to compare the functional outcome and cost of surgery for tuberculosis (TB) of the thoracic spine between two commonly used fixation modalities “pedicular screws and rods” and “Hartshill loop rectangle and sublaminar wires.”Overview of Literature:TB is a common ailment in Asia. Surgical indications have remained almost unchanged since the middle-path regimen was advocated by Tuli. Pedicle screws and Hartshill loop rectangle with sublaminar wires are the two common fixation techniques used.Materials and Methods:This retrospective observational study was performed at a single tertiary center. Patients were divided into two groups depending on the method of fixation (pedicle screw rod/Hartshill loop rectangle and sublaminar wires). All patients were evaluated preoperatively by X-rays and magnetic resonance imaging. Patients were assessed clinically by preoperative and postoperative neurology and Visual Analog Scale score and radiologically assessed by the K angle. These variables were separately compared in both the groups.Results:The functional outcomes of Hartshill loop rectangle and sublaminar wire fixation and that of pedicular screw fixation were comparable. Hartshill loop rectangle and sublaminar wire fixation was found to be more cost-effective.Conclusion:Hartshill loop rectangle and sublaminar wire fixation gets purchase over the posterior column structures alone when compared to pedicle screws which have a 3-column hold. However, when combined with meticulous neural decompression and skillful preparation of osteogenic bed with autologous strut grafting and additional onlay grafting, it gives overall adequate stabilization of the column with functional outcome comparable to pedicular screw and rod fixation with additional benefit of cost-effectiveness. Although Hartshill loop rectangle and sublaminar wire fixation is less commonly used now, it has a special place in the management of TB, especially in a resource-poor setting like some countries of Asia.

A universal mathematical model applied to the congenital ventral penile curvature.

The macroscopic dynamic of fetal penis development presents a pattern resembling the unfolding of a spiral, so congenital ventral penile curvature could indicate that this natural sequence has been interrupted. Our aim in this article is to offer a mathematical model of congenital ventral curvature of the penis. Five individuals who presented with congenital ventral penile curvature and three who presented with acquired penile ventral curvature due to Peyronie's disease were evaluated. The penises were photographed during an induced erection test and the penile curvature patterns were compared with an equiangular spiral. When an association was found, a potential relationship to the golden spiral-a type of equiangular spiral-was also assessed. The mathematical spiral relationships were analyzed using Wolfram CDF Player® (Logarithmic Spiral) and PhiMatrix® software. The Wolfram software generated logarithmic spirals equivalent to the penile curvature with appropriate mathematical values. The PhiMatrix software, which builds any golden spirals from golden rectangles, was used to check whether the spiral was golden as well as equiangular. An equiangular spiral that was also golden was found in all cases of congenital ventral penile curvature. In contrast, none of the acquired penile ventral curvature cases showed a specific pattern. Congenital ventral penile curvature has the mathematical pattern of a golden spiral. Our results offer a mathematical algorithm for potential use in surgical reconstruction procedures, regenerative medicine, tissue engineering, robotics, and body-machine interfaces.

Shrinkage Points of Golden Rectangle, Fibonacci Spirals, and Golden Spirals

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.

Golden section based method of 3D modeling of icosahedron and dodecahedron in construction design

This paper provides an introduction of a three-dimensional modeling of the two most complex regular polyhedrons: icosahedron and dodecahedron. It is suggested to apply a method of the icosahedron and dodecahedron building by using rectangles with the sides ratio like in the golden section, and taking the icosahedron’s golden rectangles as a basis. The author gives information on the first mentioning of this building method by the Italian mathematician L. Pacioli in his Divine Proportion book. Based on this method, the building of the icosahedron and dodecahedron inscribed in the regular hexahedron and circumscribed about it is performed. This may be important for the design of facilities, based on the shapes of these geometrical figures, with inscribed regular hexahedrons for the purpose of a better organization of residential space of storeys. The examples of using the geometrical shapes of regular polygons (RP) in architecture are given. The building is performed using the AutoCAD package, but it also can be repeated in other known CAD software.