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Throughout the Anatolian Seljuk Period, numerous examples of buildings with combined functions were built. The precise proportions or design decisions that underlie the division of various functional masses in combined-function buildings are unknown. The purpose of this study is to determine whether geometric and proportional relationships are used in the process of combining multiple functional masses. Geometric analyses of the combined-function buildings in plan and section drawings were carried out to determine the proportions and rules established in the combination of masses with each other. Geometric analyses include analyses of the special triangles, golden ratio, and module systems on the section and plan drawings of the buildings. The investigation revealed that the golden rectangle determined the proportional relations between various masses in the plans. In contrast, the modular system or golden rectangle determined the proportional relations in the sections.Formal relations were only defined in the sections by special triangles. The identification of mass proportions by a golden rectangle represented that one of the main parameters considered in the design of Seljuk buildings is the golden ratio. Achieving a formal and proportionate balance, particularly between masses and critical architectural and structural elements such as portals and cones, was essential in their design process. This focus on geometry enriches our understanding of architectural history and plays a crucial role in the preservation efforts of historical edifices. The insights gained from studying the application of the golden ratio in Seljuk architecture can significantly benefit scholarly research and practical conservation methods.

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.

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.

From the pyramids and the Parthenon to the Sydney Opera House and the Bilbao Guggenheim, this book takes readers on an eye-opening tour of the mathematics behind some of the world's most spectacular buildings. Beautifully illustrated, the book explores the milestones in elementary mathematics that enliven the understanding of these buildings and combines this with an in-depth look at their aesthetics, history, and structure. Whether using trigonometry and vectors to explain why Gothic arches are structurally superior to Roman arches, or showing how simple ruler and compass constructions can produce sophisticated architectural details, Alexander Hahn describes the points at which elementary mathematics and architecture intersect. Beginning in prehistoric times, Hahn proceeds to guide readers through the Greek, Roman, Islamic, Romanesque, Gothic, Renaissance, and modern styles. He explores the unique features of the Pantheon, the Hagia Sophia, the Great Mosque of Cordoba, the Duomo in Florence, Palladio's villas, and Saint Peter's Basilica, as well as the U.S. Capitol Building. Hahn celebrates the forms and structures of architecture made possible by mathematical achievements from Greek geometry, the Hindu-Arabic number system, two- and three-dimensional coordinate geometry, and calculus. Along the way, Hahn introduces groundbreaking architects, including Brunelleschi, Alberti, da Vinci, Bramante, Michelangelo, della Porta, Wren, Gaudí, Saarinen, Utzon, and Gehry. Rich in detail, this book takes readers on an expedition around the globe, providing a deeper understanding of the mathematical forces at play in the world's most elegant buildings.

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.

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 oil and gas, industry has been on an 'elephant hunt\" off West Africa for the past decade. This elephant hunt, or search for giant offshore oil fields, is focused on geologic sweet spots in the Gulf of Guinea and is most active in the offshore sectors of Nigeria, Equatorial Guinea, and Angola. One of the richest and most prolific of these sweet spots is a 1.2 million acre offshore tract that we call the Gulf of Guinea's \"Golden Rectangle,\" where recoverable oil reserves are reported in excess of 5 billion bbl and oil production exceeds 1 million b/d. The Golden Rectangle is located near the offshore intersection of the maritime boundaries of Nigeria, Equatorial Guinea, and Cameroon. The geological and engineering expertise gained from the exploration and exploitation of the Golden Rectangle's oil reserves over the past 15 years has provided a template for hydrocarbon exploration and production into other deepwater areas of Nigeria and Equatorial Guinea. This expertise will be crucial to the successful development of the Nigeria-Sao Tome Joint Development Zone, where 6 to 12 billion bbl of oil are mooted.

The traditional remote graphic design system cannot provide users with the basis of design geometry theory, which leads to the disharmony of the design works. For this reason, a computer-aided remote graphic design system is proposed. Determine the image import direction, obtain the contour ratio, compare it with different dynamic rectangle ratios, and give the difference between the image and the standard value and the recommended adjustment value. The experimental results show that the works designed by the design system conform to geometry, are visually harmonious, and the evaluation results are good.

Consumers' reactions to rectangles have implications for package and product design. In two lab studies and an analysis of field data, the authors find that the ratio of the sides of a rectangular product or package can influence purchase intentions and preferences and is related to marketplace demand. In more exploratory inquiries, the authors also find that the impact of this ratio on consumers depends on the relative seriousness of the context in which the product is used. Furthermore, ratio can also affect product perceptions, and consumers appear to prefer a range of contiguous ratios for different contexts rather than a particular ratio.

Art and architecture can be an obvious choice to pair with science though historically this has not always been the case. This paper is an attempt to interact across disciplines, define a new genre, bioarchitecture, and present opportunities for further research, collaboration and professional cooperation. Biomimetics, or the copying of living nature, is a field that is highly interdisciplinary, involving the understanding of biological functions, structures and principles of various objects found in nature by scientists. Biomimetics can lead to biologically inspired design, adaptation or derivation from living nature. As applied to engineering, bioinspiration is a more appropriate term, involving interpretation, rather than direct copying. Art involves the creation of discrete visual objects intended by their creators to be appreciated by others. Architecture is a design practice that makes a theoretical argument and contributes to the discourse of the discipline. Bioarchitecture is a blending of art/architecture and biomimetics/bioinspiration, and incorporates a bioinspired design from the outset in all parts of the work at all scales. Herein, we examine various attempts to date of art and architecture to incorporate bioinspired design into their practice, and provide an outlook and provocation to encourage collaboration among scientists and designers, with the aim of achieving bioarchitecture. This article is part of the themed issue 'Bioinspired hierarchically structured surfaces for green science'.