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Misconceptions
about
the Golden Ratio
by George Markowsky
Misconceptions about the Golden Ratio
Abstract
The golden ratio, also called by different authors the golden section [Cox], golden number [Fi4], golden mean [Lin], divine proportion [Hun], and division in extreme and mean ratios [Smi], has captured the popular imagination and is discussed in many books and articles. Generally, its mathematical properties are correctly stated, but much of what is presented about it in art, architecture, literature, and esthetics is false or seriously misleading. Unfortunately, these statements about the golden ratio have achieved the status of common knowledge and are widely repeated. Even current high school geometry textbooks such as [Ser] make many incorrect statements about the golden ratio. It would take a large book to document all the misinformation about the golden ratio, much of which simply the repetition of the same errors by different authors. This paper discusses some of the most commonly repeated misconceptions.
... Here, we are presenting the golden ratio corresponding to plane geometry. For details, we refer to (Akhtaruzzaman & Shafie, 2011;Livio, 2002;and Markowsky, G. (1992). ...
... Here, we are presenting a few of such solids. For details, we refer to (Markowsky, 1992;Pacioli, 1509) . ...
Mathematics has a strong logical interconnection with structural design and architecture. Ratios are good examples of their interconnectivity. Nature is attractive due to the proper ratios of various components in them and between others. In nature, trees, forests, leaves, flowers, fruits, Himalayas, hills, valleys, springs, rivers, lakes, etc. show a rhythmic balance and seem harmonic and aesthetically pleasing. The golden ratio is considered the most pleasing to human visual sensation not limited only to aesthetic beauty but also found prominently in the natural world and is a fascinating topic. It can be constructed in different ways depending on the geometrical structures preferred. In this paper, our focus is on the geometry of the golden ratio.
... Also, any power of φ is equal to the sum of two immediately proceeding powers as φ n = φ n−1 + φ n−2 (9) If n = 1 then this Equation 9 reduces to φ = 1 + 1 φ . ...
... Moreover, it is even closer to π-based mathematical pyramid. Historians of science have long debated whether the Egyptian had any such knowledge, contending that its apearance in the great pyramid of Giza is the result of chance [9]. ...
Nature is attractive due to the proper ratios of various components in them and in between others. In nature, leaves, flowers, fruits, Himalayas, hills, valleys, springs, rivers, etc. show a rhythmic balance and seem harmonic and aesthetically pleasing. The golden ratio or the golden (divine) proportion is considered the most pleasing to human visual sensation and is not limited only to aesthetic beauty but also found prominently in the natural world. It is an irrational number denoted by φ and is approximately 1.618034. It is a fascinating topic for mathematicians, artists, natural scientists, philosophers, biologists, artists, architects, musicians, psychologists, designers, etc. This research work presents a panoramic view of the golden ratio; their construction algorithms, and structures; mathematical presentations, geometry, patterns, and properties from mathematical perspectives and opens the broad horizon for further research. Besides these, the paper also presents their existences, diversities, their relationships with nature, the universe, arts, design, architecture, and engineering in ancient and modern mathematics and sciences.
... and fit a polar equation for the logarithmic spiral through any two of these points [5]. It is a general property of all such spirals that the tangents to the spiral at any point make a fixed angle with the rays from the center point [ Markowsky,(1992) i. Petals of Flowers: Many flowers have a number of petals that is a Fibonacci number. The arrangement of petals often exhibits the Golden Ratio, optimizing exposure to sunlight and space. ...
The Golden Ratio, approximately 1.618, is a mathematical constant that has captivated the imagination of scholars, artists, architects, and scientists for centuries. This article delves into the mathematical underpinnings of the Golden Ratio, tracing its origins and historical significance in ancient civilizations and its evolution in mathematical theory. Known for its unique properties, the Golden Ratio is often associated with beauty, balance, and harmony, which has led to its widespread application in diverse fields. From the design of iconic architectural marvels like the Parthenon to its presence in Renaissance art, the Golden Ratio continues to inspire creativity. It also manifests in nature, evident in the arrangement of leaves, the spiral patterns of shells, and the structure of galaxies. In modern science, the Golden Ratio finds relevance in engineering, computer algorithms, and even financial models. This exploration underscores its enduring allure, showcasing its theoretical elegance and practical utility across time and disciplines
... It appears in many geometrical constructions, including the proportions of the regular pentagon and the Fibonacci sequence. See [11], [1] and [8]. ...
In this article, we present new numerical approximations involving five numerical constants, namely the Pi constant, the Euler constant, the Euler-Mascheroni constant, the Catalan constant, and the golden ratio constant. Four of these constants can be simply approximated by the others with relative accuracy. These results have potential applications in mathematics education by providing an accessible framework for exploring the connections between fundamental constants. In a sense, they also promote a deeper conceptual understanding of numerical analysis.
... Basically, the following mathematical rule is used to find the golden ratio. If a line AB is divided from any point C to give the ratio AB/AC = AC/CB, C is the golden part of AB, and the ratio AB/AC and AC/CB that creates this ratio is called the golden ratio (Markowsky, 1992;Mitchison, 1977). ...
There is evidence showing the existence of the golden ratio in many fields from architecture to medicine, from art to philosophy. The aim of this study is to investigate the relationship between the measurements taken from the face area and the golden ratio (1.618) value and to present results by comparing the determined measurement parameters with the existing literature. A total of 101 university students aged between 18-25 participated in this study on a voluntary basis. In this study, data were collected by photographing method. Photographs of each individual were taken from 3 angles: the front and both sides. After the photographs were transferred to digital media, measurement was performed by means of a software program (ImageJ). In the study, data were presented as mean±standard deviation in the form of descriptive statistics. In order to compare the measurement points within the scope of the study with the golden ratio, 23 measurement indices (ratio) were determined. Proportional Index Calculation was done as follows. Head Index Ratio=Head Height/Total Cranial Head Height. A statistically significant difference was determined when the mean value of this index was compared according to sex (p<0.05). As a result, we think that the data we obtained in the study will be useful in the fields of science and art and may be a pioneer for the golden ratio studies to be done later.
The Impacts of Golden Ratios on Visual Aesthetics in the Historic City of Masouleh*
Saeid Hasanpour Loumer1*, Ahad Nejad Ebrahimi 2, Hassan Sattari Sarbangholi3, Ali VandShoari4
1. Ph.D. Graduated in Architecture, Department of Architecture, Faculty of Art and Architecture, Islamic Azad University of Tabriz Branch, Tabriz, Iran.
2. Professor of Architecture, Department of Architecture, Faculty of Art and Architecture, Tabriz Islamic Art University, Tabriz, Iran.
3. Associate Professor of Architecture, Department of Architecture, Faculty of Art and Architecture, Islamic Azad University, Tabriz Branch, Tabriz, Iran.
4. Associate Professor of Carpet Studies, Department of Carpet, Faculty of Carpet, Tabriz Islamic Art University, Tabriz, Iran.
Abstract
The historic city of Masouleh, located in Gilan Province, is a prime example of the application of golden and geometric proportions in Iranian architecture, resulting in visual beauty and conveying spiritual concepts. This research investigates the effects of golden ratios on Masouleh's visual beauty, exploring the concepts that influenced its architectural patterns and the utilization of golden proportions in its structure.The research method is descriptive-analytical, using field studies and software like AutoCAD 2024, Atrise Golden Section 5.9.2, and PhiMatrix 1.618 to analyze the patterns and golden proportions present in Masouleh's architecture. The findings indicate that religious beliefs, respect for nature, climatic and structural needs, and the use of local materials are key factors shaping Masouleh's architectural design. Golden proportions, such as the golden circle, golden rectangle, golden spiral, and Khayyam-Pascal triangle, are extensively used in building designs and decorations.The results suggest that golden proportions create visual balance and harmony, establishing an appealing and sustainable rhythm in Masouleh's physical structure. These proportions also aid in the transmission of cultural and spiritual concepts through the design of openings, doors, and interior decorations such as religious and floral motifs. The alignment of Masouleh's stepped structure with the Fibonacci sequence and Khayyam-Pascal triangle demonstrates intelligence in harmonizing with environmental and climatic conditions, facilitating easy access in sloped terrains. Additionally, the examination of architectural decorations reveals that using small and large modules plays a significant role in creating local identity and adapting to indigenous materials. In conclusion, Masouleh's architectural structures successfully demonstrate harmony between nature, culture, and architecture. The integration of traditional and modern design principles offers a valuable model for contemporary architectural endeavors in similar regions, fostering a sustainable and culturally enriched built environment, ensuring that future designs respect and reflect the rich heritage and natural beauty of the area.
Highlights
Extensive Use of Golden Ratios in Masouleh's Architecture
Influence of Religious Beliefs and Respect for Nature on Architectural Formation
Alignment of Masouleh's Terraced Structure with the Fibonacci sequence
Transmission of Cultural and Spiritual Concepts through Architectural Decorations
Keywords: Golden ratios, visual aesthetics, architectural elements, Masouleh
*The article, derived from a doctoral dissertation titled "Typology of Motifs and Architectural Decorations in the Historic City of Masouleh," was conducted at the Department of Architecture and Art, Islamic Azad University, Tabriz Branch.
Author Corresponding
Email: stu.saeidhasanpourloumer@iaut.ac.ir
Phone: +9809119863952
Introduction: Urban aesthetics, as a fundamental aspect of urban design and planning, plays a crucial role in shaping harmonious and visually appealing spaces. Among the various principles influencing urban aesthetics, the golden ratio stands out as a key mathematical and geometric concept that enhances balance and beauty in urban environments. This ratio, approximated as 1.618, is widely found in nature, art, and architecture, and has been admired for its harmonious proportions throughout history. From ancient monuments like the Egyptian pyramids and the Parthenon to modern architectural designs, the golden ratio continues to be a guiding principle in achieving visual harmony. In urban design, the golden ratio is employed to create balance in public spaces, influencing the organization of elements such as form, color, and spatial composition. When applied in the design of streets and squares, it enhances aesthetic appeal and fosters a pleasant experience for citizens. Studies suggest that spaces designed with these proportions evoke a sense of tranquility and satisfaction, making them more inviting and comfortable. Iranian-Islamic architecture, deeply rooted in geometry and proportionality, has traditionally relied on unwritten principles passed down through generations. However, this valuable heritage has gradually faded, and today only a few artists are familiar with its foundations. In the past, Iranian architects skillfully used golden and geometric proportions to create structures that reflected cultural beliefs and artistic excellence. The historical city of Masouleh, with its unique architectural style, is a remarkable example of this tradition. Its physical structure and decorative elements are based on geometric principles and golden proportions, contributing to its distinct visual appeal and attracting researchers and tourists alike. Preserving and understanding Masouleh’s architectural heritage is essential to maintaining its historical significance. Research highlights the impact of the golden ratio not only on visual aesthetics but also on the perception and experience of urban spaces. Public areas such as parks and squares, where social interactions take place, benefit greatly from the harmony created by these proportions. Moreover, the thoughtful application of the golden ratio in urban planning can enhance the overall quality of life. However, its improper use may result in imbalanced spaces that disrupt visual harmony and user experience. Therefore, a precise understanding of its effects and applications is necessary for effective urban design. This study explores the influence of the golden ratio on the aesthetics of Masouleh’s urban spaces. Through case study analysis and literature review, it examines the role of proportionality in shaping architectural elements and its impact on user perception. The key research questions include: What concepts have influenced the formation of Masouleh’s architectural features? And which geometric patterns and proportional systems have been used in its design? The findings aim to provide insights for urban designers and planners in creating visually harmonious urban environments. With increasing urban expansion, preserving historical sites like Masouleh is more important than ever. Rapid and uncoordinated development threatens the integrity of historic urban fabrics, accelerating the decline of traditional arts. Although Masouleh has been recognized as a national heritage site and is in the final stages of UNESCO World Heritage registration, research on its geometric and proportional systems remains scarce. Additionally, due to its mountainous location, use of traditional materials, and lack of modern infrastructure, the city faces significant risks from natural and human-induced disasters. Given these concerns, this study seeks to highlight the importance of preserving Masouleh’s architectural identity and ensuring its sustainable development.
Materials and Methods: The research methodology in this study follows a descriptive-analytical approach, incorporating field studies and software-based analysis to examine the geometric and golden proportions in the architecture of Masouleh. In the initial phase, out of 564 existing buildings in Masouleh—primarily categorized into residential, market, and religious structures—a total of 151 buildings from the city's six districts were selected for an in-depth analysis of their geometric and golden proportions as well as the architectural elements shaping their ornamentation.Subsequently, through field studies, the physical proportions and architectural features of these 151 buildings were drafted using AutoCAD 2024. The collected data were then analyzed with specialized software such as Atris 5.9.2 for golden section divisions and Phi Matrix 1.618, along with manual sketches, to assess the geometric and golden proportions of Masouleh’s structures.Specialized software like Phi Matrix and Atris, which are based on mathematical and geometric principles, enabled precise analysis of these proportions. Architectural plans and decorative elements of the buildings, documented through field surveys and drafted in AutoCAD, were imported into Atris and Phi Matrix for further examination of their proportional relationships. Additionally, library research was conducted to explore the conceptual foundations of these architectural elements.This methodological approach contributes to a deeper understanding of traditional Iranian architecture and can serve as a model for future research in this field.
Discussion and Conclusion: The historic town of Masouleh, known for its unique architecture and visual harmony, exemplifies the application of the golden ratio and geometric proportions in creating aesthetic beauty. This study explores the impact of golden ratios on Masouleh’s visual appeal, examining the concepts, patterns, and proportions in its architecture and decorations. Findings indicate that golden ratios significantly contribute to harmony, balance, and visual attraction in the town.
1. Influential Concepts in Masouleh’s Architecture: Religious beliefs, respect for nature, climatic needs, and the use of local materials have shaped Masouleh's architecture, impacting geometric designs, material choice, and forms.
2. Application of Golden Proportions: Golden ratios such as the golden circle, rectangle, and spiral are evident in the design of buildings and decorations, enhancing visual balance and rhythmic beauty while conveying cultural and spiritual meanings.
3. Harmony and Balance in Design: The golden ratio, used in architectural elements from the city structure to decorative details, provides an organic, harmonious quality, contributing to both accessibility and balance, especially in the towns stepped structure.
4. Spiritual and Cultural Significance: The golden ratio conveys deeper spiritual meanings, with symbols like the eight-pointed star representing divine light. These elements add a sacred dimension to the town’s architectural spaces.
5. Coordination with Nature and Climate: Masouleh’s architecture is harmoniously adapted to its mountainous environment, utilizing local materials and modular proportions to create sustainable, climate-responsive structures.
6. Modular Proportions in Design: Modular proportions, used in architectural components like doors, windows, and decorations, help maintain visual harmony and contribute to Masouleh’s aesthetic unity and cultural identity.
7. Comparison with Other Studies: Research on the golden ratio in Persian architecture supports the findings of this study, particularly in places like Tabriz and Qaen. This research focuses on Masouleh’s unique use of these proportions in its historical and cultural context.
8. Visual Aesthetics and Balance: The golden ratio creates visual balance, enhancing the viewer's experience. It also improves the sensory experience of urban spaces and conveys cultural meanings, enriching the visual environment.
Declarations
Conflict of Interest:
We, Saeid Hasanpour Loumer and our colleagues, Dr. Ahad Nejad Ebrahimi, Dr. Hassan Sattari Sarbangholi, and Dr. Ali Vand Shoari, declare that there is no conflict of interest in conducting this research, and this research has been conducted independently and without any pressure or interference from related institutions.
Funding:
This research did not use any financial grants from governmental or private organizations to advance the research.
Authors' Contributions:
Conceptualization and study design: Saeid Hasanpour Loumer, Ahad Nejad Ebrahimi, Hassan Sattari Sarbangholi, and Ali Vand Shoari. Data collection: Saeid Hasanpour Loumer Data analysis: Saeid Hasanpour Loumer Writing the initial draft: Saeid Hasanpour Loumer, Ahad Nejad Ebrahimi, Hassan Sattari Sarbangholi, Ali Vand Shoari Reviewing and editing the article: Saeid Hasanpour Loumer. Final approval: All authors have approved the final version of the article.
Acknowledgments:
I would like to thank my esteemed professors, Dr. Ahad Nejad Ebrahimi, Dr. Hassan Sattari Sarbangholi, and Dr. Ali Vand Shoari, for their guidance and scientific support throughout this research. I would also like to thank my esteemed professor, Dr. Alireza Sadeghi, as the editor-in-chief of the Journal of Urban Design Studies of Iran, for his meticulous attention and scientific support in accepting and publishing this article. I sincerely thank Shiraz University for accepting this article in their journal.
Keywords: Golden ratios, visual aesthetics, architectural elements, Masouleh
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The Fibonacci numbers were first described in Indian mathematics as early as 200 BC; however, they are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence in his 1202 book Liber Abaci. Interesting patterns in the digital roots and endings of the Fibonacci and Lucas sequences can be easily observed. Divisibility of Fibonacci numbers and characteristic prime factors, along with Fibonacci and fibonorial primes, are discussed in detail. Some unusual and amazing applications of Fibonacci numbers, such as in binary strings, multiple reflections, partitioning of integers, and finding the equivalent resistance of a ladder of resistors, are introduced.
Fibonacci numbers are related to the golden ratio, which has a unique characteristic in that it differs from its reciprocal by 1. This characteristic leads to several fascinating properties, which are discussed.
Properties of the golden ratio, along with its geometrical connections and misconceptions about the golden ratio, are dealt with in this chapter. The application of Benford law in Fibonacci and Lucas numbers is illustrated. The chapter ends with the amazing fallacy of a missing square, which is covered in detail.
1. INTRODUCCIÓN El presente trabajo surgió en el contexto de la búsqueda de un tema que afrontara la idea del uso de la geometría en la construcción de los edificios religiosos. Era obvio que el primer modelo al que nos íbamos a acercar no era otro que el aportado por tantos estudios acerca de la presencia de la razón áurea en el ámbito de la arquitectura, pero también de la escultura o de la pintura. Pero esa búsqueda nos llevó a tropezar-nos con una información no suficientemente divulgada ni conocida. Es-tamos hablando del descubrimiento que en 1944 se realizó en la ciudad de Córdoba al tratar de probar la hipótesis de que en dicha ciudad do-minaría la proporción áurea en sus construcciones. Dicha hipótesis partía del hecho de que los Elementos de Euclides apa-recieron y se tradujeron en esta ciudad, y es en ellos donde nos encon-traremos con la geometría que sustenta la razón áurea. En 1944, la Universidad de Madrid, hoy conocida como Universidad Complutense de Madrid, se propuso demostrar la presencia constante de la proporción áurea en la arquitectura a lo largo de la historia. Para ello, llevó a cabo un experimento seleccionando Córdoba, una ciudad con una rica tradición y vinculada a los Elementos de Euclides. Se creía que las diversas culturas que habían dejado su huella en Córdoba, como los romanos, los árabes, los cristianos y los judíos, habían empleado sistemáticamente esta proporción. Este estudio se inspiró en el trabajo previo realizado por Gustav Fechner en 1876, cuyo objetivo era demos-trar que la proporción áurea era el estándar absoluto de la belleza.
O artigo analiza o uso da coñecida como Proporción Harmónica ou Proporción Áurea nun tipo concreto de esculturas no noroeste da Iberia da Idade do Ferro procedentes do sur de Gallaecia. Proponse que a aplicación dun concepto tan elaborado na solución dunha serie de problemas técnicos implica a existencia de artesáns especializados. Considérase que as esculturas de guerreiros foron un instrumento de lexitimación dun grupo social dominante nun contexto dunha sociedade de xefatura complexa. Estas esculturas, como artefactos ideolóxicos, foron usadas na Gallaecia da tardía Idade do Ferro como unha representación material do poder social.
This article could be subtitled ‘Thoughts on contemplating a model of five tetrahedra inscribed in a dodecahedron’; it is an attempt to communicate the pleasure that ensues when logical reasoning is combined with a visual delight in geometrical figures.