From the Golden Rectangle and Fibonacci to Pedagogy and Problem Posing

1976 ◽  
Vol 69 (3) ◽  
pp. 180-188
Author(s):  
Stephen I. Brown
Keyword(s):  
Number Theory ◽  
Thirteenth Century ◽  
Golden Section ◽  
Golden Ratio ◽  
Mathematical Journal ◽  
Problem Posing ◽  
Total Height ◽  
Regular Pentagon ◽  
Golden Rectangle ◽  
Relative Placement

There is a wealth of valuable material on the golden section, and “fallout” from it as well, Indeed, the problem of constructing the golden rectangle with straightedge and a pair of compasses, and its relationship to constructing the regular pentagon with similar tools, is documented in Euclid's Elements (Euclid 1956; vol. 1, bk. 2, prop. 11; vol. 2, bk. 4. prop. 11). In the thirteenth century, the Italian mathematician Leonardo Fibonacci found connections between that aspect of geometry and number theory, and today there is a mathematical journal that is devoted exclusively to such issues. The golden section has appeared not only in mathematics but in architecture and art as well. The Parthenon in Greece, for example, has the proportions of the golden rectangle, and various portions of Michelangelo's David, from the joints of the fingers to the relative placement of the navel with respect to the total height, exemplify the golden ratio.

Application of the Golden Section and Golden Rectangle to Fountain Location in Landscape Design

2012 ◽  
Vol 174-177 ◽  
pp. 2554-2557
Author(s):  
Ling Xu ◽  
Chuan Gui Yang ◽  
Tong Chun Wei
Keyword(s):  
Golden Section ◽  
Golden Ratio ◽  
Landscape Design ◽  
Garden Design ◽  
Golden Rectangle

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.

scholarly journals Golden Ratio, Silver Ratio and other Metallic Means ; Geometric Substantiation of all Metallic Ratios

2021 ◽  
Vol 20 ◽  
pp. 174-187
Author(s):  
Chetansing Rajput
Keyword(s):  
Golden Ratio ◽  
Geometric Features ◽  
Regular Pentagon

This paper introduces the concept of special right angled triangles those epitomize the different Metallic Ratios. These right triangles not only have the precise Metallic Means embedded in all their geometric features, but they also provide the most accurate geometric substantiation of all Metallic Means. These special right triangles manifest the corresponding Metallic Ratios more holistically than the regular pentagon, octagon or tridecagon, etc

The Grand Periodic Function

2018 ◽  
Author(s):  
Jan C. A. Boeyens
Keyword(s):  
Number Theory ◽  
Periodic Function ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Magic Number ◽  
Self Similarity ◽  
Covalent Interaction ◽  
Great Pyramid ◽  
Relationship Of ◽  
The Relationship

The discovery of material periodicity must rank as one of the major achievements of mankind. It reveals an ordered reality despite the gloomy pronouncements of quantum philosophers. Periodicity only appears in closed systems with well-defined boundary conditions. This condition excludes an infinite Euclidean universe and all forms of a chaotic multiverse. Manifestations of cosmic order were observed and misinterpreted by the ancients as divine regulation of terrestrial events, dictated by celestial intervention. Analysis of observed patterns developed into the ancient sciences of astrology, alchemy and numerology, which appeared to magically predict the effects of the macrocosm on the microcosm. The sciences of astronomy and chemistry have by now managed to outgrow the magic connotation, but number theory remains suspect as a scientific pursuit. The relationship between Fibonacci numbers and cosmic self-similarity is constantly being confused with spurious claims of religious and mystic codes, imagined to be revealed through the golden ratio in the architecture of the Great Pyramid and other structures such as the Temple of Luxor. The terminology which is shared by number theory and numerology, such as perfect number, magic number, tetrahedral number and many more, contributes to the confusion. It is not immediately obvious that number theory does not treat 3 as a sacred number, 13 as unlucky and 666 as an apocalyptic threat. The relationship of physical systems to numbers is no more mysterious nor less potent than to differential calculus. Like a differential equation, number theory does not dictate, but only describes physical behavior. The way in which number theory describes the periodicity of matter, atomic structure, superconductivity, electronegativity, bond order, and covalent interaction was summarized in a recent volume. The following brief summary of these results is augmented here by a discussion of atomic and molecular polarizabilities, as derived by number theory, and in all cases specified in relation to the grand periodic function that embodies self-similarity over all space-time.

scholarly journals PROPORTIONS IN ARCHITECTURE

2018 ◽  
Vol 6 (3) ◽  
pp. 126-133
Author(s):  
Minakshi Rajput Singh
Keyword(s):  
Complex Geometry ◽  
Golden Section ◽  
Golden Ratio ◽  
Building Blocks ◽  
Life Forms ◽  
Big Bang ◽  
Celestial Bodies ◽  
To Come ◽  
The Big Bang ◽  
The Universe

The uniqueness of His creation is reflected in different fields of life by the great masters throughout the ages that have born with the formation of the universe, from the big bang, till times still to come. Special ratio that can be used to describe the proportions of everything from nature’s smallest building blocks, such as atoms, to the most advanced patterns in the universe, such as unimaginably large celestial bodies. One of the key evidences presented for creation is the recurring appearance of the Divine proportion, or golden section, throughout the design of the human body and other life forms. An attempt has been made to relate Sri yantra and golden ratio and the various forms that seem to exemplify in the plan and elevation of the Indian temple. The yantra which is a complex geometry has been perfected to be used for the development of temple forms of different eras. The following paper will be a tool for the researchers to use the yantra in deriving the spaces of Indian temples.

scholarly journals Aesthetic preference of object position on pictures

Psihologija ◽  
2003 ◽  
Vol 36 (3) ◽  
pp. 313-330 ◽  
Author(s):  
Oliver Toskovic ◽  
Slobodan Markovic
Keyword(s):  
Aesthetic Experience ◽  
Golden Section ◽  
Aesthetic Preference ◽  
Golden Rectangle ◽  
Black Circle ◽  
Different Shapes ◽  
The Aesthetic ◽  
The Right ◽  
The Given ◽  
Optimal Effect

In this study three hypothesis were evaluated. The first claims that the golden section position is an ideal position of an object on a picture and that this position does not depend on picture shape, or on the number of objects on it. According to the second hypothesis, the aesthetically optimal effect is achieved when the focus is on the right side of the picture ( for asymmetrically composed pictures). According to the third hypothesis, there is an influence of previous stimulation on aesthetic experience; that is, because of the monotony, the aesthetic preference of observers will change. An experiment was done, with two sections. In the first section, subjects were asked to put a little black circle, on three different shapes of cards (square, golden rectangle and rectangle), in a such way that the given configuration is the most beautiful one in their own opinion. The second section of the experiment was almost identical to the first one, with the exception that the subjects were asked to put two circles on each of the cards. Each one of the three hypothesis was confirmed by the results of this experiment. The preferred position of the circle is the same as the position of the golden section and it does not change with the change of card shape and number of objects. There is a clear preference of the upper-right corner of cards. The preferred position of an object is changed with repetition of the same stimulation (the same shape of cards and the same number of circles).

FINDING HARMONY IN CHAOS: THE ROLE OF THE GOLDEN RECTANGLE IN DECONSTRUCTIVE ARCHITECTURE

2018 ◽  
Vol 12 (3) ◽  
pp. 183
Author(s):  
Luai Aljubori ◽  
Chaham Alalouch
Keyword(s):  
Case Studies ◽  
Delphi Method ◽  
Golden Ratio ◽  
The Other ◽  
Free Form ◽  
Proportional Systems ◽  
Other Hand ◽  
Golden Rectangle

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.

Investigate, because of the golden rectangle

2020 ◽  
pp. 187-194
Author(s):  
Susan D'Agostino
Keyword(s):  
Golden Ratio ◽  
Mathematics Students ◽  
Greek Letter ◽  
Golden Rectangle ◽  
Mathematical Play

“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.

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