GOLDEN RATIO AND ITS APPLICATIONS

Dr Manoj Sharma1 & Rishabh Shukla

2

1 HEAD Department of Mathematics, Rustamji Institute of Technology, Gwalior, India.

2 Department of Electronics and Communication, Rustamji Institute of Technology, Gwalior,

India.

Abstract

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient

Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance

astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger

Penrose, have spent endless hours over this simple ratio and its properties. But the fascination

with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians,

historians, architects, psychologists, and even mystics have pondered and debated the basis of

its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired

thinkers of all disciplines like no other number in the history of mathematics. Thus the

primary focus of this paper is to present a general view about the Golden Ratio and its

various application s in Engineering and science.

Keywords: Golden Ratio, Fibonacci Series, Phidias, Architecture, Art,

Geometry.

I. INTRODUCTION

In mathematics, two quantities are in the golden ratio if the ratio of the sum of the quantities

to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden

ratio is an irrational mathematical constant, approximately 1.61803398874989. Other names

frequently used for the golden ratio are the golden section and golden mean. Other terms

encountered include extreme and mean ratio, medial section, divine proportion, divine

section, golden proportion, golden cut, golden number, and mean of Phidias. Golden ratio is

denoted by the Greek lowercase letter phi (φ). The figure below illustrates the geometric

relationship that defines this constant.

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This equation has one positive solution in the set of algebraic irrational numbers:

I. APPLICATIONS OF GOLEDN RATIO

a. Aesthetics

Do we surround ourselves with the Golden Ratio because we find it aesthetically pleasing, or

do we find it aesthetically pleasing because we are surrounded by it. If you were going to

design a rectangular TV screen or swimming pool, would one shape be more pleasing to the

eye than others? Since the early Greeks, a ratio of length to width of approximately 1.618, has

been considered the most visually appealing. This ratio, called the golden ratio, not only

appears in art and architecture, but also in natural structures. In the 1930's, New York's Pratt

Institute laid out rectangular frames of different proportions, and asked several hundred art

students to choose which they found most pleasing. The winner was one with Golden Ratio

proportions. You can carry out a test for yourself at www.jimloy.com/poll/poll.htm. So we

find that the objects related with the Golden Ratio are more aesthetically pleasing.

b. Architecture

Golden Ratio plays an important role in the architecture, ranging from appropriate

descriptions of designs to guiding the designer’s intuition. It is a number that has fascinated

humans throughout the centuries. As we take a look at modern and ancient architecture we

soon realize that an increasing number of buildings with exotic shapes fall in the league of

Golden Ratio. The ratio of the width and height of the Parthenon in Greece is approximately

equal to the golden ratio; see Fig. 1. The Parthenon's facade as well as elements of its facade

and elsewhere are said by some to be circumscribed by golden rectangles. In the same way the

geometrical analysis of the Great Mosque of Kairouan reveals a consistent application of the

golden ratio throughout the design, according to Boussora and Mazouz. It is found in the

overall proportion of the plan and in the dimensioning of the prayer space, the court, and the

minaret. Boussora and Mazouz also examined earlier archaeological theories about the

mosque, and demonstrate the geometric constructions based on the golden ratio by applying

these constructions to the plan of the mosque to test their hypothesis. In the same way The Taj

Mahal of Agra, India also shows the compatibility with the Golden Rectangles.

Fig.1 The Parthenon in Greece Fig2. The Taj Mahal in India.

c. Art

As the Golden Section is found in the design and beauty of nature, it can also be used to

achieve beauty and balance in the design of art. This is only a tool though, and not a rule, for

composition. The Golden Section was used extensively by Leonardo Da Vinci. Note in the

Fig.3 how all the key dimensions of the room and the table in Da Vinci's "The Last Supper"

were based on the Golden Section, which was known in the Renaissance period as The Divine

Proportion. In "The Sacrament of the Last Supper," Salvador Dali framed his painting in a

golden rectangle. Following Da Vinci's lead, Dali positioned the table exactly at the golden

section of the height of his painting. He positioned the two disciples at Christ's side at the

golden sections of the width of the composition. In addition, the windows in the background

are formed by a large dodecahedron. Dodecahedrons consist of 12 pentagons, which exhibit

phi relationships in their proportions.

Fig. 3 The Sacrament of the Last Supper

Fig.4 The Vitruvian Man

d. Music

James Tenney reconceived his piece For Ann (rising), which consists of up to twelve

computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start

so it is the golden ratio (in between an equal tempered minor and major sixth) below the

previous tone, so that the combination tones produced by all consecutive tones are a lower or

higher pitch already, or soon to be, produced. Ernő Lendvai analyzes Béla Bartók's works as

being based on two opposing systems, that of the golden ratio and the acoustic scale,[39]

though other music scholars reject that analysis.[2] In Bartok's Music for Strings, Percussion

and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.[40] French

composer Erik Satie used thegolden ratio in several of his pieces, including Sonneries de la

Rose+Croix.The golden ratio is also apparent in the organization of the sections in the music

of Debussy's Reflets dans l'eau(Reflections in Water), from Images (1st series, 1905), in

which "the sequence of keys is marked out by the intervals34, 21, 13 and 8, and the main

climax sits at the phi position.

e. Industrial design

Some sources claim that the golden ratio is commonly used in everyday design, for example

in the shapes of postcards, credit cards, match box, playing cards, posters, wide-screen

televisions, photographs, and light switch plates. If you take a close look at the Credit card in

your pocket you will find that the ratio of its length to breadth gives us the golden ratio.

Fig.5 Credit card Fig.6 Toyaota logo

f. Nature Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio

expressed in the arrangement of branches along the stems of plants and of veins in leaves. He

extended his research to the skeletons of animals and the branching of their veins and nerves,

to the proportions of chemical compounds and the geometry of crystals, even to the use of

proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a

universal law. In connection with his scheme for golden-ratio-based human body proportions,

Zeising wrote in 1854 of a universal law in which is contained the ground-principle of all

formative striving for beauty and completeness in the realms of both nature and art, and which

permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether

cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest

realization, however, in the human form. In 2003, Volkmar Weiss and Harald Weiss analyzed

psychometric data and theoretical considerations and concluded that the golden ratio underlies

the clock cycle of brain waves. In 2008 this was empirically confirmed by a group of

neurobiologists. In 2010, the journal Science reported that the golden ratio is present at the

atomic scale in the magnetic resonance of spins in cobalt niobate crystals. Several researchers

have proposed connections between the golden ratio and human genome DNA.

g. Geometry

The number φ turns up frequently in geometry, particularly in figures with pentagonal

symmetry. The length of a regular pentagon's diagonal is φ times its side. The vertices of a

regular icosahedron are those of three mutually orthogonal golden rectangles. There is no

known general algorithm to arrange a given number of nodes evenly on a sphere, for any of

several definitions of even distribution. However, a useful approximation results from

dividing the sphere into parallel bands of equal area and placing one node in each band at

longitudes spaced by a golden section of the circle, i.e. 360°/φ ≅ 222.5°. This method was

used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.The golden

ratio plays an important role in the geometry of pentagrams. Each intersection of edges

sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to

the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's

center) is φ, as the four-color illustration shows.

triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer

side to the shorter side is φ. The acute triangles are golden triangles. The

triangles are golden gnomons.

Fig.7 Golden Rectangle Fig.8 Pentagram

h. Finance The golden ratio and related numbers are used in the financial markets. It is used in trading

algorithms, applications and strategies. Some typical forms include: the Fibonacci fan, the

Fibonacci arc, Fibonacci retracement and the

Thus we have observed that the implications of Golden Ratio are vast ranging from the art to

architecture to music to Industrial

that any object designed by keeping in mind the golden ratio will be mor

compared with the normal design.

[1] Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing

Number (http:/ / books. google.

Books. ISBN 0-7679-0815-5. .

color illustration shows. The pentagram includes ten isosceles

obtuse isosceles triangles. In all of them, the ratio of the longer

The acute triangles are golden triangles. The obtuse isosceles

Fig.7 Golden Rectangle Fig.8 Pentagram

ed numbers are used in the financial markets. It is used in trading

and strategies. Some typical forms include: the Fibonacci fan, the

Fibonacci arc, Fibonacci retracement and the Fibonacci time extension.

II. CONCLUSION

Thus we have observed that the implications of Golden Ratio are vast ranging from the art to

architecture to music to Industrial designs to aesthetics and Geometry. So we can conclude

that any object designed by keeping in mind the golden ratio will be mor

compared with the normal design.

IV. REFERENCES

The Golden Ratio: The Story of Phi, The World's Most Astonishing

books. google.com/books?id=w9dmPwAACAAJ). New York: Broadway

The pentagram includes ten isosceles

obtuse isosceles triangles. In all of them, the ratio of the longer

obtuse isosceles

Fig.7 Golden Rectangle Fig.8 Pentagram

ed numbers are used in the financial markets. It is used in trading

and strategies. Some typical forms include: the Fibonacci fan, the

Thus we have observed that the implications of Golden Ratio are vast ranging from the art to

we can conclude

that any object designed by keeping in mind the golden ratio will be more pleasing as

The Golden Ratio: The Story of Phi, The World's Most Astonishing

com/books?id=w9dmPwAACAAJ). New York: Broadway

[2] Piotr Sadowski, The Knight on His Quest: Symbolic Patterns of Transition in Sir Gawain

and the Green Knight, Cranbury NJ: Associated University Presses, 1996

[3] Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific

Publishing, 1997

[4] Euclid, Elements (http:/ / aleph0. clarku. edu/ ~djoyce/ java/ elements/ toc. html), Book 6,

Definition 3.

[5] Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York:

W.W. Norton, 1963) p. 37. "And the same applies in architecture, to the rectangles

representing these and other ratios (e.g. the 'golden cut'). The sole value of these ratios is that

they are intellectually fruitful and suggest the rhythms of modular design."

[6] Jay Hambidge, Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University

Press, 1920

[7] William Lidwell, Kritina Holden, Jill Butler, Universal Principles of Design: A Cross-

Disciplinary Reference, Gloucester MA: Rockport