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ISPUB.COM The Internet Journal of Plastic SurgeryVolume 9 Number 1

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The Golden Proportion: Key To The Secret Of BeautyS Saraf, P Saraf

Citation

S Saraf, P Saraf. The Golden Proportion: Key To The Secret Of Beauty. The Internet Journal of Plastic Surgery. 2013Volume 9 Number 1.

Abstract

The Golden Proportion, a mathematical ratio, represents beauty, harmony and balance in physical form. Over the centuries, thisgeometric constant has influenced architecture, biological systems, mathematics and art. This ratio is believed to hold the key tothe secret of beauty and finds its representation in innumerous natural and manmade masterpieces. The paper discussesvarious aspects of this ratio and their relevance in human aesthetics.

INTRODUCTION

The Golden Proportion (Ф) has been known as mathematicsof harmony since antiquity. This is believed to be a blueprintfor features in nature, art, architecture and humans that

conform to harmony and beauty.[1],[2],[3],[4] This proportionencompasses both organic and inorganic entities and hasbeen found represented in numerous natural andarchitectural marvels ranging from Egyptian Pyramids,famous Greek temple Parthenon, classical work of LeonardoDa Vinci “Mona Lisa” and “Last Supper”, Corbusier humanbody sketch of proportion “Le Modulor”, musicalcompositions of Mozart , Beethoven to the human formitself.

The Golden Proportion is defined geometrically as the ratios,where the ratio of the whole segment to the longer segmentis equal to the ratio of the longer segment to the shortersegment. Mathematically, the precise value of this Ratio isexpressed as 1.6180339887..., a never-ending, never-repeating number which goes to infinity. Hence this ratiocannot be expressed as a whole number or as a fraction andis considered an irrational number. Drawing algebraicallythis ratio, the point C divides the line AB in such a way thatthe ratio of AC to CB is equal to the ratio of AB to AC. Thealgebraic calculation shows that the ratio of AC to CB andAB to AC equals 1.618… while the ratio of CB to AC isequal to 0.618…

Figure 1

Expressing Golden proportion (Ф) using a line segment, theratio of total length A + B is to the longer segment A isequal to ratio of A to the shorter segment B.

Expressed algebraically:

Figure 2

Mathematically, these ratios are such that the longer segmentis 1.618054 times the length of the shorter segment, whilethe shorter is 0.618054 times the longer. This astonishingnumber is the only one in mathematics which, whensubtracted by units (1.0), yields its own reciprocal. Theprogression using the golden proportion numbers is uniqueand remarkable because, without exception, the proportionof the mathematical ratio of 0.618… to 1.000 is universal; 1divided by 0.618… equals 1.618.Conversely, 0.618…

divided by 1 equals 0.618… [5], [6] This unique proportion inthus known as golden proportion (Ф) and is a universalprinciple which occurs on all levels of creation.

This fraction has influenced the artists, musicians,

The Golden Proportion: Key To The Secret Of Beauty

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mathematicians and philosophers throughout the history.The discovery date of the Golden Proportion is largelyunknown as the proportion was rediscovered many times inthe history. The earliest recorded application of GoldenProportion probably goes as back as 2500 BC to theEgyptian pyramids where it was probably used for the layoutof the great pyramids of Giza. [Figures 1, 2]

Figure 3

Figure 1. Great pyramids of Giza

Figure 4

Figure 2.Golden Pyramid

The length of each side of the base of Pyramid of Giza is756 feet and have a height of 481 feet making ratio of thebase to height very close to the Golden Ratio Ф(756/481=1.571). The Pyramid’s slope of 51° 52' isextremely close to the “golden” pyramid inclination of 51°50' and the π-based pyramid inclination of 51° 51'; otherpyramids at Giza (Chephren, 52° 20', and Mycerinus, 50°

47') are also quite close to the golden ratio.[7],[8]

The concept of Golden Proportion was well known toancient Greeks and had tremendous influence on their artand architecture. The majority of the ancient Greekbuildings, including the Parthenon, considered beingantiquity's most perfect structure was constructed upon theprinciple of the Golden Proportion. The Greeks attributedthe discovery of golden proportion to Pythagoras (560-480BC), who was a great Greek geometer of the 5th centuryB.C. He showed that the human body is built with each partin a definite golden proportion to all the other parts. A littlelater, Euclid, a great Greek mathematician (365BC - 300BC)defined the same as a proportion derived from a division of aline at the 0.6180399… into what he called its “extreme andmean ratio” in his book “Elements”. Euclid's mentions: “Astraight line is said to have been cut in extreme and meanratio when, as the whole line is to the greater segment, so is

the greater to the lesser.” [9]

The Golden Rectangle, an embodiment of the Golden Ratio,is also made on the same principle and was also usedfrequently in Greek architecture. The Golden Rectangle’sratio of the height to width equals the Golden Ratio 1.618…which makes it one of the most visually satisfying geometricforms. [Figure 3]

Figure 5

Figure 3. Golden Rectangle

This has the unique property that when a square is removed(square 1 in the diagram below) smaller rectangles of thesame shape remains. Then again if a smaller square isremoved (2), a smaller rectangle of the same shape remains,again a process that seemingly could go on indefinitely. Allthe squares are in Golden (Ф) relationship, with each quadrantbeing 1.618… times longer than the one preceding it. A‘golden spiral’ phi (Ф) can also be generated by linking the


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quadrants (quarter circles) of each square of the GoldenRectangle.

The Phidias, a famous Greek sculptor and mathematicianused the Golden Proportion in his architecture so much thatit came to be known as phi (ф). He sculptured manyarchitectural masterpieces including Parthenon which wasbuilt in about 440BC at the Temple of Athena on theAcropolis in Athens. The Parthenon's facades as well aselements of its facade were built in the golden rectangleswith the altitude constructed in the proportion as 1.0 and thebase 1.618 times the altitude. The spaces between the

columns also formed the golden rectangles. [10] [Figures 4, 5]

Figure 6

Figure 4. Parthenon

Figure 7

Figure 5. Golden proportions in Parthenon

During the Hellenistic period, the human body wasconsidered as one of the most perfect example of symmetry

and eurhythmy. [11]Doryphoros statue created by Polycleitus,[Figure 6] considered an architectural marvel of the classic

Greek sculpture, is also made on the same principle.

Figure 8

Figure 6. Doryphoros statue

The figure of the young man combines the unity of beautyand balance underlying the Greek art principles. The statueis athletic, exudes confidence with unique balance of partswith many inherent Golden Proportions.

The influence of Golden Proportion secret declined with thefall of Greece, but it began to resurface in the 15th centurywhen Classical Renaissance artists namely Michelangelo,Raphael, Van Gogh , Leonardo da Vinci began using theGolden Proportion in their paintings and sculptures toachieve beauty and balance. Leonardo da Vinci (1451-1519)is believed to have incorporated Golden Proportion in hiswell-known paintings namely the “Vitruvian Man”, “TheLast Supper” and the classical masterpiece “Mona Lisa”. In“Vitruvian Man” (or Man in Action) he used the GoldenProportion to create illustrations for the mathematician LucaPacioli’s paper “De Divina Proportione” (1509).This wasprobably the earliest reference of the Golden Proportion asthe “Divine Proportion”. The drawing shows a nude maninscribed in a circle with arms and legs outstretched likespokes in a wheel [Figures 7, 8].


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Figure 9

Figure 7. “Vitruvian Man”

Figure 10

Figure 8. “De Divina Proportione”

This illustrates all of the divine proportions within the

human being. [12] The distance from the top of the man's headto the middle of his chest is 1.618… times the length of thehead alone. The distance from the top of his head to hisnavel is 1.618… times the distance from his head to themiddle of the chest, and so on.

Leonardo da Vinci also used Golden Proportions in his

classical painting “The Last Supper” starting from thedimensions of the table at which Christ and his disciples satto the proportions of the walls and windows in thebackground. [Figure 9]

Figure 11

Figure 9. “The Last Supper”

His masterpiece marvel “Mona Lisa” is also said to have the

golden ratios in its geometric equivalents. [13]The evaluationof the painting reveals many golden rectangles. Therectangle around her face is golden and further division ofthat rectangle with a line drawn across the eyes, anothergolden rectangle is obtained, making the proportion of herhead length to her eyes as golden. If a rectangle is drawnwhose base extends from the her right wrist to her left elbowand if we extend the rectangle vertically until it reaches thevery top of her head we again get a golden rectangle. If wedraw squares inside this Golden Rectangle, we will discoverthat the edges of these new squares come to all the importantfocal points of her face: chin, eye, nose, and the mouth.[Figure 10, 11]


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Figure 12

Figure 10, 11. “Mona Lisa” depicting golden ratios

Leonardo Da Vinci also incorporated the Divine Proportionin the design of famous cathedral Notre de Dame in Paris.[Figure 12]

Figure 13

Figure 12. Notre de Dame

Le Corbusier (1887–1965), the famous French architect andpainter developed a scale of proportions which he called “LeModulor”, based on a human body whose height is dividedin golden section commencing at the navel. He thensubdivided those sections in golden ratio at the knees andthroat. [Figure 13]


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Figure 14

Figure 13. “Le Modulor”

The concept was supposed to provide a standardized systemthat would automatically confer harmonious proportionsapplicable to man, mechanics and architecture. He describedthe concept of the golden ratio as “rhythms apparent to theeye and clear in their relations with one another. And theserhythms are at the very root of human activities. Theyresound in man by an organic inevitability, the same fineinevitability which causes the tracing out of the Golden

Section by children, old men, savages and the learned.”[14]

The concept also find its representation in world renownedmonument “Taj Mahal” [Figure 14,15] constructed in 1648by the Mughal Emperor Shahjahan,in Agra, India and also inmodern architecture marvels like United Nations building inNew York [Figure 16] , The CN Tower in Toronto[Figure17] and “Golden Ratio” sculpture in Jerusalem.[Figure 18]

Figure 15

Figure 14. Taj Mahal


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Figure 16

Figure 15. Golden proportions in Taj Mahal

Figure 17

Figure 16. Golden proportions in United Nations building


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Figure 18

Figure 17. Golden proportions in CN Tower


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Figure 19

Figure 18. Golden ratio sculpture in Jerusalem

Evidence suggests that the Golden proportion also presentedin classical musical compositions written by Mozart,Beethoven and Bach. Mozart divided a striking number ofhis sonatas into two parts whose lengths reflect the goldenproportion. The first movement of Mozart's sonata consistsof 100 measures that are divided into the customary twoparts; 38 in the first, 62 in the second making this ratio(38/62) 0.613 which gets closer to 0.618 in a composition of100 measures. The second movement of this sonata is alsodivided according to the Golden proportion. Whether it wasa conscious consideration or just coincidence, the mystery

persists. [15]

The representation of golden proportion is also abounding innature and animal kingdom. Probably nature also preferscertain angles and ratios in order to optimize its creations.There are numerous examples of this perfect ratio from thepentagonal symmetry of the flowers to the logarithmic spiral

of the chambered nautilus shell. [16], [17]The nautilus builds ashell to protect itself from the outside elements. As it furthergrows, it builds another chamber at the shell bigger than thepreceding one, and after moving into this bigger area closesoff the previous. It continues this process of building largerand larger chambers along a logarithmic spiral. The resultantspiral inside the shell is a Golden Proportioned spiral withspace of each successive chamber having approximately1.618 times more volume than the previous chamber. [Figure

19]. [18]

Figure 20

Figure 19. The nautilus

The proportions of different plant components including theideal distribution and positioning of leaf around a stem anddiameters of geometrical figures inside the flowers oftenshows the Golden Proportion or angle in several species.[19]The phyllotaxis is based on a Golden angle ofapproximately 222.5 degrees rotated from one leaf to thenext, which probably provides the flower petals and leaveswith maximum sun exposure and allows rain drops to flowdown to the root in the most efficient manner. [Figure 20]


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Figure 21

Figure 20. The phyllotaxis

It is known as the Golden angle as it divides the 360 degreescircle into a Golden Ratio: 222.5/360 = 0.618055… and360/222.5 = 1.617977…The individual florets of thesunflower also grow in two spirals extending out from thecentre. The first spiral contains 24 arms, while the othercontains 35, making the ratio 24 to 35 a Golden Ratio.[Figures 21, 22]

Figure 22

Figures 21, 22. Sunflower florets

The spirals of a pinecone, where spirals from the center have5 and 8 arms, respectively [or 8 and 13, depending on thesize] again represents Fibonacci sequence. [Figures 23, 24]

Figure 23

Figures 23, 24. Spirals of a pinecone

In animal kingdom there are plenty examples of the Goldenproportion being represented in dolphin, butterfly moth tothe peacock’s feather. [Figure 25, 26]

Figure 24

Figure 25. Butterfly moth


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Figure 25

Figure 26. Peacock’s feather

In a dolphin's body, the eye, fins and tail all fall at goldensections of its length. The eye-like markings of the butterflyfalls at golden sections of the lines that mark its width andlength.[Figure 27] The ratio of the lengths of the thorax andabdomen in most bees is nearly the golden ratio and thepeacock feather’s presents 12 interdependent Golden

Proportions.6.

The DNA, the basic molecule of life also contains GoldenProportion in his structure. The cross section of the DNAdouble helix forms a golden decagon which is a constituentof two pentagons rotated by 36 degrees from the otherhaving the diagonal ratios of 1:1.618.The DNA moleculemeasures 34 angstroms long by 21 angstroms wide for eachfull cycle of its double helix shape making a ratio of1.619(ratio 34/21equals 1.619…), which is very close to1.618.The ratio of the major to the minor groove (21angstroms to 13 angstroms) in DNA is also golden (21/13

equals 1.619).[Figure 27] [20]

Figure 26

Figure 27. The DNA

In Physics of atoms, despite of four fundamentalasymmetries namely, structure of atomic nuclei, distributionof fission fragments, distribution of numbers of isotopes, andthe distribution of emitted particles “the numerical values ofall of these asymmetries are equal approximately to thegolden ratio. The electrons also follow Fibonacci sequencewith changing states of hydrogen atoms during the change of

orbits [21], [22]

In the universe there are many spiral galaxies representingthe golden ratio in their structures. [Figure 28]


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Figure 27

Figure 28. Spiral galaxies

The orbital relationships between certain planets are alsogolden proportional. The distance from Mercury to Venusbeing approximately 1.618 times the distance from the Sunto Mercury. The distance from Earth to Mars beingapproximately 1.618 times the distance from Venus to Earth.There is an emergent theory which suggests that the universemay actually be in the shape of a dodecahedron based on theGolden Proportion. Johannes Kepler, (1571-1630) thediscoverer of the elliptical nature of the orbits of the planetsaround the sun, described the Golden Proportion as:“Geometry has two great treasures: one is the other, thedivision of a line into extreme and mean ratio. The first wemay compare to a measure of gold; the second we may namea precious jewel.’‘

No consideration of the golden proportion however can becomplete without the mention of the great mathematician ofthe middle ages, Leonardo Pisano Fibonacci (1170–1250). Inthe 12th century he discovered a mathematical series thatfind its representation throughout the nature. The classicalproblem of “rabbits reproduction” is one of the most knownamong the many problems formulated by Fibonacci resultingin the discovery of the numerical sequence of 0, 1, 1, 2, 3, 5,8, 13, 21, 34, 55, 89, 144, 233, 377, 610... to infinity, knownas Fibonacci numbers. [Figure 29]

Figure 28

Figure 29. Fibonacci numbers

In this sequence each new number is simply the sum of thetwo before it. However, if it’s divided by the numberpreceding it, than the number obtained is very close to

Golden ratio. [1, 5]The farther the sequence goes, the closer itgets to the Golden Proportion. The Fibonacci numbers arefrequently found in nature from the leaf arrangements inplants, the pattern of florets in flowers, the logarithmicspirals of shells, the bracts of pinecones making the goldenratio inherent in it.

The golden proportion also finds its representationthroughout the human body and is well described in Greekcanons. [Figure 30]


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Figure 29

Figure 30. Golden proportions in human body

The height of the human body in comparison to the distancefrom the head to the hand is 1.618. The ratio of the forearmto the hand is 1.618. The combined length of the hand andthe forearm divided by the length of the forearm results in1.618.Similarly the ratio of upper arm to hand + forearm isin the same ratio of 1: 1.618. The ratio of successivephalanges of the digits and the metacarpal bone also

approximates the golden ratio of 1.618.24

The golden proportion finds its representation in innumerousancient Greek canons and defines the relationships between

various areas of the head and face. [11], [23] [Figures 31, 32]

Figure 30

Figure 31. Greek canon

{image:31}

The head forms a golden rectangle with the eyes at itsmidpoint. The width of the face is a golden section of thelength of the face beginning from the top of the head to thementon.The mouth and nose are placed at the goldensections of the distance between the eyes and the menton. Inthe golden ratio taken from the total facial height, the ratio ofthe eye to menton is 1.618. A reverse measurement frommenton to the ala of the nose is golden to the forehead. Thewidth of the nose is golden to the width of the mouth and theeyes are golden to the mouth. The head width at the templeis golden to the eyes’ width. Given the upper lip length fromala to mouth as 1.0, the eye to the ala of the nose is golden toit and the mouth to chin is golden to it. With reference to thedistance from eyebrow to ala of nose the cheek prominenceis located at the 1.618 relationship. The lateral canthus of theeye is also golden to the eyebrow to the cheek prominence.The width of the Cupid’s bow peaks is golden to the distance

from one peak to either commissure. [6] The human ear alsorepresents the golden rectangle with its architecture closelyresembling the shape of a Fibonacci spiral. The cochlea inthe inner ear too has a logarithmic spiral shape containingthe golden proportion. [Figure 33]


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{image:32}

The ideal dentition also represents the golden proportion.The frontal four teeth from central incisor to premolar, themost significant part of the dental aesthesis are in GoldenProportion to each other. [Figure 34]

{image:33}

The ratio of the width of the central incisor is in Golden ratioto the width of the lateral incisor. The width of the lateralincisor is Golden to the width of the canine and the width ofthe canine is Golden to that of the first premolar creating arhythmic normal occlusion. The height of the central incisoris in the Golden Proportion to the width of the two central

incisors. [Figure 35]. [6], [24] Many studies done to investigatethe relation between the golden proportion and dentalaesthetics have varying opinions. Many studies have beenreported in literature echoing golden proportion in dentalaesthetics. However they have varying opinions regardingthe relation between the golden proportion and dental

aesthetics.[24],[25],[26],[27],[28],[29],[30],[31[,[32],[33]

DISCUSSION

From time immemorial, the enigma of beauty exists. Thebeauty is defined as a characteristic that provides apleasurable perceptual experience to the eye and humanbrain. The most debatable issue is what makes a personbeautiful? Is beauty really in the eye of the beholder or is itdetermined by some objective parameters governed by amathematical ratio? Since antiquity, artists and scientistshave tried to quantify the form of the ideal or most perfectand beautiful face using subjective and objective criteriaswhich are well represented in ancient Greek canonicals.Subjective factors because of the large diversity inperception of beauty in different cultures varied fromcontinent to continent and country to country. Some of thewomen and men being considered attractive in onegeographical area are not considered the same in the other.Additionally, extensive variability in the human face,experience, insight and personal values also plays animportant role in assessment. Hence subjective factors arenot enough to define ideal beauty and the objective analysisof the face is imperative. In practical it is an important stepin the approach to the patient desirous of improved facialaesthesis. Various objective methods of determining idealfacial aesthetics have been discussed in the literature rangingfrom using cephalometric / anthropometric analysis,

photogrammetry, application of ideal golden mask [Figure36], computer simulations, optical-surface scanning and 3-Dfacial scan [Figure 37] and imaging but are not conclusive.[34], [35] [, 36], [37], [38], [39, [40]

{image:34}

{image:35}

However they represent one thing in common - symmetryand proportion. Hence even though the enigma of idealfacial beauty persists, the Golden Ratio remains thefoundation for all the past, present and future facial analysis

systems. [41]The principle needs to be applied in conjunctionwith other factors including prevalent racial and culturalcharacteristics, respect for the patient's individuality andsurgical possibilities to get the desired result.

The Golden Proportion is a fascinating ratio which appearsto be a universal constant of harmony and beauty, with itscreation being represented throughout the universe rangingfrom nature, art, architecture and even human form itself.Why is this particular ratio and its representations seemmore appealing to the human eye and mind? Is this theNature's perfect number? Is this the language of theUniverse? We do not have any answer yet. But undoubtedlythis mathematical law quantitatively defines beauty, balanceand harmony, the understanding of which is an essentialprerequisite for achieving aesthetically pleasing results.

SUMMARY

Beauty is a quality which gives pleasure to the senses and ischaracterized by balance of proportions. The “Golden ratio”which gets succinctly expressed in the ratio of the number“1” to the irrational “1.6180339887...” represents idealmeasured relationships and encourages a scientificappreciation of symmetry and beauty’ and has a definiteimpact on human aesthetics. Further understanding of thisconcept with its possible recreation in human aestheticsmight help in achieving what we always dream… the beautybeyond perfection.

References

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Author Information

Sanjay Saraf, MS, MCh. (Plastic Surgery)DNB (Plastic Surgery)MNAMS, FICS, FACSSpecialist Plastic Surgeon, NMC Specialty Hospital

Praveena Saraf, MS (Gyn/Obs)Specialist Gyn/Obs, NMC Specialty Hospital

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