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The Internet Journal of Plastic Surgery ISPUB.COM Volume 9 Number 1

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

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

Abstract The Golden Proportion, a mathematical , represents beauty, and balance in physical form. Over the centuries, this geometric has influenced , biological systems, and . This ratio is believed to hold the key to the secret of beauty and finds its in innumerous natural and manmade masterpieces. The paper discusses various aspects of this ratio and their relevance in human .

INTRODUCTION Expressing Golden proportion (Ф) using a , the The Golden Proportion (Ф) has been known as mathematics ratio of total length A + B is to the longer segment A is of harmony since antiquity. This is believed to be a blueprint equal to ratio of A to the shorter segment B. for features in , art, architecture and humans that conform to harmony and beauty.[1],[2],[3],[4] This proportion Expressed algebraically: encompasses both organic and inorganic entities and has Figure 2 been found represented in numerous natural and architectural marvels ranging from Egyptian , famous Greek temple , classical work of Leonardo “” and “”, Corbusier sketch of proportion “Le ”, musical compositions of Mozart , Beethoven to the human form itself.

The Golden Proportion is defined geometrically as the , where the ratio of the whole segment to the longer segment is equal to the ratio of the longer segment to the shorter segment. Mathematically, the precise value of this Ratio is expressed as 1.6180339887..., a never-ending, never- repeating number which goes to . Hence this ratio Mathematically, these ratios are such that the longer segment cannot be expressed as a whole number or as a and is 1.618054 times the length of the shorter segment, while is considered an . algebraically the shorter is 0.618054 times the longer. This astonishing this ratio, the point C divides the line AB in such a way that number is the only one in mathematics which, when the ratio of AC to CB is equal to the ratio of AB to AC. The subtracted by units (1.0), yields its own reciprocal. The algebraic calculation shows that the ratio of AC to CB and progression using the golden proportion numbers is unique AB to AC equals 1.618… while the ratio of CB to AC is and remarkable because, without exception, the proportion equal to 0.618… of the mathematical ratio of 0.618… to 1.000 is universal; 1 divided by 0.618… equals 1.618.Conversely, 0.618… Figure 1 divided by 1 equals 0.618… [5], [6] This unique proportion in thus known as golden proportion (Ф) and is a universal principle which occurs on all levels of creation.

This fraction has influenced the , musicians,

1 of 16 The Golden Proportion: Key To The Secret Of Beauty and philosophers throughout the history. The concept of Golden Proportion was well known to The discovery date of the Golden Proportion is largely ancient and had tremendous influence on their art unknown as the proportion was rediscovered many times in and architecture. The majority of the the history. The earliest recorded application of Golden buildings, including the Parthenon, considered being Proportion probably goes as back as 2500 BC to the antiquity's most perfect structure was constructed upon the Egyptian pyramids where it was probably used for the layout principle of the Golden Proportion. The Greeks attributed of the great pyramids of Giza. [Figures 1, 2] the discovery of golden proportion to (560-480 BC), who was a great Greek geometer of the 5th century Figure 3 B.C. He showed that the human body is built with each part Figure 1. Great pyramids of Giza in a definite golden proportion to all the other parts. A little later, , a great Greek (365BC - 300BC) defined the same as a proportion derived from a of a line at the 0.6180399… into what he called its “extreme and mean ratio” in his book “Elements”. Euclid's mentions: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.” [9]

The Golden , an embodiment of the , is also made on the same principle and was also used frequently in Greek architecture. The ’s Figure 4 ratio of the height to width equals the Golden Ratio 1.618… Figure 2.Golden which makes it one of the most visually satisfying geometric forms. [Figure 3]

Figure 5 Figure 3. Golden Rectangle

The length of each side of the base of Pyramid of Giza is This has the unique property that when a is removed 756 feet and have a height of 481 feet making ratio of the (square 1 in the diagram below) smaller of the base to height very close to the Golden Ratio Ф same shape remains. Then again if a smaller square is (756/481=1.571). The Pyramid’s slope of 51° 52' is removed (2), a smaller rectangle of the same shape remains, extremely close to the “golden” pyramid inclination of 51° again a process that seemingly could go on indefinitely. All 50' and the π-based pyramid inclination of 51° 51'; other the are in Golden (Ф) relationship, with each quadrant pyramids at Giza (Chephren, 52° 20', and Mycerinus, 50° being 1.618… times longer than the one preceding it. A 47') are also quite close to the golden ratio.[7],[8] ‘golden ’ (Ф) can also be generated by linking the

2 of 16 The Golden Proportion: Key To The Secret Of Beauty quadrants (quarter circles) of each square of the Golden Greek , is also made on the same principle. Rectangle. Figure 8 The , a famous Greek sculptor and mathematician Figure 6. statue used the Golden Proportion in his architecture so much that it came to be known as phi (ф). He sculptured many architectural masterpieces including Parthenon which was built in about 440BC at the Temple of Athena on the Acropolis in Athens. The Parthenon's facades as well as elements of its facade were built in the golden rectangles with the altitude constructed in the proportion as 1.0 and the base 1.618 times the altitude. The spaces between the columns also formed the golden rectangles. [10] [Figures 4, 5]

Figure 6 Figure 4. Parthenon

The figure of the young man combines the unity of beauty and balance underlying the Greek art principles. The statue is athletic, exudes confidence with unique balance of parts with many inherent Golden Proportions.

The influence of Golden Proportion secret declined with the fall of Greece, but it began to resurface in the 15th century when Classical artists namely Michelangelo, , Van Gogh , began using the Golden Proportion in their and to achieve beauty and balance. Leonardo da Vinci (1451-1519) is believed to have incorporated Golden Proportion in his well-known paintings namely the “”, “The Figure 7 Last Supper” and the classical masterpiece “Mona Lisa”. In Figure 5. Golden proportions in Parthenon “Vitruvian Man” (or Man in Action) he used the Golden Proportion to create illustrations for the mathematician ’s paper “De ” (1509).This was probably the earliest reference of the Golden Proportion as the “Divine Proportion”. The drawing shows a nude man inscribed in a circle with arms and legs outstretched like spokes in a wheel [Figures 7, 8].

During the Hellenistic period, the human body was considered as one of the most perfect example of and eurhythmy. [11]Doryphoros statue created by Polycleitus, [Figure 6] considered an architectural marvel of the classic

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Figure 9 classical “The Last Supper” starting from the Figure 7. “Vitruvian Man” dimensions of the table at which Christ and his disciples sat to the proportions of the walls and windows in the background. [Figure 9]

Figure 11 Figure 9. “The Last Supper”

Figure 10 His masterpiece marvel “Mona Lisa” is also said to have the [13] Figure 8. “De Divina Proportione” golden ratios in its geometric equivalents. The evaluation of the painting reveals many golden rectangles. The rectangle around her face is golden and further division of that rectangle with a line drawn across the eyes, another golden rectangle is obtained, making the proportion of her head length to her eyes as golden. If a rectangle is drawn whose base extends from the her right wrist to her left elbow and if we extend the rectangle vertically until it reaches the very top of her head we again get a golden rectangle. If we draw squares inside this Golden Rectangle, we will discover that the edges of these new squares come to all the important focal points of her face: chin, eye, nose, and the mouth. [Figure 10, 11]

This illustrates all of the divine proportions within the human being. [12] The distance from the top of the man's head to the middle of his chest is 1.618… times the length of the head alone. The distance from the top of his head to his navel is 1.618… times the distance from his head to the middle of the chest, and so on.

Leonardo da Vinci also used Golden Proportions in his

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Figure 12 Figure 13 Figure 10, 11. “Mona Lisa” depicting golden ratios Figure 12. Notre de Dame

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

Le Corbusier (1887–1965), the famous French architect and painter developed a of proportions which he called “Le Modulor”, based on a human body whose height is divided in golden section commencing at the navel. He then subdivided those sections in golden ratio at the knees and throat. [Figure 13]

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Figure 14 Figure 15 Figure 13. “Le Modulor” Figure 14.

The concept was supposed to provide a standardized system that would automatically confer harmonious proportions applicable to man, mechanics and architecture. He described the concept of the golden ratio as “rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability 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 renowned monument “Taj Mahal” [Figure 14,15] constructed in 1648 by the Mughal Emperor Shahjahan,in Agra, and also in modern architecture marvels like United Nations building in New York [Figure 16] , The CN Tower in Toronto[Figure 17] and “Golden Ratio” sculpture in Jerusalem.[Figure 18]

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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 20 Figure 18. Golden ratio sculpture in Jerusalem Figure 19. The

Evidence suggests that the Golden proportion also presented in classical musical compositions written by Mozart, Beethoven and Bach. Mozart divided a striking number of his sonatas into two parts whose lengths reflect the golden proportion. The first movement of Mozart's sonata consists of 100 measures that are divided into the customary two parts; 38 in the first, 62 in the second making this ratio (38/62) 0.613 which gets closer to 0.618 in a composition of 100 measures. The second movement of this sonata is also divided according to the Golden proportion. Whether it was a conscious consideration or just coincidence, the mystery persists. [15] The proportions of different plant components including the The representation of golden proportion is also abounding in ideal distribution and positioning of around a stem and nature and animal kingdom. Probably nature also prefers diameters of geometrical figures inside the flowers often certain and ratios in order to optimize its creations. shows the Golden Proportion or in several species. [19] There are numerous examples of this perfect ratio from the The is based on a of pentagonal symmetry of the flowers to the approximately 222.5 degrees rotated from one leaf to the of the chambered nautilus shell. [16], [17]The nautilus builds a next, which probably provides the flower petals and shell to protect itself from the outside elements. As it further with maximum sun exposure and allows rain drops to flow grows, it builds another chamber at the shell bigger than the down to the root in the most efficient manner. [Figure 20] preceding one, and after moving into this bigger closes off the previous. It continues this process of building larger and larger chambers along a logarithmic spiral. The resultant spiral inside the shell is a Golden Proportioned spiral with space of each successive chamber having approximately 1.618 times more volume than the previous chamber. [Figure 19]. [18]

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Figure 21 Figure 23 Figure 20. The phyllotaxis Figures 23, 24. of a pinecone

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

Figure 24 Figure 25. Butterfly moth

It is known as the Golden angle as it divides the 360 degrees circle into a Golden Ratio: 222.5/360 = 0.618055… and 360/222.5 = 1.617977…The individual florets of the sunflower also grow in two spirals extending out from the centre. The first spiral contains 24 arms, while the other contains 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 have 5 and 8 arms, respectively [or 8 and 13, depending on the size] again represents sequence. [Figures 23, 24]

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Figure 25 Figure 26 Figure 26. Peacock’s feather Figure 27. The DNA

In a dolphin's body, the eye, fins and tail all fall at golden In Physics of atoms, despite of four fundamental sections of its length. The eye-like markings of the butterfly asymmetries namely, structure of atomic nuclei, distribution falls at golden sections of the lines that mark its width and of fission fragments, distribution of numbers of isotopes, and length.[Figure 27] The ratio of the lengths of the thorax and the distribution of emitted particles “the numerical values of abdomen in most bees is nearly the golden ratio and the all of these asymmetries are equal approximately to the peacock feather’s presents 12 interdependent Golden golden ratio. The electrons also follow Fibonacci sequence Proportions.6. with changing states of hydrogen atoms during the change of orbits [21], [22] The DNA, the basic molecule of life also contains Golden Proportion in his structure. The of the DNA In the universe there are many spiral galaxies representing double helix forms a golden which is a constituent the golden ratio in their structures. [Figure 28] of two rotated by 36 degrees from the other having the ratios of 1:1.618.The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix shape making a ratio of 1.619(ratio 34/21equals 1.619…), which is very close to 1.618.The ratio of the major to the minor groove (21 angstroms to 13 angstroms) in DNA is also golden (21/13 equals 1.619).[Figure 27] [20]

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Figure 27 Figure 28 Figure 28. Spiral galaxies Figure 29. Fibonacci numbers

In this sequence each new number is simply the sum of the two before it. However, if it’s divided by the number preceding it, than the number obtained is very close to [1, 5] The orbital relationships between certain planets are also Golden ratio. The farther the sequence goes, the closer it golden proportional. The distance from Mercury to Venus gets to the Golden Proportion. The Fibonacci numbers are being approximately 1.618 times the distance from the Sun frequently found in nature from the leaf arrangements in to Mercury. The distance from Earth to Mars being plants, the of florets in flowers, the logarithmic approximately 1.618 times the distance from Venus to Earth. spirals of shells, the bracts of pinecones making the golden There is an emergent theory which suggests that the universe ratio inherent in it. may actually be in the shape of a based on the The golden proportion also finds its representation Golden Proportion. , (1571-1630) the throughout the human body and is well described in Greek discoverer of the elliptical nature of the orbits of the planets canons. [Figure 30] around the sun, described the Golden Proportion as: “ has two great treasures: one is the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of ; the second we may name a precious jewel.’‘

No consideration of the golden proportion however can be complete without the mention of the great mathematician of the , Leonardo Pisano Fibonacci (1170–1250). In the 12th century he discovered a mathematical series that find its representation throughout the nature. The classical problem of “rabbits reproduction” is one of the most known among the many problems formulated by Fibonacci resulting in 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, known as Fibonacci numbers. [Figure 29]

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Figure 29 approximates the golden ratio of 1.618.24 Figure 30. Golden proportions in human body The golden proportion finds its representation in innumerous ancient Greek canons and defines the relationships between various 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 its midpoint. The width of the face is a golden section of the length of the face beginning from the top of the head to the menton.The mouth and nose are placed at the golden sections of the distance between the eyes and the menton. In the golden ratio taken from the total facial height, the ratio of the eye to menton is 1.618. A reverse measurement from menton to the ala of the nose is golden to the forehead. The width of the nose is golden to the width of the mouth and the eyes are golden to the mouth. The head width at the temple is golden to the eyes’ width. Given the upper lip length from ala to mouth as 1.0, the eye to the ala of the nose is golden to it and the mouth to chin is golden to it. With reference to the distance from eyebrow to ala of nose the cheek prominence is located at the 1.618 relationship. The lateral canthus of the eye is also golden to the eyebrow to the cheek prominence. The height of the human body in comparison to the distance The width of the Cupid’s bow peaks is golden to the distance from the head to the hand is 1.618. The ratio of the forearm from one peak to either commissure. [6] The human ear also to the hand is 1.618. The combined length of the hand and represents the golden rectangle with its architecture closely the forearm divided by the length of the forearm results in resembling the shape of a Fibonacci spiral. The cochlea in 1.618.Similarly the ratio of upper arm to hand + forearm is the inner ear too has a logarithmic spiral shape containing in the same ratio of 1: 1.618. The ratio of successive the golden proportion. [Figure 33] phalanges of the digits and the metacarpal bone also

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{image:32} photogrammetry, application of ideal golden mask [Figure 36], computer simulations, optical- scanning and 3-D The ideal dentition also represents the golden proportion. facial scan [Figure 37] and imaging but are not conclusive. [34], [35] [, 36], [37], [38], [39, [40] The frontal four teeth from central incisor to premolar, the most significant part of the dental aesthesis are in Golden {image:34} Proportion to each other. [Figure 34] {image:35} {image:33} However they represent one thing in common - symmetry The ratio of the width of the central incisor is in Golden ratio and proportion. Hence even though the enigma of ideal to the width of the lateral incisor. The width of the lateral facial beauty persists, the Golden Ratio remains the incisor is Golden to the width of the canine and the width of foundation for all the past, present and future facial analysis the canine is Golden to that of the first premolar creating a systems. [41]The principle needs to be applied in conjunction rhythmic normal occlusion. The height of the central incisor with other factors including prevalent racial and cultural is in the Golden Proportion to the width of the two central characteristics, respect for the patient's individuality and [6], [24] incisors. [Figure 35]. Many studies done to investigate surgical possibilities to get the desired result. the relation between the golden proportion and dental aesthetics have varying opinions. Many studies have been The Golden Proportion is a fascinating ratio which appears reported in literature echoing golden proportion in dental to be a universal constant of harmony and beauty, with its aesthetics. However they have varying opinions regarding creation being represented throughout the universe ranging the relation between the golden proportion and dental from nature, art, architecture and even human form itself. aesthetics.[24],[25],[26],[27],[28],[29],[30],[31[,[32],[33] Why is this particular ratio and its representations seem more appealing to the human eye and mind? Is this the DISCUSSION Nature's perfect number? Is this the language of the From time immemorial, the enigma of beauty exists. The Universe? We do not have any answer yet. But undoubtedly beauty is defined as a characteristic that provides a this mathematical law quantitatively defines beauty, balance pleasurable perceptual experience to the eye and human and harmony, the understanding of which is an essential brain. The most debatable issue is what makes a person prerequisite for achieving aesthetically pleasing results. beautiful? Is beauty really in the eye of the beholder or is it SUMMARY determined by some objective parameters governed by a mathematical ratio? Since antiquity, artists and scientists Beauty is a which gives pleasure to the senses and is have tried to quantify the form of the ideal or most perfect characterized by balance of proportions. The “Golden ratio” and beautiful face using subjective and objective criterias which gets succinctly expressed in the ratio of the number which are well represented in ancient Greek canonicals. “1” to the irrational “1.6180339887...” represents ideal Subjective factors because of the large diversity in measured relationships and encourages a scientific of beauty in different cultures varied from appreciation of symmetry and beauty’ and has a definite continent to continent and country to country. Some of the impact on human aesthetics. Further understanding of this women and men being considered attractive in one concept with its possible in human aesthetics geographical area are not considered the same in the other. might help in achieving what we always dream… the beauty Additionally, extensive variability in the human face, beyond perfection. experience, insight and personal values also plays an References important role in assessment. Hence subjective factors are 1. Dunlap RA, The Golden Ratio and Fibonacci Numbers, not enough to define ideal beauty and the objective analysis 1st ed.Singapore: World Scientific Publishing; 1998. of the face is imperative. In practical it is an important step 2. Livio M. The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Broadway in the approach to the patient desirous of improved facial Books; 2002. aesthesis. Various objective methods of determining ideal 3. Lidwell W, Holden K, Butler J. Universal principles of facial aesthetics have been discussed in the literature ranging Design: A Cross-Disciplinary Reference, Gloucester MA: Rockport Publishers; 2003. from using cephalometric / anthropometric analysis, 4. Stakhov A. Virtual; Museum of Harmony and Golden Section Mathematical connections in nature, science, and art.

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Author Information Sanjay Saraf, MS, MCh. (Plastic Surgery)DNB (Plastic Surgery)MNAMS, FICS, FACS Specialist Plastic Surgeon, NMC Specialty Hospital

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

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