The Golden Ratio - I

Golden Ratio - A Brief History

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887.

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.

The golden ratio can be expressed as a mathematical constant, usually denoted by the Greek letter . The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically:

(a+b)/a = a/b = The Golden Ratio

This equation has as its unique positive solution the algebraic irrational number

The Golden ratio = [1 + sqrt(5)]/2 = 1.6180339887...

Other names frequently used for or closely related to the golden ratio are golden section (Latin: sectio aurea), golden mean, golden number, and the Greek letter phi (φ). Other terms encountered include extreme and mean ratio, medial section, divine proportion (Italian: proporzione divina), divine section (Latin: sectio divina), golden proportion, golden cut, and mean of Phidias.

Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The ratio is important in the geometry of regular pentagrams and pentagons. The Greeks usually attributed discovery of the ratio to Pythagoras or to the Pythagoreans. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.

Euclid's Elements (Greek: Στοιχεῖα) gives the first known written definition of what is now called the golden ratio: "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 less." Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e. the golden ratio. Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio. Some of these propositions show that the golden ratio is an irrational number. The name "extreme and mean ratio" was the principal term used from the 3rd century BC until about the.

The modern history of the golden ratio starts with Luca Pacioli's Divina Proportione of 1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio.

Since the twentieth century, the golden ratio has been represented by the Greek letter (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by τ (tau, the first letter of the ancient Greek root τομή– meaning cut).

Golden Ratio in Architecture

Some studies of the Acropolis, including the Parthenon, conclude that many of its proportions approximate the golden ratio. The Parthenon's facade as well as elements of its facade and elsewhere can be circumscribed by golden rectangles. To the extent that classical buildings or their elements are proportioned according to the golden ratio, this might indicate that their architects were aware of the golden ratio and consciously employed it in their designs. Alternatively, it is possible that the architects used their own sense of good proportion, and that this led to some proportions that closely approximate the golden ratio. On the other hand, such retrospective analyses can always be questioned on the ground that the investigator chooses the points from which measurements are made or where to superimpose golden rectangles, and that these choices affect the proportions observed.

Some scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Midhat J. Gazalé says, "It was not until Euclid, however, that the golden ratio's mathematical properties were studied. In the Elements (308 B.C.) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular pentagons and decagons was duly observed, as well as in the dodecahedron (a regular polyhedron whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties." And Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 B.C., showed how to calculate its value." Near-contemporary sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

A 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.

The Swiss architect called Le Corbusier was famous for his contributions to the modern international style. Systems of harmony and proportion were at the centre of his design philosophy. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described 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."
Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took Leonardo's suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.

Golden Ratio in Art

Leonardo da Vinci's illustrations in De Divina Proportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his own paintings. Some suggest that his Mona Lisa, for example, employs the golden ratio in its geometric equivalents. Whether Leonardo proportioned his paintings according to the golden ratio has been the subject of intense debate. The secretive Leonardo seldom disclosed the bases of his art, and retrospective analysis of the proportions in his paintings can never be conclusive.

Salvador Dalí explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, with edges in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.
Mondrian used the golden section extensively in his geometrical paintings.
A statistical study on 565 works of art of different great painters was performed in 1999 and it was found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).

Golden Ratio in Sculpture
Australian sculptor Andrew Rogers's 50-ton stone and gold sculpture, entitled Golden Ratio, is installed outdoors in Jerusalem. Rogers donated the sculpture. The height of each stack of stones, beginning from either end and moving toward the center, is the beginning of the Fibonacci sequence: 1, 1, 2, 3, 5, 8.

Golden Ratio in 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. 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. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry.
The golden ratio is also apparent in the organisation of the sections in the music of Debussy's Image, Reflections in Water, in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position."
This Binary Universe, an experimental album by Brian Transeau (aka BT), includes a track entitled "1.618" in homage to the golden ratio. The track features musical versions of the ratio and the accompanying video displays various animated versions of the golden mean.

Golden Ratio in 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 branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, as well as the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law. Zeising wrote in 1854:
[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.

The Vitruvian Man - Da Vinci's love with Proportions and Analogies

This image exemplifies the blend of art and science during the Renaissance and provides the perfect example of Leonardo's keen interest in proportion. In addition, this picture represents a cornerstone of Leonardo's attempts to relate man to nature. Leonardo envisaged the great picture chart of the human body he had produced through his anatomical drawings and Vitruvian Man as a "cosmografia del minor mondo (cosmography of the microcosm)". He believed the workings of the human body to be an analogy for the workings of the universe." It is also believed by some that Leonardo symbolised the material existence by the square and spiritual existence by the circle. Thus he attempted to depict the correlation between these two aspects of human existence. According to Leonardo's notes in the accompanying text, it was made as a study of the proportions of the (male) human body as described in a treatise by the ancient Roman architect Vitruvius.

According to Vitruvius' De Architectura 3.1.3, which reads, The navel is naturally placed in the centre of the human body, and, if in a man lying with his face upward, and his hands and feet extended, from his navel as the centre, a circle be described, it will touch his fingers and toes. It is not alone by a circle, that the human body is thus circumscribed, as may be seen by placing it within a square. For measuring from the feet to the crown of the head, and then across the arms fully extended, we find the latter measure equal to the former; so that lines at right angles to each other, enclosing the figure, will form a square.

Leonardo's drawing combines a careful reading of the ancient text, combined with his own observation of actual human bodies. In drawing the circle and square he correctly observes that the square cannot have the same center as the circle, the navel, but is somewhat lower in the anatomy. This adjustment is the innovative part of Leonardo's drawing and what distinguishes it from earlier illustrations. He also departs from Vitruvius by drawing the arms raised to a position in which the fingertips are level with the top of the head, rather than Vitruvius' much higher angle, in which the arms form lines passing through the navel. The drawing itself is often used as an implied symbol of the essential symmetry of the human body, and by extension, to the universe as a whole.

This extension is what we come to appreciate through Leonardo's works. If studied, almost all of them point towards the same theory, that everything in this universe is linked to one another. The characteristic features that we may observe in one creature definitely appear in everything else around us. It my not be apparent to the eyes, but if we delve deeper the string that binds us all will make itself seem so obvious.

Let us have a look at Leonardo's point of view of Cosmography of Microcosm -

The word 'Cosmography' according to the dictionary means a science that describes and maps the main features of the heavens and the earth, including astronomy, geography, and geology.

And 'Microcosm' means a small, representative system having analogies to a larger system in constitution, configuration or development.

Such an analogy between human life and the universe is obvious. Just like humans, celestial bodies have a well defined life cycle in which they are born, they mature and eventually they die. Hence, a direct similarity is present which can be considered as a scientific and rational support for Leonardo's theory. Similarly, a philosophical support can also be drawn. If the whole universe is considered to be the earth then Milky Way, our galaxy, can be likened to the continent in which the solar system (our country) resides. Different planets are states of the country and the organisms their citizens. And on the basis of such an analogy we can very well conclude that everything in this universe has life. Everything goes through the phases of Nativity to Demise. From celestial bodies to human beings, individual organisms to whole species. Even this web page was created on a particular day and time (birth) and will be removed from the server (death) someday. From the toothbrush we use everyday in the morning to the car we drive, everything was born and will, surely, become useless for us and die. Even our daily routine works on a similar theory. Getting up in the morning = Birth. Going to bed in the night (to sleep, ;-) ) = Death.

This school of thought (and other related ones) has bred many branches of science and mathematics. Observation and studies of effects of a new drug is tested on guinea pigs or rats or monkeys... why? Because every organism is similar. The effects, hence, will be similar as well. Many theories of ratio and proportions are direct results of studies in this field. Even "Set Theory", which is considered to be the basis of arithmetic, is the mathematical expression of Leonardo's theory.

So, everything is similar. Everybody is equal. This unity is what will protect us and provide a deeper understanding of the universe around us. And to guide us on this endeavour is THE VITRUVIAN MAN.....