The definition of golden ratio

The definition of golden ratio was given around the VI century BC by the Pythagorean scholars (the followers of Pythagoras) in South Italy.

The Classical Greece, and especially Pythagoras’ and Plato’s traditions, tried to unify all the arts and sciences following harmonic relationships that, in the opinion of Greek scholars, were related to the universe.

Greek artists and architects used freely golden rectangles, that is rectangles where the ratio of the long to the short sides is represented by the golden number. They thought that this figure was welcomed by the soul. If we cut a square out of a corner of the golden rectangle, the remaining rectangle is a golden rectangle, too. Golden rectangles were used to design the plan of the floor and the façade of temples. The Parthenon, on the Acropolis of Athens, confirms this rule. Greek vases and statues, representing human beings, were also built according to the divine proportion. The navel of a statue, for example, divided the body height into two golden segments. Then, the upper segment was further divided – at the level of the neck – into two more segments of the same kind. Finally, eyes divided in the same way the head.

LEONARDO DA PHInci

In a darkened parchment the renowned male nude by Leonardo da Vinci could be seen, it is the Vitruvian man – called after Marco Vitruvio, the great Roman architect who praised the divine proportion in his book De Architectura.

“No one could understand better than Leonardo da Vinci the divine structure of the human body. Leonardo dug out corps in order to measure the exact proportions of the human bone structure. He was the first one to show that the human body is literally composed by elements that are bound together by the phi ratio”. But not only … rumor says that he was obsessively fascinated by the golden section and we seem to find evidence of this in some of his most famous pictures:

St. Jerome, The Virgin of the Rocks, Head o fan Old Man, and the celebrated Mona Lisa.

In the latter case, the presence of the golden section seems to be more plausible, given his cooperation with Luca Pacioli for the writing of De Divina Proportione, but some of the abovementioned pictures come before this period of cooperation between the two classical scholars, a period that can be traced back to 1496 in Milan at the court of Ludovico il Moro; except for the Gioconda about which the debate is still open and quite controversial. Furthermore, the golden ratio should be detected within a golden rectangle the references of which haven’t been well defined yet.

The RenaiPHIance

Starting from the Renaissance, the European tradition of fine arts has also referred frequently and deliberately to the divine proportion in the shape of canvases, in the size of figures and in other details. Even composers have used this proportion in their scores. In this case, time replaces space as the dimension to be divided. To our knowledge, the use of divine proportion in music was unintentional until the Twentieth century. This confirms the notion that the proportion occurs naturally thanks to its pleasantness.

In the Nineteenth century a high portion of common objects – such as playing cards, windows, book covers and folders – was discovered to approach golden rectangles. Since then, designers have consciously used golden proportions to design packaging, shop-windows and advertisements.

A similar geometrical figure, the golden spiral, is another means through which the divine proportion of many object can be observed. In order to obtain this spiral, a set of decreasing golden rectangles must be drawn one inside the other. The drawing will also show a set of decreasing squares. Now, you can draw through these squares a set of circular arcs having as their radius the sides of the squares. The curve obtained approaches the golden spiral, also known as log spiral. The exact equation of the golden spiral includes the golden number as a factor.

MuPHIc

Music has several connections with Mathematics and many people think that the golden section plays a key role in it. In support of this theory, some structural characteristics of certain instruments, such as violin and piano, are highlighted.

As regards the violin, they believed that the pleasantness of its sound derives from the special skills of lutists who are able to build its sounding board following certain geometries; for example, the arc at its base would have, in many cases, its centre of curvature exactly in the golden position compared to whole length of the instrument. Furthermore, they said that surely even Stradivari himself tried to locate the eyelets of the violin always in that position. However, there is no evidence that these solutions can actually provide a “better” sound for this instrument than the mastery in the processing of materials or in their choice.

On the other hand, in the piano the structure of the keyboard is given particular prominence, especially by drawing parallels between its numbers and those of Fibonacci. The thirteen keys of the octaves, distinguished between eight white keys and five black keys, in turn divided into groups of two or three keys each – 2, 3, 5, 8, 13 – belong to the Fibonacci sequence, but also in this case, much more than in the previous one, it is by sheer coincidence and it can’t be either ascribed to its manufacturer’s will, given that it is a solution justified only by the structural evolution of the instrument.

There are many other cases in the history of music.

For example, even rock music, especially the so-called progressive rock, has confronted itself with the mystical and esoteric aspects of the golden section, and more precisely with Fibonacci sequence. The most typical example is the musical production of Genesis who have frequently made use of Fibonacci’s numbers in the harmonic-temporal creation of their tracks: Firth of Fifth is entirely based on the golden numbers, for examples there are solos made up of 55, 34 and 13 beats, of which some are composed by 144 notes, etc. Besides Genesis, other rock bands have used, even if sporadically, golden numbers in their compositions. Among these we can mention Deep Purple with their piece Child in Time and Dream Theater in the album Octavarium, entirely created following the ratio of the number 8 to the number 5 and the consecutive terms of Fibonacci sequence. In 2001 was released the Lateralus, an album of the American band Tool containing the homonymous single Lateralus that faithfully reproduced Fibonacci’s series.

Only the artist who is aware can change with awareness this ratio to make sense out of disequilibrium.

PHIgures in Nature

The golden spiral can be found in the artistic production of several cultures and very often in nature. Many species of common marine organisms, from plankton to slug and nautilus, have golden spirals in their developmental stages or in their shells. The lower part of sea waves forms golden spirals so leading shipbuilders to give the same shape to anchors. Most animal horns, fangs, tusks, beaks and claws also have a shape that approach the golden spiral, and the same is true for the spiral arms of the Milky Way and many other galaxies. Again, the golden spiral can be seen in the tails of comets and in the spiral of some spiders. Golden spirals can be also found in the distribution of seeds in many species of flowers, and in the arrangement of the pineapple skin.

It has been found that these and other botanical examples have another connection with the divine proportion emerging from the Fibonacci sequence.

On the head of a typical sunflower, for example, the number of spirals falls within the following pattern: 89 spirals that radiate steeply clockwise; 55 spirals moving counter-clockwise and 34 moving clockwise but less steeply. These are three adjacent numbers in Fibonacci sequence. The biggest sunflower ever known had 144, 89 and 55 spirals. In many plant species, first of all in Asteraceae (sunflowers, daisies, etc.), the number of petals in each flower is usually one of Fibonacci’s numbers, such as 5, 13, 55 or even 377, as in the case of diaccola. The bracts of cones are arranged in two set of spirals from the branch to the external part – one clockwise and the other counter-clockwise. A study on more than 4000 cones of ten species of pine has showed that over the 98 percent of them contained one of Fibonacci’s numbers in their spirals that branched off in every direction. Furthermore, the two numbers were adjacent, or adjacent by jumping one, in the Fibonacci sequence – for example, 8 spirals in one direction and 13 in the other direction, or 8 spirals in a direction and 21 in the other direction. The pineapple skin shows a closer relationship with the phenomena described by Fibonacci: No exceptions were found in a test carried out on 2000 pineapples.

Fibonacci’s numbers can be also observed in phyllotaxis, that is the arrangement of leaves on the plant stem. In many types of trees, leaves are arranged following a pattern that includes two numbers of Fibonacci. Starting from any leave, after one, two, three or five rotations from the spiral you can always find a leave that is aligned with the first one and, based on the species, this can be the second, the third, the fifth, the eighth or the thirteenth leave. Even in beehives, by dividing the number of females and that of males the result is always a golden number.

PHInally, the Human Being

It is the ratio of your height to the distance of your navel from the ground.

It is the ratio of the distance between your shoulder and the fingertip to the distance between the elbow and the fingertip.

PHI is in proportion, in time, in the cosmos, in the peacefulness coming from being in love and harmony with what is around us and, above all, what is inside us….

Article from www.phitofilos.it