THE SCIENCE OF FORMLESS FLOW by Monya Gorelik, M.Sc. : Golden Ratio and its hidden meaning

Tuesday, February 3, 2015

Golden Ratio and its hidden meaning

 The term Golden Ratio has very ancient roots. These two words, Golden Ratio, became the universal symbol of beauty – of the human body, in the architecture, music, gardens, natural landscapes and everything in the Nature. In this article we will try to understand the real reasons for our intuitive feeling of beauty. We will provide scientific basis to our feeling of harmony and beauty and point to the direction of harmonious life. The importance of this understanding is in its practical value, in the real and simple method which takes us to this harmonious world.


The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. According to Mario Livio:
"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.[11]
Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagonsEuclid's Elements (Greek: Στοιχεῖα) provides 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 lesser."[12] Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e., the golden ratio.[13] Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio.[14]
The golden ratio is explored in Luca Pacioli's book De divina proportione of 1509.
The first known approximation of the (inverse) golden ratio by a decimal fraction, stated as "about 0.6180340", was written in 1597 by Michael Maestlin of the University of Tübingen in a letter to his former student Johannes Kepler.[15].
Since the 20th century, the golden ratio has been represented by the Greek letter φ (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). ".
The source of the chapter "History": Wikipedia

Vitruvian Man

The Vitruvian Man is a world-renowned drawing created by Leonardo da Vinci around the year 1487. It is accompanied by notes based on the work of Vitruvius. The drawing, which is in pen and ink on paper, depicts a nude male figure in two superimposed positions with his arms and legs apart and simultaneously inscribed in a circle and square. The drawing and text are sometimes called the Canon of Proportions or, less often, Proportions of Man. It is stored in the Gallerie dell’Accademia in Venice, Italy, and, like most works on paper, is displayed only occasionally.

Pic. 1:  Leonardo da Vinci, Vitruvian Man, 1487, 34.4 × 25.5 cm (13.5 × 10.0 in)

A passage from Roman architect Vitruvius (Marcus Vitruvius Pollio), describing the perfect human form in geometrical terms, was the source of inspiration for numerous renaissance artists. Only one of these, the incomparable Leonardo da Vinci, was successful in correctly illustrating the proportions outlined in Vitruvius’ work De Architectura, and the result went on to become the most recognized drawings in the world, and came to represent the standard of human physical beauty. It was the version produced by Leonardo da Vinci, whose vast knowledge of both anatomy and geometry made him uniquely suited to the task.

The drawing is based on the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise De Architectura. Vitruvius described the human figure as being the principal source of proportion among the Classical orders of architecture. Other artists had attempted to depict the concept, with less success. The drawing is traditionally named in honour of the architect.

The Vitruvian Man 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. Encyclopaedia Britannica online states, “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 symbolized the material existence by the square and spiritual existence by the circle. [ Source: ]

Vitruvius, De ArchitecturaTHE PLANNING OF TEMPLES, Book 3, Chapter I
1. The planning of temples depends upon symmetry: and the method of thisarchitects must diligently apprehend. It arises from proportion (which in Greek is called analogia). Proportion consists in taking a fixed module, in each case, both for the parts of a building and for the whole, by which the method of symmetry is put to practice. For without symmetry and proportion no temple can have a regular plan; that is, it must have an exact proportion worked out after the fashion of the members of a finely-shaped human body.
2. For Nature has so planned the human body that the face from the chin to the top of the forehead and the roots of the hair is a tenth part; also the palm of the hand from the wrist to the top of the middle finger is as much; the head from the chin to the crown, an eighth part; from the top of the breast with the bottom of the neck to the roots of the hair, a sixth part; from the middle of the breast to the crown, a fourth parta third part of the height of the face is from the bottom of the chin to the bottom of the nostrils; the nose from the bottom of the nostrils to the line between the brows, as much; from that line to the roots of the hair, the forehead is given as the third part. The foot is a sixth of the height of the body; the cubit a quarter, the breast also a quarter. The other limbs also have their own proportionate measurements. And by using these, ancient painters and famous sculptors have attained great and unbounded distinction.
3. In like fashion the members of temples ought to have dimensions of their several parts answering suitably to the general sum of their whole magnitude. Now the navel is naturally the exact centre of the body. For if a man lies on his back with hands and feet outspread, and the centre of a circle is placed on his navel, his figure and toes will be touched by the circumference. Also a square will be found described within the figure, in the same way as a round figure is produced. For if we measure from the sole of the foot to the top of the head, and apply the measure to the outstretched hands, the breadth will be found equal to the height, just like sites which are squared by rule.
4. Therefore if Nature has planned the human body so that the members correspond in their proportions to its complete configuration, the ancients seem to have had reason in determining that in the execution of their works they should observe an exact adjustment of the several members to the general pattern of the plan. Therefore, since in all their works they handed down orders, they did so especially in building temples, the excellences and the faults of which usually endure for ages. [Source:]

The source of the chapter "Vitruvian Man":

Golden Ratio and functionality

Golden Ratio is a synonym of the beauty and harmony of the human body, of the ideal proportions of its body parts. As usual, we take our relation to the human body and extent it to all other objects of external world. Thus our perception of Golden Ratio proportions bears global character. 
In order to understand the reason for this global phenomena we have to understand why we intuitively catch Golden Ratio as an ideal proportion for the parts of the human body. We know that beauty and functionality strongly correlate each other. So let's try to find explanation in the functionality of the body.
One of the most important functions of the body is a motion in the space of the usual habitat of the organism, its ecological niche. This kind of motion is called locomotion. Actually the shape of the body, its internal mechanisms and the nerve system were developed in the course of the evolution in order to make the locomotion possible and most efficient in the present ecological niche and to prepare itself for the future conquests of the other ecological niches.
The efficiency of the motion expresses itself first of all in the energetic efficiency, it means that more efficient motion - the less energy it costs. Another very important feature of the great motion is its variability and ability to continue any other motion and to transform itself to any other motion at every moment. 

The play of the energies

Every motion of living body is a result of complex game of the great number of forces. It is not only external forces and internal muscular forces, but also the forces of reaction and inertia, which have non-muscular nature.   
The founder of Soviet biomechanics, the brilliant scientist Prof. N.A. Bernstein was the first who realized that in any skilled movement, from simplest walking along the street to most complicated acrobatic performances, any person or animal is taking advantage of these non-muscular forces. These forces not fighting each other, but rather working together and support each other, like musicians playing music in perfectly conducted orchestra, in order to produce motion of outstanding stability, vitality, charm and beauty. 
The active internal forces are forces produced by muscle contractions. Reaction forces act as result of the application of active forces and inertia. The more significant role playing these forces in our motion the more efficient it becomes. Just imagine that we can apply short and minute physical force and move long time after it without any effort, similar to walking on the Moon.
These inertial forces must be strong enough and be able to provide swift, strong motions if we want them to be useful. However we have to be able to change these motions swiftly. We can change or even stop them by means of strong muscle contractions. This method demands considerable muscular effort; costs a lot of energy and results in fast tiring; harms the joints, ligaments and muscles; slows down the motions and brings the practitioner into excited state of mind, which can affect his or her judgment. The best and the most efficient way to move is to produce such inertial motions that can annihilate themselves by using joint reaction forces, like whip that terminates its own whiplashes.
The Nature or Creator provides our bodies with this wonderful natural ability.

The fractal structure of human body

Most of the structures in Nature build of parts similar to the bigger parts they are parts of and similar to the structure in whole. The good example of such structures is a tree.
The smallest branches have the same general shape as a bigger branches and the tree in whole. The same can be said about rivers, blood vessels, waves in the sea, sand dunes in the desert and so on. This kind of structures is called fractals. By definition, the fractal is a structure, build of the structures, similar to itself.  
The human body is not different from any other creations of the Nature. It has a fractal structure - see attached picture:

Pic. 2: Empty-full-empty model

The Golden Ratio Law for human body

Let's recall the definition of Golden Ratio with minor adaptation to our future needs:

"A straight line is said to have been cut in Golden Ratio when, as greater segment is to the whole line, so is the lesser to the greater".

Let's consider the arm including shoulder blade (AS) as a whole, upper arm with a shoulder (UAS) as a greater segment, the forearm with the palm (FAP) as a lesser segment. We assume that distribution of the masses along the arm is in according to the relation between the lengths of its parts.

Let's consider following motion of the arm. UAS starts to move first. It will be followed by arm's flection or extension in the elbow. At some moment contraction of muscles, flexors or extensors, will slow down or stop for a split of second the elbow's opening or closing. This maneuver will be followed by conversion of the active motion of UAS into inertial motion of the AS. Almost immediately after that, the muscles will relax and the elbow can rotate freely again. Additional contraction of the other group of muscles will cause additional deceleration of UAS and this deceleration will cause inertial rotation of FAP in the elbow. This inertial rotational motion of FAP in the elbow will annihilate the motion of the UAS by means of joint reaction forces. This is the most efficient and natural way to move the arm.

This motion is possible when and only when the arm has proportions of Golden Ratio, which means that UAS/AS = FAP/UAS. This is the biomechanical meaning of Golden Ratio, the key to effortless natural movement.

Of course this is only a simplified and partial description of what happens during the complicated process of arm rotation, but it can provide the reader with a general understanding of the process and of the role of the Golden Ratio in our body construction.  For more complete picture we have to consider the way how the arm is constructed, the mass distribution, the angle and linear velocities and so on, but all these matters are quite too complicated for this article.

The fractal nature of the human body appears also in the structures of forearm, torso, head, legs and all the body in whole. These fractals doesn't act as separate entities but rather form consequent fractal chains so every body's motion spreads along these chains.

The Golden Ratio Law for human body by Monya Gorelik:

When and only when there are Golden Ratio proportions within human body, there is a possibility to apply such dynamic sequence of muscle contractions which will cause inertial motion that annihilates itself - for any fractal or fractal chain of the skeletal structure.

We can see in attached picture of the arm motion how this annihilation happens:

Pic. 3: Golden Ratio in the human arm and annihilation of inertial motion

We can notice that the rotation of the upper arm and the forearm must be done in the same plane, otherwise annihilation will not happen. This plane itself moves and rotes in quite sophisticated manner so external observer will see rather complicated motion of upper and fore-arms, but if observer will locate himself on some spot of this plane he will watch quite simple motion of the arm. It means that not any possible body motion has this outstanding quality, but rather very small part of them. The criteria here is an ability to produce self-annihilating inertial motions (SAIM). The most efficient natural motions will follow limited number of routes in the space in order to follow this criteria. These motions and their routes will create unique forms in the space which we call Fish Play. The Central Nerve System (CNS) of performer learns these routes as he becomes more skillful.  
Decades ago Prof. N.A. Bernstein formulated one of the most important problems of motor-control:  It is clear that the basic difficulties for co-ordination consist precisely in the extreme abundance of degrees of freedom, with which the [nervous] center is not at first in a position to deal.
Until now this problem seems remained unresolved even there are some very interesting theories that offer their approach to understanding and resolving the problem.
I feel that SAIM criteria can provide the best and most reasonable answer to this important problem.


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