Starting with two identical squares we can generate the 1.118 ratio (√5/2) and 1.618 ratio (√5/2 + 1/2)

Melancoli I and the Magic Square

This is etching from Albrech Durer called Melancoli I. It is Durer's psychological self-portrait, portrait of his mental state.

Durer was very close to solving the riddle of the Renaissance perspective. He had been to Italy twice and must have brought some information. Here he expresses the frustration over inability to decipher the secret geometric system. Behind the winged person is the so-called Magic Square, where rows, columns, diagonals and other combinations add up to the number 34.

What's so special about number 34? The secret of the magic square is related to perspective, particularly the distance point. 34.34° is the angle that occurs when we are at the distance of 1,618 from a cube with a side of 1. It is the most used distance point by the old masters. It is the sacred geometry behind the masterpieces.

The face of the mysterious polyhedron in Melancolia I could be the outline of the geometrical scheme. Below is the Vermeer's foreshortening geometry (green) for the Music Lesson.

Yet there are still pieces missing to complete the puzzle. Durer's muse got wings, but can not fly.

 GOLDEN RATIO

I do not want to repeat what has already been said many times but let me remind you of a bit of history. The golden ratio has captured the attention of intellectuals and thinkers from all disciplines like no other number in the history of mathematics. Some of the greatest minds of all ages, from Pythagoras and Euclid in ancient Greece; the medieval mathematician Leonardo Fibonacci and the Renaissance astronomer Johannes Kepler; to modern molecular scientists or astrophysicists have examined this simple, yet irrational ratio. However, the fascination with the golden ratio is not limited to mathematics. Many others spent endless hours over this ratio and its properties. Biologists, artists, musicians, historians, architects and philosophers have inspected and speculated its importance and its ubiquity at the same time. The golden ratio inspired thinkers of all scientific disciplines like no other number, perhaps with the exception of Pí.

The golden ratio used to be called tau, which is the Greek letter meaning "slice" or "portion". The golden ratio is a naturally occurring phenomenon in the physical world. We can find it in our bodies, in bone proportions and in the branching of veins and nerves. The golden ratio is also recognizable in the arrangement of plant seeds, flower petals or leaves along the stem. It may even be seen in the ratio of chemical compounds and the geometry of crystals. On a macrocosmic level, we can observe the golden ratio in the shape of Venus' orbit. The golden ratio permeates all structures and forms, whether microcosmic or macrocosmic; organic or inorganic; acoustic or optical. Because of its unique features, we can reasonably assume that it is a sacred or divine proportion and view it as a door to a deeper understanding of beauty and spirituality in life. The beauty of nature manifests itself in many various ways throughout this ratio, that I could devote a whole book just to this subject.

The attraction of the golden ratio stems from the fact that it has an almost supernatural ability to appear where it is least expected.
It has been known from ancient times to the present as the golden section, golden number, golden ratio, divine cut, Phi, Tau; all these names have been used. The first recorded definition of the golden ratio goes back to the time of the Greek mathematician Euclid (c. 325 BC - 265 BC). He described it as "an extreme and mean ratio". Translated into our language, it means the ratio of the entire segment to the larger part equals the ratio of the larger part to the smaller part.

Geometrically and mathematically speaking, although asymmetrical, it's a perfectly logical division that can fractally continue to infinity in both directions, to macrocosm or microcosm. By the way, it was the ancient philosopher Pythagoras, who came up with the concepts of macrocosm and microcosm.

In this acticle I will try to reduce mathematics to a minimum, but let's see how we get the golden ratio in numerical form. Expressed algebraically, for quantities a and b with a > b > 0. Then a+b/a = a/b. By further multiplying a/b we get the quadratic equation (a/b)2 - (a/b) - 1 = 0, then we get x = 1+ √5/2 = 1,618033988749894848 and the digits just keep going. After all it is an irrational number.

There are several geometrical construction methods of the golden ratio. One of them is:

        Construction of the golden section from the square.

It is generally accepted that some Greek monuments were built in accordance with this attractive number.
However, the dimensions of the Great Pyramid of Giza show a much clearer relationship to this sacred ratio than those in the Parthenon.
It is quite obvious that the Egyptians (if it was them who built the pyramids) knew the properties of this ratio. They supposedly called it - The Sacred section.

The Golden triangle

Probing the outline of this relic I finally realized how is it incorporated into the design of the perspective system.
It was this golden ratio triangle that enlightened my thought processes - cracked open the door of insight and sent me on a long but fascinating journey that led to deciphering long-forgotten secrets. 

Egyptian stylized art did not use any perspective system, but the Egyptian concept of looking at the laws of nature has been transferred to the Renaissance through the lens of Greek culture. Renaissance artists respected proportional elegance, harmony and beauty, propounded by Plato whose legacy appeals to us even today. Plato considered geometry and numbers as reduced and therefore an ideal language for philosophy, a language immaculate; to clearly describe the metaphysical realm. Platonists believed that geometrical knowledge is inherent, incorporated in our immortal souls.

Platon drew from the intellectual influence of his predecessor Pythagoras of Samos.

 PYTHAGORAS

Only limited historical information remains about Pythagoras but his legend still lives on today. He was born in Samos sometime around 580 BC. In his youth he travelled besides other places, to Egypt where he was initiated into the mysteries of geometry. After returning to his native island, he founded an eminent school. His successful method of teaching brought him legions of students. Later, his fame grew so great that the ruler Polykratés forced him into exile. Pythagoras settled in southern Italy (then part of Greece) where his teachings experienced the same success. Pythagoreans considered the pentagram as a symbol of life, harmony and health. The pentagon and inscribed pentagram abounds in the golden section and Pythagoreans regarded this divine ratio as the creative archetype of the world.

 In Pythagorean times the practicing of geometry was a discipline to find out and study how the universe works. Pythagoras came up with the notion that the entire universe is composed of vibrations. This is associated with the theory of musical harmony and the musical scale, another of his enormously important inventions. The musical scale is based on closely related geometric shapes and proportions. Pythagoras also considered geometry and numbers as an ideal philosophical language, albeit reduced.
But let's get back to the unique properties of the golden ratio. The ratio began to be rediscovered and promoted in the 15th century when the aesthetics of the Hellenistic culture was an important part of Renaissance art. In the perspective system, the golden ratio and sacred geometry play a major role.

In the chapters which follow I'll show you how the golden ratio is firmly integrated in the selected works of great art. The golden ratio occurs where it is least expected.  When we look at some of these masterpieces there is something that works on a subconscious level, at a level that makes us feel deeply connected. It is a feeling that it's the way it is supposed to be, that it is the correct way. The sacred geometry is the building block of the Renaissance perspective system. A system which delivers a rendition of space with such accuracy that linear perspective can never achieve: a precise mathematical definition of space.

We will look together beneath the surface of these ancient, but in terms of message, still current works. Let's uncover the hidden harmony in some of the best paintings of all time.

The geometry has endless possibilities. The fivefold symetry of pentagon.