Golden section

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Golden Section

The Divine Proportion

In the 16th Century, Luca Pacioli (1445-1514), geometer and friend of the great Renaissance painters, rediscovered the "golden secret".Luca Pacioli, however, was a great admirer of the Golden Section, as evidenced by the name of his treatise, Divina proportione, which actually comprises three independent works ( 1509 ). At the beginning Pacioli places the Compendium de divina proportione, the book about the Golden Section, which Pacioli dedicated to Duke Lodovico Sforza of Milan in December 1498.

The learned Franciscan Pacioli finds five attributes of God in this special proportion, the first four of which are unity and uniqueness, trinity, the impossibility of defining in terms of human ratio, and the immutability. In the fifth, functional comparison Pacioli sets the "divine proportion" in relation to the Platonic quintessence:

As God confers being to the celestial virtue, called by the other name 'fifth essence', and through that one to the other four simple bodies, that is, to the four earthly elements...and so through these to every other thing in nature. Thus this our proportion is the formal being of (according to Timaeus) heaven, attributing to it the figure of the solid called Duodecahedron, otherwise known as the solid of twelve pentagons.

In the rest of the first book, Pacioli describes in detail the geometrical attributes of the "proportion having a middle and two extreme ends", in medieval words used also by Pacioli. He does not advance the field of research in his discipline, does not give any advice or thought about the practical application in the pictorial arts. In fact Pacioli's achievement consists more in diligent compilation than in scientific originality.

Of course the Compendium is not architectural theory, but Pacioli himself wrote probably by c.1509 in Venice a small Tractato de l'architectura, published in the Divina porportione as the second book. His connection with architecture dates from his time in Rome, when he was Alberti's guest for months. Later in Urbino between 1472 and 1474, he had an occasion to meet Francesco di Giorgio Martini and Bramante, and in Milan (1494-1499/1500) he collaborated with Leonardo. At Milan Pacioli and Leonardo quickly became close friends. Mathematics and art were topics which they discussed at length, both gaining greatly from the other. At this time Pacioli began work on the second of his two famous works, Divina proportione and the figures for the text were drawn by Leonardo.

Leonardo da Vinci (1451-1519) was one of the greatest inventor-scientist of recorded history. His genius was unbounded by time and technology, and was driven by his insatiable curiosity, and his intuitive sense of the laws of nature. Da Vinci was dedicated to discovery of truth and the mysteries of nature, and his insightful contributions to science and technology were legendary. As the archetypical Renaissance man, Leonardo helped set an ignorant and superstitous world on a course of reason, science, learning, and tolerance. He was an internationally renowned inventor, scientists, engineer, architect, painter, sculptor, musician, mathematician, anatomist, astronomer, geologists, biologist, and philosopher in his time.
Ever the perfectionist, Leonardo turned to science in the quest to improve his artwork. His study of nature and anatomy emerged in his stunningly realistic paintings, and his dissections of the human body paved the way for remarkably accurate figures. He was the first artist to study the physical proportions of men, women and children and to use these studies to determine the "ideal" human figure. Unlike many of his contemporaries -- Michelangelo for example -- he didn't get carried away and paint ludicrously muscular bodies, which he referred to as "bags of nuts." All in all, Leonardo believed that the artist must know not just the rules of perspective, but all the laws of nature. The eye, he believed, was the perfect instrument for learning these laws, and the artist the perfect person to illustrate them.

"...no human inquiry can be called science unless it pursues its path through mathematical exposition and demonstration." -Leonardo Da Vinci

Leonardo had for a long time displayed an ardent interest in the mathematics of art and nature.He had earlier, like Pythagoras, made a close study of the human figure and had shown how all its different parts were related by the golden section. Leonardo's unfinished canvas Saint Jerome shows the great scholar with a lion lying at his feet.A golden rectangle fits so neatly around the central figure that it is often said the artist deliberately painted the figure to conform to those proportions. Knowing Leonardo's love of "geometrical recreations" as he described them, this is quite likely.

Hacking back to classical themes and techniques for their inspiration, artists of the Renaissance like Michelangelo (1475-1564) and Raphael (1483-1530) once more began to construct their compositions on the golden ratio. The proportions of Michelangelo's David conform to the golden ratio from the location of the navel with respect to the height to the placement of the joints in the fingers.

The Medieval builders of churches and cathedrals approached the design of their buildings in much the same way as the Greeks. A good geometric structure was their aim. Inside and out, their buildings were intricate constructions based on the golden section.



        

        

        

        

        

 © 2001-2003 Radoslav Jovanovic              translated: D.Filipovic   created:  January 2003.