Golden Section in Greek's Art The idea of harmony based on the "golden section" became one of the fruitful ideas of the Greek art. The nature taken in a broad sense included also of the person creative patterns, art, music, where the same laws of a rhythm and harmony act. Let's give a word to Aristotle: "The Nature aims to the contrasts and from them, instead of from similar things, it forms a consonance ... It combined a male with female and thus the first public connection is formed through the connection of contrasts, instead of by means of similar. As well the art, apparently, by imitating to the nature acts in the same way. Namely the painting makes the pictures conforming to the originals by admixing white, black, yellow and red paints. Music creates the unified harmony by mixing of different voices, high and low, lingering and short, in a congregational singing. The grammar created the whole art from the mixture of vowels and consonants". To take a material and to eliminate all superfluous is the aphoristically embodied schedule of artist incorporated all gravity of philosophical wisdom of the antique thinker. And this is the main idea of the Greek art, for which the "golden section" became some aesthetic canon. Theory of proportions is the basis of art. And, certainly, the problems of proportionality could not pass past Pythagor. Among the Greek's philosophers Pythagor was the first one who attempted mathematically to understand an essence of musical harmonic proportions. Pythagor knew, that the intervals of the octave can be expressed by numbers, which fit to the corresponding oscillations of the cord, and these numerical relations were put by Pythagor in the basis of his musical harmony. It is assigned to Pythagor knowledge of arithmetical, geometrical and harmonic proportions, and also the law of the "golden section". Pythagor gave a special, outstanding attention to the "golden section" by making the pentagon or pentagram as distinctive symbol of the "Pythagorean Union". By borrowing the Pythagorean doctrine about harmony Plato used five regular polyhedrons ("Platonic solids") and emphasis their "ideal" beauty. Importance of proportions is emphasized by Plato in the following words: " Two parts or values can not be satisfactorily connected among themselves without third part; the most beautiful link is that, which together with two initial values gives the perfect unit. It is reached in the best way by proportion (analogy), in which among three numbers, planes or bodies, the mean one so concerns to the second one, as the first one to the mean one, and also the second one to the mean one as the mean one to the first one. This implies, that the mean one can exchange the first one and the second one, the first one and the second one can exchange the mean one and all things together thus makes a indissoluble unit". As the main requirements of beauty Aristotle puts forward an order, proportionality and limitation in the sizes. The order arises then, when between parts of the whole there are definite ratios and proportions. In music Aristotle recognizes the octave as the most beautiful consonance taking into consideration that a number of oscillations between the basic ton and the octave is expressed by the first numbers of a natural series: 1:2. In poetry, in his opinion, the rhythmic relations of a verse are based on small numerical ratio, thanks to this it is reached a beautiful impression. Except for a simplicity based on a commensurability of separate parts and the whole, Aristotle as well as Plato recognizes the highest beauty of the regular figures and proportions based on the "golden section". Not only the philosophers of Ancient Greece, but also many Greek artists and architects gave considerable attention to achievement of proportionality. And it is confirmed by the analysis of architectural monuments of the Greek architects. The antique Parthenon, "Canon" by Policlet, and Afrodita by Praksitle, the perfect Greek theatre in Epidavre and the most ancient theatre of Dionis in Athens - all this are bright art examples executed by steep harmony on the basis of the golden section. The theatre in Epidavre is constructed by Poliklet to the 40th Olympiad. It was counted on 15 thousand persons. Theatron (the place for the spectators) was divided into two tiers: the first one had 34 rows of places, the second one 21 (Fibonacci numbers)! The angle between theatron and scene divides a circumference of the basis of an amphitheater in ratio: 137°,5 : 222°,5 = 0.618 (the golden proportion). This ratio is realized practically in all ancient theatres. Theatre of Dionis in Athens has three tiers. The first tier has 13 sectors, the second one 21 sectors (Fibonacci numbers)!. The ratio of angles dividing a circumference of the basis into two parts is the same, the golden proportion. Three adjacent numbers from the initial fragment of Fibonacci series: 5, 8, 13 are values of differences between radiuses of circumferences lying in the basis of the schedule of construction of the majority of the Greek theatres. The Fibonacci series served as the scale, in which each number corresponds to integer units of Greek's foot, but at the same time these values are connected among themselves by unified mathematical regularity. At construction of temples a man is considered as a "measure of all things: in temple he should enter with a "proud raised head ". His growth was divided into 6 units (Greek foots), which were sidetracked on the ruler, and on it the scale was put, the latter was connected hardly with sequence of the first six Fibonacci numbers: 1, 2, 3, 5, 8, 13 (their sum is equal to 32=2 As to the Greek sculpture, here again searches of proportionality of a human body are doubtless. Still Diodor mentions about two sculptors from the island Samos, Telecle and Tiodor, who for the first time transferred the norms of human body developed in Egypt to the Greek sculpture. Plinij testifies that sculptor Policlet written the article about regular proportions of a human body and molded the famous statue of Dorifor (Fig.1), which a long time served as a canon.
Harmonic analysis of the Dorifor statue given in the book "Proportionality in the architecture" (1933) by the Russian architect G.D. Grimm indicates the following connections of the famous statue with the golden section - The first section of the Dorifor figure or its overall height
*M*^{0}= 1 in the proportion of the golden section*M*^{1}= t^{ -1}and*M*^{2}= t^{ -2}passes through a navel. - The second section of bottom part of a trunk
*M*^{1}= t^{ -1}and*M*^{2}= t^{ -2}passes through the line of his knee. - The third section
*M*^{3}= t^{ -3}and*M*^{4}= t^{ -4}passes through the line of his neck.
The theory of measurement of harmony by a principle of division of the whole in middle and extreme ratio (the "golden section") developed by antique mathematicians became as that foundation, that launch pad, on which concepts of harmony in science and art of European culture subsequently were constructed. But about this we will tell at the next sections of our Museum. Follow us! |