Pythagoras (560-480 BC), the Greek geometer, was especially interested in the Golden Section, and proved that it was the basis for the proportions of the human figure. He showed that the human body is built with each part in a definite Golden Proportion to all the other parts.
Pythagoras' discoveries of the proportions of the human figure had a tremendous effect on Greek art. Every part of their major buildings, down to the smallest detail of decoration, was constructed upon this proportion.
The Parthenon was perhaps the best example of a mathematical approach to art.
Throughout history, the ratio for length width of rectangles of 1.6180339887... has been considered the most pleasing to the eye.The Greek sculptor Phidias sculptured many things including the bands of sculpture that run above the columns of the Parthenon. There are golden rectangles throughout this structure which is found in Athens, Greece. Phidia widely used the Golden Ratio in his works of sculpture.
The second half of the 5th century B.C. was the Golden Age of Greece. This was the period of her most beautiful art and architecture, and some of her wisest thinkers besides. Both owed much to the popular new study of geometry.
By the start of the next century, geometry itself was entering its own classic age with a series of great developments, including the Golden Mean. The times were glorious in many ways. The Persian invaders had been driven out of Hellas forever, and Pericles was rebuilding Athens into the most beautiful city in the world. At his invitation, Greek mathematicians from elsewhere flocked into the new capital. From Ionia came Anaxa- goras, nicknamed "the mind." From southern Italy and Sicily came learned Pythagoreans and the noted Zeno of Elea. And their influence was felt over all Athens.
High on the hill of the Acropolis rose new marble temples and bronze and painted statues. Crowds thronged the vast new open-air theater nearby, to hear immortal tragedies and comedies by the greatest Greek playwrights. These splendid public works were completed under the direction of the sculptor Phidias and several architects, all of whom knew and used the principles of geometry and optics. "Success in art," they insisted, "is achieved by meticulous accuracy in a multitude of mathematical proportions." And their buildings had a dazzling perfection never seen before-the beauty of calculated geometric harmony.
Elsewhere in the city, the impact of the new geometry took another form. On the narrow streets of Athens walked world-famous philosophers, talking to the people, lecturing on mathematics, geography, rhetoric, how to live the good life. Socrates and others asked, "What is beauty? What is virtue?"- and tried to teach men to think out the answers.
Their method was borrowed from the geometers. They called it dialectics, and it was patterned after the deductive reasoning and proofs of geometry. "For geometry," they said, "will lead the soul toward truth and create the spirit of philosophy."
And geometry itself made tremendous strides in the Golden Age and the darker time that followed. Even after Athenian democracy collapsed in the war with Sparta, geometry continued to flourish in the Athens of the restored aristocracy.
But now, in the 4th century, the study was carried on in schools with grounds and buildings of their own. The first and most famous of these was the Academy, headed by the great philosopher Plato. It was located in an olive grove a half-mile outside of town, and over ifs gate was this inscription:
LET NONE IGNORANT OF GEOMETRY ENTER HERE.
Plato's Academy was the earliest institution of higher learning. Its curriculum was frankly inspired by the old program of the Secret Brotherhood. Studies were broader now- the highest branch was moral and political philosophy. But the ideal was still pure wisdom, and the basic training was still in the "Mathemata." Plato was partly a Pythagorean.
When his teacher, Socrates, was put to death by the Athenian government, Plato had fled to Sicily. There he studied mathematics under noted Pythagoreans, picked up mystical ideas, and
dabbled in aristocratic politics. Finally, he came home to Athens to found his own school and make it the great mathematical center of the Greek world. Most of the mathematicians of that era were his friends or associated with his Academy.
Perhaps the most gifted geometer to study there was Eudoxus of Cnidus, who finally broke the deadlock of the irrationals, and freed geometry for the great advances that were to come. How he did this-with his work on the Golden Mean and his new theory of proportion-is an exciting story. And if we add a bit of imagination, it gives us a fascinating glimpse of Athens and the Academy in Plato's time.
At the age of twenty-four, Exodus came to Athens from his home town of Cnidus on the Black Sea, in order to study at Plato's Academy. He was so poor that he could not afford lodgings in the city, but lived in the small seaport of Piraeus and walked to school every day. Of course, he had already studied some geometry; it was the entrance requirement. But at the Academy he got particularly interested in the matter of an irrational number of a geometric figure. For in Athens the problem was in plain sight every day, in a concrete, or rather, a marble form.
On the high Acropolis, against the shimmering sky, stood the beautiful temple called the
Parthenon-the most wonderful monument of the Age of Pericles, the "perfect" building whose
ruins enthrall us even today. The Parthenon had been designed by Ictinus and Callicrates
according to mathematical principles Its surrounding pillars were an example of "number"
applied: 8 pillars in front, an even number, as Pythagoras had advised, so no central
posts would block the view; but 17 pillars on each side, where it was all right to have
an odd number.
And some of its lines were deliberately curved and slanted to correct optical distortions.
But above all, the Parthenon was a crowning example of proportion in architecture. Scholars still marvel at the logical and harmonious ratios in the whole building and its various parts. And this beauty was achieved with one of the "dynamic rectangles" then in vogue.
Like many Greek temples of time, the Parthenon used the "root five rectangle," a rectangle with an irrational side the square root of 5. How did this root five rectangle come to be used? How was it constructed and shown to be irrational? How did Eudoxus analyze in it the most beautiful of all linear proportions, the Golden Section, or Golden Mean? That is our story.
The development was natural in the architecture of the Golden Age. Greek builders, we
must remember, did not have a minutely graduated measuring rod, in inches or centimeters, like ours. Ground plans were still laid out in the old way, with string (rope), straightedge, level, and carpenter's right angle or "set square." And some of the older temples, and even a few new ones, were quite carelessly designed.
Selections from Julia E. Diggins, String, Straightedge, and Shadow Viking Press, New York , 1965
The great creations of the Greek sculptors are considered as the standards of human beauty and the samples of a harmonic body. In those works the principle of the Golden Proportion are used. Dorifor's statue created by Policlet is considered to be the highest achievements of the classic Greek sculpture. The figure of the young man expresses the unity of beauty and valor underlying the Greek art principles.In the same way Afrodita's sculpture created by Agesander is considered to be the masterpiece of woman's beauty:
The Roman architect and author is far away from the development of a convincing system of proportions. However, his precepts of design are a notable attempt to unify different methods apparently drawn from the Greek and Roman traditions. His knowledge of musical theory seems to exceed his skills in geometry, in which he takes less interest.