Golden Mean

From Academic Kids

For those mathematically inclined -- See Golden ratio.

The Golden Mean is a mathematical expression of proportion that ancient Greek philosophers observed throughout the natural world. Several names have been invented, ranging from the mystical to the mathematical: the "divine proportion", the "golden section", and the golden ratio. The Latin poet Horace coined the term aurea mediocritas. The simplest geometrical example of the Ratio is the golden rectangle, whose sides have unequal length but whose shape the Greeks found particularly pleasing.

Greek philosophers considered the Golden Mean to be the middle between two extremes, one of excess and the other of deficiency. Therefore, to the Greek mentality, it was an attribute of beauty. Both ancients and moderns realized that "there is a close association in mathematics between beauty and truth". (1)

The Poet Keats in his "Ode on a Grecian Urn" composed these lines:

"Beauty is truth, truth beauty, that is all"
"Ye know on earth, and all ye need to know."

The Golden Mean is the middle between two extremes; the one of excess and the other of deficiency. This golden mean is an attribute of beauty. There are three concomitants of Beauty: symmetry, proportion, and harmony. This triad of principles infused their life. They were very much attuned to beauty as an object of love and something that was to be imitated and reproduced in their lives, architecture, Paideia and politics. They judged life by this mentality. This mentality is called aestheticism.

The two parts of the Golden Mean which can be seen in the Golden rectangle, has two parts; the major and minor parts which are unequal. These parts are opposites united in a harmonious proportion. This is a pattern that repeats itself throughout nature. This pattern-forming process is the union of opposites. György Doczi coins the term dinergy for this.

Contents

Pythagoras

According to legend, the Greek Philosopher Pythagoras discovered the concept of harmony when he began his studies of proportion while listening to the different sounds given off when the blacksmith’s hammers hit their anvils. The weights of the hammers and of the anvils all gave off different sounds. From here he moved to the study of stringed instruments and the different sounds they produced. He started with a single string and produced a monochord in the ratio of 1:1 called the Unison. By varying the string, he produced other chords: a ratio of 2:1 produced notes an octave apart.(Modern music theory calls a 5:4 ratio a "major third" and an 8:5 ratio a "major sixth".) In further studies of nature, he observed certain patterns and numbers reoccurring. Pythagoras believed that beauty was associated with the ratio of small integers.

Astonished by this discovery and awed by it, the Pythagoreans endeavored to keep this a secret; declaring that anybody that broached the secret would get the death penalty. With this discovery, the Pythagoreans saw the essence of the cosmos as numbers and numbers took on special meaning and significance.

The symbol of the Pythagorean brotherhood was the pentagram, in itself embodying several Golden Means.

Golden mean in philosophy

The earliest representation of this idea in culture is probably in the mythological Cretan tale of Daedalus and Icarus. Daedalus, a famous artist of his time, built feathered wings for himself and his son so that they might escape the clutches of King Minos. Daedalus warns his son to "fly the middle course", between the sea spray and the sun's heat. Icarus did not listen to his father and flew up and up till the sun melted the wax of his wings and he fell to his death.

Another early elaboration is the pithy laconic Doric saying carved on the front of the temple at Delphi: "Nothing in Excess".

Socrates teaches that a man "must know how to choose the mean and avoid the extremes on either side, as far as possible". (2)

In education, Socrates asks what effect an exclusive devotion to gymnastic or the exclusive devotion to music. It either "produced a temper of hardness and ferocity, (or) the other of softness and effeminacy". (3) But having both will produce harmony, hence beauty and the good. Furthermore, he stresses the importance of mathematics in education as teaching beauty and truth in men.

Something that was disproportionate was evil and something to be despised. They hated extremes. Plato says, "If we disregard due proportion by giving anything what is too much for it; (i.e.) too much canvas to a boat, too much nutriment to a body, too much authority to a soul, the consequence is always shipwreck." (4)

Plato in the Laws, uses this in his critique of governments. In his opus magnum of the perfect state, he says, "Conducted in this way, the election will strike a mean between monarchy and democracy,…". (5)

In the Eudemian Ethics, Aristotle writes on the virtues. His constant phrase is, "… is the Middle state between …". His psychology of the soul and its virtues is based on the golden mean between the extremes. (6) In the Politics, Aristotle critizes the Spartan Polity by critiquing the disproportionate elements of the constitution; i.e. they trained the men and not the women and they trained for war but not peace. (7) This disharmony produced difficulties which he elaborates on.

Golden mean in art

In architecture, the golden mean is the ideal relationships of mass and line which the Greeks perfected over time. Moreover, they found that architecture and art that incorporate this feature are more pleasing to people. This finds its perfection in the Parthenon. This can be compared to one of the first examples of Greek temple building, the temple of Poseidon at Paestum, Italy as it is squat and unelegant. The front of the Parthenon with its triangular pediment fits inside a golden rectangle. The divine proportion and its related figures were incorporated into every piece and detail of the Parthenon.

The Triumphal Arch of Constantine and the Colosseum, both in Rome, are great examples ancient use of golden relationships in architecture.

Phidias, a famous ancient Greek sculptor, incorporated the Golden Mean in all his work.

Golden mean in the Renaissance

Kepler was fascinated by the mystery of the Golden Mean and also coined it as the "divine proportion". Luca Pacioli, in 1509, wrote a dissertation called De Divina Proportione which was illustrated by Leonardo da Vinci.

Golden mean in psychology

The famous German psychologist, Gustav Fechner, inspired by Adolf Zeising’s book, Der goldene Schnitt, began a serious inquiry to see if the golden rectangle had psychological aesthetic impact. It was published in 1876. With German zeal of thoroughness, Fechner made thousands of measurements of commonly seen rectangles, such as writing pads, books, playing cards, windows, and found that most were close to Phi. He also tested people’s preferences and found most people prefer the shape of the golden rectangle. His experiments were repeated by Witmar (1894), Lalo (1908) and Thorndike (1917).

Quotations

  • "In many things the middle have the best / Be mine a middle station."
    Phocylides (8)
  • "Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel."
    Johannes Kepler
  • "When Coleridge tried to define beauty, he returned always to one deep thought; beauty, he said, is unity in variety! Science is nothing else than the search to discover unity in the wild variety of nature,—or, more exactly, in the variety of our experience. Poetry, painting, the arts are the same search, in Coleridge’s phrase, for unity in variety."
    — J. Bronowski (9)
  • "…but for harmony beautiful to contemplate, science would not be worth following."
    Henri Poincaré. (10)

Miscellanea

  • Jacques Maritain in his Introduction to Philosophy uses the idea of the golden mean to place Aristotelian-Thomist Philosophy, throughout his book, between the deficiencies and extremes of other philosophers and systems.

References

  1. The Divine Proportion, p. 75
  2. Republic 619, Jowett p. 394.
  3. Laws, 691c,756e-757a .
  4. Eudemian Ethics, 1233b15; Loeb Classical Library, p. 351-355.
  5. Politics, Aristotle, 1270af and 1271b; Loeb p. 137 and p. 147.

See also

Bibliography

  • The Greek Way, Edith Hamilton, W. W. Norton & Co., NY, l993.
  • Sailing the Wine-Dark Sea, Why the Greeks Matter, Thomas Cahill, Nan A. Talese an imprint of Doubleday, NY, 2003.
  • The Divine Proportion, A Study in Mathematical Beauty, H. E. Huntley, Dover Publications, Inc., NY, l970.
  • The Power of Limits, Proportional Harmonies in Nature, Art, and Architecture, György Doczi, Shambala Publications, Inc., Boston & London, l981.
  • Der goldene Schnitt, Adolf Zeising, (1884)

Leonardo Fibonacci (filius Bonacci), alias Leonardo of Pisa,

sl:Zlato razmerje

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