【Design】狐狸太太日記

The golden ratio was first studied by ancient mathematicians because of its frequent appearance in geometry. 
It is related to regular pentagons and pentagrams, which are known in Sumerian tablets as early as 3200 BC. 
It has been speculated since 1859 by Taylor that the Egyptians embodied the golden ratio in the dimensions of pyramids, but according to a thorough modern analysis there is absolutely no evidence that the Egyptians either knew about the golden ratio or used it in the dimensions of the pyramids.

The ancient Greeks usually attributed its discovery to Pythagoras (or to the Pythagoreans, notably Theodorus) or to Hippasus of Metapontum. Hellenistic mathematician Euclid spoke of the "golden mean" this way, "a straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". The golden ratio is represented by the Greek letter (phi, after Phidias, a sculptor who commonly employed it) or less commonly by τ (tau, the first letter of the ancient Greek root τ(ε/ο)μ– meaning cut).

It has been claimed that the ancient Egyptians knew the golden ratio because ratios close to the golden ratio may be found in the positions or proportions of the Pyramids of Giza, but most likely, it was not until the Ancient Greeks that the Golden Ratio was fully understood and used.

The ancient Greeks knew the golden ratio from their investigations into geometry, 
but there is no evidence they thought the number warranted special attention above that for numbers like π (pi), 
for example. Studies by psychologists have been devised to test the idea that the golden ratio plays a role in human perception of beauty. 
They are, at best, inconclusive. Despite this, a large corpus of beliefs about the aesthetics of the golden ratio has developed. These beliefs include the mistaken idea that the purported aesthetic properties of the ratio was known in antiquity. For instance, the Acropolis, including the Parthenon, is often claimed to have been constructed using the golden ratio. This has encouraged modern artists, architects, photographers, and others, during the last 500 years, to incorporate the ratio in their work. As an example, a rule of thumb for composing a photograph is called the rule of thirds; it is said to be roughly based on the golden ratio.

In 1509, Luca Pacioli published the Divina Proportione, which explored not only the mathematics of the golden ratio, but also its use in architectural design. This was a major influence on subsequent generations of artists and architects. Leonardo Da Vinci drew the illustrations, leading many to speculate that he himself incorporated the golden ratio into his work. It has been suggested for example that Da Vinci's painting of the Mona Lisa employs the Golden Ratio in its geometric equivalents.

Da Vinci suggested that the human body has proportions close to the golden ratio. The Swiss architect Le Corbusier took this suggestion to an extreme, not only splitting the body's height at the navel into two sections in the golden ratio, but splitting those each again, at the knees and at the throat.

Le Corbusier also used the golden ratio as the basis of his Modulor system of Architecture.

The ratio is sometimes used in modern human-made constructions, such as stairs and buildings, woodwork, and in paper sizes; however, the series of standard sizes that includes A4 is based on a ratio of and not on the golden ratio. The average ratio of the sides of great paintings, according to a recent analysis, is 1.34. [1]. Credit cards are generally 3 3/8 by 2 1/8 inches in size, which is less than 2 % from the golden ratio.

The ratios of justly tuned octave, fifth, and major and minor sixths are ratios of consecutive numbers of the Fibonacci sequence, making them the closest low integer ratios to the golden ratio. James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.

Ernő Lendvai (1971) analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry.