Golden ratio

:This article is about the mathematical ratio. For the Aristotelian concept of "golden mean" see Nicomachean Ethics.

Aesthetic uses

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.

Related Topics:
Ancient Egyptians - Pyramids of Giza

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The ancient Greeks already 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 pi (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. http://plus.maths.org/issue22/features/golden/ 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.

Related Topics:
Ancient Greeks - Geometry - Pi - Beauty - Acropolis - Parthenon - Rule of thumb - Rule of thirds

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It is also claimed that the human body has proportions close to the golden ratio.

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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, although there is no evidence supporting this.

Related Topics:
Luca Pacioli - Leonardo Da Vinci

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

Related Topics:
Architect - Le Corbusier - Modulor - Architecture

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The ratio is sometimes used in modern man-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 sqrt{2} and not on the golden ratio. The average ratio of the sides of great paintings, according to a recent analysis, is 1.34. http://arxiv.org/abs/physics/9908036/. Credit cards are generally 3 3/8 by 2 1/8 inches in size, which is less than 2 percent from the golden ratio.

Related Topics:
Woodwork - Paper size - A4

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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.

Related Topics:
Justly tuned - Octave - Fifth - Sixth - Fibonacci sequence - James Tenney - For Ann (rising) - Glissando - Equal tempered - Minor - Major sixth

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French composer Erik Satie used the Golden Section in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry.

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The construction of a pentagram is based on the golden ratio. The pentagram can be seen as a geometric shape consisting of 5 straight lines arranged as a star with 5 points. The intersection of the lines naturally divides each length into 3 parts. The smaller part (which forms the pentagon inside the star) is proportional to the longer length (which form the points of the star) by a ratio of 1:1.618... It is thought by some that this fact may be a reason why the ancient philosopher Pythagoras chose the pentagram as the symbol of the secret fraternity of which he was both leader and founder.

Related Topics:
Pentagram - Pythagoras

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