 | Golden ratio: Encyclopedia II - Golden ratio - Aesthetic uses
Golden ratio - 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.
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 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.
It is also claimed that the human body has proportions close to 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.
The Architect Le Corbusier used the golden ratio as the basis of his Modulor system of Architecture.
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 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 percent 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) analyses Béla Bartók's works as being based on two opposing systems, that of the golden section and the acoustic scale. 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.
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.
Other related archivesA4, Acropolis, Architect, Architecture, Béla Bartók, Dynamic symmetry, Egyptians, Erik Satie, Euclid, Fibonacci number, Fibonacci numbers, Fibonacci sequence, For Ann (rising), Golden angle, Golden function, Golden ratio base, Golden rectangle, Golden section (page proportion), Golden section search, Greek letter, Hippasus of Metapontum, James Tenney, Lagrange, Le Corbusier, Leonardo Da Vinci, Logarithmic spiral, Luca Pacioli, Modulor, OEIS, Parthenon, Penrose tiles, Phidias, Pi, Pisot-Vijayaraghavan number, Plastic number, Pyramids of Giza, Pythagoras, Pythagoreans, Sacred geometry, Shepard tone, The Roses of Heliogabalus, Theodorus, Venus, Vitruvian man, acoustic scale, aesthetically, algebraic number field, ancient Egyptians, ancient Greeks, approximation theorem, beauty, continued fraction, equal tempered, fifth, geometry, glissandoing, golden mean base, golden rectangles, icosahedron, integers, irrational number, justly tuned, limit, lowest terms, major sixth, minor, numeral system, octave, paper sizes, pentagon, pentagram, phi, phyllotaxis, quadratic equation, ratio, reciprocal, recurrence, rule of thirds, rule of thumb, sixths, square root, symmetry, tau, woodwork
 Adapted from the Wikipedia article "Aesthetic uses", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
