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summary of classical construction
upon a unit square
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), 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. [1] 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. This has encouraged modern artists, architects, and others, during the last 500 years, to incorporate the ratio in their work.
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.
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. It's also interesting to note that the average ratio of the sides of great paintings, according to a recent analysis, is 1.34. [2]
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.
The golden ratio turns up in nature as a result of the dynamics of some systems - for instance, in the angular spacing of trees around a trunk, or sunflower seeds. In both cases, the problem is "wedge this next one into the biggest available space".
You can draw a nice sunflower by plotting the points
The shape of the shell of the chambered nautilus ( Nautilus pompilius) is often claimed to be related to the golden ratio. (Please see about the "Golden ratio" page!)
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