Talk:Golden ratio/sandbox
Number, approximately 1.618 / From Wikipedia, the free encyclopedia
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities and with ,
Representations | |
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Decimal | 1.618033988749894...[1] |
Algebraic form | |
Continued fraction |
where the Greek letter phi ( or ) denotes the golden ratio.[lower-alpha 1] The constant satisfies the quadratic equation and is an irrational number with a value of[1]
The golden ratio was called the extreme and mean ratio by Euclid,[2] and the divine proportion by Luca Pacioli,[3] and also goes by several other names.[lower-alpha 2]
Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron.[7] A golden rectangle—that is, a rectangle with an aspect ratio of —may be cut into a square and a smaller golden rectangle.
The value is also known as the most irrational number,[8][9][10] because it has the greatest possible difference between it and any rational approximation, in proportion to the square of the approximation's denominator. A more careful but less catchy phrasing would be "least accurately approximated by rationals". The fact that multiples of avoid integers by the greatest possible distances leads to numerous applications where avoiding even multiples is desired, including some patterns in nature such as the spiral arrangement of leaves, seeds, and other parts of vegetation.
Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle.
The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data.[11]