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Fibonacci and the Golden Mean

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Fibonacci and the Golden Mean

From Wikipedia

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The figure on the right illustrates the geometric relationship. Expressed algebraically:

    frac{a+b}{a} = frac{a}{b} equiv varphi,

where the Greek letter phi (varphi) represents the golden ratio. Its value is:

    varphi = frac{1+sqrt{5}}{2} = 1.61803,39887ldots.

At least since the 20th century, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing (see Applications and observations below). A golden rectangle can be cut into a square and a smaller rectangle with the same aspect ratio. Mathematicians since Euclid have studied the golden ratio because of its unique and interesting properties. The golden ratio is also used in the analysis of financial markets, in strategies such as Fibonacci retracement.

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