In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887 (from the quadratic formula). At least since the Renaissance, 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. Mathematicians have studied the golden ratio because of its unique and interesting properties. The golden ratio can be expressed as a mathematical constant, usually denoted by the Greek letter varphi (phi). The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically:

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