golden ratio (φ)
The golden ratio is an irrational number of the type known as an algebraic number (in contrast with pi and e, which are transcendental) and is represented by the Greek letter φ (phi). It can be defined in various ways. For example, it is the only number equal to its own reciprocal plus 1, i.e. φ = (1/φ so that φ2 = φ + 1. From this comes the quadratic equation φ2 - φ - 1 = 0 of which the golden ratio is the positive solution, (1 + √(5))/2 = 1.6180339887... The golden ratio is also approximated by the ratio of successive terms in the Fibonacci sequence; in fact, F(n+1)/F(n) gets closer and closer to phi as n tends to infinity. Because 1/(1 - φ) = φ, the continued fraction representation of φ is
Two quantities are said to be in the golden ratio, if the ratio of the larger one, a, to the smaller one, b, is the same as the ratio of the smaller one to their difference, i.e., a/b = b/(a-b). The so-called golden rectangle is one whose sides a and b stand in the golden ratio. It is famously said to have great aesthetic appeal and is closely approximated by the dimensions of the front of the Parthenon in Rome. Leonardo da Vinci's masterpiece the Mona Lisa is said to have a face that is framed by a golden rectangle; what is certain is that Leonardo was a
The dodecahedron, which according to Plato is the solid "which the god used for embroidering the constellations on the whole heaven," is intimately related to the golden ratio – both the surface area and the volume of a dodecahedron of unit edge length are simple functions of the golden ratio. In fact, φ turns up frequently in figures that have pentagonal symmetry. For instance the ratio of a regular pentagon's side and diagonal is equal to φ, and the vertices of a regular icosahedron are located on three orthogonal golden rectangles. The golden ratio is also related to Penrose tiling and to the plastic number.
Related category NOTABLE NUMBERS
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