16 Facts About Golden ratio

Golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and goes by several other names.

The golden ratio has been used to analyze the proportions of natural objects as well as artificial systems such as financial markets, in some cases based on dubious fits to data.

The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.

The first known decimal approximation of the golden ratio was stated as "about " in 1597 by Michael Maestlin of the University of Tubingen in a letter to Kepler, his former student.

Golden ratio is an algebraic number and even an algebraic integer.

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Which has roots and As the root of a quadratic polynomial, the golden ratio is a constructible number.

Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers:.

Golden ratio appears prominently in the Penrose tiling, a family of aperiodic tilings of the plane developed by Roger Penrose, inspired by Johannes Kepler's remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together.

The decimal expansion of the golden ratio has been calculated to an accuracy of ten trillion digits.

Golden ratio appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is .

Golden ratio appears in the theory of modular functions as well.

Golden ratio saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture.

The golden ratio is apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau, from Images, in which "the sequence of keys is marked out by the intervals and and the main climax sits at the phi position".

Psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law.

Golden ratio is a critical element to golden-section search as well.

From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries.