golden ratio

also known as the golden section, golden mean , or divine proportionin mathematics, the irrational number (1 + √5)/2, often denoted by the Greek letters τ or ϕ, and approximately equal to 1.618. The origin of this number and its name may be traced back to about 500 BC and the investigation in Pythagorean geometry (Pythagoreanism) of the regular pentagon, in which the five diagonals form a fivepointed star. On each such diagonal lie two points of intersection with other diagonals, and either of those points divides the whole diagonal into two segments of unequal lengths so that the ratio of the whole diagonal to the larger segment equals the ratio of the larger segment to the smaller one. In terms of present day algebra, letting the length of the shorter segment be one unit and the length of the larger segment be x units gives rise to the equation (x + 1)/x = x/1; this may be rearranged to form the quadratic equation x^{2} – x – 1 = 0, for which the positive solution is x = (1 + √5)/2, the golden ratio.The ancient Greeks recognized this “dividing” or “sectioning” property and described it generally as “the division of a line into extreme and mean ratio,” a phrase that was ultimately shortened to simply “the section.” It was more than 2,000 years later that both “ratio” and “section” were designated as “golden” in references by the German astronomer Johannes Kepler (Kepler, Johannes) and others. The Greeks also had observed that the golden ratio provided the most aesthetically pleasing proportion of sides of a rectangle, a notion that was enhanced during the Renaissance by, for example, work of the Italian polymath Leonardo da Vinci and the publication of De divina proportione (1509; Divine Proportion) by the Italian mathematician Luca Pacioli, and illustrated by Leonardo (see the photograph—>).The golden ratio occurs in many mathematical contexts. It is geometrically constructible by straightedge and compass, and it occurs in the investigation of the Archimedean and Platonic solids (Platonic solid). It is the limit of the ratios of consecutive terms of the Fibonacci number (Fibonacci numbers) sequence 1, 1, 2, 3, 5, 8, 13, …, in which each term beyond the second is the sum of the previous two, and it is also the value of the most basic of continued fractions, namely 1 + 1/(1 + 1/(1 + 1/(1 + ….In modern mathematics, the golden ratio occurs in the description of fractals (fractal), figures that exhibit selfsimilarity and play an important role in the study of chaos (chaos theory) and dynamical systems (analysis).Stephan C. Carlson
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Universalium. 2010.
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Golden ratio — For the Ace of Base album, see The Golden Ratio (album). Not to be confused with Golden number. The golden section is a line segment divided according to the golden ratio: The total length a + b is to the length of the longer segment a as the… … Wikipedia
golden ratio — aukso pjūvis statusas T sritis informatika apibrėžtis Atkarpos dalijimas į dvi nevienodo ilgio dalis taip, kad mažesnės ir didesnės dalių santykis būtų lygus didesnės dalies ir visos atkarpos santykiui. Šį santykis lygus iracionaliajam skaičiui.… … Enciklopedinis kompiuterijos žodynas
golden ratio — noun The irrational number (approximately 1·618), usually denoted by the Greek letter φ (phi), which is equal the sum of its own reciprocal and 1, or, equivalently, is such that the ratio of 1 to the number is equal to the ratio of its reciprocal … Wiktionary
Golden ratio base — is a non standard positional numeral system that uses the golden ratio (an irrational number ≈1.61803... symbolized by the Greek letter φ) as its base. It is sometimes referred to as base φ, golden mean base, phi base, or, colloquially, phinary.… … Wikipedia
golden ratio/rectangle/section — Numerical proportion considered to be an aesthetic ideal in classical design. It refers to the ratio of the base to the height of a rectangle or to the division of a line segment into two in such a way that the ratio of the shorter part to the… … Universalium
List of works designed with golden ratio — Works designed with the golden ratio are works of human design that are proportioned according to the golden ratio, an irrational number that is approximately 1.618; it is often denoted by the Greek letter φ (phi).Early historyIt is claimed that… … Wikipedia
Golden mean — may refer to: * Doctrine of the Golden Mean ((Chinese: 中庸; pinyin: Zhōng Yóng), a chapter in Li Ji , one of the Four Books of Confucianism *Golden mean (philosophy), the felicitous middle between the extremes of excess and deficiency *Golden… … Wikipedia
Golden — means made of, or relating to gold. Golden can refer to:Color*Gold (color), an orangish yellow colorMathematics*The Golden RatioPhilosophy*Golden mean (philosophy), the felicitous middle between the extremes of excess and… … Wikipedia
Golden Triangle — may refer to: * Geographical areas: ** Golden Triangle (Southeast Asia), named for its opium production ** Golden Triangle (China), named for its rapid economic development ** Golden Triangle (New South Wales), named for its agricultural… … Wikipedia
Golden number — may mean: *The golden numbers (sometimes capitalized) that denote a year s position in a 19 year Metonic cycle, used in the computus and the Runic calendar. * Occasionally, to refer to the golden ratio … Wikipedia