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Many works of art are claimed to have been designed using the golden ratio. However, many of these claims are disputed, or refuted by measurement. [1] The golden ratio, an irrational number, is approximately 1.618; it is often denoted by the Greek letter φ .
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The golden ratio φ and its negative reciprocal −φ −1 are the two roots of the quadratic polynomial x 2 − x − 1. The golden ratio's negative −φ and reciprocal φ −1 are the two roots of the quadratic polynomial x 2 + x − 1. The golden ratio is also an algebraic number and even an algebraic integer.
The reader would be inclined to believe that the phrases daśa-tāla, paṅcha-tāla and ēkatāl mean lengths equal to ten, five and one tāla respectively, but unfortunately this interpretation does not seem to agree with the actual measurements; for example, the total length of an image made according to the Uttama-daśa-ālc measurement is ...
Some examples of organisms in published studies of the golden ratios: The nautilus, the human body proportion, the reproduction of rabbits (Fibonacci started by observing the rabbits), the reproduction of cows (Henry E. Dudeney), the number of petals in flowers, the sunflower spirals, the pine cones, species of Radiolaria (shaped like polyhedra ...
Other scholars argue that until Pacioli's work in 1509, the golden ratio was unknown to artists and architects. [53] For example, the height and width of the front of Notre-Dame of Laon have the ratio 8/5 or 1.6, not 1.618. Such Fibonacci ratios quickly become hard to distinguish from the golden ratio. [54]
The Fibonacci numbers, often presented in conjunction with the golden ratio, are a popular theme in culture. They have been mentioned in novels, films, television shows, and songs. The numbers have also been used in the creation of music, visual art, and architecture.
In reality, the navel of the Vitruvian Man divides the figure at 0.604 and nothing in the accompanying text mentions the golden ratio. [23] In his conjectural reconstruction of the Canon of Polykleitos, art historian Richard Tobin determined √ 2 (about 1.4142) to be the important ratio between elements that the classical Greek sculptor had ...