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The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data. [8] The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.
Other scholars question whether the golden ratio was known to or used by Greek artists and architects as a principle of aesthetic proportion. [11] Building the Acropolis is calculated to have been started around 600 BC, but the works said to exhibit the golden ratio proportions were created from 468 BC to 430 BC.
A golden triangle. The ratio a/b is the golden ratio φ. The vertex angle is =.Base angles are 72° each. Golden gnomon, having side lengths 1, 1, and .. A golden triangle, also called a sublime triangle, [1] is an isosceles triangle in which the duplicated side is in the golden ratio to the base side:
The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio. In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs such that the ratio of the length of the smaller arc to the length of the larger arc is the same as ...
Golden spirals are self-similar. The shape is infinitely repeated when magnified. In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. [1] That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.
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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 ...
The Greek letter phi, symbol for the golden ratio. Barr was a friend of William Schooling, and worked with him in exploiting the properties of the golden ratio to develop arithmetic algorithms suitable for mechanical calculators. [15] According to Theodore Andrea Cook, Barr gave the golden ratio the name of phi (ϕ).