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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.
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.
The golden ratio is approximately 1.6180339887. At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.
The golden ratio budget echoes the more widely known 50-30-20 budget that recommends spending 50% of your income on needs, 30% on wants and 20% on savings and debt. The “needs” category covers ...
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:
Adolf Zeising (24 September 1810 – 27 April 1876) was a German psychologist, whose main interests were mathematics and philosophy.. Among his theories, Zeising claimed to have found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves.
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The ratio of the slant height to half the base length of the Great Pyramid of Giza is less than 1% from the golden ratio. [51] If this was the design method, it would imply the use of Kepler's triangle (face angle 51°49'), [51] [52] but according to many historians of science, the golden ratio was not known until the time of the Pythagoreans. [53]