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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.
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:
For if the algorithm requires N steps, then b is greater than or equal to F N+1 which in turn is greater than or equal to φ N−1, where φ is the golden ratio. Since b ≥ φ N−1, then N − 1 ≤ log φ b. Since log 10 φ > 1/5, (N − 1)/5 < log 10 φ log φ b = log 10 b. Thus, N ≤ 5 log 10 b.
A golden spiral with initial radius 1 is the locus of points of polar coordinates (,) satisfying = /, where is the golden ratio. The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor b: [10] = or = (/), with e being the base of natural logarithms, a being the ...
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 ...
Ancient Greeks had a method for evaluating sexual attractiveness based on the Golden ratio, part of which included measurements of the head. [4] Headhunting is the practice of taking and preserving a person's head after killing the person. Headhunting has been practiced across the Americas, Europe, Asia, and Oceania for millennia. [5]
The Acropolis of Athens (468–430 BC), including the Parthenon, according to some studies, has many proportions that approximate the golden ratio. [10] Other scholars question whether the golden ratio was known to or used by Greek artists and architects as a principle of aesthetic proportion. [ 11 ]
geometrical representation of a division Transferring the ratio of a partition of line segment AB to line segment HG: | | | | = | | | |. The theorem of the gnomon can be used to construct a new parallelogram or rectangle of equal area to a given parallelogram or rectangle by the means of straightedge and compass constructions.