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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.
In geometry, a golden rectangle is a rectangle with side lengths in golden ratio +:, or :, with approximately equal to 1.618 or 89/55. Golden rectangles exhibit a special form of self-similarity : if a square is added to the long side, or removed from the short side, the result is a golden rectangle as well.
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 ...
The Farnsworth House, designed by Ludwig Mies van der Rohe, has been described as "the proportions, within the glass walls, approach 1:2" [43] and "with a width to length ratio of 1:1.75 (nearly the golden section)" [44] and has been studied with his other works in relation to the golden ratio.
A root rectangle is a rectangle in which the ratio of the longer side to the shorter is the square root of an integer, such as √ 2, √ 3, etc. [2] The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a square to the length of the square's diagonal. The root-3 rectangle is constructed by extending the two ...
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:
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In mathematics, the supergolden ratio is a geometrical proportion close to 85/58. Its true value is the real solution of the equation x 3 = x 2 + 1. The name supergolden ratio results from analogy with the golden ratio, the positive solution of the equation x 2 = x + 1. A triangle with side lengths ψ, 1, and 1 ∕ ψ has an angle of exactly ...
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