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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 golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (pitch angle about 17.03239 degrees). It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to Fibonacci numbers .
Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance , Luca Pacioli wrote the influential treatise De divina proportione (1509), illustrated with woodcuts by Leonardo da Vinci , on the use of the golden ratio in art.
The Julia set is the common boundary of the filled-in Julia set and the attractive basin of infinity = ... c=1−φ=−0.618033988749…, where φ is the Golden ratio.
Therefore, the ratio must be the unique positive solution to this equation, the golden ratio, and the triangle must be a Kepler triangle. [ 1 ] The three edge lengths 1 {\displaystyle 1} , φ {\displaystyle {\sqrt {\varphi }}} and φ {\displaystyle \varphi } are the harmonic mean , geometric mean , and arithmetic mean , respectively, of the two ...
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 ratio has the slowest convergence of any irrational number. [5] It is, for that reason, one of the worst cases of Lagrange's approximation theorem and it is an extremal case of the Hurwitz inequality for diophantine approximations. This may be why angles close to the golden ratio often show up in phyllotaxis (the growth of plants). [6]
The root-3 rectangle is also called sixton, [6] and its short and longer sides are proportionally equivalent to the side and diameter of a hexagon. [7] Since 2 is the square root of 4, the root-4 rectangle has a proportion 1:2, which means that it is equivalent to two squares side-by-side. [7] The root-5 rectangle is related to the golden ratio ...