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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.
A golden rectangle—that is, a rectangle with an aspect ratio of —may be cut into a square and a smaller rectangle with the same aspect ratio. 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 diagonal of a half square forms the basis for the geometrical construction of a golden rectangle.. The golden ratio φ is the arithmetic mean of 1 and . [4] The algebraic relationship between , the golden ratio and the conjugate of the golden ratio (Φ = − 1 / φ = 1 − φ) is expressed in the following formulae:
A supergolden rectangle is a rectangle whose side lengths are in a : ratio. Compared to the golden rectangle , the supergolden rectangle has one more degree of self-similarity . Given a rectangle of height 1 , length ψ {\displaystyle \psi } and diagonal length ψ 3 {\displaystyle {\sqrt {\psi ^{3}}}} (according to 1 + ψ 2 = ψ ...
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 ...
Other theorems and methods have allowed paperfolders to get other shapes from a square, such as equilateral triangles, pentagons, hexagons, and special rectangles such as the golden rectangle and the silver rectangle. Methods for folding most regular polygons up to and including the regular 19-gon have been developed. [36]
Macaulay2 is built around fast implementations of algorithms useful for computation in commutative algebra and algebraic geometry. This core functionality includes arithmetic on rings, modules, and matrices, as well as algorithms for Gröbner bases, free resolutions, Hilbert series, determinants and Pfaffians, factoring, and similar.
See #Numerical linear algebra for linear equations. Root-finding algorithm — algorithms for solving the equation f(x) = 0 General methods: Bisection method — simple and robust; linear convergence Lehmer–Schur algorithm — variant for complex functions; Fixed-point iteration
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