Search results
Results from the WOW.Com Content Network
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 ]
Its true value is the real solution of the equation x 3 = 2x 2 + 1. The name supersilver ratio results from analogy with the silver ratio, the positive solution of the equation x 2 = 2x + 1, and the supergolden ratio.
The golden ratio follows from this application of Ptolemy's theorem. A more interesting example is the relation between the length a of the side and the (common) length b of the 5 chords in a regular pentagon. By completing the square, the relation yields the golden ratio: [4]
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 ...
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 ...
Consider a rectangle such that the ratio of its length L to its width W is the n th metallic ratio. If one remove from this rectangle n squares of side length W, one gets a rectangle similar to the original rectangle; that is, a rectangle with the same ratio of the length to the width (see figures).
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]