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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 with long side a + b and short side a can be divided into two pieces: a similar golden rectangle (shaded red, right) with long side a and short side b and a square (shaded blue, left) with sides of length a. This illustrates the relationship a + b / a = a / b = φ.
Both volumes have formulas involving the golden ratio but are taken to different powers. [1] Golden rectangle may also related to both regular icosahedron and regular dodecahedron. The regular icosahedron can be constructed by intersecting three golden rectangles perpendicularly, arranged in two-by-two orthogonal, and connecting each of the ...
A root-phi rectangle divides into a pair of Kepler triangles (right triangles with edge lengths in geometric progression). The root-φ rectangle is a dynamic rectangle but not a root rectangle. Its diagonal equals φ times the length of the shorter side. If a root-φ rectangle is divided by a diagonal, the result is two congruent Kepler triangles.
The Sacrament of the Last Supper (1955): The canvas of this surrealist masterpiece by Salvador Dalí is a golden rectangle. A huge dodecahedron, with edges in golden ratio to one another, is suspended above and behind Jesus and dominates the composition. [11] [40]
Geometrically, the Borromean rings may be realized by linked ellipses, or (using the vertices of a regular icosahedron) by linked golden rectangles. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space.
Golden ratio: In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is a mathematical constant, usually denoted by the Greek letter φ .
The golden ratio φ, here shown as the long side of a golden rectangle, is a solution to an algebraic equation and thus not a transcendental number. In 2004, the Bogdanovs published a commercially successful popular science book, Avant Le Big Bang ( Before the Big Bang ), based on a simplified version of their theses, where they also presented ...