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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 ]
For example, a golden spiral can be approximated by first starting with a rectangle for which the ratio between its length and width is the golden ratio. This rectangle can then be partitioned into a square and a similar rectangle and this rectangle can then be split in the same way. After continuing this process for an arbitrary number of ...
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 .
Derek Haylock [60] claims that the opening motif of Ludwig van Beethoven's Symphony No. 5 in C minor, Op. 67 (c. 1804–08), occurs exactly at the golden mean point 0.618 in bar 372 of 601 and again at bar 228 which is the other golden section point (0.618034 from the end of the piece) but he has to use 601 bars to get these figures. This he ...
According to the Spirit, the golden rectangle has influenced both ancient and modern cultures in many ways. Donald then learns how the golden rectangle appears in many ancient buildings, such as the Parthenon and the Notre Dame cathedral. Paintings such as the Mona Lisa and various sculptures such as the Venus de Milo contain several golden ...
A Penrose tiling with rhombi exhibiting fivefold symmetry. A Penrose tiling is an example of an aperiodic tiling.Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches.
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 ...