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 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.
In each step, a square the length of the rectangle's longest side is added to the rectangle. Since the ratio between consecutive Fibonacci numbers approaches the golden ratio as the Fibonacci numbers approach infinity, so too does this spiral get more similar to the previous approximation the more squares are added, as illustrated by the image.
Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.
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.
According to author Leon Harkleroad, "Some of the most misguided attempts to link music and mathematics have involved Fibonacci numbers and the related golden ratio." [61] With few exceptions, numerators for the meter signatures (over 100) in Karlheinz Stockhausen's Klavierstück IX are either Fibonacci or Lucas numbers. [62]
The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive Fibonacci numbers, which leads to the exact unit area in the thin parallelogram. Many other geometric dissection puzzles are based on a few simple properties of the Fibonacci sequence.
where F n is the nth Fibonacci number. The ratio of numbers of kites to darts in any sufficiently large P2 Penrose tiling pattern therefore approximates to the golden ratio φ. [47] A similar result holds for the ratio of the number of thick rhombs to thin rhombs in the P3 Penrose tiling. [45]