NationMaster - Encyclopedia: Golden spiral

Approximate and true golden spirals: the green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a golden spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio.

A Fibonacci spiral approximates the golden spiral; unlike the "whirling rectangle diagram" based on the golden ratio, above, this one uses squares of integer Fibonacci-number sizes, shown for square sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.

In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to φ, the golden ratio.^[1] Specifically, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes. Fake spiral made of circle quarters, logarithmic spiral. ... Fake spiral made of circle quarters, logarithmic spiral. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... A tiling with squares whose sides are successive Fibonacci numbers in length In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci. ... For other uses, see Geometry (disambiguation). ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... Not to be confused with Golden mean (philosophy), the felicitous middle between two extremes, Golden numbers, an indicator of years in astronomy and calendar studies, or the Golden Rule. ...

1 Formula
2 Approximations of the golden spiral
3 Spirals in nature
4 References
5 See also

Formula

The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of b:^[2] The polar coordinate system is a two-dimensional coordinate system in which points are given by an angle and a distance from the pole, called the origin in the Cartesian coordinate system. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ...

$r = ae^{btheta},$

$theta = frac{1}{b} ln(r/a),$

with e being the base of natural logarithms, a being an arbitrary positive real constant, and b such that when θ is a right angle (a quarter turn in either direction): e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... -1... This article is about angles in geometry. ...

$e^{btheta_mathrm{right}}, = varphi$

Therefore, b is given by

$b = {ln{varphi} over theta_mathrm{right}}$

The numerical value of b depends on whether the right angle is measured as 90 degrees or as π/2 radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of b (that is, b can also be the negative of this value):

$|b| = {ln{varphi} over 90} = 0.0053468,$ for θ in degrees;

$|b| = {ln{varphi} over pi/2} = 0.306349,$ for θ in radians.

An alternate formula for a logarithmic and golden spiral is:^[3]

$r = ac^{theta},$

where the constant c is given by:

$c = e^b,$

which for the golden spiral gives c values of:

$c = varphi ^ frac{1}{90} = 1.0053611$

and

$c = varphi ^ frac{2}{pi} = 1.358456$

Approximations of the golden spiral

There are several similar spirals that approximate, but do not exactly equal, a golden spiral.^[4] These are often confused with the golden spiral.

For example, a golden spiral can be approximated by a "whirling rectangle diagram," in which the opposite corners of squares formed by spiraling golden rectangles are connected by quarter-circles. The result is very similar to a true golden spiral (See image on top right).

Another approximation is a Fibonacci spiral, which is not a true logarithmic spiral. Every quarter turn a Fibonacci spiral gets wider not by φ, but by a changing factor related to the ratios of consecutive terms in the Fibonacci sequence. The ratios of consecutive terms in the Fibonacci series approach φ, so that the two spirals are very similar in appearance. (See image on bottom right). A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above â€“ see golden spiral. ... In mathematics, the Fibonacci numbers form a sequence defined recursively by: In words: you start with 0 and 1, and then produce the next Fibonacci number by adding the two previous Fibonacci numbers. ...

Spirals in nature

Approximate logarithmic spirals can occur in nature (for example, the arms of spiral galaxies or sunflower heads). It is sometimes stated that nautilus shells get wider in the pattern of a golden spiral, and hence are related to both φ and the Fibonacci series. In truth, nautilus shells exhibit logarithmic spiral growth, but at a rate distinctly different from that of the golden spiral.^[5] The reason for this growth pattern is that it allows the organism to grow at a constant rate without having to change shape. Spirals are common features in nature; golden spirals are but one special case of these. A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... A spiral galaxy is a type of galaxy in the Hubble sequence which is characterized by the following physical properties: Spiral Galaxy M74 presents a face-on view of its spiral arms. ... Genera Allonautilus Nautilus Nautilus (from Greek Î½Î±Ï…Ï„Î¯Î»Î¿Ï‚, sailor) is the common name of any marine creatures of the cephalopod family Nautilidae, the sole family of the suborder Nautilina. ...

References

^ "Golden Spiral" by Yu-Sung Chang, The Wolfram Demonstrations Project.
^ Priya Hemenway (2005). Divine Proportion: Φ Phi in Art, Nature, and Science. Sterling Publishing Co, 127–129. ISBN 1402735227.
^ Klaus Mainzer (1996). Symmetries of Nature: A Handbook for Philosophy of Nature and Science. Walter de Gruyter, 45, 199–200. ISBN 3110129906.
^ Charles B. Madden (1999). Fractals in Music: introductory mathematics for musical analysis. High Art Press, 14–16. ISBN 0967172764.
^ Oberon Zell-Ravenheart (2004). Grimoire for the Apprentice Wizard. Career Press, 274. ISBN 1564147118.

Not to be confused with Golden mean (philosophy), the felicitous middle between two extremes, Golden numbers, an indicator of years in astronomy and calendar studies, or the Golden Rule. ... The large rectangle BA is a golden rectangle; that is, the proportion b:a is 1:. If we remove square B, what is left, A, is another golden rectangle. ... The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden section... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ...

Results from FactBites:

Golden spiral (617 words)
	From the bone structure of human beings to the seed pattern of a sunflower to the spiral of a sea shell, the phi proportion is there, underlying all biological structures, seeming to be a geometrical blueprint for life itself.
	The other method of determining the Golden Section is by dividing a line segment, AB, at a point C, in such a way that the whole line is longer than the first part in the same proportion as the first part being longer than the remainder.
	This spiral is almost identical to the graph of the logarithmic spiral of the phi ratio, known as the Golden Mean Spiral (The difference is that the Fibonacci spiral is a whole number interpretation of the arithmetically impossible Golden Mean spiral, which has no beginning or end.

Golden spiral - Wikipedia, the free encyclopedia (425 words)
	In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to φ, the golden ratio.
	A Fibonacci spiral that approximates the Golden Spiral.
	The polar equation for a golden spiral is

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents

Approximations of the golden spiral

Spirals in nature

References

See also