Information about 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 is not shown, but could be constructed from a similar "whirling rectangle diagram", in which the ratios of the rectangles were based on the terms in the Fibonacci series, rather than phi.)
In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to φ, the golden ratio. Specifically, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter-turn it makes.
Formula
The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of b:[1]
or
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):
Therefore, b is given by
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):
for θ in degrees;
for θ in radians.
An alternate formula for a logarithmic and golden spiral is:[2]
where the constant c is given by:
which for the golden spiral gives c values of:
and
Approximations of the golden spiral
There are several similar spirals that approximate, but do not exactly equal, a golden spiral.[3] 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).
Golden spiral in nature
Although it is often suggested that the golden spiral occurs repeatedly in nature (e.g. the arms of spiral galaxies or sunflower heads) , this claim is rarely valid except perhaps in the most contrived of circumstances. For example, it is commonly believed 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.[4] The reason for this growth pattern is that it allows the animal to grow at a constant rate without having to change shape. Spirals are common features in nature, but there is no evidence that a single number dictates the shape of every one of these spirals. The greatest misconception in the mystification of the golden spiral is the incorrect assumption that all spirals in nature are in fact the golden spiral. While logarithmic spirals are often observed, they may be of differing pitches, and therefore there is no single "spira mirabilis".References
1. ^ Priya Hemenway (2005). Divine Proportion: Φ Phi in Art, Nature, and Science. Sterling Publishing Co, 127–129. ISBN 1402735227.
2. ^ Klaus Mainzer (1996). Symmetries of Nature: A Handbook for Philosophy of Nature and Science. Walter de Gruyter, 45, 199–200. ISBN 3110129906.
3. ^ Charles B. Madden (1999). Fractals in Music: introductory mathematics for musical analysis. High Art Press, 14–16. ISBN 0967172764.
4. ^ Oberon Zell-Ravenheart (2004). Grimoire for the Apprentice Wizard. Career Press, 274. ISBN 1564147118.
2. ^ Klaus Mainzer (1996). Symmetries of Nature: A Handbook for Philosophy of Nature and Science. Walter de Gruyter, 45, 199–200. ISBN 3110129906.
3. ^ Charles B. Madden (1999). Fractals in Music: introductory mathematics for musical analysis. High Art Press, 14–16. ISBN 0967172764.
4. ^ Oberon Zell-Ravenheart (2004). Grimoire for the Apprentice Wizard. Career Press, 274. ISBN 1564147118.
See also
Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
..... Click the link for more information.
..... Click the link for more information.
A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it
..... Click the link for more information.
..... Click the link for more information.
golden section is a line segment sectioned into two according to the golden ratio. The total length a+b is to the longer segment a as a
..... Click the link for more information.
..... Click the link for more information.
polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. The polar coordinate system is especially useful in situations where the relationship between two points is most easily expressed in terms of
..... Click the link for more information.
..... Click the link for more information.
A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it
..... Click the link for more information.
..... Click the link for more information.
e is the unique real number such that the value of the derivative (slope of the tangent line) of f(x) = ex at the point x = 0 is exactly 1.
..... Click the link for more information.
..... Click the link for more information.
logarithm (to base b) of a number x is the exponent y that satisfies x = by. It is written logb(x) or, if the base is implicit, as log(x).
..... Click the link for more information.
..... Click the link for more information.
right angle is an angle of 90 degrees, corresponding to a quarter turn (that is, a quarter of a full circle). It can be defined as the angle such that twice that angle amounts to a half turn, or 180° [1].
..... Click the link for more information.
..... Click the link for more information.
Fibonacci numbers form a sequence defined by the following recurrence relation:
..... Click the link for more information.
..... Click the link for more information.
Fibonacci numbers form a sequence defined by the following recurrence relation:
..... Click the link for more information.
..... Click the link for more information.
spiral galaxy is a galaxy belonging to one of the three main classes of galaxy originally described by Edwin Hubble in his 1936 work “The Realm of the Nebulae”[1] and, as such, forms part of the Hubble sequence.
..... Click the link for more information.
..... Click the link for more information.
Nautilina
Agassiz, 1847
Family: Nautilidae
Blainville, 1825
Genera
Allonautilus
Nautilus
Nautilus (from Greek ναυτίλος
..... Click the link for more information.
Agassiz, 1847
Family: Nautilidae
Blainville, 1825
Genera
Allonautilus
Nautilus
Nautilus (from Greek ναυτίλος
..... Click the link for more information.
A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it
..... Click the link for more information.
..... Click the link for more information.
A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it
..... Click the link for more information.
..... Click the link for more information.
golden section is a line segment sectioned into two according to the golden ratio. The total length a+b is to the longer segment a as a
..... Click the link for more information.
..... Click the link for more information.
A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: (one-to-phi), that is, approximately 1:1.618.
A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle, that is,
..... Click the link for more information.
A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle, that is,
..... Click the link for more information.
golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden section; that is, into two arcs such that the ratio of the length of the larger arc to the smaller is the same as the ratio of the full circumference to the
..... Click the link for more information.
..... Click the link for more information.
A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it
..... Click the link for more information.
..... Click the link for more information.
This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.
Herod_Archelaus
