Golden spiral 1

Golden spiral

Approximate and true golden spirals: the greenspiral is made from quarter-circles tangent to theinterior of each square, while the red spiral is a

golden spiral, a special type of logarithmic spiral.Overlapping portions appear yellow. The lengthof the side of a larger square to the next smaller

square is in the golden ratio.

In geometry, a golden spiral is a logarithmic spiral whose growthfactor is φ, the golden ratio.[1] That is, a golden spiral gets wider (orfurther 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 otherlogarithmic spirals, but with a special value of the growth factor b:[2]

or

with e being the base of natural logarithms, a being an arbitrarypositive real constant, and b such that when θ is a right angle (a quarterturn 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 radians; and since theangle can be in either direction, it is easiest to write the formula for the absolute value of (that is, b can also be thenegative of this value):

A Fibonacci spiral approximates the goldenspiral; unlike the "whirling rectangle diagram"based on the golden ratio, above, this one uses

quarter-circle arcs inscribed in squares of integerFibonacci-number side, shown for square sizes 1,

1, 2, 3, 5, 8, 13, 21, and 34.

for θ in degrees;

for θ in radians.

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

where the constant c is given by:

which for the golden spiral gives c values of:

if θ is measured in degrees, and

if θ is measured in radians.

Golden spiral 2

Approximations of the golden spiral

Lithuanian coin.

There are several similar spirals that approximate, but do not exactly equal, a goldenspiral.[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 areconnected by quarter-circles. The result is very similar to a true golden spiral (Seeimage on top right).Another approximation is a Fibonacci spiral, which is not a true logarithmic spiral. Itis made up of a series of quarter-circular arcs whose radii are consecutivelyincreasing Fibonacci numbers. Every quarter turn a Fibonacci spiral gets wider notby φ, but by a changing factor that equals the ratio of a term in the Fibonacci sequence to its predecessor. The ratiosof consecutive terms in the Fibonacci series approach φ, so that the two spirals are very similar in appearance. (Seeimage on top right).

Spirals in natureApproximate logarithmic spirals can occur in nature (for example, the arms of spiral galaxies[5]). It is sometimesstated that spiral galaxies and nautilus shells get wider in the pattern of a golden spiral, and hence are related to bothφ and the Fibonacci series.[6] In truth, spiral galaxies and nautilus shells (and many mollusk shells) exhibitlogarithmic spiral growth, but at a variety of angles usually distinctly different from that of the golden spiral.[7][8][9]

This pattern allows the organism to grow without changing shape. Approximate logarithmic spirals are commonfeatures in nature; golden spirals are one special case of these.

References[1] Chang, Yu-sung, " Golden Spiral (http:/ / demonstrations. wolfram. com/ GoldenSpiral/ )", The Wolfram Demonstrations Project.[2] Priya Hemenway (2005). Divine Proportion: Φ Phi in Art, Nature, and Science. Sterling Publishing Co. pp. 127–129. ISBN 1-4027-3522-7.[3] Klaus Mainzer (1996). Symmetries of Nature: A Handbook for Philosophy of Nature and Science (http:/ / books. google. com/

books?id=rqzaQo6CaA0C& pg=PA200& dq="golden+ spiral"+ log). Walter de Gruyter. pp. 45, 199–200. ISBN 3-11-012990-6. .[4] Charles B. Madden (1999). Fractals in Music: introductory mathematics for musical analysis (http:/ / books. google. com/

books?id=JhnERQLm4lUC& dq=rectangles+ approximate+ golden-spiral). High Art Press. pp. 14–16. ISBN 0-9671727-6-4. .[5] Midhat Gazale (1999). Gnomon: From Pharaohs to Fractals (http:/ / books. google. com/ books?id=R0d76m-Be10C& pg=PR17). Princeton

University Press. p. 3. ISBN 9780691005140. .[6] For example, these books: Jan C. A. Boeyens (2009). Chemistry from First Principles (http:/ / books. google. com/

books?id=aSRqUgllec8C& pg=PA261). Springer. p. 261. ISBN 9781402085451. ., P D Frey (2011). Borderlines of Identity: A Psychologist'sPersonal Exploration (http:/ / books. google. com/ books?id=0MiKZdub8CQC& pg=PT135). Xlibris Corporation. ISBN 9781465355850. .,Russell Howell and James Bradley (2011). Mathematics Through the Eyes of Faith (http:/ / books. google. com/ books?id=TosVluTfLOEC&pg=PA162). HarperCollins. p. 162. ISBN 9780062024473. ., Charles Seife (2000). Zéro: The Biography of a Dangerous Idea (http:/ / books.google. com/ books?id=0xNvJqQEEvMC& pg=PT40#v=onepage& q& f=false). Penguin. p. 40. ISBN 9780140296471. ., Sandra Kynes(2008). Sea Magic: Connecting With the Ocean's Energy (http:/ / books. google. com/ books?id=lvaAG8HCzVEC& pg=PA100). LlewellynWorldwide. p. 100. ISBN 9780738713533. ., Bruce Burger (1998). Esoteric Anatomy: The Body as Consciousness (http:/ / books. google.com/ books?id=Pjx21e0a4BEC& pg=PA144). North Atlantic Books. p. 144. ISBN 9781556432248. .

[7] David Darling (2004). The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes (http:/ / books. google. com/books?id=nnpChqstvg0C& pg=PA188). John Wiley & Sons. p. 188. ISBN 9780471270478. .

[8] Devlin, Keith (2007-05). "The myth that will not go away" (http:/ / www. maa. org/ devlin/ devlin_05_07. html). .[9] Peterson, Ivars (2005-04-01). "Sea Shell Spirals" (http:/ / www. sciencenews. org/ view/ generic/ id/ 6030/ title/ Sea_Shell_Spirals). Science

News. Society for Science & the Public. .

Article Sources and Contributors 3

Article Sources and ContributorsGolden spiral  Source: http://en.wikipedia.org/w/index.php?oldid=521411033  Contributors: Bento00, Charles Matthews, Count3r, Dan Hetherington, DarcNessX, Dicklyon, Doniago, Egg, EricBurnett, Flewis, Frietjes, Goldbren, Hairy Dude, Hephaestos, HopeMaker, Hyacinth, Icairns, Icey, Iulius, Josh Grosse, Jyar, Kauffner, Lechatjaune, Lemonflash, Lucyintheskywithdada,Mckaysalisbury, Melchoir, Nae'blis, OlEnglish, Optimist on the run, Pleasantville, Preston47, Robma, Ronhjones, Svick, THF, Tamfang, The Thing That Should Not Be, The stuart, Tide rolls,Tumble, Ukexpat, Versus22, Wiki alf, Wknight94, WoollyMind, Zaslav, 99 anonymous edits

Image Sources, Licenses and ContributorsImage:FakeRealLogSprial.svg  Source: http://en.wikipedia.org/w/index.php?title=File:FakeRealLogSprial.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors:FakeRealLogSpiral.png: Pau derivative work: silverhammermbaImage:Fibonacci spiral 34.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Fibonacci_spiral_34.svg  License: Public Domain  Contributors: User:DicklyonImage:History of Gold. Rersum 2007.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:History_of_Gold._Rersum_2007.jpg  License: Public Domain  Contributors: uploaded by arz

LicenseCreative Commons Attribution-Share Alike 3.0 Unported//creativecommons.org/licenses/by-sa/3.0/