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| [[Image:FakeRealLogSpiral.svg|right|thumb|Approximate and true golden spirals: the <span style="color:#008000;">green</span> spiral is made from quarter-circles tangent to the interior of each square, while the <span style="color:#800000;">red</span> spiral is a golden spiral, a special type of [[logarithmic spiral]]. Overlapping portions appear <span style="color:#D9B900;">yellow</span>. The length of the side of a larger square to the next smaller square is in the [[golden ratio]].]]
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| In [[geometry]], a '''golden spiral''' is a [[logarithmic spiral]] whose growth factor is [[Phi|{{math|φ}}]], the [[golden ratio]].<ref>Chang, Yu-sung, "[http://demonstrations.wolfram.com/GoldenSpiral/ Golden Spiral]", [[The Wolfram Demonstrations Project]].</ref> That is, a golden spiral gets wider (or further from its origin) by a factor of {{math|φ}} for every quarter turn it makes.
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| ==Formula==
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| The [[polar equation]] for a golden spiral is the same as for other [[logarithmic spiral]]s, but with a special value of the growth factor {{math|b}}:<ref>{{cite book | title = Divine Proportion: {{math|Φ}} Phi in Art, Nature, and Science | author = Priya Hemenway | isbn = 1-4027-3522-7 | publisher = Sterling Publishing Co | year = 2005 | pages = 127–129}}</ref>
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| :<math>r = ae^{b\theta}\,</math>
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| or
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| :<math>\theta = \frac{1}{b} \ln(r/a),</math>
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| with [[e (mathematical constant)|{{math|e}}]] being the base of natural [[logarithm]]s, {{math|a}} being an arbitrary positive real constant, and {{math|b}} such that when {{math|θ}} is a [[right angle]] (a quarter turn in either direction):
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| :<math>e^{b\theta_\mathrm{right}}\, = \varphi</math>
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| Therefore, {{math|b}} is given by
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| :<math>b = {\ln{\varphi} \over \theta_\mathrm{right}}.</math>
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| The numerical value of {{math|b}} depends on whether the right angle is measured as 90 degrees or as <math>\textstyle\frac{\pi}{2}</math> radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of <math>b</math> (that is, {{math|b}} can also be the negative of this value):
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| [[Image:Fibonacci spiral 34.svg|right|thumb|A [[Fibonacci number|Fibonacci spiral]] approximates the golden spiral using quarter-circle arcs inscribed in squares of integer Fibonacci-number side, shown for square sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.]]
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| :<math>|b| = {\ln{\varphi} \over 90} = 0.0053468\,</math> for {{math|θ}} in degrees;
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| :<math>|b| = {\ln{\varphi} \over \pi/2} = 0.306349\,</math> for {{math|θ}} in radians.
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| An alternate formula for a logarithmic and golden spiral is:<ref>{{cite book | title = Symmetries of Nature: A Handbook for Philosophy of Nature and Science | author = Klaus Mainzer | pages = 45, 199–200 | year = 1996 | url = http://books.google.com/books?id=rqzaQo6CaA0C&pg=PA200&dq=%22golden+spiral%22+log| isbn = 3-11-012990-6 | publisher = Walter de Gruyter }}</ref>
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| :<math>r = ac^{\theta}\,</math>
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| where the constant {{math|c}} is given by:
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| :<math>c = e^b\,</math>
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| which for the golden spiral gives {{math|c}} values of:
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| :<math>c = \varphi ^ \frac{1}{90} \doteq 1.0053611</math>
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| if {{math|θ}} is measured in degrees, and
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| :<math>c = \varphi ^ \frac{2}{\pi} \doteq 1.358456.</math>
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| if {{math|θ}} is measured in radians.
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| ==Approximations of the golden spiral==
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| [[Image:History of Gold. Rersum 2007.jpg|thumb|right|Lithuanian coin.]]
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| There are several similar spirals that approximate, but do not exactly equal, a golden spiral.<ref>{{cite book | title = Fractals in Music: introductory mathematics for musical analysis | author = Charles B. Madden | isbn = 0-9671727-6-4 | publisher = High Art Press | year = 1999 | pages = 14–16 | url = http://books.google.com/books?id=JhnERQLm4lUC&dq=rectangles+approximate+golden-spiral}}</ref> These are often confused with the golden spiral.
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| 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 the rectangle can then be split in the same way. After continuing this process for an arbitrary amount of steps, the result will be an almost complete partitioning of the rectangle into squares. The corners of these squares can be connected by quarter-circles. The result, though not a true logarithmic spiral, approximates a golden spiral (See image on top right).
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| Another approximation is a [[Fibonacci number|Fibonacci spiral]], which is constructed similarly to the above method except that you start with a rectangle partitioned into 2 squares and then in each step add to the rectangle's longest side a square of the same length. 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. (See image on middle right).
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| ==Spirals in nature==
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| Approximate [[logarithmic spiral]]s can occur in nature (for example, the arms of [[spiral galaxies]]<ref>
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| {{cite book
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| | title = Gnomon: From Pharaohs to Fractals
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| | author = Midhat Gazale
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| | publisher = Princeton University Press
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| | year = 1999
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| | isbn = 9780691005140
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| | page = 3
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| | url = http://books.google.com/books?id=R0d76m-Be10C&pg=PR17
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| }}</ref> or [[phyllotaxis]] of leaves); golden spirals are one special case of these.
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| It is sometimes stated that spiral galaxies and [[nautilus]] shells get wider in the pattern of a golden spiral, and hence are related to both {{math|φ}} and the Fibonacci series.<ref>
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| For example, these books:
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| {{cite book
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| | title = Chemistry from First Principles
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| | author = Jan C. A. Boeyens
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| | publisher = Springer
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| | year = 2009
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| | isbn = 9781402085451
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| | page = 261
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| | url = http://books.google.com/books?id=aSRqUgllec8C&pg=PA261
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| }},
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| {{cite book
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| | title = Borderlines of Identity: A Psychologist's Personal Exploration
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| | author = P D Frey
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| | publisher = Xlibris Corporation
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| | year = 2011
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| | isbn = 9781465355850
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| | url = http://books.google.com/books?id=0MiKZdub8CQC&pg=PT135
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| }},
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| {{cite book
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| | title = Mathematics Through the Eyes of Faith
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| | author = Russell Howell and James Bradley
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| | publisher = HarperCollins
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| | year = 2011
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| | isbn = 9780062024473
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| | page = 162
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| | url = http://books.google.com/books?id=TosVluTfLOEC&pg=PA162
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| }},
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| {{cite book
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| | title = Zéro: The Biography of a Dangerous Idea
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| | author = Charles Seife
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| | publisher = Penguin
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| | year = 2000
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| | isbn = 9780140296471
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| | page = 40
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| | url = http://books.google.com/books?id=0xNvJqQEEvMC&pg=PT40#v=onepage&q&f=false
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| }},
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| {{cite book
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| | title = Sea Magic: Connecting With the Ocean's Energy
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| | author = Sandra Kynes
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| | publisher = Llewellyn Worldwide
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| | year = 2008
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| | isbn = 9780738713533
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| | page = 100
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| | url = http://books.google.com/books?id=lvaAG8HCzVEC&pg=PA100
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| }},
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| {{cite book
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| | title = Esoteric Anatomy: The Body as Consciousness
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| | author = Bruce Burger
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| | publisher = North Atlantic Books
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| | year = 1998
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| | isbn = 9781556432248
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| | page = 144
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| | url = http://books.google.com/books?id=Pjx21e0a4BEC&pg=PA144
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| }}</ref>
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| In truth, spiral galaxies and nautilus shells (and many [[mollusk]] shells) exhibit logarithmic spiral growth, but at a variety of angles usually distinctly different from that of the golden spiral.<ref>
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| {{cite book
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| | title = The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes
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| | author = David Darling
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| | publisher = John Wiley & Sons
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| | year = 2004
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| | isbn = 9780471270478
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| | page = 188
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| | url = http://books.google.com/books?id=nnpChqstvg0C&pg=PA188
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| }}</ref><ref>{{cite web|last=Devlin|first=Keith|title=The myth that will not go away|url=http://www.maa.org/external_archive/devlin/devlin_05_07.html|date=May 2007}}</ref><ref>{{cite web|last=Peterson|first=Ivars|title=Sea Shell Spirals|url=http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals|work=Science News|publisher=Society for Science & the Public|date=2005-04-01}}</ref> This pattern allows the organism to grow without changing shape.
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| ==See also==
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| * [[Golden angle]]
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| * [[Golden ratio]]
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| * [[Golden rectangle]]
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| * [[Logarithmic spiral]]
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| ==References==
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| {{reflist|2}}
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| [[Category:Spirals]]
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| [[Category:Golden ratio]]
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The individual who wrote the post is called Jayson Hirano and he totally digs that name. Invoicing is what I do for a residing but I've usually needed my own business. Alaska is the only place I've been residing in but now I'm contemplating other choices. As a lady what she really likes is style and she's been performing it for fairly a while.
Here is my blog :: free online tarot card readings