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| [[File:Spiral of Theodorus.svg|thumb|right|400px|The spiral of Theodorus up to the triangle with a hypotenuse of √17]]
| | e - Shop Word - Press is a excellent cart for your on the web shopping organization. Affilo - Theme is the guaranteed mixing of wordpress theme that Mark Ling use for his internet marketing career. Change the site's theme and you have essentially changed the site's personality. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. The top 4 reasons to use Business Word - Press Themes for a business website are:. <br><br> |
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| In [[geometry]], the '''spiral of Theodorus''' (also called ''square root spiral'', ''Einstein spiral'' or ''Pythagorean spiral'')<ref name=KAHN2>. Also, this was used to visualize certain things in nature
| | If you treasured this article and also you would like to collect more info about [http://dinky.in/?WordpressBackupPlugin814392 backup plugin] nicely visit our web-site. Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. By using Word - Press, you can develop very rich, user-friendly and full-functional website. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Are you considering getting your website redesigned. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. Websites using this content based strategy are always given top scores by Google. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. The templates are designed to be stand alone pages that have a different look and feel from the rest of your website. Word - Press is the most popular open source content management system (CMS) in the world today. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. If this is not possible you still have the choice of the default theme that is Word - Press 3. Get started today so that people searching for your type of business will be directed to you. |
| {{cite arxiv
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| |last=Hahn
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| |first=Harry K.
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| |title=The Ordered Distribution of Natural Numbers on the Square Root Spiral
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| |eprint=0712.2184
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| }}</ref> is a [[spiral]] composed of [[contiguous]] [[right triangle]]s. It was first constructed by [[Theodorus of Cyrene]].
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| ==Construction==
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| The spiral is started with an [[isosceles]] right triangle, with each [[Cathetus|leg]] having unit [[length]]. Another right triangle is formed, an [[automedian triangle|automedian right triangle]] with one leg being the [[hypotenuse]] of the prior triangle (with length [[square root of 2|√2]]) and the other leg having length of 1; the length of the hypotenuse of this second triangle is [[square root of 3|√3]]. The process then repeats; the ''i''th triangle in the sequence is a right triangle with side lengths √''i'' and 1, and with hypotenuse {{sqrt|''i'' + 1}}.
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| ==History==
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| Although all of Theodorus' work has been lost, [[Plato]] put Theodorus into his dialogue ''[[Theaetetus (dialogue)|Theaetetus]]'', which tells of his work. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are [[Irrational number|irrational]] by means of the Spiral of Theodorus.<ref>
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| {{citation
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| |last=Nahin joe mo
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| |first=Paul J.
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| |title=An Imaginary Tale: The Story of [the Square Root of Minus One<nowiki>]</nowiki>
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| |publisher=Princeton University Press
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| |page=33
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| |url=http://books.google.com/?id=WvcfqBgZDWQC&printsec=frontcover
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| |isbn=0-691-02795-1
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| |year=1998
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| }}</ref>
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| Plato does not attribute the irrationality of the [[square root of 2]] to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.<ref>
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| {{citation
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| |last=Plato
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| |last2=Dyde
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| |first2=Samuel Walters
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| |title=The Theaetetus of Plato
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| |publisher=J. Maclehose
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| |pages=86–87.
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| |url=http://books.google.com/?id=wt29k-Jz8pIC&printsec=titlepage
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| |year=1899
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| }}</ref>
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| ==Hypotenuse==
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| Each of the triangles' hypotenuses ''h<sub>i</sub>'' gives the [[square root]] of the corresponding [[natural number]], with ''h''<sub>1</sub> = √2.
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| Plato, tutored by Theodorus, questioned why Theodorus stopped at √17. The reason is commonly believed to be that the √17 hypotenuse belongs to the last triangle that does not overlap the figure.<ref name=LONG>
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| {{cite web
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| |last=Long
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| |first=Kate
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| |title=A Lesson on The Root Spiral
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| |url=http://courses.wcupa.edu/jkerriga/Lessons/A%20Lesson%20on%20Spirals.html
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| |accessdate=2008-04-30
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| }}</ref>
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| ===Overlapping===
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| In 1958, Erich Teuffel proved that no two hypotenuses will ever coincide, regardless of how far the spiral is continued. Also, if the sides of unit length are extended into a [[line (geometry)|line]], they will never pass through any of the other vertices of the total figure.<ref>Erich Teuffel, Eine Eigenschaft der Quadratwurzelschnecke, ''Math.-Phys. Semesterber.'' 6 (1958), pp. 148-152.</ref><ref name=LONG/>
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| ==Extension==
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| [[File:Colored extended spiral of roots.pdf|thumb|Colored extended spiral of Theodorus with 120 triangles]]
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| Theodorus stopped his spiral at the triangle with a hypotenuse of √17. If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.
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| ===Growth rate===
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| ====Angle====
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| If φ<sub>''n''</sub> is the angle of the ''n''th triangle (or spiral segment), then:
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| :<math>\tan\left(\varphi_n\right)=\frac{1}{\sqrt{n}}.</math>
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| Therefore, the growth of the angle φ<sub>''n''</sub> of the next triangle ''n'' is:<ref name=KAHN2/>
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| :<math>\varphi_n=\arctan\left(\frac{1}{\sqrt{n}}\right).</math>
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| The sum of the angles of the first ''k'' triangles is called the total angle φ(''k'') for the ''k''th triangle, and it equals:<ref name=KAHN2/>{{Clarify|date=June 2012|reason=It's not clear what c_2(k) is; giving its limit at infinity is not enough}} | |
| :<math>\varphi\left (k\right)=\sum_{n=1}^k\varphi_n = 2\sqrt{k}+c_2(k)</math>
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| with
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| :<math>\lim_{k \to \infty} c_2(k)= - 2.157782996659\ldots.</math>
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| [[File:Spiral of Theodorus triangle.svg|thumb|A triangle or section of spiral]]
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| ====Radius====
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| The growth of the radius of the spiral at a certain triangle ''n'' is
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| :<math>\Delta r=\sqrt{n+1}-\sqrt{n}.</math>
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| ===Archimedean spiral===
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| The Spiral of Theodorus [[approximate]]s the [[Archimedean spiral]].<ref name=KAHN2/> Just as the distance between two windings of the Archimedean spiral equals [[mathematical constant]] [[pi]], as the number of spins of the spiral of Theodorus approaches [[infinity]], the distance between two consecutive windings quickly approaches π.<ref>
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| {{cite arxiv
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| |last=Hahn |first=Harry K.
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| |year=2008
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| |title=The distribution of natural numbers divisible by 2, 3, 5, 7, 11, 13, and 17 on the Square Root Spiral
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| |eprint=0801.4422
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| }}</ref>
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| The following is a table showing the distance of two windings of the spiral approaching pi:
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| {| class="wikitable"
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| !Winding No.:
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| !width=200px|Calculated average winding-distance
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| !width=200px|Accuracy of average winding-distance in comparison to π
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| |-
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| |2
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| |3.1592037
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| |99.44255%
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| |-
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| |3
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| |3.1443455
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| |99.91245%
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| |-
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| |4
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| |3.14428
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| |99.91453%
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| |-
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| |5
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| |3.142395
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| |99.97447%
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| |-
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| |[[Limit of a function#Limits involving infinity|→]] ∞
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| |→ π
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| |→ 100%
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| |}
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| As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to π.<ref name=KAHN2/>
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| ==Continuous curve==
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| The question of how to [[interpolation|interpolate]] the discrete points of the spiral of Theodorus by a smooth curve was proposed and answered in {{Harv|Davis|1993|loc = pp. 37–38 }} by analogy with Euler's formula for the [[gamma function]] as an [[Interpolation|interpolant]] for the [[factorial]] function. [[Philip J. Davis|Davis]] found the function
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| :<math>T(x) = \prod_{k=1}^\infty \frac{1 + i/\sqrt{k}}{1 + i/\sqrt{x+k}} \qquad ( -1 < x < \infty )</math>
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| which was further studied by his student [[Jeffery J. Leader|Leader]]<ref>[[Jeffery J. Leader|Leader, J.J.]] The Generalized Theodorus Iteration (dissertation), 1990, Brown University</ref> and by [[Arieh Iserles|Iserles]] (in an appendix to {{Harv|Davis|1993}} ). An axiomatic characterization of this function is given in {{Harv|Gronau|2004}} as the unique function that satisfies the [[functional equation]]
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| :<math>f(x+1) = \left( 1 + \frac{i}{\sqrt{x+1} }\right) \cdot f(x),</math>
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| the initial condition <math>f(0) = 1,</math> and [[Monotonic function|monotonicity]] in both [[Argument (complex analysis)|argument]] and [[Absolute value|modulus]]; alternative conditions and weakenings are also studied therein. An alternative derivation is given in {{Harv | Heuvers | Moak | Boursaw | 2000}}.
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| Some have suggested a different interpolant which connects the spiral and an alternative inner spiral, as in {{Harv|Waldvogel|2009}}.
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| ==See also==
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| *[[Fermat's spiral]]
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| * {{Citation | title = Spirals from Theodorus to Chaos | first = P. J. | last = Davis | authorlink = Philip J. Davis |year = 1993 }}
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| * {{Citation | doi = 10.2307/4145130 | title = The Spiral of Theodorus | first = Detlef| last = Gronau | journal = [[The American Mathematical Monthly]] | volume = 111 |date=March 2004 | pages = 230–237 | issue = 3 | publisher = Mathematical Association of America | jstor = 4145130 }}
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| * {{Citation | first1 = J. | last1 = Heuvers | first2 = D. S. | last2 = Moak | first3 = B | last3 = Boursaw | chapter = The functional equation of the square root spiral | title = Functional Equations and Inequalities | editor = T. M. Rassias | year = 2000 | pages = 111–117 }}
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| * {{Citation | title = Analytic Continuation of the Theodorus Spiral | first = Jörg | last = Waldvogel | year = 2009 | url = http://www.math.ethz.ch/~waldvoge/Papers/theopaper.pdf }}
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| {{refend}}
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| {{Greek mathematics}}
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| [[Category:Spirals]]
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