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{{short description|Polygonal curve made from right triangles}}
[[File:Spiral of Theodorus.svg|thumb|right|400px|The spiral of Theodorus up to the triangle with a hypotenuse of <math>\sqrt{17}</math>]]

In [[Geometry|geometry]], the '''spiral of Theodorus''' (also called the '''square root spiral''', '''Pythagorean spiral''', or '''Pythagoras's snail'''){{r|KAHN2}} is a [[Spiral|spiral]] composed of [[Right triangle|right triangle]]s, placed edge-to-edge. It was named after [[Biography:Theodorus of Cyrene|Theodorus of Cyrene]].

==Construction==
The spiral is started with an isosceles right triangle, with each [[Cathetus|leg]] having unit [[Physics:Length|length]]. Another right triangle is formed, an [[Automedian triangle|automedian right triangle]] with one leg being the [[Hypotenuse|hypotenuse]] of the prior triangle (with length the [[Square root of 2|square root of 2]]) and the other leg having length of 1; the length of the hypotenuse of this second triangle is the [[Square root of 3|square root of 3]]. The process then repeats; the <math>n</math>th triangle in the sequence is a right triangle with the side lengths <math>\sqrt{n}</math> and 1, and with hypotenuse <math>\sqrt{n+1}</math>. For example, the 16th triangle has sides measuring <math>4=\sqrt{16}</math>, 1 and hypotenuse of <math>\sqrt{17}</math>.

==History and uses ==
Although all of Theodorus' work has been lost, [[Biography:Plato|Plato]] put Theodorus into his 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.{{r|nahin}}

Plato does not attribute the irrationality of the [[Square root of 2|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.{{r|plato}}

==Hypotenuse==
Each of the triangles' hypotenuses <math>h_n</math> gives the [[Square root|square root]] of the corresponding [[Natural number|natural number]], with <math>h_1=\sqrt{2}</math>.

Plato, tutored by Theodorus, questioned why Theodorus stopped at <math>\sqrt{17}</math>. The reason is commonly believed to be that the <math>\sqrt{17}</math> hypotenuse belongs to the last triangle that does not overlap the figure.{{r|LONG}}

===Overlapping===
In 1958, Kaleb Williams 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.{{r|LONG|teuffel}}

==Extension==
[[File:Spiral of Theodorus extended.svg|thumb|Colored extended spiral of Theodorus with 110 triangles]]
Theodorus stopped his spiral at the triangle with a hypotenuse of <math>\sqrt{17}</math>. If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.

===Growth rate===

====Angle====
If <math>\varphi_n</math> is the angle of the <math>n</math>th triangle (or spiral segment), then:
<math display=block>\tan\left(\varphi_n\right)=\frac{1}{\sqrt{n}}.</math>
Therefore, the growth of the angle <math>\varphi_n</math> of the next triangle <math>n</math> is:{{r|KAHN2}}
<math display=block>\varphi_n=\arctan\left(\frac{1}{\sqrt{n}}\right).</math>

The sum of the angles of the first <math>k</math> triangles is called the total angle <math>\varphi(k)</math> for the <math>k</math>th triangle. It grows proportionally to the square root of <math>k</math>, with a [[Bounded function|bounded]] correction term <math>c_2</math>:{{r|KAHN2}}
<math display=block>\varphi\left (k\right)=\sum_{n=1}^k\varphi_n = 2\sqrt{k}+c_2(k)</math>
where
<math display=block>\lim_{k \to \infty} c_2(k)= - 2.157782996659\ldots</math>
({{OEIS2C|A105459}}).

[[File:Spiral of Theodorus triangle.svg|thumb|A triangle or section of spiral]]

====Radius====
The growth of the radius of the spiral at a certain triangle <math>n</math> is
<math display=block>\Delta r=\sqrt{n+1}-\sqrt{n}.</math>

===Archimedean spiral===
The Spiral of Theodorus approximates the [[Archimedean spiral]].{{r|KAHN2}} Just as the distance between two windings of the Archimedean spiral equals [[Mathematical constant|mathematical constant]] <math>\pi</math>, as the number of spins of the spiral of Theodorus approaches [[Infinity|infinity]], the distance between two consecutive windings quickly approaches <math>\pi</math>.{{r|hahn}}

The following table shows successive windings of the spiral approaching pi:

{| class="wikitable"
!Winding No.:
!width=200px|Calculated average winding-distance
!width=200px|Accuracy of average winding-distance in comparison to π
|-
|2
|3.1592037
|99.44255%
|-
|3
|3.1443455
|99.91245%
|-
|4
|3.14428
|99.91453%
|-
|5
|3.142395
|99.97447%
|-
|<math>\to\infty</math>
|<math>\to\pi</math>
|<math>\to 100\%</math>
|}

As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to <math>\pi</math>.{{r|KAHN2}}

==Continuous curve==

[[File:Theodorus Wiki.svg|thumb|right|400px|Davis' analytic continuation of the Spiral of Theodorus, including extension in the opposite direction from the origin (negative nodes numbers).]]

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|2001|loc = pp. 37–38 }} by analogy with Euler's formula for the [[Gamma function|gamma function]] as an [[Interpolation|interpolant]] for the [[Factorial|factorial]] function. [[Biography:Philip J. Davis|Davis]] found the function
<math display=block>T(x) = \prod_{k=1}^\infty \frac{1 + i/\sqrt{k}}{1 + i/\sqrt{x+k}} \qquad ( -1 < x < \infty )</math>
which was further studied by his student Leader{{r|leader}} and by [[Biography:Arieh Iserles|Iserles]] (in an appendix to {{Harv|Davis|2001}} ). An axiomatic characterization of this function is given in {{Harv|Gronau|2004}} as the unique function that satisfies the [[Functional equation|functional equation]]
<math display=block>f(x+1) = \left( 1 + \frac{i}{\sqrt{x+1} }\right) \cdot f(x),</math>
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}}.

An analytic continuation of Davis' continuous form of the Spiral of Theodorus which extends in the opposite direction from the origin is given in {{Harv|Waldvogel|2009}}.

In the figure the nodes of the original (discrete) Theodorus spiral are shown as small green circles. The blue ones are those, added in the opposite direction of the spiral.
Only nodes <math>n</math> with the integer value of the polar radius <math>r_n=\pm\sqrt{|n|}</math> are numbered in the figure.
The dashed circle in the coordinate origin <math>O</math> is the circle of curvature at <math>O</math>.

==See also==
*[[Fermat's spiral]]
*[[List of spirals]]

== References ==
{{reflist|refs=

<ref name=hahn>
{{citation
 |last=Hahn |first=Harry K.
 |year=2008
 |title=The distribution of natural numbers divisible by 2, 3, 5, 7, 11, 13, and 17 on the square root spiral
 |arxiv=0801.4422}}</ref>

<ref name=KAHN2>{{citation
 |last=Hahn
 |first=Harry K.
 |title=The ordered distribution of natural numbers on the square root spiral
 |year=2007
 |arxiv=0712.2184}}</ref>

<ref name=leader>{{citation
 | id = {{ProQuest|303808219}}
 | mr = 2685516
 | page = 173
 | publisher = Brown University
 | title = The generalized Theodorus iteration
 | type = PhD thesis
 | year = 1990}}</ref>

<ref name=LONG>
{{citation
 |last=Long
 |first=Kate
 |title=A Lesson on The Root Spiral
 |url=http://courses.wcupa.edu/jkerriga/Lessons/A%20Lesson%20on%20Spirals.html
 |access-date=30 April 2008
|archive-url=https://web.archive.org/web/20130411230043/http://courses.wcupa.edu/jkerriga/Lessons/A%20Lesson%20on%20Spirals.html
 |archive-date=11 April 2013}}</ref>

<ref name=nahin>{{citation
 |last=Nahin
 |first=Paul J.
 |title=An Imaginary Tale: The Story of <math>\sqrt{-1}</math>
 |publisher=Princeton University Press
 |page=33
 |isbn=0-691-02795-1 
 |year=1998}}</ref>

<ref name=plato>{{citation
 |last1=Plato
 |last2=Dyde
 |first2=Samuel Walters
 |title=The Theaetetus of Plato
 |publisher=J. Maclehose
 |pages=86–87
 |url=https://books.google.com/books?id=wt29k-Jz8pIC
 |year=1899}}</ref>

<ref name=teuffel>{{citation
 | last = Teuffel | first = Erich
 | journal = Mathematisch-Physikalische Semesterberichte zur Pflege des Zusammenhangs von Schule und Universität
 | mr = 96160
 | pages = 148–152
 | title = Eine Eigenschaft der Quadratwurzelschnecke
 | volume = 6
 | year = 1958}}</ref>

}}

==Further reading==
* {{Citation | title = Spirals from Theodorus to Chaos | first = P. J. | last = Davis | publisher = A K Peters/CRC Press | year = 2001 }}
* {{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 | jstor = 4145130 }}
* {{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 }}
* {{Citation | title = Analytic Continuation of the Theodorus Spiral | first = Jörg | last = Waldvogel | year = 2009 | url = http://www.math.ethz.ch/~waldvoge/Papers/theopaper.pdf }}

{{Ancient Greek mathematics}}
{{Spirals}}

[[Category:Spirals|Theodorus]]

{{Sourceattribution|Spiral of Theodorus}}