Spiral of Theodorus: Difference between revisions
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[[File:Spiral of Theodorus.svg|thumb|right|400px|The spiral of Theodorus up to the triangle with a hypotenuse of <math>\sqrt{17}</math>]] |
[[File:Spiral of Theodorus.svg|thumb|right|400px|The spiral of Theodorus up to the triangle with a hypotenuse of <math>\sqrt{17}</math>]] |
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In [[geometry]], the '''spiral of Theodorus''' (also called ''square root spiral'', '' |
In [[geometry]], the '''spiral of Theodorus''' (also called the '''square root spiral''', '''Pythagorean spiral''', or '''Pythagoras's snail'''){{r|KAHN2}} is a [[spiral]] composed of [[right triangle]]s, placed edge-to-edge. It was named after [[Theodorus of Cyrene]]. |
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{{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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|year=2007 |
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|class=math.GM |
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|eprint=0712.2184 |
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}}</ref> is a [[spiral]] composed of [[right triangle]]s, placed edge-to-edge. It was named after [[Theodorus of Cyrene]]. |
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==Construction== |
==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 |
The spiral is started with an [[isosceles]] right triangle, with each [[Cathetus|leg]] having unit [[length]]. Another right triangle (which is the ''only'' [[automedian triangle|automedian right triangle]]) is formed, with one leg being the [[hypotenuse]] of the prior right triangle (with length the [[square root of 2]]) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the [[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>. |
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==History and uses == |
==History and uses == |
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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. |
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.{{r|nahin}} |
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{{citation |
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|last=Nahin |
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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=https://books.google.com/books?id=WvcfqBgZDWQC |
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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. |
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.{{r|plato}} |
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{{citation |
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|last1=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=https://books.google.com/books?id=wt29k-Jz8pIC |
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|year=1899 |
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}}</ref> |
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==Hypotenuse== |
==Hypotenuse== |
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Each of the triangles' hypotenuses |
Each of the triangles' hypotenuses <math>h_n</math> gives the [[square root]] of the corresponding [[natural number]], with <math>h_1=\sqrt{2}</math>. |
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Plato, tutored by Theodorus, questioned why Theodorus stopped at { |
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}} |
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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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|access-date=30 April 2008 |
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|archive-url=https://web.archive.org/web/20130411230043/http://courses.wcupa.edu/jkerriga/Lessons/A%20Lesson%20on%20Spirals.html |
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|archive-date=11 April 2013 |
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}}</ref> |
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===Overlapping=== |
===Overlapping=== |
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In 1958, |
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}} |
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==Extension== |
==Extension== |
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[[File:Spiral of Theodorus extended.svg|thumb|Colored extended spiral of Theodorus with 110 triangles]] |
[[File:Spiral of Theodorus extended.svg|thumb|Colored extended spiral of Theodorus with 110 triangles]] |
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Theodorus stopped his spiral at the triangle with a hypotenuse of { |
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. |
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===Growth rate=== |
===Growth rate=== |
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====Angle==== |
====Angle==== |
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If |
If <math>\varphi_n</math> is the angle of the <math>n</math>th triangle (or spiral segment), then: |
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<math display=block>\tan\left(\varphi_n\right)=\frac{1}{\sqrt{n}}.</math> |
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Therefore, the growth of the angle |
Therefore, the growth of the angle <math>\varphi_n</math> of the next triangle <math>n</math> is:{{r|KAHN2}} |
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<math display=block>\varphi_n=\arctan\left(\frac{1}{\sqrt{n}}\right).</math> |
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The sum of the angles of the first |
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}} |
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<math display=block>\varphi\left (k\right)=\sum_{n=1}^k\varphi_n = 2\sqrt{k}+c_2(k)</math> |
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where |
where |
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<math display=block>\lim_{k \to \infty} c_2(k)= - 2.157782996659\ldots</math> |
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({{OEIS2C|A105459}}). |
({{OEIS2C|A105459}}). |
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====Radius==== |
====Radius==== |
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The growth of the radius of the spiral at a certain triangle |
The growth of the radius of the spiral at a certain triangle <math>n</math> is |
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<math display=block>\Delta r=\sqrt{n+1}-\sqrt{n}.</math> |
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===Archimedean spiral=== |
===Archimedean spiral=== |
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The Spiral of Theodorus [[approximate]]s the [[Archimedean spiral]]. |
The Spiral of Theodorus [[approximate]]s the [[Archimedean spiral]].{{r|KAHN2}} Just as the distance between two windings of the Archimedean spiral equals [[mathematical constant]] <math>\pi</math>, as the number of spins of the spiral of Theodorus approaches [[infinity]], the distance between two consecutive windings quickly approaches <math>\pi</math>.{{r|hahn}} |
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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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|class=math.GM |
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|eprint=0801.4422 |
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}}</ref> |
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The following |
The following table shows successive windings of the spiral approaching pi: |
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{| class="wikitable" |
{| class="wikitable" |
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|99.97447% |
|99.97447% |
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|- |
|- |
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|<math>\to\infty</math> |
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|[[Limit of a function#Limits involving infinity|→]] ∞ |
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|<math>\to\pi</math> |
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|→ π |
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|<math>\to 100\%</math> |
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|} |
|} |
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As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to |
As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to <math>\pi</math>.{{r|KAHN2}} |
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==Continuous curve== |
==Continuous curve== |
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[[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).]] |
[[File:Theodorus Wiki.svg|thumb|right|400px|Philip J. Davis' analytic continuation of the Spiral of Theodorus, including extension in the opposite direction from the origin (negative nodes numbers).]] |
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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 |
The question of how to [[interpolation|interpolate]] the discrete points of the spiral of Theodorus by a smooth curve was proposed and answered by [[Philip J. Davis]] in 2001 by analogy with Euler's formula for the [[gamma function]] as an [[Interpolation|interpolant]] for the [[factorial]] function. Davis found the function{{sfnp|Davis|2001|pp=37–38}} |
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<math display=block>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]] |
which was further studied by his student [[Jeffery J. Leader|Leader]]{{r|leader}} and by [[Arieh Iserles|Iserles]].<ref>In an appendix to {{Harv|Davis|2001}}</ref> This function can be characterized axiomatically as the unique function that satisfies the [[functional equation]] |
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<math display=block>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]] |
the initial condition <math>f(0) = 1,</math> and [[Monotonic function|monotonicity]] in both [[Argument (complex analysis)|argument]] and [[Absolute value|modulus]].<ref>{{Harvtxt|Gronau|2004}}. An alternative derivation is given in {{Harvtxt|Heuvers|Moak|Boursaw|2000}}.</ref> |
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An analytic continuation of Davis' continuous form of the Spiral of Theodorus |
An analytic continuation of Davis' continuous form of the Spiral of Theodorus extends in the opposite direction from the origin.{{sfnp|Waldvogel|2009}} |
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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. |
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. |
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== References == |
== References == |
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{{reflist |
{{reflist|refs= |
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<ref name=hahn> |
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{{citation |
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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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|arxiv=0801.4422}}</ref> |
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<ref name=KAHN2>{{citation |
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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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|year=2007 |
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|arxiv=0712.2184}}</ref> |
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<ref name=leader>{{citation |
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| last = Leader | first = Jeffery James | author-link = Jeffery J. Leader |
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| id = {{ProQuest|303808219}} |
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| mr = 2685516 |
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| page = 173 |
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| publisher = Brown University |
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| title = The generalized Theodorus iteration |
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| type = PhD thesis |
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| year = 1990}}</ref> |
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<ref name=LONG> |
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{{citation |
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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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|access-date=30 April 2008 |
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|archive-url=https://web.archive.org/web/20130411230043/http://courses.wcupa.edu/jkerriga/Lessons/A%20Lesson%20on%20Spirals.html |
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|archive-date=11 April 2013}}</ref> |
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<ref name=nahin>{{citation |
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|last=Nahin |
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|first=Paul J. |
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|title=An Imaginary Tale: The Story of <math>\sqrt{-1}</math> |
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|publisher=Princeton University Press |
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|page=33 |
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|isbn=0-691-02795-1 |
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|year=1998}}</ref> |
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<ref name=plato>{{citation |
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|last1=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=https://books.google.com/books?id=wt29k-Jz8pIC |
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|year=1899}}</ref> |
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<ref name=teuffel>{{citation |
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| last = Teuffel | first = Erich |
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| journal = Mathematisch-Physikalische Semesterberichte zur Pflege des Zusammenhangs von Schule und Universität |
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| mr = 96160 |
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| pages = 148–152 |
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| title = Eine Eigenschaft der Quadratwurzelschnecke |
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| volume = 6 |
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| year = 1958}}</ref> |
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}} |
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==Further reading== |
==Further reading== |
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* {{Citation | title = Spirals from Theodorus to Chaos | first = P. J. | last = Davis | publisher = A K Peters/CRC Press | author-link = Philip J. Davis |year = 2001 }} |
* {{Citation | title = Spirals from Theodorus to Chaos | first = P. J. | last = Davis | publisher = A K Peters/CRC Press | author-link = Philip J. Davis |year = 2001 }} |
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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 |
* {{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 }} |
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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 }} |
* {{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 }} |
* {{Citation | title = Analytic Continuation of the Theodorus Spiral | first = Jörg | last = Waldvogel | year = 2009 | url = http://www.math.ethz.ch/~waldvoge/Papers/theopaper.pdf }} |
Latest revision as of 18:15, 6 August 2024
In geometry, the spiral of Theodorus (also called the square root spiral, Pythagorean spiral, or Pythagoras's snail)[1] is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.
Construction
[edit]The spiral is started with an isosceles right triangle, with each leg having unit length. Another right triangle (which is the only automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3. The process then repeats; the th triangle in the sequence is a right triangle with the side lengths and 1, and with hypotenuse . For example, the 16th triangle has sides measuring , 1 and hypotenuse of .
History and uses
[edit]Although all of Theodorus' work has been lost, 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 by means of the Spiral of Theodorus.[2]
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.[3]
Hypotenuse
[edit]Each of the triangles' hypotenuses gives the square root of the corresponding natural number, with .
Plato, tutored by Theodorus, questioned why Theodorus stopped at . The reason is commonly believed to be that the hypotenuse belongs to the last triangle that does not overlap the figure.[4]
Overlapping
[edit]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, they will never pass through any of the other vertices of the total figure.[4][5]
Extension
[edit]Theodorus stopped his spiral at the triangle with a hypotenuse of . If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.
Growth rate
[edit]Angle
[edit]If is the angle of the th triangle (or spiral segment), then: Therefore, the growth of the angle of the next triangle is:[1]
The sum of the angles of the first triangles is called the total angle for the th triangle. It grows proportionally to the square root of , with a bounded correction term :[1] where (OEIS: A105459).
Radius
[edit]The growth of the radius of the spiral at a certain triangle is
Archimedean spiral
[edit]The Spiral of Theodorus approximates the Archimedean spiral.[1] Just as the distance between two windings of the Archimedean spiral equals mathematical constant , as the number of spins of the spiral of Theodorus approaches infinity, the distance between two consecutive windings quickly approaches .[6]
The following table shows successive windings of the spiral approaching pi:
Winding No.: | Calculated average winding-distance | 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% |
As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to .[1]
Continuous curve
[edit]The question of how to interpolate the discrete points of the spiral of Theodorus by a smooth curve was proposed and answered by Philip J. Davis in 2001 by analogy with Euler's formula for the gamma function as an interpolant for the factorial function. Davis found the function[7] which was further studied by his student Leader[8] and by Iserles.[9] This function can be characterized axiomatically as the unique function that satisfies the functional equation the initial condition and monotonicity in both argument and modulus.[10]
An analytic continuation of Davis' continuous form of the Spiral of Theodorus extends in the opposite direction from the origin.[11]
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 with the integer value of the polar radius are numbered in the figure. The dashed circle in the coordinate origin is the circle of curvature at .
See also
[edit]References
[edit]- ^ a b c d e Hahn, Harry K. (2007), The ordered distribution of natural numbers on the square root spiral, arXiv:0712.2184
- ^ Nahin, Paul J. (1998), An Imaginary Tale: The Story of , Princeton University Press, p. 33, ISBN 0-691-02795-1
- ^ Plato; Dyde, Samuel Walters (1899), The Theaetetus of Plato, J. Maclehose, pp. 86–87
- ^ a b Long, Kate, A Lesson on The Root Spiral, archived from the original on 11 April 2013, retrieved 30 April 2008
- ^ Teuffel, Erich (1958), "Eine Eigenschaft der Quadratwurzelschnecke", Mathematisch-Physikalische Semesterberichte zur Pflege des Zusammenhangs von Schule und Universität, 6: 148–152, MR 0096160
- ^ Hahn, Harry K. (2008), The distribution of natural numbers divisible by 2, 3, 5, 7, 11, 13, and 17 on the square root spiral, arXiv:0801.4422
- ^ Davis (2001), pp. 37–38.
- ^ Leader, Jeffery James (1990), The generalized Theodorus iteration (PhD thesis), Brown University, p. 173, MR 2685516, ProQuest 303808219
- ^ In an appendix to (Davis 2001)
- ^ Gronau (2004). An alternative derivation is given in Heuvers, Moak & Boursaw (2000).
- ^ Waldvogel (2009).
Further reading
[edit]- Davis, P. J. (2001), Spirals from Theodorus to Chaos, A K Peters/CRC Press
- Gronau, Detlef (March 2004), "The Spiral of Theodorus", The American Mathematical Monthly, 111 (3): 230–237, doi:10.2307/4145130, JSTOR 4145130
- Heuvers, J.; Moak, D. S.; Boursaw, B (2000), "The functional equation of the square root spiral", in T. M. Rassias (ed.), Functional Equations and Inequalities, pp. 111–117
- Waldvogel, Jörg (2009), Analytic Continuation of the Theodorus Spiral (PDF)