Editing Heptadecagon (section)

=== Construction ===
[[File:Gauß 17-Eck.gif|thumb|Publication by C.&nbsp;F.&nbsp;Gauss in ''Intelligenzblatt der allgemeinen Literatur-Zeitung'']]
As 17 is a [[Fermat prime]], the regular heptadecagon is a [[constructible polygon]] (that is, one that can be constructed using a [[straightedge and compass|compass and unmarked straightedge]]): this was shown by [[Carl Friedrich Gauss]] in 1796.<ref name="Jones">Arthur Jones, Sidney A. Morris, Kenneth R. Pearson, ''Abstract Algebra and Famous Impossibilities'', Springer, 1991, {{ISBN|0387976612}}, [https://books.google.com/books?id=6dSIBBW87b8C&pg=PA178 p. 178.]</ref> This proof represented the first progress in regular polygon construction in over 2000 years.<ref name="Jones"/> Gauss's proof relies firstly on the fact that constructibility is equivalent to expressibility of the [[trigonometric function]]s of the common angle in terms of [[arithmetic]] operations and [[square root]] extractions, and secondly on his proof that this can be done if the odd prime factors of <math>N</math>, the number of sides of the regular polygon, are distinct Fermat primes, which are of the form <math>F_n = 2^{2^n} + 1</math> for some nonnegative integer <math>n</math>. Constructing a regular heptadecagon thus involves finding the cosine of <math>2\pi/17</math> in terms of square roots.  Gauss's book ''[[Disquisitiones Arithmeticae]]''<ref>Carl Friedrich Gauss "[https://edoc.hu-berlin.de/bitstream/handle/18452/1163/h284_gauss_1801.pdf?sequence=1&isAllowed=y#page=682&zoom=auto,-129,503 Disquisitiones Arithmeticae]" eod books2ebooks, p. 662 item 365.</ref> gives this (in modern notation) as<ref name=Callagy>Callagy, James J. "[https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/6738-the-central-angle-of-the-regular-17gon/81DAC320D40B8C6D850C1ACC15E870B3 The central angle of the regular 17-gon]", ''Mathematical Gazette'' 67, December 1983, 290–292.</ref>
:<math>
 \begin{align}\cos\frac{2\pi}{17} = & \frac{1}{16}\left(\sqrt{17}-1+\sqrt{34-2\sqrt{17}}\right)\\
                                                     & + \frac{1}{8}\left(\sqrt{17+3\sqrt{17}-
                                                        \sqrt{34-2\sqrt{17}}-
                                                       2\sqrt{34+2\sqrt{17}}} \right).\\
\end{align}</math>
[[File:01-Siebzehneck-Formel Gauss-2.svg|center|thumb|990x990px|Gaussian construction of the regular heptadecagon.]]

Constructions for the [[equilateral triangle|regular triangle]], [[pentagon]], [[pentadecagon]], and polygons with ''2''<sup>''h''</sup> times as many sides had been given by Euclid, but constructions based on the Fermat primes other than 3 and 5 were unknown to the ancients. (The only known Fermat primes are ''F<sub>n</sub>'' for ''n'' = 0, 1, 2, 3, 4. They are 3, 5, 17, 257, and 65537.)

The explicit construction of a heptadecagon was given by [[Herbert William Richmond]] in 1893. The following method of construction uses [[Carlyle circle]]s, as shown below. Based on the construction of the regular 17-gon, one can readily construct ''n''-gons with ''n'' being the product of 17 with 3 or 5 (or both) and any power of 2: a regular 51-gon, 85-gon or 255-gon and any regular ''n''-gon with ''2''<sup>''h''</sup> times as many sides.

[[File:Regular Heptadecagon Using Carlyle Circle.gif|512px|left]]
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[[File:01-Heptadecagon-Carlyle circle.gif|thumb|600px|left|Construction according to Duane W. DeTemple with Carlyle circles,<ref>Duane W. DeTemple "Carlyle Circles and the Lemoine Simplicity of Polygon Constructions" in ''The American Mathematical Monthly, Volume 98, Issuc 1 (Feb. 1991), 97–108.'' [https://web.archive.org/web/20151221113614/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf#6#6 "4. Construction of the Regular Heptadecagon (17-gon)"] pp. 101–104, p.103, web.archive document, selected on 28 January 2017</ref> animation 1 min 57 s]]
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Another construction of the regular heptadecagon using straightedge and compass is the following:
[[File:Regular Heptadecagon Inscribed in a Circle.gif|509px|left]]
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T. P. Stowell of Rochester, N. Y., responded to Query, by W.E. Heal, Wheeling, Indiana in ''The Analyst'' in the year 1877:<ref name=Hendricks>{{cite journal|last=Hendricks|first=J. E.|title=Answer to Mr. Heal's Query; T. P. Stowell of Rochester, N. Y.|journal=The Analyst: A Monthly Journal of Pure and Applied Mathematicus Vol.1|date=1877|pages=94–95|url=https://books.google.com/books?id=ovhZAAAAYAAJ&pg=PA94}} [https://books.google.com/books?id=ovhZAAAAYAAJ&pg=PA64 Query, by W. E. Heal, Wheeling, Indiana] p. 64; accessdate 30 April 2017</ref>

''"To construct a regular polygon of seventeen sides in a circle.''
''Draw the radius CO at right-angles to the diameter AB: On OC and OB, take OQ  equal to the half, and OD equal to the eighth part of the radius: Make DE and DF each equal to DQ and EG and FH respectively equal to EQ and FQ; take OK a mean proportional  between OH and OQ, and through K, draw KM parallel to AB, meeting the semicircle described on OG in M; draw MN parallel to OC, cutting the given circle in N – the arc AN is the seventeenth part of the whole circumference."''
[[File:01 Siebzehneck-1806.svg|400px|thumb|left|Construction according to<br />
''"sent by T. P. Stowell, credited to Leybourn's Math. Repository, 1818"''.<br />
Added: ''"take OK a [[Geometric mean theorem|mean proportional]] between OH and OQ"'']]
[[File:01 Siebzehneck-1818-Animation.gif|535px|thumb|center|Construction according to<br />
''"sent by T. P. Stowell, credited to Leybourn's Math. Repository, 1818"''.<br />
Added: ''"take OK a mean proportional between OH and OQ"'', animation]]
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The following simple design comes from Herbert William Richmond from the year 1893:<ref>Herbert W. Richmond, [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN600494829_0026&DMDID=DMDLOG_0030&LOGID=LOG_0035&PHYSID=PHYS_0218 description "A Construction for a regular polygon of seventeen side"] [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN600494829_0026&DMDID=DMDLOG_0040&LOGID=LOG_0046&PHYSID=PHYS_0301 illustration (Fig. 6)], The Quarterly Journal of Pure and Applied Mathematics 26: pp. 206–207. Retrieved 4 December 2015</ref>
 
::''"LET OA, OB (fig. 6) be two perpendicular radii of a circle. Make OI one-fourth of OB, and the angle OIE one-fourth of OIA; also find in OA produced a point F such that EIF is 45°. Let the circle on AF as diameter cut OB in K, and let the circle whose centre is E and radius EK cut OA in N<sub>3</sub> and N<sub>5</sub>; then if ordinates N<sub>3</sub>P<sub>3</sub>, N<sub>5</sub>P<sub>5</sub> are drawn to the circle, the arcs AP<sub>3</sub>, AP<sub>5</sub> will be 3/17 and 5/17 of the circumference."''

*The point N<sub>3</sub> is very close to the center point of [[Thales' theorem]] over AF.
[[File:01-Siebzehneck-Richmond.svg|400px|thumb|left|Construction according to H. W. Richmond]][[File:01.Siebzehneck-Animation-Richmond.gif|400px|thumb|center|Construction according to H. W. Richmond as animation]]
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The following construction is a variation of H. W. Richmond's construction.

The differences to the original:
*The circle k<sub>2</sub> determines the point H instead of the bisector w<sub>3</sub>.
*The circle k<sub>4</sub> around the point G' (reflection of the point G at m) yields the point N, which is no longer so close to M, for the construction of the tangent.
*Some names have been changed.
[[File:01-Siebzehneck-Variation.svg|400px|thumb|left|Heptadecagon in principle according to H.W. Richmond, a variation of the design regarding to point N]]
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Another more recent construction is given by Callagy.<ref name=Callagy/>