Editing Heptadecagon

{{short description|Polygon with 17 edges}}
{{Use dmy dates|date=January 2020}}
{{Regular polygon db|Regular polygon stat table|p17}}
In [[geometry]], a '''heptadecagon''', '''septadecagon''' or '''17-gon''' is a seventeen-sided [[polygon]].

== Regular heptadecagon==
A ''[[regular polygon|regular]] heptadecagon'' is represented by the [[Schläfli symbol]] {17}.

=== 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]]
{{clear}}

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/>

=== Trigonometric derivation using nested quadratic equations===
Combine nested double-angle formula with supplementary-angle formula to get the nested quadratic polynomial below.
:<math>\cos\frac{2m\pi}{17} = 2\cos^2\frac{m\pi}{17} - 1</math>, AND 
:<math>\cos\frac{16\pi}{17} = \cos({\pi - \frac{\pi}{17}}) = - \cos\frac{\pi}{17} = -X</math>
Therefore,
:<math>-X = - \cos\frac{\pi}{17} = \cos\frac{16\pi}{17} = 2 \cos^2\frac{8\pi}{17} - 1 = 2 \times {(2 \cos^2\frac{4\pi}{17} - 1)}^2 - 1</math>
:<math>\cos\frac{4\pi}{17} = 2 \cos^2\frac{2\pi}{17} - 1 = 2 \times {(2 \cos^2\frac{\pi}{17} - 1)}^2 - 1 = 2 (2 X^2-1)^2 - 1</math>
On simplifying and solving for X,
:<math>32768 X^{16} - 131072 X^{14} + 212992 X^{12} - 180224 X^{10} + 84480 X^8 - 21504 X^6 + 2688 X^4 - 128 X^2 + 1 = -X</math>
:<math>\cos\frac{\pi}{17} = X = \frac{\sqrt{34-\sqrt{68}}-\sqrt{17}+1 + 2\sqrt{\sqrt{34-\sqrt{68}}+\sqrt{17}-1} \sqrt{\sqrt{17+\sqrt{272}}}}{16}</math>

===Exact value of sin and cos of {{sfrac|m{{pi}}|(17 &times; 2<sup>n</sup>)}}===

If <math>A = \sqrt{2(17\pm\sqrt{17})}</math>, <math>B = (\sqrt{17}\pm1)</math> and <math>C = 17\mp4\sqrt{17}</math> then, depending on any integer m
:<math>\cos\frac{m\pi}{17} = \pm\frac{(A \pm B) \pm 2\sqrt{(A \mp B)\sqrt{C}}}{16}</math>
:<math> = \pm\frac{\sqrt{34\pm\sqrt{68}}\pm(\sqrt{17}\pm1) \pm 2\sqrt{\sqrt{34\pm\sqrt{68}} \mp(\sqrt{17}\pm1)} \sqrt{\sqrt{17\mp\sqrt{272}}}}{16}</math>

For example, if m = 1
:<math>\cos\frac{\pi}{17} = \frac{\sqrt{34-\sqrt{68}}-\sqrt{17}+1 + 2\sqrt{\sqrt{34-\sqrt{68}}+\sqrt{17}-1} \sqrt{\sqrt{17+\sqrt{272}}}}{16}</math>

Here are the expressions simplified into the following table.

{| class="wikitable"
|+ Cos and Sin (m π / 17) in first quadrant, from which other quadrants are computable.
|-
! m !! 16 cos (m π / 17) !! 8 sin (m π / 17)
|-
| 1 || <math>+1-\sqrt{17}+\sqrt{34-\sqrt{68}} + \sqrt{68+\sqrt{2448}+\sqrt{2720+\sqrt{6284288}}}</math> || <math>\sqrt{34 - \sqrt{68} - \sqrt{136-\sqrt{1088}} - \sqrt{272+\sqrt{39168}-\sqrt{43520+\sqrt{1608777728}}}}</math>
|-
| 2 || <math>-1+\sqrt{17}+\sqrt{34-\sqrt{68}} + \sqrt{68+\sqrt{2448}-\sqrt{2720+\sqrt{6284288}}}</math> || <math>\sqrt{34 - \sqrt{68} + \sqrt{136-\sqrt{1088}} - \sqrt{272+\sqrt{39168}+\sqrt{43520+\sqrt{1608777728}}}}</math>
|-
| 3 || <math>+1+\sqrt{17}+\sqrt{34+\sqrt{68}} + \sqrt{68-\sqrt{2448}-\sqrt{2720-\sqrt{6284288}}}</math> || <math>\sqrt{34 + \sqrt{68} - \sqrt{136+\sqrt{1088}} - \sqrt{272-\sqrt{39168}+\sqrt{43520-\sqrt{1608777728}}}}</math>
|-
| 4 || <math>-1+\sqrt{17}-\sqrt{34-\sqrt{68}} + \sqrt{68+\sqrt{2448}+\sqrt{2720+\sqrt{6284288}}}</math> || <math>\sqrt{34 - \sqrt{68} - \sqrt{136-\sqrt{1088}} + \sqrt{272+\sqrt{39168}-\sqrt{43520+\sqrt{1608777728}}}}</math>
|-
| 5 || <math>+1+\sqrt{17}+\sqrt{34+\sqrt{68}} - \sqrt{68-\sqrt{2448}-\sqrt{2720-\sqrt{6284288}}}</math> || <math>\sqrt{34 + \sqrt{68} - \sqrt{136+\sqrt{1088}} + \sqrt{272-\sqrt{39168}+\sqrt{43520-\sqrt{1608777728}}}}</math>
|-
| 6 || <math>-1-\sqrt{17}+\sqrt{34+\sqrt{68}} + \sqrt{68-\sqrt{2448}+\sqrt{2720-\sqrt{6284288}}}</math> || <math>\sqrt{34 + \sqrt{68} + \sqrt{136+\sqrt{1088}} - \sqrt{272-\sqrt{39168}-\sqrt{43520-\sqrt{1608777728}}}}</math>
|-
| 7 || <math>+1+\sqrt{17}-\sqrt{34+\sqrt{68}} + \sqrt{68-\sqrt{2448}+\sqrt{2720-\sqrt{6284288}}}</math> || <math>\sqrt{34 + \sqrt{68} + \sqrt{136+\sqrt{1088}} + \sqrt{272-\sqrt{39168}-\sqrt{43520-\sqrt{1608777728}}}}</math>
|-
| 8 || <math>-1+\sqrt{17}+\sqrt{34-\sqrt{68}} - \sqrt{68+\sqrt{2448}-\sqrt{2720+\sqrt{6284288}}}</math> || <math>\sqrt{34 - \sqrt{68} + \sqrt{136-\sqrt{1088}} + \sqrt{272+\sqrt{39168}+\sqrt{43520+\sqrt{1608777728}}}}</math>
|}

Therefore, applying induction with m=1 and starting with n=0:
:<math>\cos\frac{\pi}{17 \times 2^0} = \frac{1-\sqrt{17}+\sqrt{34-\sqrt{68}} + \sqrt{68+\sqrt{2448}+\sqrt{2720+\sqrt{6284288}}}}{16}</math>
:<math>\cos\frac{\pi}{17 \times 2^{n+1}} = \frac{\sqrt{2 + 2\cos\frac{\pi}{17 \times 2^n}}}{2}</math> and <math>\sin\frac{\pi}{17 \times 2^{n+1}} = \frac{\sqrt{2 - 2\cos\frac{\pi}{17 \times 2^n}}}{2}.</math>

== Symmetry==
[[File:Symmetries of heptadecagon.png|thumb|200px|Symmetries of a regular heptadecagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.]]

The ''regular heptadecagon'' has [[dihedral symmetry|Dih<sub>17</sub> symmetry]], order 34. Since 17 is a [[prime number]] there is one subgroup with dihedral symmetry: Dih<sub>1</sub>, and 2 [[cyclic group]] symmetries: Z<sub>17</sub>, and Z<sub>1</sub>.

These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon. [[John Horton Conway|John Conway]] labels these by a letter and group order.<ref>John H. Conway, Heidi Burgiel, [[Chaim Goodman-Strauss]], (2008) The Symmetries of Things, {{ISBN|978-1-56881-220-5}} (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)</ref> Full symmetry of the regular form is '''r34''' and no symmetry is labeled '''a1'''. The dihedral symmetries are divided depending on whether they pass through vertices ('''d''' for diagonal) or edges ('''p''' for perpendiculars), and '''i''' when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as '''g''' for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the '''g17''' subgroup has no degrees of freedom but can be seen as [[directed edge]]s.

{{Clear}}

==Related polygons==
===Heptadecagrams===
A heptadecagram is a 17-sided [[star polygon]]. There are seven regular forms given by [[Schläfli symbol]]s: {17/2}, {17/3}, {17/4}, {17/5}, {17/6}, {17/7}, and {17/8}. Since 17 is a prime number, all of these are regular stars and not compound figures.
{| class=wikitable
|- align=center
! Picture
|[[File:Regular star polygon 17-2.svg|120px]]<BR>{17/2}
|[[File:Regular star polygon 17-3.svg|120px]]<BR>{17/3}
|[[File:Regular star polygon 17-4.svg|120px]]<BR>{17/4}
|[[File:Regular star polygon 17-5.svg|120px]]<BR>{17/5}
|[[File:Regular star polygon 17-6.svg|120px]]<BR>{17/6}
|[[File:Regular star polygon 17-7.svg|120px]]<BR>{17/7}
|[[File:Regular star polygon 17-8.svg|120px]]<BR>{17/8}
|- align=center
! Interior angle
| ≈137.647°
| ≈116.471°
| ≈95.2941°
| ≈74.1176°
| ≈52.9412°
| ≈31.7647°
| ≈10.5882°
|}

===Petrie polygons===
The regular heptadecagon is the [[Petrie polygon]] for one higher-dimensional regular convex polytope, projected in a skew [[orthogonal projection]]:

{| class=wikitable
|- align=center
|[[File:16-simplex t0.svg|150px]]<br>[[16-simplex]] (16D)
|}

==References==
{{reflist}}

==Further reading==
*{{cite journal |last=Dunham |first=William |author-link=William Dunham (mathematician) |date=September 1996 |title=1996—a triple anniversary |journal=[[Math Horizons]] |volume=4 |pages=8–13 |doi=10.1080/10724117.1996.11974982 |url=http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3057 |access-date=6 December 2009 |archive-date=13 July 2010 |archive-url=https://web.archive.org/web/20100713044448/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3057 |url-status=dead |url-access=subscription }}
*[[Felix Klein|Klein, Felix]] et al. ''Famous Problems and Other Monographs''. – Describes the algebraic aspect, by Gauss.

==External links==
{{Commons category|17-gons}}
*{{MathWorld|title=Heptadecagon|urlname=Heptadecagon}} Contains a description of the construction.
*{{MathPages|id=home/kmath487|title=Constructing the Heptadecagon}} 
*[http://mathworld.wolfram.com/TrigonometryAnglesPi17.html Heptadecagon trigonometric functions]
*[http://news.bbc.co.uk/1/hi/england/7911406.stm BBC video] of New R&D center for SolarUK
*Archived at [https://ghostarchive.org/varchive/youtube/20211211/87uo2TPrsl8 Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20150213151420/https://www.youtube.com/watch?v=87uo2TPrsl8&gl=US&hl=en Wayback Machine]{{cbignore}}: {{cite web|last1=Eisenbud|first1=David|author-link=David Eisenbud|title=The Amazing Heptadecagon (17-gon)|date=13 February 2015 |url=https://www.youtube.com/watch?v=87uo2TPrsl8|publisher=[[Brady Haran]]|access-date=2 March 2015|format=Video}}{{cbignore}}
*{{OEIS2C|A210644}}

{{Polygons}}

[[Category:Constructible polygons]]
[[Category:Polygons by the number of sides]]
[[Category:Euclidean plane geometry]]