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{{Odd polygon db|Odd polygon stat table|p17}}
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In [[geometry]], a '''heptadecagon''' (or '''17-gon''') is a seventeen-sided [[polygon]].
 
==Regular heptadecagon construction==
The regular heptadecagon is a [[constructible polygon]] (that is, one that can be constructed using a compass and unmarked straightedge), as was shown by [[Carl Friedrich Gauss]] in 1796 at the age of 19.<ref name="Jones">Arthur Jones, Sidney A. Morris, Kenneth R. Pearson, ''Abstract Algebra and Famous Impossibilities'', Springer, 1991, ISBN 0387976612, [http://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 ''n'' are distinct Fermat primes, which are of the form <math>\scriptstyle F_{n} = 2^{2^{ \overset{n} {}}} + 1</math>. Constructing a regular heptadecagon thus involves finding the cosine of <math>2\pi/17</math> in terms of square roots, which involves an equation of degree 17—a Fermat prime.  Gauss' book ''[[Disquisitiones Arithmeticae]]'' gives this as (in modern notation):
:<math>
\begin{align} 16\,\operatorname{cos}{2\pi\over17} = & -1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \\
                                                    & 2\sqrt{17+3\sqrt{17}-
                                                        \sqrt{34-2\sqrt{17}}-
                                                      2\sqrt{34+2\sqrt{17}}}.
\end{align}</math>
 
Constructions for the regular triangle 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 first explicit construction of a heptadecagon was given by [[Johannes Erchinger]] in 1825. Another 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|center]]
 
===Petrie polygons===
The regular heptadecagon is the [[Petrie polygon]] for one higher-dimensional polytope, projected in a skew [[orthogonal projection]]:
 
{| class=wikitable
|- align=center
|[[File:16-simplex_t0.svg|150px]]<br>[[16-simplex]] (16D)
|}
 
==See also==
*[[Compass and straightedge]]
*[[Constructible polygon]]
 
==References==
{{reflist}}
 
==Further reading==
*{{cite journal |last=Dunham |first=William |authorlink=William Dunham (mathematician) |date=September 1996 |title=1996&mdash;a triple anniversary |journal=[[Math Horizons]] |pages=8&ndash;13 |url=http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3057 |accessdate=2009-12-06 }}
*[[Felix Klein|Klein, Felix]] et al. ''Famous Problems and Other Monographs''. – Describes the algebraic aspect, by Gauss.
 
==External links==
*{{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://www.SolarUK.net heptadecagon building] New R&D center for SolarUK
*[http://news.bbc.co.uk/1/hi/england/7911406.stm BBC video] of New R&D center for SolarUK
 
{{Polygons}}
 
[[Category:Polygons]]
[[Category:Euclidean plane geometry]]

Latest revision as of 15:47, 9 January 2015

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