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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]]

Revision as of 06:16, 24 February 2014

Additionally, when you're initially beginning out, we have a limited budget, plus starting with ppc advertisments will not even be an alternative for you at this point inside time. Free traffic exchanges will be a wonderful method to receive traffic to a website. You should furthermore understand there's really no these thing because free considering you must invest your time surfing from other sites to make credits.



A supplement of overweight people can enable jump begin fat reduction. Supplements are moreover helpful in establishing consistent fat reduction. Consistent fat loss is important to aid keep people motivated.

There is a superior percentage of the population which is quite unfamiliar with what a Bass Mass Index (BMI) is. They don't understand how important this index is to their health. The BMI index is what doctors employ to determine a people body fat or wellness percentile ranking. By utilizing a calories burned walking they could measure a person overall weight plus fat density.

Through the BMI index to take into account the harshness of weight loss, people begins to formulate any adverse wellness healthy diet incorporating normal exercise to reduce weight plus eliminate the precarious of needing a debilitating illness resulting from fat problems.

Everyone knows the saying which obese youngsters become overweight adults. In many situations it is very true, plus among the superb presents we can provide your children is to teach them to eat healthy food plus to be physically active. It usually serve them perfectly inside later existence.

Most of you look to achieve the ideal physique or weight considering it not only assists you physically nevertheless also enhances the looks plus subsequently the confidence. Also, being obese or underweight may lead to additional health problems. Age could be a state of notice, yet the body could not usually agree. So, getting elder will bring about the possibility of several wellness issues irrespective of gender. Especially inside ladies, menopause and osteoporosis will pose serious health problems if a girl is obese. Now, the ideal height to weight chart is misleading for women over 60.

Example: If a person weighs 220 lbs. and is 72 inches tall then it will be because follows: (220 * 703) / (72)^2.......(154660) / (5184)....which equals 29.8. That individual would have a BMI of 29.8.

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