Heptadecagon: Difference between revisions

Content deleted Content added

Inline

Revision as of 03:47, 15 December 2013

Template:Odd polygon db 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 1.^[1] This proof represented the first progress in regular polygon construction in over 2000 years.^[1] Gauss's proof relies firstly on the fact that constructibility is equivalent to expressibility of the trigonometric functions 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 $\scriptstyle F_{n}=2^{2^{\overset {n}{}}}+1$ . Constructing a regular heptadecagon thus involves finding the cosine of $2\pi /17$ 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):

{\begin{aligned}16\,\operatorname {cos} {2\pi  \over 17}=&-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{aligned}}

Constructions for a regular triangle and polygons with 2^h 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_n 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 circles, 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^h times as many sides.

Petrie polygons

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

16-simplex (16D)

References

^ ^a ^b Arthur Jones, Sidney A. Morris, Kenneth R. Pearson, Abstract Algebra and Famous Impossibilities, Springer, 1991, ISBN 0387976612, p. 178.

External links

Weisstein, Eric W. "Heptadecagon". MathWorld. Contains a description of the construction.
"Constructing the Heptadecagon". MathPages.com.
Heptadecagon trigonometric functions
heptadecagon building New R&D center for SolarUK
BBC video of New R&D center for SolarUK

[Jones-1] Arthur Jones, Sidney A. Morris, Kenneth R. Pearson, Abstract Algebra and Famous Impossibilities, Springer, 1991, ISBN 0387976612, p. 178.

[1]

@@ Line 3: / Line 3: @@
 ==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 56.<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):
+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 1.<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}}+ \\
@@ Line 11: / Line 11: @@
  \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.)
+Constructions for a 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.