Heptadecagon
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In geometry, a heptadecagon (or 17-gon) is a seventeen-sided polygon. A regular heptadecagon has internal angles each measuring 158.823529411765 degrees.
The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796. Gauss was so pleased by this that he asked for one to be inscribed on his tombstone.
Constructibility implies that trigonometric functions of 2π/17 can be expressed with basic arithmetic and square roots alone. Gauss' book Disquisitiones contains the following equation, given here in modern notation:
- <math>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}}}.<math>
See also
External links
You can see how to construct a regular 17-gon geometrically at either of
- http://www.showmath.co.kr/const/polygon/rpoly17.html (Korean, flash)
- http://www.geocities.com/RainForest/Vines/2977/gauss/formulae/heptadecagon.html
- http://mathworld.wolfram.com/Heptadecagon.html
- http://www.jimloy.com/geometry/17-gon.htm
And you can see the algebraic aspect (by Gauss) in this book :
'Famous Problems and Other Monographs' by F.Klein et al.