Heptadecagon

A regular heptadecagon.

In geometry, a heptadecagon (or 17-gon) is a seventeen-sided polygon. A regular heptadecagon has internal angles each measuring ${\frac {2700}{17}}=158{\frac {14}{17}}\approx 158.82$ degrees.

Heptadecagon construction

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. The stonemason declined, stating that the difficult construction would essentially look like a circle.

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:

$16\,\operatorname {cos} {2\pi \over 17}=-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+2{\sqrt {17+3{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {34+2{\sqrt {17}}}}}}.$

The first actual method of construction was devised by Johannes Erchinger, a few years after Gauss' work, as shown step-by-step in the animation below. It takes 64 steps.

Heptadecagon Construction Animation

See also

• Compass and straightedge

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.

http://www.mathlove.org/bbs/data/mathfb/alg17gon.ppt