Heptadecagon

Regular heptadecagon
A regular heptadecagon
Type	Regular polygon
Edges and vertices	17
Schläfli symbol	{17}
Coxeter–Dynkin diagram
Symmetry group	Dihedral (D17)
Internal angle (degrees)	$\approx 158.82$°
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a heptadecagon (or 17-gon) is a seventeen-sided polygon.

Heptadecagon construction

The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796 at the age of 19.

Constructibility implies that trigonometric functions of ^2π⁄₁₇ can be expressed with basic arithmetic and square roots alone. Gauss' book Disquisitiones Arithmeticae contains the following equation, given here in modern notation:

$\begin{align} 16\,\operatorname{cos}{2\pi\over17} = & -1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \\ & 2\sqrt{17+3\sqrt{17}- \sqrt{34-2\sqrt{17}}- 2\sqrt{34+2\sqrt{17}}}. \end{align}$

The first actual method of construction was devised by Johannes Erchinger, a few years after Gauss' work.

Thanks to the construction above it is easy to obtain multiples of 17 by 3 and 5 and any power of 2, for example a 34-gon, 51-gon, 85-gon or 255-gon.

Carl Friedrich Gauss proved – as a 19-year-old student at Göttingen University – that the regular heptadecagon (a 17-sided polygon) is constructible with a pair of compasses and a straightedge. His proof relies on the property of irreducible polynomial equations that roots composed of a finite number of square root extractions only exist when the order of the equation is a product of the forms $\scriptstyle (F_{k})\times(2^{h})$. There are distinct primes of the form: $\scriptstyle F_{n} = 2^{2^{ \overset{n} {}}} + 1$, known as Fermat primes. Constructions for the regular triangle, square, pentagon, hexagon et al. 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, 65537.) The first explicit construction of a heptadecagon was given by Erchinger (see above).

Another method of construction uses Carlyle circles, as shown below:

Petrie polygons

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

16-simplex (16D)

See also

• Compass and straightedge
• Constructible polygon

Further reading

• Dunham, William (September 1996). "1996—a triple anniversary". Math Horizons: 8–13. http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3057. Retrieved 2009-12-06. 
• Klein, Felix et al. Famous Problems and Other Monographs. – Describes the algebraic aspect, by Gauss.

External links

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