Heptadecagon

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Regular heptadecagon
Regular polygon 17 annotated.svg
A regular heptadecagon
Type Regular polygon
Edges and vertices 17
Schläfli symbol {17}
Coxeter diagram CDel node 1.pngCDel 17.pngCDel node.png
Symmetry group Dihedral (D17), order 2×17
Internal angle (degrees) \approx 158.82°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

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

Regular heptadecagon construction[edit]

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.[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{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}

Constructions for the regular triangle and polygons with 2h 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 Fn 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 2h times as many sides.

Regular Heptadecagon Using Carlyle Circle.gif

Petrie polygons[edit]

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

16-simplex t0.svg
16-simplex (16D)

See also[edit]

References[edit]

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

Further reading[edit]

External links[edit]