Jan 19, 2010

Heptadecagon - Wikipedia, the free encyclopedia


Heptadecagon construction

The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796.

Constructibility implies that trigonometric functions of 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, as shown step-by-step in the animation below. It takes 64 steps.

Heptadecagon Construction Animation

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 (Fk) * (2h). There are distinct primes of the form :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 Fn 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).

The following construction is adapted from the one first given by H. W. Richmond in 1893.

Draw the large circle, centre O.
Draw a diameter AB.
Construct a perpendicular bisector to that diameter.
Bisect one of the radii on this line.
Bisect it again, to get point C in the diagram.
Draw line AC.
With C as a centre, draw an arc with radius CA, from A to the vertical diameter in the diagram.
Bisect this arc.
Bisect it again, to get point D in the diagram.
Draw line CD, which then intersects line AB at point E.
Construct line CF at ? to line CE, as in the diagram (so F is on AB).
Bisect line AF and draw the circle with AF as its diameter. This circle intersects the vertical diameter at a point G.
Draw the circle with centre E and radius EG. This intersects line AB at H and I.
Draw lines perpendicular to AB, at points H and I. These intersect the big circle at J and K.
Bisect angle JOK, producing point L.
Points J, K, L, and A are vertices of the heptadecagon. From these points, the rest of the vertices may be constructed.

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