## Heptadecagon construction

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

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:

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.

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 **( F_{k}) * (2^{h})**. There are distinct primes of the form :, 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).

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.

## [edit] See also

## [edit] 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.

## [edit] External links

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

## No comments:

Post a Comment