Heptadecagon - Wikipedia, the free encyclopedia

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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

• Compass and straightedge
• Constructible polygon

[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