Constructing a 17-Sided Polygon

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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One of the most unexpected discoveries in mathematics was the discovery by the young Karl Friedrich Gauss that it was possible, using the rules of the ancient Greeks, to construct a regular 17-sided polygon using a ruler and compass alone. He showed that only certain polygons could be constructed, and in the process showed that all others could not. One of the polygons that cannot be constructed is a 9-sided polygon. Since it would be possible to construct a 9-sided polygon if you could trisect a 120-degree angle, therefore trisecting an angle using a ruler and compass alone cannot be done.

One construction for a 17-sided polygon is shown below.

• Construct a circle with diameter AOB and radius OC perpendicular to AOB
• Construct OD = OC/4
• Construct DA
• Bisect angle ODA twice to construct angle ODE (= ODA/4)
• Construct angle EDF = 45 degrees
• Construct a semicircle with AF as diameter, intersecting OC at G
• Using E as center and EG as radius, construct a semicircle intersecting AB at H and K.
• Construct HL and KM, both perpendicular to AB
• Angle LOM = 720/17. Bisect LOM to get 360/17.

Here is a "particularly simple" construction. If this doesn't meet your definition of "particularly simple," maybe it would be wise to admit that trying to construct angles with ruler and compass when we have perfectly good protractors lying around is a waste of time.

• Construct a circle with diameter POB (extended) and radius OA perpendicular to POB
• Construct C with OC = OB/4 (The reason this feature appears in both constructions is that AC = AOsqrt(17).)
• Using C as center and CA as radius, construct a semicircle meeting POB at D and E.
• Using E as center, draw arc AF.
• Using D as center, draw arc AG.
• Construct H, the midpoint of BF
• Using H as center, draw semicircle BJF.
• Construct K, the midpoint of OG
• Using OK as radius and J as center, draw an arc meeting OP at L
• KL = PQ = the side of an inscribed 34-gon.

The side of a regular 17-gon inscribed in a unit circle (radius = 1) is:

(1/4)sqrt{34 - sqrt(17) - sqrt[34 - 2sqrt(17)] - 2sqrt(17 + 3sqrt(17) + sqrt[170 - 26sqrt(17)] - 4sqrt[34 + 2sqrt(17)])}

References

The constructions are found in, respectively,

Squaring the circle, and other monographs, New York, Chelsea Pub. Co. 1953. 

Robin Hartshorne, Geometry: Euclid and Beyond, Springer, 2000

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Created 21 January, 2003,  Last Update 22 September, 2003

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