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.

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.

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)])}


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 02 June, 2010

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