Steven Dutch, Natural and Applied Sciences, Universityof Wisconsin - Green Bay
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One of the most unexpected discoveries in mathematics was the discovery bythe young Karl Friedrich Gauss that it was possible, using the rules of theancient Greeks, to construct a regular 17-sided polygon using a ruler andcompass alone. He showed that only certain polygons could be constructed, and inthe process showed that all others could not. One of the polygons that cannot beconstructed is a 9-sided polygon. Since it would be possible to construct a9-sided polygon if you could trisect a 120-degree angle, therefore trisectingan 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 meetyour definition of "particularly simple," maybe it would be wise toadmit that trying to construct angles with ruler and compass when we haveperfectly 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 24 May, 2020
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