Nine-Sided PolygonDate: 06/11/97 at 19:10:49 From: Allan Semenoff Subject: Very important How do you make a 9-sided polygon inside a circle using only a compass and a straight-edge? Date: 06/12/97 at 10:24:05 From: Doctor Wilkinson Subject: Re: Very important I suppose you mean a regular 9-sided polygon. (If you just want any old 9-sided polygon, you can just mark any 9 points on the circle and join them up with the straight-edge). It is not possible to construct a 9-sided regular polygon with only compass and straight-edge. It was shown by Gauss in the eighteenth century that the only regular polygons that can be constructed using only a straight-edge and compass are those for which the number of sides is of the form: 2^n * p where p is either 1 or a so-called Fermat prime, which means a prime of the form: 2^(2^m) + 1 The first few Fermat primes are 3, 5, 17, and 257. So you can construct a regular polygon with 12 sides, for example, since this is 3 * 4, or with 34 sides, since this is 2 * 17, but not with 9 sides. I believe Gauss has a 17-sided regular polygon on his tombstone. -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 06/16/97 at 20:09:39 From: Allan Semenoff Subject: very important So are the only angles that you can trisect with a compass and straight-edge: 90, 45, 180, and 360? Thank you. Date: 06/17/97 at 11:30:29 From: Doctor Wilkinson Subject: Re: very important No, as I mentioned, for example, Gauss showed that you can construct a regular polygon with 17 sides. This means you can construct an angle of 360/17 degrees, so you can trisect an angle of 3*360/17 or 3*360/34 or 3*360/68, etc. Similarly, adding to your list, you could also trisect an angle of 45/2 = 22 1/2 degrees, 11 1/4 degrees, and so on. Nobody knows the complete answer, since nobody knows how many Fermat primes there are. -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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