Nine-Sided Polygon

Date: 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/