Constructible Angles and Regular Polygons

Date: 04/17/98 at 22:20:26
From: Danny Dawson
Subject: Euclidean Construction of Polygons

I've found that you can create angles (using Euclidean Geometry) that 
are multiples of 9 and/or 15 degrees. I've used the construction of 
regular polygons to help me, but that's all I can construct. Are there 
any other angles that are constructible, and what regular polygons are 
constructible? I'd appreciate a prompt respose because I've been 
collaborating with a reporter on an article for National Mathematics 
week. Thank you.

Date: 04/20/98 at 11:56:10
From: Doctor Wilkinson
Subject: Re: Euclidean Construction of Polygons

It was very clever of you to notice the connection between 
construction of regular polygons and construction of angles. 

The problem of which regular polygons can be constructed was solved 
completely by Gauss around 1800, although there are still some 
mysteries connected with it. The angles you mention correspond to the 
regular hexagon and the regular pentagon. But there are other 
constructible regular polygons, notably the ones with 15 and 17 sides. 

The rule Gauss discovered was that a regular polygon with n sides can 
be constructed if and only if n is the product of a power of 2 and of 
distinct primes of the form 2^2^k + 1.  For k = 0, 1, 2, 3, 4, this 
gives the primes 3, 5, 17, 257, and 65537.  2^2^5 + 1 is not prime, 
however, and nobody has ever discovered another prime of this form, 
but nobody has ever proved that there aren't any more either.

Looking just at the numbers from 3 to 20, we can build the following 
table of which regular polygons can and can't be constructed:

     3    yes    3 = 2^2^0 + 1
     4    yes    4 = 2^2
     5    yes    5 = 2^2^1 + 1
     6    yes    6 = 2 * 3
     7    no
     8    yes    8 = 2^3
     9    no 
    10    yes    10 = 2 * 5
    11    no
    12    yes    12 = 2^2 * 3
    13    no
    14    no
    15    yes    15 = 3 * 5
    16    yes    16 = 2^4
    17    yes    17 = 2^2^2 + 1
    18    no
    19    no
    20    yes    20 = 2^2 * 5

Gauss was the first to discover that a regular polygon with 17 sides 
could be constructed. The construction of a regular polygon with 15 
sides was known to the ancient Greeks.

For more about constructable regular polygons, see the Dr. Math FAQ:

 http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html   

-Doctors Wilkinson and Sarah,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/