From: ken@straton.demon.co.uk (Ken Starks)
Newsgroups: sci.math
Subject: Re: Trisection of angle
Date: Thu, 24 Sep 1998 18:34:20 BST

graham_fyffe@hotmail.com wrote:

> AFAIK, the original problem of trisecting the angle does not allow any
> kind of constructions to help you do it.  This is kind of vague, I know.
>  Anyway, some guy whose name has slipped my mind had come up with a way
> to construct a bunch of curves on a sheet of paper, using the allowed
> tools, which enabled him to trisect arbitrary angles.  The math
> community said "that doesn't count, since you had to construct that big
> ugly thing first."

> So, it all depends on how you define "construct", and where you draw the
> line (pardon the pun)

Perhaps you should try a book by George E Martin:
"Geometric Constructions" ( Springer-Verlag ISBN 0-387-98276-0)

This covers, in modern notation, many kinds of construction including
the trisection of an angle using a 'tomohawk'

The chapter headings will give you some idea.
   Euclidean Constructions
   The Ruler and compass
   The Compass and the Mohr-Mascheroni Theorem
   The ruler
   The ruler and dividers
   The Poncelot-steiner Theorem and double rulers
   The ruler and rusty compass
   Sticks
   The marked ruler
   Paperfolding
   
The result you need for the smallest constructable angle
( with straight edge and compass ) as an integral number of
degrees is:

The Gauss-Wantzel Theorem.
--------------------------
   A regular n-gon is constructable with ruler and compass iff 
   n is an integer greater than 2 such that the greatest odd
   factor of n is either 1 or a product of distinct Fermat 
   primes.
   
   A 'Fermat prime' is one of the form  ((2^(2^n))-1), and there
   are 4 known:
   
   3, 5, 17, 257, 65537
   
   
   For a whole number of degrees in your n-gon  you therefore need 
   a to factorise  360 into one of
   n x  3 x E 
   n x  5 x E
or n x 15 x E   where E is a power of two.
   
   The Largest n is obviously 120, giving a three degree angle.
   
By the way, Gauss proved that a 17-gon is construcable when he was 19.

As well as a construction, you also get a formula:
16 cos(360/17)° = 
    -1 + root17 + root(34 - 2root17)
    +2root( 17 + 3root17 - root(34 - 2root17) - 2root(34+2root17) )