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) )