Date: Thu, 26 Oct 95 15:22:12 CDT
From: rusin (Dave Rusin)
To: u01iea@abdn.ac.uk
Subject: Re: Constructability of the regular N-gon
My knowledge of the history of math is just like everyone else's:
full of anecdotes which are untrue and attributions which are misapplied.
So I'd best not help you there.
But the mathematics of this problem is well known and of course rather
pretty. It should be covered in a second-semester abstract algebra course.
I would encourage you to consult some texts and present the details.
You can construct an N-gon with compass and _unmarked_ straightedge iff
N=2^k * N1 where N1 is a product of distinct Fermat primes. These
include 2^(2^n)+1 for n=0,1,2,3,4; no others are known, although
several other n's have been checked and found to give composite numbers.
(I would say most people would rather wager there are no more if they had
to choose one position over the other.) It's easy enough to give drawing
instructions for the N-gon if you know how to draw the p-gon for each
odd p | N (In fact, if by divine intervention you have an N1-gon and
an N2-gon you can draw the gcd(N1,N2)-gon.) I suppose you need no
help drawing a regular triangle, and a pentagon is also easy (although
the most interesting way to draw it is to take a narrow strip of paper,
perhaps ripped off a page of formfeed paper, tie an overhand knot into
it, and then force it to lie flat). I have seen a 17-gon, and I could
probably recreate the process used to draw it. I have heard it repeatedly
asserted that someone actually set to paper the instructions for
drawing 257- and 65537-gons; the manuscripts are usually said to be
collecting dust in some German university library. (Oops -- we're back
to anecdotes.) I can draw a nonagon (9 sides) by trisecting a 60-degree angle,
a feat which requires cheating by using a marked straightedge. I don't
how much bending of the rules is needed to draw a 7-gon (may be
possible with the same cheat), 11-gon, 25-gon, etc.
Your talk might branch off in other directions, of course, taking in
connections with other constructibility questions (circle-squaring, e.g.),
other polygonal matters (e.g. Poncelot's theorem), regular polyhedra
(e.g. the classification of the Platonic solids), tesselations (e.g.
Penrose tilings), or other topics of local interest.
Best of luck.
dave