From: ksbrown@seanet.com (Kevin Brown)
Newsgroups: sci.math
Subject: Re: HEPTADECAGON, regular:   Help please!
Date: Thu, 13 Aug 1998 07:49:00 GMT

On 12 Aug 1998 mathwft@math.canterbury.ac.nz (Bill Taylor) wrote:
> There is a particularly slick construction of a regular 17-gon that 
> I would like to see again...  the construction of the regular 
> heptadecagon I am looking for... may have even been so simple 
> as the above [constrtuction of pentagon], but with quarter-sections 
> replacing the bisections...

You may be thinking of Richmond's construction (1893), as reproduced
in Stewart's "Galois Theory".  The proof begins with two perpindicular
radii OA and OB in a circle centered at O.  Then locate point I on OB
such that OI is 1/4 of OB.  Then locate the point E on OA such that
angle OIE is 1/4 the angle OIA.  Then find the point F on OA
(extended) such that EIF is half of a right angle.

Let K denote the point where the circle on AF cuts OB.  Now draw a
circle centered at E through the point K, and let N3 and N5 denote
the two points where this circle strikes OA.  Then, if perpindiculars
to OA are drawn at N3 and N5 they strike the main circle (the one
centered at O through A and B) at points P3 and P5.

The points A, P3, and P5 are the zeroth, third, and fifth verticies
of a regular heptadecagon, from which the remaining verticies are
easily found (i.e., bisect angle P3 O P5 to locate P4, etc.).


On 12 Aug 1998 mathwft@math.canterbury.ac.nz (Bill Taylor) wrote:
> Dunno if it was Gauss' original, I tend to doubt it.  

Gauss's Disquisitiones gives only the algebraic expression for the
cosine of 2pi/17 in terms of nested square roots, i.e.,

  cos(2pi/17)  =  -1/16 + 1/16 sqrt(17) + 1/16 sqrt[34 - 2sqrt(17)]

 + 1/8 sqrt[17 + 3sqrt(17) - sqrt(34-2sqrt(17)) - 2sqrt(34+2sqrt(17)]

which is just the solution of three nested quadratic equations.
Interestingly, although Gauss states in the strongest terms (all caps)
that his criteria for constructibility (based on Fermat primes) is
necessary as well as sufficient, he never published a proof of the
necessity, nor has any evidence of one ever been found in his papers
(according to Buhler's biography).


On 12 Aug 1998 mathwft@math.canterbury.ac.nz (Bill Taylor) wrote:
> I think it was REALLY mean of the town council of Gottingen, or
> the executors of his will, or whoever, to NOT put a regular 17-gon
> on his gravestone, as he requested.  Especially as that was his 
> most proud theorem; rather like Archimedes and his cylinder-sphere 
> grave icon.

I've heard that this story is apochryphal (about Gauss, not about
Archimedes), but it's apparently true that Gauss's discovery of 
the 17-gon's constructibility (which had been an open problem from
antiquity) at or before the age of 19 led to his decision to follow 
a career in mathematics rather than philology.