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.