Constructing the Heptadecagon

The ancient Greek geometers devoted considerable thought to the
question of which regular n-gons could be constructed by straightedge
and compass. They knew how to construct an equilateral triangle
(3-gon), a square (4-gon), and a regular pentagon (5-gon), and of
course they could double the number of sides of any polygon simply
by bisecting the angles, and they could construct the 15-gon by
combining a triangle and a pentagon. For over 2000 years no other
constructible n-gons were known.

Then, on 30 March 1796, the 19 year old Gauss discovered that it
was possible to construct the regular heptadecagon (17-gon). (This
discovery apparently convinced him to pursue a career in mathematics
rather than philology.) The result was announced in the "New
Discoveries" column of the journal "Intellegenzblatt der allgemeinen
Litteraturzeitung" on 1 June 1796 by A. W. Zimmermann, a professor
at the Collegium Carolinum and an early mentor of the young Gauss.

Subsequently Gauss presented this result at the end of Disquistiones
Arithmeticae, in which he proves the constructibility of the n-gon
for any n that is a prime of the form 2^(2^k) + 1, also known as
Fermat primes. 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).

One of the nicest actual constructions of the 17-gon is Richmond's
(1893), as reproduced in Stewart's "Galois Theory". Draw a circle
centered at O, and choose one vertex V on the circle. Then locate
the point A on the circle such that OA is perpindicular to OV, and
locate point B on OA such that OB is 1/4 of OA. Then locate the
point C on OV such that angle OBC is 1/4 the angle OBV. Then find
the point D on OV (extended) such that DBC is half of a right angle.

Let E denote the point where the circle on DV cuts OA. Now draw a
circle centered at C through the point E, and let F and G denote
the two points where this circle strikes OV. Then, if perpindiculars
to OV are drawn at F and G they strike the main circle (the one
centered at O through V) at points V3 and V5, as shown below:

The points V, V3, and V5 are the zeroth, third, and fifth vertices
of a regular heptadecagon, from which the remaining vertices are
easily found (i.e., bisect angle V3 O V5 to locate V4, etc.).

Gauss was clearly fond of this discovery, and there's a story that
he asked to have a heptadecagon carved on his tombstone, like the
sphere incribed in a cylinder on Archimedes' tombstone. The story
is probably apochryphal, because if Gauss had seriously wanted such
a monument located in the proximity of his actual remains, it would
have to be placed, not at his grave site, but above the jar in the
anatomical collection of the University of Gottingen where his brain
has been preserved (rather goulishly, in my opinion). On the other
hand, if proximity to the actual remains is not important, then the
heptadecagon on the monument to Gauss in his native town of Brunswick,
or even the figure above, may suffice.