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.

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