Feb 9, 2010

Constructing the Heptadecagon


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:  heptadecagon  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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