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© Copyright 2000, Jim Loy
Karl Friedrich Gauss proved which
Geometric Constructions were possible and which were
impossible. In the process, he proved that a regular nonagon was impossible to
construct, and he showed how to construct a 17-gon (heptadecagon). I don't know
who came up with the method that I show here. I have modified it slightly.
Draw the large circle O. Draw a diameter AB. Draw a diameter perpendicular to that diameter. Bisect one of the radii on this line. Bisect it again, to get point C in the diagram. Draw line AC. With C as a center, draw an arc with radius CA, from A to the vertical diameter in the diagram. Bisect this arc. Bisect it again, to get point D in the diagram. Draw line CD, which then intersects line AB at point E. Draw line CF at 45 degrees to line CE, as in the diagram (so F is on AB). Bisect line AF and draw the circle with AF as its diameter. This circle intersects the vertical diameter of the big circle at a point G. Draw the circle with center E and radius EG. This intersects line AB at H and I. Draw lines perpendicular to AB, at points H and I. These intersect the big circle at J and K. Bisect arc JK, producing point L. J, K, L, and A are vertices of the 17-gon. You can use these points to find the rest of the vertices, and connect them to produce the 17-gon.
Above is one of the equations (slightly modified) that Gauss derived, which apparently led to the above construction. Pi/17 is the angle in radians. There are 2 pi radians in a circle. Gauss deduced a similar equation for the cos (pi/17). In our construction, angle JOK is 4pi/17. So apparently, a somewhat different equation was used to do the construction.
Does the above equation look
daunting? First of all, constructing the square root of 17 is easy (see the
diagram on the left). That is why we bisected one of the radii twice. And there
is a great deal of repetition in the above equation. if you construct
sqr(17-sqr(17)), where sqr() is the square root, you won't have to do it again
later. Each step is simple. It's just that there are a lot of steps. Of course,
deriving the above equation was the hard part.
Also see Trisection Of An Angle. In Braunschweig (Brunswick) Germany, where Gauss was born, there is a statue of Gauss, standing on a 17 pointed star.