I have just completed my first construction of the regular heptadecagon — a construction that even the ancient Greeks were never able to figure out. They did figure out how to construct a regular pentadecagon (by combining the constructions for the regular pentagon and triangle), and I once replicated that discovery, meaning that I figured it out independently.
The regular heptadecagon construction, however, I did not figure out independently. I used instructions found here (http://www.mathpages.com/home/kmath487.htm), which built on the work of Carl Friedrich Gauss, who, in 1796, at the age of 19, became the first person in history to determine that such a construction is possible with the traditional Euclidean tools.
A word of warning, if you attempt to replicate this construction yourself: points M and G are merely close together, but are not in the same place. Point M is the center of the circle which passes through points D and V17, while point G is one of the two points of intersection of (1) the line passing through points O and V17, and (2) the circle centered at C, and passing through E.
Gauss (and other mathematicians, building on his work) also showed, later, that constructions are possible for regular polygons with 257 sides, as well as 65,537 sides. I might, someday, replicate the construction of the regular polygon with 257 sides.
A man named Johann Gustav Hermes once spent ten years completing a 200-page manuscript showing how to construct the regular polygon with 65,537 sides, and I believe he actually performed the construction, as well. I will not be constructing this polygon — ever. I will, however, figure out a proper name for it. Let’s see . . . it’s the heptakaitriacontakaipentacosioikaipentachilikaihexamyriagon. Try saying that five times in a row, quickly!