#13-gon

"A pretty good approximate geometric construction of the triskaidecagon, or 13 pointed star. Constructing the triskaidecagon exactly is impossible using the traditional rules of compass and straightedge construction, but this construction is close - it has a cumulative relative error of .7%, or 2.51 degrees. As far as I'm aware, I've invented this. Or, as Plato would have it, I'm the first one to remember it. I wouldn't actually be surprised to find out that someone else has used this construction, because it's very simple - but then again, there probably aren't many people who go around trying to figure this kind of thing out. I'm also sure there are more precise constructions since one can do this with arbitrary precision, but they're probably more complex. First, draw a circle centered at A. Then construct the square BCDE inside the circle. Bisect the segment AF at G. The bisection will mark point H on the circle. Construct a circle centered at H with radius HG. This circle will encompass 5 points of the triskaidecagon. Construct a circle with center at J and radius JH, then draw line AK by connecting the points where circles H and J intersect. HL and JL are sides of the triskaidecagon. You can construct the remaining sides by stepping this radius around the circle. If you want to distribute the error more effectively than just stepping around the circle 13 times, you can use the point where an extension of HG crosses the circle and the point where the vertical diameter of the circle crosses the circumference as reference points to step from. The angle HAL is 27.885567º, the angle subtended by the side of a perfect triskaidecagon is 27.692308."