I'm looking at fractals created by a point moving inside a polygon with vx vertices. The point chooses a vertex at random and then jumps a/b of the distance towards it. For example, if vx = 3 and a/b = 1/2, the point creates the Sierpinski triangle, i.e. a 3-ring of contingent triangles, each of which is a 3-ring of contingent triangles, etc.
If vx = 5 and a/b = 1/φ (I think), the point creates a 5-ring of contingent pentagons, etc. You can see the 5-ring at this link (the colors simply record how many times a pixel has been visited by the moving point):
If vx = 6 and a/b = 2/3, the points creates a 6-ring of contingent hexagons, etc:
If a/b doesn't take these values, the polygons aren't contingent, i.e. they overlap or fail to touch at all.
My question is: what is the formula to derive a/b from vx to create the required vx-ring?
Bob Strichartz provides a formula on page 323 of this paper:
$$\frac{\sin \left(\pi/n\right)}{\sin \left(2 \pi \left\lfloor n/4\right\rfloor/n+\pi/n\right)+ \sin \left(\pi/n\right)}.$$
The first dozen values are
$$ \begin{array}{c|c|c} n & r(n) & \approx \\ \hline 2 & 1/2 & 0.5 \\ \hline 3 & 1/2 & 0.5 \\ \hline 4 & 1/2 & 0.5 \\ \hline 5 & \frac{2}{3+\sqrt{5}} & 0.381966 \\ \hline 6 & 1/3 & 0.333333 \\ \hline 7 & \frac{2}{2+\csc \left(\frac{\pi }{14}\right)} & 0.307979 \\ \hline 8 & \frac{1}{1+\cot \left(\frac{\pi }{8}\right)} & 0.292893 \\ \hline 9 & \frac{2}{2+\csc \left(\frac{\pi }{18}\right)} & 0.257773 \\ \hline 10 & \sqrt{5}-2 & 0.236068 \\ \hline 11 & \frac{2}{2+\csc \left(\frac{\pi }{22}\right)} & 0.221566 \\ \hline 12 & \frac{1}{6} \left(3-\sqrt{3}\right) & 0.211325 \\ \hline 13 & \frac{2}{2+\csc \left(\frac{\pi }{26}\right)} & 0.194246 \\ \hline \end{array} $$
Leading to the following sequence of images: