Chaos game and fractal polygons

Question

Let $V=\{v_1,v_2,\cdots,v_n\}$ the set of $n$ vertices of a regular polygon. For $n=3$ the chaos game implies that with an arbitrary point on the plane $x_0$, and by applying the recursive relation $$x_{k+1}=\frac{x_k+v_r}{2}$$ where $1\le r\le n$ is a random integer at each step, then $x_k$ will ultimately lie on a Sierpiński triangle for sufficiently large $k$.

Without any prior knowledge about the technical terms and proofs (and just as a fun way of passing time), I tried to generalize this formula for $n$-polygon, i.e. $x_{k+1}=(x_k+v_r)/c(n)$ where $c(n)$ is a constant depending on $n$. It is evident that $c(n)$ has some kind of a "threshold value" for which if a lesser value is applied then the polygons will overlap on each other and vice versa. This picture illustrates my point:

After a lot of trial and error, I came up with this formula: $$c(n) =\begin{cases} a/\tan(a\pi/n)+1, & \text{if }\; n\equiv 0 \mod 4 \\ a/\sin(a\pi/n)+1, & \text{otherwise} \end{cases}\tag{*}\label{*}$$ where $a=\frac12$ for odd $n$ and $1$ if $n$ is even. I tested this for $n$ up to $20$ and it looks like a precise estimate of the threshold value of $c(n)$. Here is a simple Mathematica code for testing the results:

coeff[n_Integer]:= With[{f = If[#~Divisible~4, Tan, Sin]&, a = 2^-Mod[n, 2]},
  a/f[n][a Pi/n]+ 1.];
iterate[polygon_List]:= With[{c = coeff@Length[polygon]}, (#1 + #2)/c &];
pointset[polygon_List, n_: 10^6]:= With[{f = iterate[polygon]},
  NestList[f[RandomChoice@polygon, #]&, RandomReal/@CoordinateBounds[polygon], n]];

With[{set = pointset[CirclePoints[5]]}, (* n = 5 in this case *)
  Graphics[{AbsolutePointSize[1/2], Blue, Point[set]}, ImageSize -> Large]]

Question

I have no idea whether $\eqref{*}$ is correct in general. If so, can it be proven? I checked the wikipedia page for $n$-flake and it does mention some formulas unrelated to this, but I couldn't find anything similar to the one I came up with.

Answer 1

Let the length of one side of the $n$-sided regular polygon be 1. By the scaling transformation, the polygon must be shrunk by factor $s$ (for your notation, $c(n) = 1 / s$).

For $n=3$, $s = 1 /2$ as you can see $2s = 1$ in the following figure:

From $n=5$, $s \cos (2\pi / n)$ should be taken into consideration ($2\pi / n$ is the exterior angle):

At $n=9$, another contribution $s \cos (4\pi / n)$ starts to appear:

In general, for $n \ge 3$, \begin{equation} 2 \left[ s + \sum_{k=1}^\infty s \cos\biggl( \frac{2\pi k}{n} \biggr) \Theta\biggl( \frac{2\pi k}{n} \le \frac{\pi}{2} \biggr)\right] = 2 s \left[ 1 + \sum_{k=1}^{\lfloor n/4 \rfloor} \cos\biggl( \frac{2\pi k}{n} \biggr) \right] = 1, \end{equation} where $\Theta(b)$ is $1$ if $b$ is true and otherwise $0$, or \begin{equation} c(n) = \frac{1}{s} = 2 \sum_{k=0}^{\lfloor n/4 \rfloor} \cos\biggl( \frac{2\pi k}{n} \biggr) = 1 + \frac{\sin\Bigl(\frac{\pi}{n} \bigl(1+2\lfloor \frac{n}{4} \rfloor \bigr)\Bigr)}{\sin\Bigl(\frac{\pi}{n}\Bigr)}. \end{equation} Though it seems a bit tedious to prove your equation is equivalent to the above formula, Mathematica can check it for you up to a certain $n$:

coeff[n_] := With[{f = If[#~Divisible~4, Tan, Sin]&, a = 2^-Mod[n, 2]}, a/f[n][a Pi/n]+ 1];
coeff2[n_] := 1 + Sin[Pi / n (1 + 2 Floor[n / 4])] / Sin[Pi / n];
Table[{n, coeff[n] == coeff2[n]}, {n, 3, 100}] // FullSimplify // MatrixForm  (* all True *)