Identification the type of curve (fractal?)

Question

First we have a square inscribed in a circle with radius $1$. By connection vertexes of this square we have two diagonals, which divides square for $4$ rectangular triangles with (at least) one corner equals $\frac{\pi}{4}$. Then we draw bisector to each center right angle. By connection vertexes of square with two nearest intersections of bisector and circle, we have octagon. Ignoring divide by bisector first $4$ rectangular triangles (which form square), we have also new $8$ rectangular triangles with one corner equals $\frac{\pi}{8}$. Repeating this operation (animation) to $2^n$-gons we have $$C=\lim\limits_{n\to\infty}2^{n+1}\sin\left(\frac{\pi}{2^n}\right)=2\pi$$ $$S=\sum\limits_{k=2}^{n}2^k\sin\left(\frac{\pi}{2^{k-1}}\right)\sin^2\left(\frac{\pi}{2^k}\right)= 2^{n-1}\sin\left(\frac{\pi}{2^{n-1}}\right)$$ $$\lim\limits_{n\to\infty}2^{n-1}\sin\left(\frac{\pi}{2^{n-1}}\right)=\pi$$ where $C$ is a perimeter of $2^n$-gon and $S$ is area (as sum areas of $n$ new rectangular triangles from each iteration). By multiplying expression under the sum with $(-1)^{n}$ we have curve with same perimeter, but another area (animation). What is the type of curve is it? Looks like fractal with changing angle and alternate type of expansion (inside and outside).