\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}[scale=4,thick]
  \draw (0,0) circle (1);
  \draw (-1,0) -- (1,0);
  \foreach \x/\y in {90/1, 45/0.4142, 22.5/0.1989, 11.25/0.0985, 5.625/0.0491} {
   %                 { ..., 180/2^n / tan(pi/2^{n+1}),... }
    \foreach \z in {0,\x,...,360} {
      \draw (\z:1) arc(\z + 90:\z + 270 - \x:\y);
    }
  }
\end{tikzpicture}
\end{document}