TikZ - Recursive Arc Drawing

Without numbers:

\documentclass[tikz]{standalone}
\begin{document}
    \begin{tikzpicture}
        \draw [ultra thick] (-8,0) -- (8,0);
        \draw [ultra thick] (0,0) circle (8);
        \foreach \i in {0,1,2,3} {%
            \draw [ultra thick] (90*\i:8) arc (270+90*\i:180+90*\i:8);}
        \foreach \i in {0,1,...,7} {%
            \draw [very thick] (45*\i:8) arc (270+45*\i:135+45*\i:3.3);}
        \foreach \i in {0,1,...,15} {%
            \draw [thick] (22.5*\i:8) arc (270+22.5*\i:112.5+22.5*\i:1.6);}
        \foreach \i in {0,1,...,31} {%
            \draw [thin] (11.25*\i:8) arc (270+11.25*\i:101.25+11.25*\i:0.8);}
        \foreach \i in {0,1,...,63} {%
            \draw [ultra thin] (5.625*\i:8) arc (270+5.625*\i:95.625+5.625*\i:0.4);}
    \end{tikzpicture}
\end{document}

It is not any different than the examples given by other languages. Only a few places where expansion needs to be taken care of. I didn't really go for the code golf but it seems working. And it gets fainter as the recursion depth increases.

\documentclass[tikz]{standalone}
\newcount\recurdepth
\begin{document}
\begin{tikzpicture}[scale=2]
\draw[style=help lines] (0,0) grid[step=0.1cm] (1,0.5);

\def\myrecur#1#2#3#4#5{
\recurdepth=#5
\ifnum\the\recurdepth>1\relax
  \advance\recurdepth by-1\relax
  \edef\tempnum{\number\numexpr#1+#3\relax}%a+b
  \edef\tempden{\number\numexpr#2+#4\relax}%c+d
  \pgfmathparse{\tempnum/\tempden}\edef\temp{\pgfmathresult}%(a+b)/(c+d)

  \node[below=\the\recurdepth*1pt,scale=0.1*\the\recurdepth]at({(\temp)*1cm},0){$\frac{\tempnum}{\tempden}$};
  \draw[ultra thin,opacity=\the\recurdepth/10] ({(\temp)*1cm},0) arc (180:0:{((#3/#4)-\temp)*0.5cm});
  \draw[ultra thin,opacity=\the\recurdepth/10] ({(\temp)*1cm},0) arc (0:180:{(\temp-(#1/#2))*0.5cm});
  \begingroup
    \edef\ttempup{\noexpand\myrecur{\tempnum}{\tempden}{#3}{#4}{\the\recurdepth}}
    \edef\ttempdown{\noexpand\myrecur{#1}{#2}{\tempnum}{\tempden}{\the\recurdepth}}
    \ttempup\ttempdown
  \endgroup
\fi
}

\myrecur{0}{1}{1}{1}{6}

\end{tikzpicture}
\end{document}

I was wondering the same thing yesterday in the comments of this answer on a close subject. For recursive drawings as this one are (relatively!) easy to do with languages closely related to (La)TeX, but external to it, as MetaPost or Asymptote. For example, here is my "quick and dirty" attempt with MetaPost on the Farey diagram illustrated in the original post. It shows how natural it is with this language to implement a recursive drawing:

input latexmp;
setupLaTeXMP(packages="amsmath", options = "12pt", textextlabel = enable, mode = rerun);

numeric u, m; 
u = 20cm; % scale
m = 8; % maximal denominator

% [a/b, c/d]: diameter, n: recursion level
def farey_diagram(expr a, b, c, d, n) = 
  draw halfcircle scaled ((c/d-a/b)*u) shifted (u*0.5[a/b,c/d], 0);
  if (n > 1) and (b+d <= m):
    label.bot("$\dfrac{" & decimal(a+c) & "}{"& decimal(b+d) & "}$", u*((a+c)/(b+d), 0));
    farey_diagram(a, b, a+c, b+d, n-1); farey_diagram(a+c, b+d, c, d, n-1);
  fi;
enddef;

beginfig(1);
  draw origin -- (u, 0);
  label.bot("$0$", origin); label.bot("$1$", (u, 0));
  % starting with 0/1 and 1/1; m levels of recursion needed
  farey_diagram(0, 1, 1, 1, m); 
endfig;
end.

So does it mean that we should better revert to external programs as MetaPost or Asymptote each time we have to draw something recursively? It would be a bit of a surprise, since I know how powerful (La)TeX packages as PSTricks, Tikz, or mfpic (the one among the three that I regularly use for my personal work) are.

EDIT Having read a bit more about Farey diagrams and series, I've tried to refine my code, only allowing the fractions with denominators lower than or equal to the number of recursion levels. Thus farey_diagram(0, 1, 1, 1, m) typesets the numbers of the Farey series Fm (and only them) and draws the corresponding semicircles.