FordCircles.tex

\documentclass[tikz]{standalone}
\usetikzlibrary{math}
\tikzmath{
  function FordCircles(\a,\b,\n){
    int \p, \q; % ------------------------------ p and q are integers
    for \q in {1,...,\n}{ % -------------------- 0 < q <= n
      for \p in {\a*\q,...,\b*\q}{ % ----------- a < p/q < b <=> [aq] < p < [bq]
        if gcd(\p,\q) == 1 then { % ------------ if the fraction is irreducible
          \f = \p/\q; % ------------------------ evaluate the tuch point f = p/q
          \r = 1/(2*\q*\q); % ------------------ evaluate the radius r = 1/2q^2
          {
            \draw[red] (\f,\r) circle(\r); % --- and draw the Ford circle at (f,r)
          };
        };
      };
    };
  };
}
\begin{document}
  \begin{tikzpicture}
    \draw[help lines,step=2mm] (-1,-0.1) grid (4,1.1);
    % Draw the Ford circles
    \tikzmath{FordCircles(-.9,3.9,8);}
  \end{tikzpicture}
\end{document}