FordCirclesExercice.tex

\documentclass[tikz]{standalone}
\usetikzlibrary{math,spy}
\tikzmath{
  let \startcolor=blue; let \endcolor=red; % --- the start and end colors
  function FordCircles(\a,\b,\n){
    int \p, \q; % ------------------------------ p and q are integers
    for \q in {1,...,\n}{ % -------------------- 0 < q <= n
      int \mix; \mix = 100*(\q-1)/(\n-1); % ---- color mix parameter in [0,100]
      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
          {
            \filldraw[ultra thin,fill=\startcolor!\mix!\endcolor,fill opacity=.77, text opacity=1]
              (\f,\r) circle(\r) % --- and draw the Ford circle at (f,r)
                  node[white,scale=3*\r]{$\frac{\p}{\q}$};
          };
        };
      };
    };
  };
}
\begin{document}
  \begin{tikzpicture}[spy using outlines={rectangle, size=10mm, connect spies}]
    \draw[help lines,step=2mm] (-1,-0.1) grid (4,1.1);
    % Draw the Ford circles
    \tikzmath{FordCircles(-.9,3.9,8);}

    % Magnification on some regions
    \spy[magnification=3,green]
      on (0.41,0.1) in node [left] at (1,-.7);
    \spy[magnification=7,green]
      on (pi,0) in node [left] at (3,-.7);
  \end{tikzpicture}
\end{document}