documentclass[tikz]{standalone}
    begin{document}
    begin{tikzpicture}[scale=12]
    draw (0,0) -- (1,0);
    draw (0,0) -- (0,.618);
    draw (1,0) -- (1,.618);
    draw (1,0) arc (0:180:.5);

    draw [dotted] (0,0) -- (0,-.1) node[below]{$frac{0}{1}$};
    draw [dotted] (1,0) -- (1,-.1) node[below]{$frac{1}{1}$};

    draw (1,0) arc (0:180:.25);
    draw [dotted] (.5,0) -- (.5,-.1) node[below]{$frac{1}{2}$};
    draw (.5,0) arc (0:180:.25);

    draw (1,0) arc (0:180:1/6);
    draw [dotted] (2/3,0) -- (2/3,-.1) node[below]{$frac{2}{3}$};
    draw (2/3,0) arc (0:180:1/12);

    draw (1/3,0) arc (0:180:1/6);
    draw [dotted] (1/3,0) -- (1/3,-.1) node[below]{$frac{1}{3}$};
    draw (1/2,0) arc (0:180:1/12);

    end{tikzpicture}
    end{document}

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

defmyrecur#1#2#3#4#5{
recurdepth=#5
ifnumtherecurdepth>1relax
  advancerecurdepth by-1relax
  edeftempnum{numbernumexpr#1+#3relax}%a+b
  edeftempden{numbernumexpr#2+#4relax}%c+d
  pgfmathparse{tempnum/tempden}edeftemp{pgfmathresult}%(a+b)/(c+d)

  node[below=therecurdepth*1pt,scale=0.1*therecurdepth]at({(temp)*1cm},0){$frac{tempnum}{tempden}$};
  draw[ultra thin,opacity=therecurdepth/10] ({(temp)*1cm},0) arc (180:0:{((#3/#4)-temp)*0.5cm});
  draw[ultra thin,opacity=therecurdepth/10] ({(temp)*1cm},0) arc (0:180:{(temp-(#1/#2))*0.5cm});
  begingroup
    edefttempup{noexpandmyrecur{tempnum}{tempden}{#3}{#4}{therecurdepth}}
    edefttempdown{noexpandmyrecur{#1}{#2}{tempnum}{tempden}{therecurdepth}}
    ttempupttempdown
  endgroup
fi
}

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

end{tikzpicture}
end{document}

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}

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.

last_level = 10
labelled_level = 3

label_dist = 0.1
tikz_scale = 12

axis_height = 0.618

def farey_sequence(n):
    """Returns the nth Farey sequence in ascending order."""
    farey_list = []
    a, b, c, d = 0, 1, 1, n
    farey_list.append("%d/%d" % (a, b))
    while c <= n:
        k = int((n + b) / d)
        a, b, c, d = c, d, k * c - a, k * d - b
        farey_list.append("%d/%d" % (a, b))

    return farey_list

def tikz_farey_sequence(n):
    """Returns the nth Farey sequence, suitable for processing in TikZ."""
    farey = farey_sequence(n)
    twinned_pairs = [farey[i] + '/' + farey[i+1] for i in xrange(len(farey) - 1)]
    return str(twinned_pairs)[1:-1].replace("'", "")

tikz_lines = [
    '\begin{tikzpicture}[scale=%d]' % tikz_scale,
    '',
    '\draw (0,%f) -- (0,0) -- (1,0) -- (1,%f);' % (axis_height, axis_height),
    ''
]

for lvl in range(last_level):
    new_lines = [
        '\foreach \a/\b/\c/\d in { %s } {' % tikz_farey_sequence(lvl + 1),
        '  \draw (\a/\b, 0) arc (180:0:\c / \d / 2 - \a / \b / 2);']

    if lvl == labelled_level:
        new_lines.extend([
        '  \draw [dotted] (\a/\b, 0) -- (\a/\b, -%f) node [below] {$\frac{\a}{\b}$};' % label_dist,
        '}',
        '\draw [dotted] (1/1, 0) -- (1/1, -%f) node [below] {$\frac{1}{1}$};' % label_dist,
        ''])
    else:
        new_lines.extend(['}',''])


    tikz_lines.extend(new_lines)

tikz_lines.append('\end{tikzpicture}')

print 'n'.join(tikz_lines)

documentclass[border=5]{standalone}
usepackage{pgffor}
{catcode`%=11gdefpc{%}}
directlua{
Rational = {}
Rational.__index = Rational

function Rational.new(p, q)
  local a, b, object
  a, b = p, q
  while b > 0 do
    a, b = b, a pc b
  end
  object = {p=p/a, q=q/a}
  setmetatable(object, Rational)
  return object
end

function Rational:toString()
  return "" .. self.p .. "/" .. self.q
end

function Rational.__eq(P, Q)
  return (P.p == Q.p) and (P.q == Q.q)
end

function Rational.__add(P, Q)
  return Rational.new(P.p*Q.q + Q.p*P.q, P.q*Q.q)
end

function Rational.__sub(P, Q)
  return Rational.new(P.p*Q.q - Q.p*P.q, P.q*Q.q)
end

Sequence = {}
Sequence.__index = Sequence

function Sequence.new()
  local object = {data={}, n=0}
  setmetatable(object, Sequence)
  return object
end

function Sequence:get(i)
  return self.data[i]
end

function Sequence:append(v)
  table.insert(self.data, v)
  self.n = self.n + 1
end

function Sequence:toString()
  local s, i
  s = ""
  if self.n > 0 then
    s = s .. self.data[1]:toString()
    if self.n > 1 then
      for i = 2,self.n do
        s = s .. "," .. self.data[i]:toString()
      end
    end
  end
  return s
end

function farey_sequence(n)
  local f1, f2, t1, t2, i, j
  f1 = Sequence.new()
  f1:append(Rational.new(0,1))
  f1:append(Rational.new(1,1))
  if n > 1 then
    for i = 1,n do
      f2 = f1
      f1 = Sequence.new()
      for j = 1, f2.n-1 do
        t1 = f2:get(j)
        t2 = f2:get(j+1)
        f1:append(t1)
        if (t2-t1 == Rational.new(1, t1.q*t2.q)) and t1.q+t2.q <= n then
          f1:append(Rational.new(t1.p+t2.p, t1.q+t2.q))
        end
      end
      f1:append(t2)
    end
  end
  return f1
end
}

defgetfareysequence#1#2{%
  edef#1{directlua{tex.print(farey_sequence(#2):toString())}}%
}
begin{document}
begin{minipage}{4in}
foreach i in {1,...,8}{
 getfareysequenceA{i}
 $F_{i}=left{foreach n/d [count=k] in A {ifnumk>1,fifrac{n}{d}}right}$
\[.5ex]
}
end{minipage}
end{document}