Btukfyl

How can I plot the following diagram for a Farey series?

The curvilinear triangles which are characteristic for this type of plot are called hypocycloid curves. We can use the parametric equations on Wikipedia to plot these, like so:

x[a_, b_, t_] := (b - a) Cos[t] + a Cos[(b - a)/a t]

y[a_, b_, t_] := (b - a) Sin[t] - a Sin[(b - a)/a t]

hypocycloid[n_] := ParametricPlot[

  {x[1/n, 1, t], y[1/n, 1, t]},

  {t, 0, 2 Pi},

  PlotStyle -> {Thickness[0.002], Black}

  ]



Show[

 Graphics[Circle[{0, 0}, 1]],

 hypocycloid[2],

 hypocycloid[4],

 hypocycloid[8],

 hypocycloid[16],

 hypocycloid[32],

 hypocycloid[64],

 ImageSize -> 500

 ]

I've previously written about an application of hypocycloids here, and I showed how to visualize epicycloids here.

How to generate the labels is described here (also linked to by moo in a comment). I will simply provide the code.

mediant[{a_, b_}, {c_, d_}] := {a + c, b + d}

recursive[v1_, v2_, depth_] := If[

  depth > 2,

  mediant[v1, v2], {

   recursive[v1, mediant[v1, v2], depth + 1],

   mediant[v1, v2],

   recursive[mediant[v1, v2], v2, depth + 1]

   }]



computeLabels[v1_, v2_] := Module[{numbers},

  numbers = 

   Cases[recursive[v1, v2, 0], {_Integer, _Integer}, Infinity];

  StringTemplate["``/``"] @@@ numbers

  ]

computeLabelsNegative[v1_, v2_] := Module[{numbers},

  numbers = 

   Cases[recursive[v1, v2, 0], {_Integer, _Integer}, Infinity];

  StringTemplate["-`2`/`1`"] @@@ numbers

  ]



labels = Reverse@Join[

    {"1/0"},

    computeLabels[{1, 0}, {1, 1}],

    {"1/1"},

    computeLabels[{1, 1}, {0, 1}],

    {"0/1"},

    computeLabelsNegative[{1, 0}, {1, 1}],

    {"-1,1"},

    computeLabelsNegative[{1, 1}, {0, 1}]

    ];



coords = CirclePoints[{1.1, 186 Degree}, 64];



Show[

 Graphics[Circle[{0, 0}, 1]],

 hypocycloid[2],

 hypocycloid[4],

 hypocycloid[8],

 hypocycloid[16],

 hypocycloid[32],

 hypocycloid[64],

 Graphics@MapThread[Text, {labels, coords}],

 ImageSize -> 500

 ]

I looked up the Farey sequence on Wikipedia, out of curiosity, because I had not heard of it before. The Farey sequence of order $n$ is "the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to $n$, arranged in order of increasing size".

On that basis, you can generate the sequence as follows, for instance:

ClearAll[farey]

farey[n_Integer] := (Divide @@@ Subsets[Range[n], {2}]) ~ Join ~ {0, 1} //DeleteDuplicates //Sort

So for instance:

farey[5]

{0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1}

I am not sure how these sequences are connected with the figure you showed though.