Crthuky

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

The curvilinear triangles which are characteristic for this type of plot are called hypocyloid 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.