# How can I plot a Farey diagram?

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

• From the beautiful book A. Hatcher Topology of numbers Apr 8, 2019 at 21:16
• Could you perhaps expand a bit on how the curves are calculated etc? Apr 8, 2019 at 21:40
• pi.math.cornell.edu/~hatcher/TN/TNch1.pdf
– Moo
Apr 8, 2019 at 23:17
• Technically this is not a Farey series/sequence $F_n$ of order $n$, which is defined to be all fractions (sometimes restricted to the interval between 0 and 1) with denominator at most $n$. For example 3/8 is present but not 1/8. It's a recursive mediant subdivision. It's related in that in any three successive terms of a Farey sequence, the middle one is the mediant of the other two. Apr 9, 2019 at 17:44
• If it wasn't for the very good answers you got, I would have voted to close this question as it gives no details, no definitions no code and shows no personal effort. Please, next time try asking good questions. Apr 11, 2019 at 9:18

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],
ImageSize -> 500
]


Using Graph with a bit of coding:

addPoint[{p : h_[a_,b_], q : h_[c_,d_]}, i_] :=
With[{np = h[a + c, b + d]}, Sow[{p \[UndirectedEdge] np, np \[UndirectedEdge] q}]; Sow[{i, i}, "Depth"]; {p, np, q}]

addPoint[{p : h_[a_,b_], q : h_[-1][c_,d_]}, i_] :=
With[{np = h[-1][a + c, b + d]}, Sow[{p \[UndirectedEdge] np, np \[UndirectedEdge] q}]; Sow[{i, i}, "Depth"]; {p, np, q}]

addPoint[{p : h_[-1][a_,b_], q : h_[c_,d_]}, i_] :=
With[{np = h[-1][a + c, b + d]}, Sow[{p \[UndirectedEdge] np, np \[UndirectedEdge] q}]; Sow[{i, i}, "Depth"]; {p, np, q}]

addPoint[{p : h_[-1][a_,b_], q : h_[-1][c_,d_]}, i_] :=
With[{np = h[-1][a + c, b + d]}, Sow[{p \[UndirectedEdge] np, np \[UndirectedEdge] q}]; Sow[{i, i}, "Depth"]; {p, np, q}]

fLabel[fr_, angle_] :=
With[{tangle=ArcTan@@angle}, Placed[fLabel[fr], AngleVector[{1/2, 1/2}, {.7, #}] & /@{tangle, tangle+Pi}]]

fLabel[h_[a_, b_]] := ToString[a] ~~ "/" ~~ ToString[b]
fLabel[h_[-1][a_, b_]] := "-" ~~ ToString[a] ~~ "/" ~~ ToString[b]

FareyDiagram[n_Integer, d_Integer: 1, opts___?OptionQ] :=
Block[{fr, top, bottom, stedges, toppart, bottompart, vert, edges, coords, labels, labpos, cfunc, i, edgestyle, dstyle, nopts},
cfunc = ColorFunction /. Flatten[{opts}] /. ColorFunction -> Automatic;
nopts = FilterRules[Flatten[{opts}], Options[Graph]];
top = {fr[0,1], fr[1,1], fr[1,0]};
bottom = {fr[1,0], fr[-1][1,1], fr[0,1]};
stedges = UndirectedEdge@@@Join[Partition[top, 2, 1], Partition[bottom, 2, 1], {{fr[0, 1],fr[1, 0]}}];
i = 0;toppart = Reap[Nest[(i++; Split[Flatten[addPoint[#, i] & /@ Partition[#, 2, 1],1]][[All,1]])&, top, n]];
i = 0;bottompart = Reap[Nest[(i++; Split[Flatten[addPoint[#, i] & /@ Partition[#,2,1],1]][[All,1]])&,bottom, n]];
vert = Join[toppart[[1]], bottompart[[1,  2;;-2]]];
edges = Flatten[{stedges, toppart[[2, 1]], bottompart[[2, 1]]}];
coords = CirclePoints[{1,0},Length[vert]];
labpos = Range[1, Length[vert], 2 ^ (d - 1)];
edgestyle = Black;
dstyle = Black;
If[cfunc =!= Automatic,
edgestyle = Flatten[{Table[0, Length[stedges]], toppart[[2, 2]], bottompart[[2, 2]]}];
edgestyle = edgestyle / Max[edgestyle];
edgestyle = Thread[edges -> Flatten[cfunc[1 - #] & /@ edgestyle]];
dstyle = cfunc[1]
];
Graph[vert, edges, nopts, VertexCoordinates->CirclePoints[{1,0},Length[vert]], VertexLabels->labels,
EdgeShapeFunction->(BSplineCurve[{#1[[1]],{0,0},#1[[2]]}, SplineWeights->{2,EuclideanDistance@@#,2}]&),
PerformanceGoal->"Speed", Epilog->{dstyle, Circle[]}, VertexShapeFunction -> "Point", EdgeStyle -> edgestyle, VertexStyle -> dstyle]
]


Example:

FareyDiagram[4]


FareyDiagram[6, 4, ColorFunction -> Hue,
VertexLabelStyle -> Darker[Red]]


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.

• Thanks to C.E., it is a concrete answer Apr 9, 2019 at 12:58
• Built in too: FareySequence Jan 10, 2020 at 20:24
• @KennyColnago Oh man! How did I miss that?! Thank you :-) Jan 10, 2020 at 22:08

While reading an excellent post on chord diagrams, I realized that this type of figure is a Poincaré hyperbolic disk. The lines are Poincaré arcs. There is MathWorld article with code here.

So, copying the definitions of PoincareArc and PoincareDisk from MathWorld, we can draw the figure like this:

PoincareArc[l0_List] :=
Module[{l = Sort[l0], dt, t, t1, t2, r, R, c},
dt = Abs[l[[1]] - l[[2]]];
t = Plus @@ l/2;
If[
dt == Pi,
Line[{
{Cos[l[[1]]], Sin[l[[1]]]},
{Cos[l[[2]]], Sin[l[[2]]]}
}], c = {Cos[t], Sin[t]};
r = Tan[dt/2];
R = Sec[dt/2];
t1 = ArcTan @@ ({Cos[l[[2]]], Sin[l[[2]]]} - R c);
t2 = ArcTan @@ ({Cos[l[[1]]], Sin[l[[1]]]} - R c);
If[t2 < t1, t2 += 2 Pi];
Circle[R c, r, {t1, t2}]]]

PoincareDisk[l_List] :=
Module[{i}, {{Thickness[.001], Circle[{0, 0}, 1]},
Table[PoincareArc[l[[i]]], {i, Length[l]}]}]

angles[n_] := Partition[Table[th, {th, 0, 2 Pi, 2 Pi/n}], 2, 1]

Graphics[{
PoincareDisk@angles[2],
PoincareDisk@angles[4],
PoincareDisk@angles[8],
PoincareDisk@angles[16],
PoincareDisk@angles[32],
PoincareDisk@angles[64]
}]


An alternative way to use Graph: Using CycleGraphs with 2^k vertices and the internal edge shape function GraphComputationGraphChartDumppEdge:

Quiet[GraphComputationGraphPropertyChart[]];
eSF = GraphComputationGraphChartDumppEdge[#2[[1]], blah, blah, ##] &;

Show[CycleGraph[2^#, VertexSize -> 0, EdgeShapeFunction -> eSF,
EdgeStyle -> Black] & /@ Range[5], Epilog -> Circle[]]


Alternatively, use a function to define the edge list to be used in a single Graph:

ClearAll[vPairs]
vPairs[k_Integer] := DeleteDuplicates[Join @@
(Partition[Append[#, 1], 2, 1] & /@ Range[1, 2^k, 2^Range[0, k-1]])]

Graph[Range[2^k], vPairs[k],
EdgeStyle->Black,
VertexShape -> Graphics[{}],
Epilog -> Circle[],
EdgeShapeFunction -> eSF,
GraphLayout -> "CircularEmbedding",
ImageSize -> 300]


same picture

With colored edges:

Row[(colors = RotateRight @ ColorData[97, "ColorList"];
Show[CycleGraph[2^#, VertexSize -> 0, EdgeShapeFunction -> eSF,
EdgeStyle -> Directive[Thick, First[colors = RotateLeft[colors]]]] & /@ Range[#],
Epilog -> Circle[], ImageSize -> 300]) & /@ Range[2, 5]]


grupo[n_] := Show[{Graphics[{Thin, Red,
Circle[{0, 0}, 1, {0, Pi/2}]}]}, {Graphics[{Thin,
Map[{BSplineCurve[{#1[[1]], {0, 0}, #1[[2]]},
SplineWeights -> {2, EuclideanDistance @@
#,2}]}&,
Partition[ReIm[Exp[Pi/2 I #]] & /@
FareySequence[n], 2, 1]]}]}, {Map[Graphics[{Blue,
Point[{ReIm[Exp[Pi/2 I #]]}]}] &,
FareySequence[n]]}, PlotRange -> All]

Show[Table[grupo[n], {n, 2, 7}]]
`

• the true farey diagram based on the answers given above Apr 16, 2019 at 23:52