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How can I plot a Farey diagram?


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2












$begingroup$


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



enter image description here










share|improve this question









New contributor




Gustavo Rubiano is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    From the beautiful book A. Hatcher Topology of numbers
    $endgroup$
    – Gustavo Rubiano
    7 hours ago










  • $begingroup$
    Could you perhaps expand a bit on how the curves are calculated etc?
    $endgroup$
    – MarcoB
    6 hours ago










  • $begingroup$
    pi.math.cornell.edu/~hatcher/TN/TNch1.pdf
    $endgroup$
    – Moo
    5 hours ago
















2












$begingroup$


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



enter image description here










share|improve this question









New contributor




Gustavo Rubiano is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    From the beautiful book A. Hatcher Topology of numbers
    $endgroup$
    – Gustavo Rubiano
    7 hours ago










  • $begingroup$
    Could you perhaps expand a bit on how the curves are calculated etc?
    $endgroup$
    – MarcoB
    6 hours ago










  • $begingroup$
    pi.math.cornell.edu/~hatcher/TN/TNch1.pdf
    $endgroup$
    – Moo
    5 hours ago














2












2








2


1



$begingroup$


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



enter image description here










share|improve this question









New contributor




Gustavo Rubiano is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




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



enter image description here







graphics number-theory






share|improve this question









New contributor




Gustavo Rubiano is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|improve this question









New contributor




Gustavo Rubiano is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|improve this question




share|improve this question








edited 1 hour ago









Michael E2

150k12203482




150k12203482






New contributor




Gustavo Rubiano is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 7 hours ago









Gustavo RubianoGustavo Rubiano

113




113




New contributor




Gustavo Rubiano is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Gustavo Rubiano is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Gustavo Rubiano is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • $begingroup$
    From the beautiful book A. Hatcher Topology of numbers
    $endgroup$
    – Gustavo Rubiano
    7 hours ago










  • $begingroup$
    Could you perhaps expand a bit on how the curves are calculated etc?
    $endgroup$
    – MarcoB
    6 hours ago










  • $begingroup$
    pi.math.cornell.edu/~hatcher/TN/TNch1.pdf
    $endgroup$
    – Moo
    5 hours ago


















  • $begingroup$
    From the beautiful book A. Hatcher Topology of numbers
    $endgroup$
    – Gustavo Rubiano
    7 hours ago










  • $begingroup$
    Could you perhaps expand a bit on how the curves are calculated etc?
    $endgroup$
    – MarcoB
    6 hours ago










  • $begingroup$
    pi.math.cornell.edu/~hatcher/TN/TNch1.pdf
    $endgroup$
    – Moo
    5 hours ago
















$begingroup$
From the beautiful book A. Hatcher Topology of numbers
$endgroup$
– Gustavo Rubiano
7 hours ago




$begingroup$
From the beautiful book A. Hatcher Topology of numbers
$endgroup$
– Gustavo Rubiano
7 hours ago












$begingroup$
Could you perhaps expand a bit on how the curves are calculated etc?
$endgroup$
– MarcoB
6 hours ago




$begingroup$
Could you perhaps expand a bit on how the curves are calculated etc?
$endgroup$
– MarcoB
6 hours ago












$begingroup$
pi.math.cornell.edu/~hatcher/TN/TNch1.pdf
$endgroup$
– Moo
5 hours ago




$begingroup$
pi.math.cornell.edu/~hatcher/TN/TNch1.pdf
$endgroup$
– Moo
5 hours ago










2 Answers
2






active

oldest

votes


















3












$begingroup$

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
]


Mathematica graphics



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
]


Mathematica graphics






share|improve this answer











$endgroup$





















    0












    $begingroup$

    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.






    share|improve this answer









    $endgroup$














      Your Answer





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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      3












      $begingroup$

      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
      ]


      Mathematica graphics



      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
      ]


      Mathematica graphics






      share|improve this answer











      $endgroup$


















        3












        $begingroup$

        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
        ]


        Mathematica graphics



        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
        ]


        Mathematica graphics






        share|improve this answer











        $endgroup$
















          3












          3








          3





          $begingroup$

          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
          ]


          Mathematica graphics



          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
          ]


          Mathematica graphics






          share|improve this answer











          $endgroup$



          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
          ]


          Mathematica graphics



          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
          ]


          Mathematica graphics







          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited 33 mins ago

























          answered 51 mins ago









          C. E.C. E.

          51.1k3101206




          51.1k3101206























              0












              $begingroup$

              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.






              share|improve this answer









              $endgroup$


















                0












                $begingroup$

                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.






                share|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  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.






                  share|improve this answer









                  $endgroup$



                  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.







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                  answered 6 hours ago









                  MarcoBMarcoB

                  38.6k557115




                  38.6k557115






















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