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How can I plot a Farey diagram?
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$begingroup$
How can I plot the following diagram for a Farey series?
graphics number-theory
edited 1 hour ago
Michael E2
150k12203482
asked 7 hours ago
Gustavo RubianoGustavo Rubiano
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.
$endgroup$
add a comment |
$begingroup$
How can I plot the following diagram for a Farey series?
graphics number-theory
edited 1 hour ago
Michael E2
150k12203482
asked 7 hours ago
Gustavo RubianoGustavo Rubiano
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.
$endgroup$
$begingroup$
From the beautiful book A. Hatcher Topology of numbers
$endgroup$
– Gustavo Rubiano
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
4 hours ago
add a comment |
$begingroup$
How can I plot the following diagram for a Farey series?
graphics number-theory
edited 1 hour ago
Michael E2
150k12203482
asked 7 hours ago
Gustavo RubianoGustavo Rubiano
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.
$endgroup$
How can I plot the following diagram for a Farey series?
graphics number-theory
graphics number-theory
edited 1 hour ago
Michael E2
150k12203482
asked 7 hours ago
Gustavo RubianoGustavo Rubiano
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.
edited 1 hour ago
Michael E2
150k12203482
asked 7 hours ago
Gustavo RubianoGustavo Rubiano
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.
edited 1 hour ago
Michael E2
150k12203482
edited 1 hour ago
Michael E2
150k12203482
edited 1 hour ago
Michael E2
150k12203482
150k12203482
asked 7 hours ago
Gustavo RubianoGustavo Rubiano
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.
asked 7 hours ago
Gustavo RubianoGustavo Rubiano
113
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
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
4 hours ago
add a comment |
$begingroup$
From the beautiful book A. Hatcher Topology of numbers
$endgroup$
– Gustavo Rubiano
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
4 hours ago
$begingroup$
From the beautiful book A. Hatcher Topology of numbers
$endgroup$
– Gustavo Rubiano
6 hours ago
$begingroup$
From the beautiful book A. Hatcher Topology of numbers
$endgroup$
– Gustavo Rubiano
6 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
4 hours ago
$begingroup$
pi.math.cornell.edu/~hatcher/TN/TNch1.pdf
$endgroup$
– Moo
4 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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
]
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
]
answered 46 mins ago
C. E.C. E.
51.1k3101206
$endgroup$
add a comment |
$begingroup$
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.
answered 6 hours ago
MarcoBMarcoB
38.6k557115
$endgroup$
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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
]
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
]
answered 46 mins ago
C. E.C. E.
51.1k3101206
$endgroup$
add a comment |
$begingroup$
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
]
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
]
answered 46 mins ago
C. E.C. E.
51.1k3101206
$endgroup$
add a comment |
$begingroup$
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
]
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
]
answered 46 mins ago
C. E.C. E.
51.1k3101206
$endgroup$
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
]
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
]
answered 46 mins ago
C. E.C. E.
51.1k3101206
edited 28 mins ago
answered 46 mins ago
C. E.C. E.
51.1k3101206
answered 46 mins ago
C. E.C. E.
51.1k3101206
answered 46 mins ago
C. E.C. E.
51.1k3101206
51.1k3101206
add a comment |
add a comment |
$begingroup$
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.
answered 6 hours ago
MarcoBMarcoB
38.6k557115
$endgroup$
add a comment |
$begingroup$
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.
answered 6 hours ago
MarcoBMarcoB
38.6k557115
$endgroup$
add a comment |
$begingroup$
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.
answered 6 hours ago
MarcoBMarcoB
38.6k557115
$endgroup$
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.
answered 6 hours ago
MarcoBMarcoB
38.6k557115
answered 6 hours ago
MarcoBMarcoB
38.6k557115
answered 6 hours ago
MarcoBMarcoB
38.6k557115
answered 6 hours ago
MarcoBMarcoB
38.6k557115
38.6k557115
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Gustavo Rubiano is a new contributor. Be nice, and check out our Code of Conduct.
Gustavo Rubiano is a new contributor. Be nice, and check out our Code of Conduct.
Gustavo Rubiano is a new contributor. Be nice, and check out our Code of Conduct.
Gustavo Rubiano is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
From the beautiful book A. Hatcher Topology of numbers
$endgroup$
– Gustavo Rubiano
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
4 hours ago