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How can I plot a Farey diagram?
How to make this beautiful animationPlotting an epicycloidGenerating a topological space diagram for an n-element setMathematica code for Bifurcation DiagramHow to draw a contour diagram in Mathematica?How to draw timing diagram from a list of values?Expressing a series formulaBifurcation diagram for Piecewise functionHow to draw a clock-diagram?How can I plot a space time diagram in mathematica?Plotting classical polymer modelA problem in bifurcation diagram
$begingroup$
How can I plot the following diagram for a Farey series?
graphics number-theory
New contributor
$endgroup$
add a comment |
$begingroup$
How can I plot the following diagram for a Farey series?
graphics number-theory
New contributor
$endgroup$
$begingroup$
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
New contributor
$endgroup$
How can I plot the following diagram for a Farey series?
graphics number-theory
graphics number-theory
New contributor
New contributor
edited 1 hour ago
Michael E2
150k12203482
150k12203482
New contributor
asked 6 hours ago
Gustavo RubianoGustavo Rubiano
113
113
New contributor
New contributor
$begingroup$
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
]
$endgroup$
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.
$endgroup$
add a comment |
Your Answer
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2 Answers
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2 Answers
2
active
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active
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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
]
$endgroup$
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
]
$endgroup$
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
]
$endgroup$
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
]
edited 20 mins ago
answered 37 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.
$endgroup$
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.
$endgroup$
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.
$endgroup$
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
38.6k557115
add a comment |
add a comment |
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