Mixing
An enigmatic pattern in division graphs
$begingroup$
Draw the numbers $1,2,dots,N$ on a circle and draw a line from $n$ to $m>n$ when $n$ divides $m$:
For larger $N$ some kind of stable structure emerges
which remains perfectly in place for ever larger $N$, even though the points on the circle get ever closer, i.e. are moving.
This really astonishes me, I wouldn't have guessed. Can someone explain?
In its full beauty the case $N=1000$ (cheating a bit by adding also lines from $m$ to $n$ when $(m-N)%N$ divides $(n-N)%N$ thus symmetrizing the picture):
Note that a similar phenomenon – stable asymptotic patterns, esp. cardioids, nephroids, and so on – can be observed in modular multiplication graphs $M:N$ with a line drawn from $n$ to $m$ if $Mcdot n equiv m pmod{N}$.
For the graphs $M:N$, $N > M$ for small $M$
But not for larger $M$
For $M:(3M -1)$
It would be interesting to understand how these two phenomena relate.
Note that one can create arbitrary large division graphs with circle and compass alone, without even explicitly checking if a number $n$ divides another number $m$:
Create a regular $2^n$-gon.
Mark an initial corner $C_1$.
For each corner $C_k$ do the following:
Set the radius $r$ of the compass to $|C_1C_k|$.
Draw a circle around $C_{k_0} = C_k$ with radius $r$.
On the circle do lie two other corners, pick the next one in counter-clockwise direction, $C_{k_1}$.
If $C_1$ does not lie between $C_{k_0}$ and $C_{k_1}$ (in counter-clockwise direction) or equals $C_{k_1}$:
Draw a line from $C_k$ to $C_{k_1}$.
Let $C_{k_0} = C_{k_1}$ and proceed with 5.
Else: Stop.
There are three equivalent ways to create the division graph for $N$ edge by edge:
For each $n = 1,2,...,N$: For each $mleq N$ draw an edge between $n$ and $m$ when $n$ divides $m$.
For each $n = 1,2,...,N$: For each $k = 1,2,...,N$ draw an edge between $n$ and $m = kcdot n$ when $m leq N$.
For each $k = 1,2,...,N$: For each $n = 1,2,...,N$ draw an edge between $n$ and $m = kcdot n$ when $m leq N$.
elementary-number-theory divisibility visualization
asked Jan 11 at 16:31
Hans StrickerHans Stricker
6,24343988
$endgroup$
|
show 7 more comments
$begingroup$
Draw the numbers $1,2,dots,N$ on a circle and draw a line from $n$ to $m>n$ when $n$ divides $m$:
For larger $N$ some kind of stable structure emerges
which remains perfectly in place for ever larger $N$, even though the points on the circle get ever closer, i.e. are moving.
This really astonishes me, I wouldn't have guessed. Can someone explain?
In its full beauty the case $N=1000$ (cheating a bit by adding also lines from $m$ to $n$ when $(m-N)%N$ divides $(n-N)%N$ thus symmetrizing the picture):
Note that a similar phenomenon – stable asymptotic patterns, esp. cardioids, nephroids, and so on – can be observed in modular multiplication graphs $M:N$ with a line drawn from $n$ to $m$ if $Mcdot n equiv m pmod{N}$.
For the graphs $M:N$, $N > M$ for small $M$
But not for larger $M$
For $M:(3M -1)$
It would be interesting to understand how these two phenomena relate.
Note that one can create arbitrary large division graphs with circle and compass alone, without even explicitly checking if a number $n$ divides another number $m$:
Create a regular $2^n$-gon.
Mark an initial corner $C_1$.
For each corner $C_k$ do the following:
Set the radius $r$ of the compass to $|C_1C_k|$.
Draw a circle around $C_{k_0} = C_k$ with radius $r$.
On the circle do lie two other corners, pick the next one in counter-clockwise direction, $C_{k_1}$.
If $C_1$ does not lie between $C_{k_0}$ and $C_{k_1}$ (in counter-clockwise direction) or equals $C_{k_1}$:
Draw a line from $C_k$ to $C_{k_1}$.
Let $C_{k_0} = C_{k_1}$ and proceed with 5.
Else: Stop.
There are three equivalent ways to create the division graph for $N$ edge by edge:
For each $n = 1,2,...,N$: For each $mleq N$ draw an edge between $n$ and $m$ when $n$ divides $m$.
For each $n = 1,2,...,N$: For each $k = 1,2,...,N$ draw an edge between $n$ and $m = kcdot n$ when $m leq N$.
For each $k = 1,2,...,N$: For each $n = 1,2,...,N$ draw an edge between $n$ and $m = kcdot n$ when $m leq N$.
elementary-number-theory divisibility visualization
asked Jan 11 at 16:31
Hans StrickerHans Stricker
6,24343988
$endgroup$
3
$begingroup$
Very pretty. The envelopes stay in place because each corresponds to connecting angle point $theta$ (measured anticlockwise from the bottom) to angle point $ktheta$ for given integer $k ge 2$ and all possible up to $0 lt theta le frac{2pi}{k}$. For given $N$ you only get lines when $theta = 2pi frac{n}{N}$ and $0 lt n le frac{N}{k}$, but that has little effect on the envelopes.
$endgroup$
– Henry
Jan 11 at 16:51
$begingroup$
@Henry. Thanks! Can you tell if the "main" envelope is a half cardoid?
$endgroup$
– Hans Stricker
Jan 11 at 16:55
5
$begingroup$
Yes it is: Wikipedia says this is a result of Luigi Cremona
$endgroup$
– Henry
Jan 11 at 16:57
1
$begingroup$
Rephrasing, these curves are arising as the envelopes of pairs of numbers whose division gives $2,3,4,...$, hence the resemblance to cardioids, where the second coordinate "moves faster" than the first. With some care you can probably also show a much lower density of near-perpendicular curves through each cardioid,
$endgroup$
– Alex R.
Jan 11 at 21:03
1
$begingroup$
@AlexRavsky: Not quite the same when it comes to the order in which edges are created - which of course may be neglected. (Note that the first to ways create the edges in the same order.)
$endgroup$
– Hans Stricker
Jan 16 at 13:28
|
show 7 more comments
$begingroup$
Draw the numbers $1,2,dots,N$ on a circle and draw a line from $n$ to $m>n$ when $n$ divides $m$:
For larger $N$ some kind of stable structure emerges
which remains perfectly in place for ever larger $N$, even though the points on the circle get ever closer, i.e. are moving.
This really astonishes me, I wouldn't have guessed. Can someone explain?
In its full beauty the case $N=1000$ (cheating a bit by adding also lines from $m$ to $n$ when $(m-N)%N$ divides $(n-N)%N$ thus symmetrizing the picture):
Note that a similar phenomenon – stable asymptotic patterns, esp. cardioids, nephroids, and so on – can be observed in modular multiplication graphs $M:N$ with a line drawn from $n$ to $m$ if $Mcdot n equiv m pmod{N}$.
For the graphs $M:N$, $N > M$ for small $M$
But not for larger $M$
For $M:(3M -1)$
It would be interesting to understand how these two phenomena relate.
Note that one can create arbitrary large division graphs with circle and compass alone, without even explicitly checking if a number $n$ divides another number $m$:
Create a regular $2^n$-gon.
Mark an initial corner $C_1$.
For each corner $C_k$ do the following:
Set the radius $r$ of the compass to $|C_1C_k|$.
Draw a circle around $C_{k_0} = C_k$ with radius $r$.
On the circle do lie two other corners, pick the next one in counter-clockwise direction, $C_{k_1}$.
If $C_1$ does not lie between $C_{k_0}$ and $C_{k_1}$ (in counter-clockwise direction) or equals $C_{k_1}$:
Draw a line from $C_k$ to $C_{k_1}$.
Let $C_{k_0} = C_{k_1}$ and proceed with 5.
Else: Stop.
There are three equivalent ways to create the division graph for $N$ edge by edge:
For each $n = 1,2,...,N$: For each $mleq N$ draw an edge between $n$ and $m$ when $n$ divides $m$.
For each $n = 1,2,...,N$: For each $k = 1,2,...,N$ draw an edge between $n$ and $m = kcdot n$ when $m leq N$.
For each $k = 1,2,...,N$: For each $n = 1,2,...,N$ draw an edge between $n$ and $m = kcdot n$ when $m leq N$.
elementary-number-theory divisibility visualization
asked Jan 11 at 16:31
Hans StrickerHans Stricker
6,24343988
$endgroup$
Draw the numbers $1,2,dots,N$ on a circle and draw a line from $n$ to $m>n$ when $n$ divides $m$:
For larger $N$ some kind of stable structure emerges
which remains perfectly in place for ever larger $N$, even though the points on the circle get ever closer, i.e. are moving.
This really astonishes me, I wouldn't have guessed. Can someone explain?
In its full beauty the case $N=1000$ (cheating a bit by adding also lines from $m$ to $n$ when $(m-N)%N$ divides $(n-N)%N$ thus symmetrizing the picture):
Note that a similar phenomenon – stable asymptotic patterns, esp. cardioids, nephroids, and so on – can be observed in modular multiplication graphs $M:N$ with a line drawn from $n$ to $m$ if $Mcdot n equiv m pmod{N}$.
For the graphs $M:N$, $N > M$ for small $M$
But not for larger $M$
For $M:(3M -1)$
It would be interesting to understand how these two phenomena relate.
Note that one can create arbitrary large division graphs with circle and compass alone, without even explicitly checking if a number $n$ divides another number $m$:
Create a regular $2^n$-gon.
Mark an initial corner $C_1$.
For each corner $C_k$ do the following:
Set the radius $r$ of the compass to $|C_1C_k|$.
Draw a circle around $C_{k_0} = C_k$ with radius $r$.
On the circle do lie two other corners, pick the next one in counter-clockwise direction, $C_{k_1}$.
If $C_1$ does not lie between $C_{k_0}$ and $C_{k_1}$ (in counter-clockwise direction) or equals $C_{k_1}$:
Draw a line from $C_k$ to $C_{k_1}$.
Let $C_{k_0} = C_{k_1}$ and proceed with 5.
Else: Stop.
There are three equivalent ways to create the division graph for $N$ edge by edge:
For each $n = 1,2,...,N$: For each $mleq N$ draw an edge between $n$ and $m$ when $n$ divides $m$.
For each $n = 1,2,...,N$: For each $k = 1,2,...,N$ draw an edge between $n$ and $m = kcdot n$ when $m leq N$.
For each $k = 1,2,...,N$: For each $n = 1,2,...,N$ draw an edge between $n$ and $m = kcdot n$ when $m leq N$.
elementary-number-theory divisibility visualization
elementary-number-theory divisibility visualization
asked Jan 11 at 16:31
Hans StrickerHans Stricker
6,24343988
asked Jan 11 at 16:31
Hans StrickerHans Stricker
6,24343988
edited Jan 16 at 15:14
Hans Stricker
asked Jan 11 at 16:31
Hans StrickerHans Stricker
6,24343988
asked Jan 11 at 16:31
Hans StrickerHans Stricker
6,24343988
asked Jan 11 at 16:31
Hans StrickerHans Stricker
6,24343988
6,24343988
3
$begingroup$
Very pretty. The envelopes stay in place because each corresponds to connecting angle point $theta$ (measured anticlockwise from the bottom) to angle point $ktheta$ for given integer $k ge 2$ and all possible up to $0 lt theta le frac{2pi}{k}$. For given $N$ you only get lines when $theta = 2pi frac{n}{N}$ and $0 lt n le frac{N}{k}$, but that has little effect on the envelopes.
$endgroup$
– Henry
Jan 11 at 16:51
$begingroup$
@Henry. Thanks! Can you tell if the "main" envelope is a half cardoid?
$endgroup$
– Hans Stricker
Jan 11 at 16:55
5
$begingroup$
Yes it is: Wikipedia says this is a result of Luigi Cremona
$endgroup$
– Henry
Jan 11 at 16:57
1
$begingroup$
Rephrasing, these curves are arising as the envelopes of pairs of numbers whose division gives $2,3,4,...$, hence the resemblance to cardioids, where the second coordinate "moves faster" than the first. With some care you can probably also show a much lower density of near-perpendicular curves through each cardioid,
$endgroup$
– Alex R.
Jan 11 at 21:03
1
$begingroup$
@AlexRavsky: Not quite the same when it comes to the order in which edges are created - which of course may be neglected. (Note that the first to ways create the edges in the same order.)
$endgroup$
– Hans Stricker
Jan 16 at 13:28
|
show 7 more comments
3
$begingroup$
Very pretty. The envelopes stay in place because each corresponds to connecting angle point $theta$ (measured anticlockwise from the bottom) to angle point $ktheta$ for given integer $k ge 2$ and all possible up to $0 lt theta le frac{2pi}{k}$. For given $N$ you only get lines when $theta = 2pi frac{n}{N}$ and $0 lt n le frac{N}{k}$, but that has little effect on the envelopes.
$endgroup$
– Henry
Jan 11 at 16:51
$begingroup$
@Henry. Thanks! Can you tell if the "main" envelope is a half cardoid?
$endgroup$
– Hans Stricker
Jan 11 at 16:55
5
$begingroup$
Yes it is: Wikipedia says this is a result of Luigi Cremona
$endgroup$
– Henry
Jan 11 at 16:57
1
$begingroup$
Rephrasing, these curves are arising as the envelopes of pairs of numbers whose division gives $2,3,4,...$, hence the resemblance to cardioids, where the second coordinate "moves faster" than the first. With some care you can probably also show a much lower density of near-perpendicular curves through each cardioid,
$endgroup$
– Alex R.
Jan 11 at 21:03
1
$begingroup$
@AlexRavsky: Not quite the same when it comes to the order in which edges are created - which of course may be neglected. (Note that the first to ways create the edges in the same order.)
$endgroup$
– Hans Stricker
Jan 16 at 13:28
3
3
$begingroup$
Very pretty. The envelopes stay in place because each corresponds to connecting angle point $theta$ (measured anticlockwise from the bottom) to angle point $ktheta$ for given integer $k ge 2$ and all possible up to $0 lt theta le frac{2pi}{k}$. For given $N$ you only get lines when $theta = 2pi frac{n}{N}$ and $0 lt n le frac{N}{k}$, but that has little effect on the envelopes.
$endgroup$
– Henry
Jan 11 at 16:51
$begingroup$
Very pretty. The envelopes stay in place because each corresponds to connecting angle point $theta$ (measured anticlockwise from the bottom) to angle point $ktheta$ for given integer $k ge 2$ and all possible up to $0 lt theta le frac{2pi}{k}$. For given $N$ you only get lines when $theta = 2pi frac{n}{N}$ and $0 lt n le frac{N}{k}$, but that has little effect on the envelopes.
$endgroup$
– Henry
Jan 11 at 16:51
$begingroup$
@Henry. Thanks! Can you tell if the "main" envelope is a half cardoid?
$endgroup$
– Hans Stricker
Jan 11 at 16:55
$begingroup$
@Henry. Thanks! Can you tell if the "main" envelope is a half cardoid?
$endgroup$
– Hans Stricker
Jan 11 at 16:55
5
5
$begingroup$
Yes it is: Wikipedia says this is a result of Luigi Cremona
$endgroup$
– Henry
Jan 11 at 16:57
$begingroup$
Yes it is: Wikipedia says this is a result of Luigi Cremona
$endgroup$
– Henry
Jan 11 at 16:57
1
1
$begingroup$
Rephrasing, these curves are arising as the envelopes of pairs of numbers whose division gives $2,3,4,...$, hence the resemblance to cardioids, where the second coordinate "moves faster" than the first. With some care you can probably also show a much lower density of near-perpendicular curves through each cardioid,
$endgroup$
– Alex R.
Jan 11 at 21:03
$begingroup$
Rephrasing, these curves are arising as the envelopes of pairs of numbers whose division gives $2,3,4,...$, hence the resemblance to cardioids, where the second coordinate "moves faster" than the first. With some care you can probably also show a much lower density of near-perpendicular curves through each cardioid,
$endgroup$
– Alex R.
Jan 11 at 21:03
1
1
$begingroup$
@AlexRavsky: Not quite the same when it comes to the order in which edges are created - which of course may be neglected. (Note that the first to ways create the edges in the same order.)
$endgroup$
– Hans Stricker
Jan 16 at 13:28
$begingroup$
@AlexRavsky: Not quite the same when it comes to the order in which edges are created - which of course may be neglected. (Note that the first to ways create the edges in the same order.)
$endgroup$
– Hans Stricker
Jan 16 at 13:28
|
show 7 more comments
2 Answers
2
active
oldest
votes
$begingroup$
To add some visual sugar to Alex R's comment (thanks for it):
answered Jan 12 at 8:41
Hans StrickerHans Stricker
6,24343988
$endgroup$
add a comment |
$begingroup$
Putting the pieces together one may explain the pattern like this:
The division graph for $N$ can be seen as the sum of the multiplication graphs $G_N^k$, $k=2,3,..,N$ with an edge from $n$ to $m$ when $kcdot n = m$. When $n > N/k$ there's no line emanating from $n$. (This relates to step 7 in the geometric construction above.)
The multiplication-modulo-$N$ graphs $H_{N}^k$ have a weaker condition: there's an edge from $n$ to $m$ when $kcdot n equiv m pmod{N}$.
So the division graph for $N$ is a proper subgraph of the sum of multiplication graphs $H_{N+1}^k$.
The multiplication graphs $H_{N}^k$ exhibit characteristic $k-1$-lobed patterns:
These patterns are truncated in the graphs $G_{N}^k$ exactly at $N/k$.
Overlaying the truncated patterns gives the pattern in question.
answered Jan 14 at 9:21
Hans StrickerHans Stricker
6,24343988
$endgroup$
add a comment |
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2 Answers
2
active
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votes
2 Answers
2
active
oldest
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$begingroup$
To add some visual sugar to Alex R's comment (thanks for it):
answered Jan 12 at 8:41
Hans StrickerHans Stricker
6,24343988
$endgroup$
add a comment |
$begingroup$
To add some visual sugar to Alex R's comment (thanks for it):
answered Jan 12 at 8:41
Hans StrickerHans Stricker
6,24343988
$endgroup$
add a comment |
$begingroup$
To add some visual sugar to Alex R's comment (thanks for it):
answered Jan 12 at 8:41
Hans StrickerHans Stricker
6,24343988
$endgroup$
To add some visual sugar to Alex R's comment (thanks for it):
answered Jan 12 at 8:41
Hans StrickerHans Stricker
6,24343988
edited Jan 12 at 9:58
answered Jan 12 at 8:41
Hans StrickerHans Stricker
6,24343988
answered Jan 12 at 8:41
Hans StrickerHans Stricker
6,24343988
answered Jan 12 at 8:41
Hans StrickerHans Stricker
6,24343988
6,24343988
add a comment |
add a comment |
$begingroup$
Putting the pieces together one may explain the pattern like this:
The division graph for $N$ can be seen as the sum of the multiplication graphs $G_N^k$, $k=2,3,..,N$ with an edge from $n$ to $m$ when $kcdot n = m$. When $n > N/k$ there's no line emanating from $n$. (This relates to step 7 in the geometric construction above.)
The multiplication-modulo-$N$ graphs $H_{N}^k$ have a weaker condition: there's an edge from $n$ to $m$ when $kcdot n equiv m pmod{N}$.
So the division graph for $N$ is a proper subgraph of the sum of multiplication graphs $H_{N+1}^k$.
The multiplication graphs $H_{N}^k$ exhibit characteristic $k-1$-lobed patterns:
These patterns are truncated in the graphs $G_{N}^k$ exactly at $N/k$.
Overlaying the truncated patterns gives the pattern in question.
answered Jan 14 at 9:21
Hans StrickerHans Stricker
6,24343988
$endgroup$
add a comment |
$begingroup$
Putting the pieces together one may explain the pattern like this:
The division graph for $N$ can be seen as the sum of the multiplication graphs $G_N^k$, $k=2,3,..,N$ with an edge from $n$ to $m$ when $kcdot n = m$. When $n > N/k$ there's no line emanating from $n$. (This relates to step 7 in the geometric construction above.)
The multiplication-modulo-$N$ graphs $H_{N}^k$ have a weaker condition: there's an edge from $n$ to $m$ when $kcdot n equiv m pmod{N}$.
So the division graph for $N$ is a proper subgraph of the sum of multiplication graphs $H_{N+1}^k$.
The multiplication graphs $H_{N}^k$ exhibit characteristic $k-1$-lobed patterns:
These patterns are truncated in the graphs $G_{N}^k$ exactly at $N/k$.
Overlaying the truncated patterns gives the pattern in question.
answered Jan 14 at 9:21
Hans StrickerHans Stricker
6,24343988
$endgroup$
add a comment |
$begingroup$
Putting the pieces together one may explain the pattern like this:
The division graph for $N$ can be seen as the sum of the multiplication graphs $G_N^k$, $k=2,3,..,N$ with an edge from $n$ to $m$ when $kcdot n = m$. When $n > N/k$ there's no line emanating from $n$. (This relates to step 7 in the geometric construction above.)
The multiplication-modulo-$N$ graphs $H_{N}^k$ have a weaker condition: there's an edge from $n$ to $m$ when $kcdot n equiv m pmod{N}$.
So the division graph for $N$ is a proper subgraph of the sum of multiplication graphs $H_{N+1}^k$.
The multiplication graphs $H_{N}^k$ exhibit characteristic $k-1$-lobed patterns:
These patterns are truncated in the graphs $G_{N}^k$ exactly at $N/k$.
Overlaying the truncated patterns gives the pattern in question.
answered Jan 14 at 9:21
Hans StrickerHans Stricker
6,24343988
$endgroup$
Putting the pieces together one may explain the pattern like this:
The division graph for $N$ can be seen as the sum of the multiplication graphs $G_N^k$, $k=2,3,..,N$ with an edge from $n$ to $m$ when $kcdot n = m$. When $n > N/k$ there's no line emanating from $n$. (This relates to step 7 in the geometric construction above.)
The multiplication-modulo-$N$ graphs $H_{N}^k$ have a weaker condition: there's an edge from $n$ to $m$ when $kcdot n equiv m pmod{N}$.
So the division graph for $N$ is a proper subgraph of the sum of multiplication graphs $H_{N+1}^k$.
The multiplication graphs $H_{N}^k$ exhibit characteristic $k-1$-lobed patterns:
These patterns are truncated in the graphs $G_{N}^k$ exactly at $N/k$.
Overlaying the truncated patterns gives the pattern in question.
answered Jan 14 at 9:21
Hans StrickerHans Stricker
6,24343988
edited Jan 16 at 11:16
answered Jan 14 at 9:21
Hans StrickerHans Stricker
6,24343988
answered Jan 14 at 9:21
Hans StrickerHans Stricker
6,24343988
answered Jan 14 at 9:21
Hans StrickerHans Stricker
6,24343988
6,24343988
add a comment |
add a comment |
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Very pretty. The envelopes stay in place because each corresponds to connecting angle point $theta$ (measured anticlockwise from the bottom) to angle point $ktheta$ for given integer $k ge 2$ and all possible up to $0 lt theta le frac{2pi}{k}$. For given $N$ you only get lines when $theta = 2pi frac{n}{N}$ and $0 lt n le frac{N}{k}$, but that has little effect on the envelopes.
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– Henry
Jan 11 at 16:51
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@Henry. Thanks! Can you tell if the "main" envelope is a half cardoid?
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– Hans Stricker
Jan 11 at 16:55
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Yes it is: Wikipedia says this is a result of Luigi Cremona
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– Henry
Jan 11 at 16:57
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Rephrasing, these curves are arising as the envelopes of pairs of numbers whose division gives $2,3,4,...$, hence the resemblance to cardioids, where the second coordinate "moves faster" than the first. With some care you can probably also show a much lower density of near-perpendicular curves through each cardioid,
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– Alex R.
Jan 11 at 21:03
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@AlexRavsky: Not quite the same when it comes to the order in which edges are created - which of course may be neglected. (Note that the first to ways create the edges in the same order.)
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– Hans Stricker
Jan 16 at 13:28