An enigmatic pattern in division graphs

Question

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?

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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):

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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 $M\cdot n \equiv m \pmod{N}$.

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.

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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 $m$:

1. Create a regular $2^n$-gon.

2. Mark an initial corner $C_1$.

3. For each corner $C_k$ do the following:

   4. Set the radius $r$ of the compass to $|C_1C_k|$.

   5. Draw a circle around $C_{k_0} = C_k$ with radius $r$.

   6. On the circle do lie two other corners, pick the next one in counter-clockwise direction, $C_{k_1}$.

   7. If $C_1$ does not lie between $C_{k_0}$ and $C_{k_1}$ (in counter-clockwise direction) or equals $C_{k_1}$:

   8. Draw a line from $C_k$ to $C_{k_1}$.

   9. Let $C_{k_0} = C_{k_1}$ and proceed with 5.

   10. Else: Stop.

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There are three equivalent ways to create the division graph for $N$ edge by edge:

1. For each $n = 1,2,...,N$: For each $m\leq N$ draw an edge between $n$ and $m$ when $n$ divides $m$.

2. For each $n = 1,2,...,N$: For each $k = 1,2,...,N$ draw an edge between $n$ and $m = k\cdot n$ when $m \leq N$.

3. For each $k = 1,2,...,N$: For each $n = 1,2,...,N$ draw an edge between $n$ and $m = k\cdot n$ when $m \leq N$.

Answer 1

To add some visual sugar to Alex R's comment (thanks for it):

Answer 2

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 $k\cdot n = m$. 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 $k\cdot n \equiv m \pmod{N}$.

• 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.