2019-01-11 16:31:31 8 Comments

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

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.

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 $m\leq N$ 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 $m = k\cdot n$ when $m \leq N$.

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

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## 2 comments

## @Hans-Peter Stricker 2019-01-14 09:21:10

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.

## @Hans-Peter Stricker 2019-01-12 08:41:24

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