24

Draw the numbers $1,2,\dots,N$ on a circle and draw a line from $n$ to $m>n$ when $n$ divides $m$:

enter image description here

For larger $N$ some kind of stable structure emerges

enter image description here

which remains perfectly in place for ever larger $N$, even though the points on the circle get ever closer, i.e. are moving.

enter image description here enter image description here

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):

![enter image description here


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 $M$

enter image description here

But not for larger $M$

enter image description here

For $M:(3M -1)$

![![enter image description here

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 $m$:

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

  2. Mark an initial corner $C_1$.

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

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

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

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

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

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

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

    7. Else: Stop.


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 $m \leq N$.

Hans-Peter Stricker
  • 17,273
  • 7
  • 54
  • 119
  • 3
    Very pretty. The envelopes stay in place because each corresponds to connecting angle point $\theta$ (measured anticlockwise from the bottom) to angle point $k\theta$ for given integer $k \ge 2$ and all possible up to $0 \lt \theta \le \frac{2\pi}{k}$. For given $N$ you only get lines when $\theta = 2\pi \frac{n}{N}$ and $0 \lt n \le \frac{N}{k}$, but that has little effect on the envelopes. – Henry Jan 11 '19 at 16:51
  • @Henry. Thanks! Can you tell if the "main" envelope is a half cardoid? – Hans-Peter Stricker Jan 11 '19 at 16:55
  • 5
    Yes it is: [Wikipedia says](https://en.wikipedia.org/wiki/Cardioid#Cardioid_as_envelope_of_a_pencil_of_lines) this is a result of Luigi Cremona – Henry Jan 11 '19 at 16:57
  • More interesting to me are the cusps and why they occur. They seem to lie on the line joining angle point $\frac{\pi}{k-1}$ to $\frac{k\pi}{k-1}$. I suspect that they may be parts of epicycloids - specifically $\frac1k$ of an $k-1$ cusp epicycloid so for example the second envelope is a third of a nephroid – Henry Jan 11 '19 at 17:04
  • @Henry: If you don't mind, please have a look at my updated question. – Hans-Peter Stricker Jan 11 '19 at 20:07
  • 1
    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, – Alex R. Jan 11 '19 at 21:03
  • @AlexR.Could you please elaborate your last phrase: "show a much lower density of near-perpendicular curves through each cardioid"? Which graph do you refer to? – Hans-Peter Stricker Jan 11 '19 at 21:06
  • @AlexR.: You might want to have a look at the "answer" I give below. – Hans-Peter Stricker Jan 12 '19 at 08:42
  • The third way to create the division graph is the same are the second, when we rename $n\leftrightarrow k$. – Alex Ravsky Jan 16 '19 at 05:14
  • This reminded me of [a video of Matologer](https://www.youtube.com/watch?v=qhbuKbxJsk8) on youtube - maybe related. – M. Winter Jan 16 '19 at 13:19
  • @M.Winter: Not by accident. This video gave me a lot of inspiration. I should give credit to Burkard Polster every now and then. – Hans-Peter Stricker Jan 16 '19 at 13:27
  • 1
    @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.) – Hans-Peter Stricker Jan 16 '19 at 13:28

2 Answers2

8

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

enter image description here enter image description here enter image description here enter image description here

Hans-Peter Stricker
  • 17,273
  • 7
  • 54
  • 119
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:

enter image description hereenter image description hereenter image description here

  • 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
  • 17,273
  • 7
  • 54
  • 119