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It's an interesting fact, that there's a straight forward way to visualize rational numbers. To each rational number – given as two integers $n<m$ – there corresponds a multiplication graph $n/m$ with a line between $a,b < m$ iff $an \equiv b\operatorname{mod} m$.

It's another interesting fact that you often can tell how a graph $n/m$ roughly looks like, just by looking at the two numbers $n$ and $m$.

To some of the following examples I give the fact of modular arithmetic that is responsible for its apparent structure.


$n/(n+1)$

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due to $a(m-1) \equiv (m-a)(m-1)\operatorname{mod} m$


$n/(2n-2)$

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due to $a(m+2)/2 \equiv (a+(m+2)/2)(m+2)/2)\operatorname{mod} m$ for even $m$


$n/(2n-1)$

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$n/2n$

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due to $am/2 \equiv 0 \operatorname{mod} m$ for even $a$ and $am/2 \equiv m/2 \operatorname{mod} m$ for odd $a$ and even $m$


$n/(2n+1)$ enter image description here


$n/(2n+2)$

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$n/(3n-1)$

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$n/3n$

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$n/(3n+1)$

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$n/7n$

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Note that for $n/kn$, i.e. multiples of $1/k$ you will always see $k$ densities around the circle, as for $n/2n$ and $n/3n$ (see above).


$n/(n^2-1)$

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Note that also quadratic relations between $n$ and $m$ give rise to obvious patterns, in this case: a structure of perfect $n-1$-fold rotation symmetry containing the regular $n$-gon. The clearness of the structure has to do with the fact that for $m=n^2 - 1$, i.e. $n = \sqrt{m+1}$, we have $an \equiv b \operatorname{mod} m$ iff $bn \equiv a \operatorname{mod} m$, i.e. we see only half of the lines - the picture is clearer than normal.


$n=23k, m = n + 22, k=1,2,\dots$

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Another interesting fact that I'd like to mention is, that you cannot only tell (roughly) how the graph looks like for given numbers $n$, $m$ (at least in the cases above) but you can also tell something about the numbers by looking at the graph $n/m$. Especially you can tell at a glance the greatest common divisor of two numbers $n$, $m$, say $n=144$ and $m=324$, when $n/m$ looks like this:

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What you see are $9$ densities, and in fact we find that the greatest common divisor of $n$ and $m$ is $\frac{m}{9} = 36$ with $n = 36\cdot 4$ and $m = 36\cdot 9$, i.e. $\frac{144}{324} = \frac{4}{9}$

Note how this graph appears in the "multiples" of $4/9$:

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Finally, note that you can tell if a number $p$ is prime just by looking at its multiplication graphs $n/p$ for $n=2,\dots p-2$.

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$p$ is prime iff all graphs $n/p$ show permutations of the numbers $1,2\dots p-1$.


My questions are:

  1. Is a general way conceivable that – given a polynomial $m = an + b$ or $m = n^2 +an + b$ – allows me to predict how the graphs $n/(an + b)$ will look like.

Of course $a$ should be small, and we may restrict ourselves to the case $b \leq a$.

(Thinking loud: I guess there could be some structure types of graphs (depending on $a$ and $b$) that can be parametrized by one or two other numbers which depend on $a$ and $b$, e.g. the number of cusps or densities, the rotation symmetry, etc.)

  1. Is there a general scheme how to explain the apparent structure of the graphs $n/(an + b)$ (their "motif" and/or periodicity or its texture)?

Find here a good explanation of the structure of $m = n^2 -1$.


Finally, here is a Cantor-like table of the first rational numbers depicted as multiplication graphs:

enter image description here

Note that you can see straight lines of slope $1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$ and $0$.

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    $\begingroup$ As utterly fascinating as this is, I think this question is not a good fit for math.stackexchange. Basically this reads like a shortened version of a very interesting expository article that might appear in a journal like The Mathematical Intelligencer or The American Mathematical Monthly. The questions asked are very broad, great for someone to pursue as a research project, but badly formed for the kind of straightforward answer that is expected of a math.stackexchange answer. $\endgroup$ – Lee Mosher Nov 17 '18 at 16:49

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