# Visualizing rational numbers as multiplication graphs

It's an interesting fact, that there's a straight forward way to visualize rational numbers. To each rational number – given as two integers $$n – there corresponds a multiplication graph $$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.

## $$n/(n+1)$$

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

## $$n/(2n-2)$$

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

## $$n/2n$$

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 $$m$$

$$n/(2n+1)$$

## $$n/7n$$

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).

## $$n/(n^2-1)$$

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.

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

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:

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 $$4/9$$:

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

$$p$$ 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:

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

• 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. – Lee Mosher Nov 17 '18 at 16:49