Consider the directed graphs $G_n^p$ with node set $\{0,1,\dots,p-1\}$, $0 \leq n < p$ with an arrow from $a$ to $b$ if $na=b\operatorname{mod}p$. These graphs are graphical multiplication tables for $\mathbb{Z}_p := \mathbb{Z}/p\mathbb{Z}$.
For example $G^{12}_n$:
My question is:
Is it just by coincidence that $G^{12}_4$ looks like a 2-dimensional drawing of a 3-dimensional cube
– or might there be something "deep" in it, assuming that there are some deep and hard to see relations between projective geometry and prime numbers?
Note that $p = 2 \cdot 2\cdot 3$, $n = 2 \cdot 2$.
But also note that $G^{12}_{10}$ does the same job, while in this case $n=2\cdot 5$.
Note especially that it is important to distribute the nodes of the graph equally on a circle – like the $p$th roots of unity – to get the "desired" graph.