Question concerning twin primes The multiplication graphs $p/q$ – with an edge between $a,b$ iff $ap \equiv b\operatorname{mod} q$  – for some twin primes $p$ and $q = p+2$ reveal an astonishing pattern:


*

*$(p+0)/(p+1)$ and $(p+1)/(p+2)$ have a seeming $2$-fold rotational symmetry. (This is per se not astonishing.)

*$(p+0)/(p+2)$ always has a seeming $3$-fold rotational symmetry.
(This is per se astonishing.)

*$(p-1)/(p+2)$ and $(p+0)/(p+3)$ have a seeming $4$-fold rotational symmetry 
Note that in each case the symmetry is only a seeming but also an obvious one.
Note further that one might try to generalize this to (for appropriate $p$)


*

*$(p+n)/(p+m)$ have a seeming $|n-m|+1$-fold rotational symmetry 


I wonder: What does this mean? 

How do $2$, $3$, $4$ come into play for arbitrary twin primes?

Now I know that the question must be (thanks to Gerhard Paseman):

How do $2$, $3$, $4$ come into play for arbitrary $6k\pm 1$ pairs?








The same pattern arises when choosing $r=6\cdot 41$ and $p = r-1$, $q = r+1$:

Note that neither $p=245$ nor $q=247$ are prime: $245 = 5\cdot 7^2$ and $247 = 13\cdot 19$.
 A: The question has an unexpected answer (at least to me). It turns out, that all multiplication graphs of the form $n/(an + b)$ have a seeming $a + b$-fold rotation symmetry. Especially, graphs of the form $n/(n + b)$ have a seeming $b + 1$-fold rotation symmetry.



Thus 


*

*all graphs of the form $n/(n+2)$ have a seeming $3$-fold symmetry

*all graphs of the form $n/(n+1)$ have a seeming $2$-fold symmetry

*all graphs of the form $(n-1)/n$ have a seeming $2$-fold symmetry

*all graphs of the form $(n-1)/(n+2)$ have a seeming $4$-fold symmetry

*all graphs of the form $n/(n+3)$ have a seeming $4$-fold symmetry
This "explains" the pictures in the question, but has nothing to do with prime numbers and prime twins at all!
What remains to be explained and understood: Why is this so? Why  do all multiplication graphs of the form $n/(an + b)$ have a seeming $a + b$-fold rotation symmetry? 
$a=2$, $b=1$

$a=2$, $b=2$

$a=3$, $b=1$

A: Following Aaron's advice I show here the graphs $n/(2n+1)$ and $n/(n + 2)$ which  have an "approximate" $3$-fold rotational symmetry first for small $n$, then for large $n$, and for $n=3k + 1$.
You don't see the approximate symmetry for small $n$, except an exact one for the relatively rare cases $n=3k + 1$, i.e. for $4/6$ and $7/9$, resp. for $4/9$ and $7/15$:


For larger $n$ the approximate symmetry becomes obvious while it still holds that only the graphs with $n=3k + 1$ are truly symmetric (which for large $n$ cannot be checked visually anymore):


Here are the first graphs with $n = 3k+1$ which can visually be checked to be truly symmetric:


