Visualized group tables for $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

Solution 1:

It turns out that the patterns can be seen more clearly when choosing another color scheme: for $k = \lfloor n/4 \rfloor$ choose red, for $k = \lfloor 3n/4 \rfloor$ choose blue, for the other values a shade of gray indicating the distance to $n/2$. For $n=128$:

One observes that the big square is divided successively into squares of sidelength $n/k$.

Furthermore one may observe that for example the red dots at the top left really lie on a hyperbola. They are placed at the grid cells $(32,1), (16,2), (8,4), (4,8), (2,16), (1,32)$ thus fullfilling $j = 32/i$.

For $n=257$, i.e. a prime number, the grid patterns vanishes, but the hyperbolic structure remains intact:

Solution 2:

Concerning the last, i.e. the second question which concerns ((sub)centers of) hyperbolas:

For $\mathbb{Z}/n\mathbb{Z}$ the visible centers of hyperbolas are placed at points $ \frac{n}{k}(i,j)$ for $k < \log_2 n$ and $0 \leq i,j \leq k$.

(Note the modular/toroidal structure of the graph.)

The "size" of the hyperbola shrinks with $1/k$, its "distinctness" is maximal along the diagonals.