Consider the matrix $\{a_{ij}\}_{i,j\in\mathbb{N}}$ with $a_{ij} = 1$ iff $i \equiv 0\,\text{mod}\,j$. Drawing it with stretched y-axis by a factor of $5$ reveals some "waves" which I referred to as "hidden patterns" in the title of this question.
The fronts of three distinguishable waves "travel" into three directions:
My question is threefold:
How can these patterns be explained in general terms of (modular) arithmetic?
How do I have to color the dots in the adjacency matrix such that dots on the same wave front have the same color? That means: For which function $f:\mathbb{N}\times \mathbb{N} \rightarrow \mathbb{N}$ and $a_{ij} = f(i,j)$ do we have $a_{ij} = a_{kl}$ iff $a_{ij}, a_{kl}$ lie on the same wave front (wherever there is a distinguished wave front).
How can the slopes of the directions of the waves (i.e. of the colored lines) be calculated? Note that even with a stretching factor of $5$ the slopes are quite small.
For larger stretching factors the wave travelling into the green direction becomes unrecognizable. Four wave fronts of the wave travelling into the orange direction are highlighted.
Without stretching you almost see nothing:
At the heart of the patterns is the identity $n^2 - k^2 \equiv 0 \mod (n\pm k)$. The right-most dots $a_{ij} = 1$ on the wave fronts along the orange line have $i = j^2$. The right-most dots on the wave fronts along the other directions have $i = 2\,j^2$ (blue) and $2\,i=j^2$ (green).
Giving those dots $a_{ij}$ with $i = n^2 - k^2$, $j = n\pm k$ the color $n$ makes the wave fronts distinguished and reveal that they are parabolas:
Since natural numbers may be written in zero to many ways as the difference of two squares, some $a_{ij}$ lie on no and some lie on several parabolas.
The orange line itself is in fact a parabola rotated by 90 degrees.