Structures emerging in a discrete plot of palindromic numbers Plotting Palindromic Numbers & Number Systems
If we decide to plot Palindromic Numbers against Number Bases, and go far enough into the number line, things start to look interesting. In fact, the deeper we go into the number line, we get similar structures but with more details.
Lets count each pixel on an image as a individual number, and start in the upper left corner with coordinates $(0,0)$. Coordinate $x$ increases as we go right, and $y$ increases as we go down. Let $x$ represent a number, and $y$ represent anumber bases. That is, let $(x,y)$ represents number $x$ written in a number base $y$.
We color point $(x,y)$ if $x$ is a palindrome in base $y$, the following way:
[$1$ - $\color{orange}{Yellow}$] [$2$ - $\color{red}{Red}$] [$3$ - $\color{limegreen}{Green}$] [$4$ - $\color{blue}{Blue}$] [$5$ - $\color{deepskyblue}{Cyan}$] [$\ge6$ - $\color{magenta}{Pink}$]
Here are numbers from $0$ to $544$, and number bases up to $99$: (click and zoom in)

What I concluded here is that $N$-digit palindromes can be connected with $N-1$ degree polynomials. Red palindromes ($2$ digits) form lines, which are polynomials of degree one. Following them, Green palindromes would form parabolas; and so on.

But the interesting things form when we go deeper; by increasing the $x$ value. Below is a plot starting at $(46000,0)$, and just some structures I highlighted below:


These are just examples of "curves" which can either be formed by a line of palindromes (highlighted orange) or by a line of black regions (highlighted yellow).

The green region seems to be the source of those curves and they seem to be able to extended both up and down. The red region forms black horizontal spaces with parabolas inside:

This is at around $(8\times10^6, 4000)$ but roughly zoomed out. Notice triangle-like structures that seem to be branching down from above.

Question

What I want to know, is how to mathematically describe this plot and structures in it?
Are there similar things examined somewhere else? I'm looking for references.



 A: All points are formed by the polynomials as you said. When the graphs of these polynomial functions are too sparse, secondary structures start to appear. You can generate these structures like this:
Take one of the polynomial and replace the coefficients by functions of x and y. The resulting function should intersect the $\Bbb{N}^2$ grid as densely enough and it should have small first and second derivations. If this is met on an interval, you will see a secondary structure in the given form.
If you start with linear functions of y, you will get most of the structures, mainly the parabolas. You can also try other polynomial functions of x and y.
There are also thick structures which are formed when a few similar structures are located close together. And there are also thick gaps which are formed when there are no structures in the given area. 
The question is still: Which functions yield the noticeable structures?
I think, this problem is similar to the research of prime numbers or the Collatz conjecture. There seems to be an order but it's really hard (if not impossible) to predict it.
