**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 a**number 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 $99$: (c*lick 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.