I wrote a Python script to plot various sets of numbers in polar coordinates.
I used two coordinate transformations to do this: $(r, \theta) = (n, n)$ and $(r, \theta) = (\sqrt{n}, \tau\sqrt{n})$, in which $r$ is the distance from the origin, $\theta$ is angle from x-axis, and $n \in \mathbb{Z}$.
According to Wikipedia, an emirp (prime spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed.
My definition explicitly includes palindromic primes, and it also includes prime numbers that when its digits in bases other than 10 is reversed results in a prime number as well.
So I have messed around with it and found something interesting:
All emirps under 100,000 in decimal, in $(n, n)$ polar coordinates:
All emirps under 1,000,000 in decimal, in $(n, n)$ polar coordinates:
All emirps under 10,000,000 in decimal, in $(n, n)$ polar coordinates:
They form several concentric rings. And more numbers do not change the shape, it merely makes the rings thicker and smoother. The images converge to something.
In $(\sqrt{n}, \tau\sqrt{n})$ coordinates, the images converge to something else:
decimal emirps under 10,000,000, in $(\sqrt{n}, \tau\sqrt{n})$
decimal emirps under 100,000,000, in $(\sqrt{n}, \tau\sqrt{n})$
This phenomenon is not limited to base-10, it occurs in other bases as well:
dozenal emirps under 2,985,984 in $(n, n)$
dozenal emirps under 2,985,984, in $(\sqrt{n}, \tau\sqrt{n})$
hexadecimal emirps under 16,777,216 in $(n, n)$
hexadecimal emirps under 16,777,216, in $(\sqrt{n}, \tau\sqrt{n})$
I am not a native English speaker and I cannot properly describe what this is, and Google search again returns nothing relevant.
Exactly what am I seeing? And what caused it?
I have found something relevant though, if the base is a prime number, these patterns do not form, so the base must be composite.
Update
I have already investigated emirps in prime bases and confirmed there are no concentric gaps, because these aren't interesting I didn't include them originally.
My program runs fast enough that it can run a few million iterations in under 10 seconds, I of course had investigated that already.
emirps in binary, under 1,048,576, in $(r, \theta) = (n, n)$
emirps in pental, under 9,765,625, in $(r, \theta) = (n, n)$
emirps in tridecimal, under 4,826,809, in $(r, \theta) = (n, n)$
They don't have gaps in them, I have also used many other prime bases, and in all cases the patterns don't emerge, using $(r, \theta) = (\sqrt{n}, \tau\sqrt{n})$ doesn't change it, in all cases more numbers make the image more disk-like, the images end up just a white disk on a black background.
Only composite bases form these patterns.
emirps in sexagesimal, under 12,960,000, $(r, \theta) = (n, n)$
emirps in sexagesimal, under 12,960,000, $(r, \theta) = (\sqrt{n}, \tau\sqrt{n})$
I have also tried to use other coordinates, namely scaling $r$ to $\log_{base}(r)$.
emirps in decimal, under 10,000,000, $(r, \theta) = (\log_{10}(n), n)$
emirps in decimal, under 10,000,000, $(r, \theta) = (\log_{10}(\sqrt{n}), \tau\sqrt{n})$
emirps in sexagesimal, under 12,960,000, $(r, \theta) = (\log_{60}(n), n)$
emirps in sexagesimal, under 12,960,000, $(r, \theta) = (\log_{60}(\sqrt{n}), \tau\sqrt{n})$
As the raw data is too large I won't share it, I didn't serialize it in the first place, though I know how to serialize it I don't see the point, it is very easy to compute if you know some programming, and serializing the data in csv or json is inefficient, it can be pickled but pickle files are not compatible across Python versions.
If you want the data, compute it yourself, or you can use the script I have written that is very efficient to compute the data, which I provided at the top of the question in a link, and it's also what I used to generate the images, it only takes 2.87 seconds to compute emirps in decimal up to 10,000,000.
Update
I have investigated prime reversals in composite bases as the comment suggested, and indeed found similar patterns, and I have also confirmed that these patterns don't emerge when using prime bases.
Though I have noticed that they fall into several spiral arms instead when using prime bases, with periodic gaps where there are no spiral arms, these gaps become thinner as the numbers become larger.
Digit reversal of primes under 10,000,000 in decimal in $(r, \theta) = (n, n)$
Digit reversal of primes under 2,985,984 in dozenal in $(r, \theta) = (n, n)$
Digit reversal of primes under 16,777,216 in hexadecimal in $(r, \theta) = (n, n)$
