Plot the primes in polar coordinates: why do spirals and rays appear?

Polar coordinates fix a point by how far it is from the centre and how far it has turned. Put the whole number $n$ at

Every point lies on the Archimedean spiral $r=\theta$, and neighbouring numbers are 1 radian (about 57.3°) apart. On the stage the primes are amber and the other numbers blue.

Close in there is only one loose spiral; the pattern appears as you zoom out.

$n$ and $n+m$ point $m$ radians apart. If $m$ is almost a whole number of turns, $n+m$ lands just outside $n$, a little off to one side, so every $m$-th number joins into one slowly curving arm, $m$ arms in all:

The smaller the miss, the straighter the arms: the 6 arms wind once every 133 numbers outwards, so they show around 150; the 44 spirals take about 15,600 to wind once; the 710 take 74 million, so below a million they look like rays.

The numbers come from the continued fraction of $2\pi$:

Its convergents are $6,\frac{19}{3},\frac{25}{4},\frac{44}{7},\frac{333}{53},\frac{710}{113}$. The larger the next term, the better that fraction and the clearer its arms: 6 is followed by 3, 44 by 7, 710 by 146. Halved, they are the famous approximations $\frac{22}{7}$ and $\frac{355}{113}$ for π.

Arm $r$ holds the numbers $mk+r$. If $r$ and $m$ share a factor $d>1$, every number on that arm is divisible by $d$, and at most $d$ itself is prime. So primes sit only on arms where $r$ and $m$ have no common factor; there are $\varphi(m)$ of them (Euler’s totient):

The 24 empty arms of the 44 are the even arms and the multiples of 11; $710=2\times5\times71$, so the empty rays hold multiples of 2, 5 or 71.

In 1837 Dirichlet proved that whenever $r$ and $m$ have no common factor, $mk+r$ contains infinitely many primes, and in the long run every such arm gets the same share.

The stage’s own numbers bear it out: the 78,498 primes below a million, sorted by remainder mod 44, put between 3874 and 3964 on each of the 20 arms; over the 280 rays of 710 they average about 280.

So the pictures do not say that primes hide a spiral structure; they say two plainer things: whole numbers form spirals in polar coordinates, and primes spread evenly over the arms they are allowed.