Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations

Plotting points (p, p) in polar coordinates—using radius r = p and angle θ = p radians—creates outward Archimedean spirals. When all integers are included, the spirals look clean. Filtering down to primes makes the picture look chaotic at first, then reveals a structured “Milky Way” of spiral arms and missing sections. The central insight is that the gaps and surviving arms come from basic divisibility constraints on primes, while the long-run balance of primes across the remaining arms is governed by Dirichlet’s theorem.

The construction turns each step of +1 in p into a rotation by 1 radian while also increasing the radius by 1. Because 6 radians is close to 2π, counting by 6 produces points whose angles nearly complete a full turn each time; visually, that near-commensurability separates the integers into six spiral arms. More precisely, the arm corresponds to a residue class modulo 6: numbers that leave the same remainder when divided by 6. When primes are selected, most arms vanish. A prime cannot be a multiple of 6, cannot be 2 more than a multiple of 6 (those are even), and cannot be 3 more than a multiple of 6 (those are divisible by 3). Only the residue classes compatible with primality remain, aside from the small exceptions 2 and 3.

Zooming out changes the scale. Now the relevant near-rotation is 44 radians, which is close to 7 full turns because 44/(2π) is just above 7. This yields 44 residue classes modulo 44, each forming its own spiral arm. Primes eliminate entire arms whenever the corresponding residue class forces a composite structure—e.g., residues that make numbers even or divisible by 11. What survives are exactly the residue classes that are coprime to 44 (share no prime factors with 44). Counting those coprime residues gives Euler’s totient function: φ(44) = 20. The picture’s “missing teeth” and the number of visible arms trace directly to this totient count.

The same mechanism repeats at an even larger scale using a striking rational approximation: 710 radians is extremely close to an integer multiple of 2π. That approximation is tied to the famous fraction 355/113 for π, since 355/113 ≈ π. Because 710 is built from prime factors 2, 5, and 71, residue classes modulo 710 that are divisible by any of these factors cannot contain primes (except for the factor itself). Visually, this produces alternating rays, occasional gaps, and clumps whose structure reflects which residue classes are ruled out by divisibility.

The most consequential part is not the geometry but the distribution. Within the residue classes that remain possible—those coprime to the modulus—primes appear to spread evenly. Dirichlet’s theorem makes that expectation precise: for any modulus n and any residue r that is coprime to n, the primes congruent to r (mod n) have asymptotic density 1/φ(n). In other words, every admissible residue class gets infinitely many primes, and in the long run they occupy an equal share among the φ(n) allowed classes. The proof’s machinery leans on analytic number theory, including complex analysis, and the theorem underpins modern work on prime gaps and related conjectures.