Why a Polar Plot of Primes Forms Spirals and Rays

The narrator explores a Math Stack Exchange question about a surprising plot of prime numbers. The picture connects rational approximations of π with the distribution of primes in residue classes.

In polar coordinates, each point has a radius r and an angle θ. The plot assigns each integer n the point (r, θ) = (n, n), with the angle measured in radians; adding a full turn to the angle does not change its direction.

As n increases, the point moves outward and rotates one radian at each step, tracing an Archimedean spiral. Keeping only primes produces apparent arms and gaps, and further zooming reveals nearly straight rays.

The first surprise is that all integers produce similar spiral arms. Their geometry comes from the choice of coordinates and approximations to a full turn; primes determine which of those arms remain populated.

Consider steps of six. Six radians is slightly less than 2π, so adding six almost returns a point to the same direction. Repeated additions drift slowly and form one visible arm.

Each arm consists of numbers with the same remainder modulo 6. Except for 2 and 3, primes can only occur in the classes 1 or 5 modulo 6; other classes are divisible by 2 or 3.

A residue class is simply a group of integers with the same remainder after division by a chosen modulus. This vocabulary lets the narrator describe each geometric arm as an arithmetic sequence.

At a larger scale, 44 radians is just over seven full turns because 44/7 approximates 2π, equivalently 22/7 approximates π. Each residue class modulo 44 therefore drifts only slightly in angle at each step.

Most classes modulo 44 cannot contain large primes: even residues are divisible by 2 and residues divisible by 11 share that factor with the modulus. The remaining visible classes are coprime to 44.

Euler’s totient φ(n) counts the positive integers up to n that are coprime to n. For 44, φ(44) = 20, explaining the twenty eligible arms in the example.

At a still larger scale, 710 radians is extremely close to 113 full turns. This is the same approximation as 355/113 for π, so successive points in a class modulo 710 stay nearly aligned.

Because 710 = 2 × 5 × 71, most residue classes can be ruled out for large primes by divisibility. Removing classes divisible by 2 or 5 creates groups of four rays, while divisibility by 71 accounts for occasional gaps.

There are φ(710) = 280 coprime residue classes. Being eligible does not mean every number in a class is prime; it only removes the immediate obstruction from a common factor with 710.

To explore distribution, the narrator switches to modulo 10. Apart from 2 and 5, primes end in 1, 3, 7 or 9, corresponding to the four residue classes coprime to 10.

A histogram suggests that the fraction of primes in each eligible class approaches one quarter. This is an asymptotic statement about larger and larger bounds, not a promise of equal counts in every finite interval.

More generally, the prime number theorem for arithmetic progressions says that for a fixed modulus n, primes are asymptotically equally distributed among the φ(n) residue classes coprime to n.

The video attributes that stronger equal-distribution result to Dirichlet, but its description corrects the history: Dirichlet proved the reciprocal sum diverges in each eligible class, establishing infinitely many primes there; the stronger asymptotic result was proved later.

The narrator notes that complex analysis became an important tool in studying prime distributions. The video does not attempt the proof, which is substantially more involved than the visualization.

The polar plot itself is an arbitrary visualization, but exploring why it looks the way it does leads to meaningful mathematics. The narrator values this kind of play because it can make later definitions and theorems feel motivated.