e^iθ = cos θ + i sin θ

Prime Helix

Primes on Euler's helix · a geometric exploration

Euler's identity, read as a three-dimensional object, is a helix. This project places the prime numbers on that helix, connects consecutive primes with straight-line chords, and asks a simple kinematic question: if two particles travel at the same speed — one along the helix, the other hopping chord to chord between primes — what is the ratio of their paths?

The answer converges to 1/√2 = cos(45°), determined by the helix geometry alone. The fluctuations around that limit carry information about prime gaps, and appear to reflect — in purely geometric terms — the same phenomena captured by the Riemann zeta function.

d_n = √(2 − 2cos(g_n) + g_n²) | R(N) = Σd_n / (p_N − 2)√2 → 1/√2

Read the paper ↗

Interactive Visualizations

3D Prime Helix

Prime Helix — 3D

Primes embedded on the helix (cos p, sin p, p) with connecting chords. Drag to orbit, scroll to zoom, hover for values. Gap coloring reveals the irregular rhythm of prime spacing.

Three.js · Interactive

Convergence Dashboard

Chord Hopper vs Helix Walker

The two-particle experiment. Watch R(N) converge toward 1/√2 as you increase the prime count. The local ratio scatter reveals how each gap size contributes to the cumulative ratio.

Convergence · Up to 8000 primes

Phasor Circle

Phasor Circle & Phase Function

Dual view: the prime helix projected onto the unit circle (revealing caustic rings from prime gap harmonics) and θ(n) = p_n mod 2π plotted against n (revealing diagonal striations — a geometric spectrogram).

Caustic rings · Up to 10000 primes

Riemann Vortex

The prime helix spiraling onto the Riemann sphere via logarithmic projection. North pole = infinity. Caustic rings become latitude bands. Up to 500k primes rendered in real-time.

Riemann sphere · Up to 500k primes