🌻 Phyllotaxis — Sunflower Spirals & the Golden Angle

Phyllotaxis is the study of how plants arrange repeated organs — seeds, leaves, florets, scales — around a stem or disk, typically in spiral patterns. This simulation reproduces the classic sunflower-seed-head pattern using Vogel's mathematical model, placing each seed n at angle θ = n·δ and distance r = c·√n from the centre, then rendering the result as a live, rotatable 3D point cloud with Three.js.

The golden angle is the angle you get by dividing a full 360° turn in golden-ratio proportion: 360° / φ² ≈ 137.50776°, where φ = (1 + √5) / 2 ≈ 1.618 is the golden ratio. Because φ cannot be well approximated by any simple fraction, rotating by this exact angle repeatedly never produces two points at the same angular position, which is why it gives the densest, gap-free packing of the seeds.

The visible spiral arms (parastichies) come in two interlocking families, one turning clockwise and one counter-clockwise. Because the golden angle placement makes the Fibonacci-number ratios (1/1, 1/2, 2/3, 3/5, 5/8, 8/13...) the best possible rational approximations of φ, the number of arms you can trace in each family is almost always a pair of consecutive Fibonacci numbers — such as 21 and 34, 34 and 55, or 55 and 89 — with larger seed counts producing larger Fibonacci pairs.

Even a tiny offset, such as 137.3° or 137.6°, breaks the near-irrational balance that prevents seeds from lining up. Instead of a smooth double spiral you get visible straight radial spokes or wide gaps, because seeds separated by a few turns now land at nearly the same angle. The simulation's δ slider (135°–140°) and "Snap to golden angle" button are built to make this contrast obvious.

Vogel's model sets radius r = c·√n specifically because the area of a ring at radius r grows proportionally to r, so scaling r by the square root of the seed index n keeps the area allotted to every seed constant as the pattern grows outward. Without the square root, seeds near the centre would be crowded together while seeds near the edge would be spaced too far apart, which does not match real sunflower or pinecone growth.