Phyllotaxis: Why Sunflower Seeds Pack at Exactly 137.5°

Phyllotaxis: how plants place new growth

Phyllotaxis — Greek for "leaf arrangement" — describes how plants position leaves, seeds and petals. Growth happens at a central tip called the meristem, which produces new features (primordia) one at a time in a spiral. The plant doesn't calculate angles; it simply follows a local rule — grow at the least-crowded position available, away from everything already there. Remarkably, that local, greedy rule turns out to be mathematically equivalent to rotating by one specific angle each time.

The golden angle: 137.5°

Split a full 360° circle in the golden ratio and you get two arcs: about 222.5° and about 137.5°. That smaller arc is the golden angle. Place each new seed rotated 137.5° from the last, and something remarkable happens: the seeds pack together with maximum density, no gaps, no clumping, and interleaved spiral arms emerge automatically — without the plant ever "knowing" a Fibonacci number.

golden angle = 360° × (1 − 1/φ) ≈ 137.50776°
Why not a "clean" angle like 90° or 120°?
  → any rational fraction of 360° lines seeds up in straight radial rows with big gaps

Vogel's model: the formula behind the seed head

In 1979, Helmut Vogel formalised the pattern into a single, elegant rule for placing seed number n:

θ(n) = n × 137.5°
r(n) = c × √n     (c = scaling constant)

Each seed sits at an angle that is just the golden angle multiplied by its index, and at a radius that grows with the square root of n — which is exactly the radial spacing needed to keep every seed's local area roughly constant as the spiral expands outward. Run this two-line formula for a few hundred points and interleaved spiral families — visible as parastichies — snap into place automatically, matching real sunflower heads almost exactly.

Why the golden angle beats every "neat" alternative

If the rotation angle were a rational fraction of 360° — say exactly 90° (360°/4) — every fourth seed would land on the same radial line, leaving obvious wedge-shaped gaps between four spokes. The golden angle avoids this because it is the least well-approximated angle by any simple fraction: its continued fraction is [0; 1, 1, 1, 1, …], the slowest-converging expansion possible. That means no matter how many seeds you place, no two ever fall on the same radial line, and the packing stays maximally dense at every stage of growth — not just at some final seed count.

This is also why sunflower spiral counts are always consecutive Fibonacci numbers — 34 and 55, or 55 and 89, or 89 and 144 depending on the variety — since the Fibonacci sequence is precisely the set of "best rational approximations" to the golden angle, and each approximation corresponds to a visible spiral family the eye picks out.

Frequently asked questions

Try it live

Everything above runs in your browser — open Phyllotaxis, drag the divergence-angle slider away from 137.5° and watch the dense, gap-free spiral pattern instantly collapse into radial rows with gaps. Nothing is installed, nothing is uploaded.