The lab

Phyllotaxis

Place seeds one at a time, each rotated by a fixed divergence angle from the last, each pushed outward as the square root of its index. At most angles, you get spokes or gaps. At the golden angle — 137.508° — you get the densely packed spirals of a sunflower head. The Fibonacci numbers are not an accident. They are the inevitable consequence of irrationality.

Vogel’s model · θ_n = n × α, r_n = c√n · α ≈ 137.508°

α = 137.508°

seeds: 500

Color

presets

Divergence angle 137.508°

Number of seeds 500

A sunflower head is not designed. It is grown. Each floret emerges at the center of the disk and is pushed outward by the ones that follow. The only parameter that matters is the angle between successive florets — the divergence angle. If that angle is a simple fraction of a full turn, you get radial spokes: 90° gives four arms, 120° gives three, 144° gives five. The seeds line up, leaving wasteful gaps between the arms.

But if the divergence angle is the golden angle — approximately 137.508° — something remarkable happens. No seed ever lines up with a previous one. The pattern fills the disk with extraordinary uniformity, and when you look at it, you see spirals. Not one family of spirals, but two: one set winding clockwise, another counterclockwise. Count them. The numbers are always consecutive Fibonacci numbers: 13 and 21, or 21 and 34, or 34 and 55.

This is not a coincidence. The golden angle is 360° × (1 − 1/φ), where φ is the golden ratio (1 + √5)/2. The golden ratio is, in a precise sense, the most irrational number — the number worst approximated by any fraction. Its continued fraction expansion is [1; 1, 1, 1, ...], all ones, converging as slowly as possible. The best rational approximations to φ are ratios of consecutive Fibonacci numbers: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21... This is why you see Fibonacci numbers in the spirals. They are the rational numbers that come closest to the golden ratio, so they are the directions along which seeds almost align.

The math

Vogel’s model (1979): θ n = n \times α r n = c \times \sqrt n where n is the seed index, α is the divergence angle, c is a scaling constant, and the \sqrt n spacing ensures equal area per seed. Golden angle: α = 360° \times (2 - φ) \approx 137.50776° where φ = (1 + \sqrt5) / 2 \approx 1.6180339...

Where this appears in nature

Sunflower heads are the most famous example, but phyllotaxis is everywhere. Pinecone scales spiral in Fibonacci numbers. Pineapple hexagons form 8, 13, and 21 spirals. Romanesco broccoli is a fractal phyllotactic pattern — each floret is a miniature copy of the whole, arranged at the golden angle. Aloe and agave succulents show the spiral clearly from above. Even the arrangement of leaves around a stem (the original meaning of phyllotaxis, from the Greek phyllon, leaf) follows this pattern: leaves are placed to minimize overlap, maximizing each leaf’s access to sunlight.

A brief history

Leonardo da Vinci noted spiral leaf patterns in his notebooks. Johannes Kepler, in 1611, observed the Fibonacci numbers in plant spirals and connected them to the golden ratio. The brothers Louis and Auguste Bravais, in 1837, gave the first mathematical treatment of phyllotaxis and identified the golden angle as the key parameter. But it was Helmut Vogel, in 1979, who wrote down the elegant model used in this simulation: polar coordinates where angle grows linearly and radius grows as the square root. That single formula — trivially simple — reproduces the entire visual complexity of a sunflower head.

The deeper question is why plants use the golden angle. The answer, worked out by Douady and Couder in 1992 with physical experiments using magnetized droplets, is that it is a dynamical attractor. When new primordia form at the growing tip and are pushed outward by growth, the system naturally converges on the golden angle because it is the divergence that minimizes the energy of repulsion between neighbors. The plant does not know about Fibonacci numbers. The physics does the work.

Try sweeping the angle slider slowly from 137° to 138°. Watch how the spiral pattern breaks and reforms. At exactly the golden angle, the pattern is as uniform as it can possibly be. Even a fraction of a degree away, visible gaps and alignments appear. Then try the “Fibonacci spirals” color mode and count the spiral arms — you will find 13, 21, 34, or 55.