Phyllotaxis (from Greek phyllon leaf + taxis arrangement) describes how leaves, seeds, petals, and scales are arranged on plant stems and heads. The most striking example is the sunflower: hundreds of seeds packed in interlocking spirals with a specific number of clockwise and anticlockwise arms, always consecutive Fibonacci numbers.
The mathematical key is the golden angle — the angle subtended at the centre of a disc such that the two arcs are in ratio φ:1 (golden ratio).
A number is well-approximated by rationals if its continued-fraction coefficients are large. φ = [1; 1, 1, 1, …] has the smallest possible coefficients, making it the hardest irrational to approximate — the "most irrational" number. Any other angle is better approximated by a rational p/q, meaning seeds would eventually cluster into q radial rows after at most q turns, leaving gaps.
With angle α, after n seeds the minimum angular separation between any two seeds is at least 1/n — guaranteed only for the golden angle at every n. This produces maximum-entropy packing at every scale.
Although no seed is truly aligned, the eye resolves pseudo-rows because the best rational approximations to 1/φ are exactly the Fibonacci ratios:
Each ratio Fₙ/Fₙ₊₁ creates a set of Fₙ clockwise and Fₙ₊₁ anticlockwise spiral arms (or vice versa). Larger sunflowers show 34&55, 55&89, or even 89&144 arms in the outermost ring.
Switch to the "137° Near-miss" preset. The angle 137° = 137/360 × 360° quickly produces 137 visible radial spokes, leaving long angular gaps. At 138° you see different spokes. Only at 137.508…° do spokes continuously shift and never lock — the pattern remains uniform at all radii.
| Angle | Behaviour | Spirals visible |
|---|---|---|
| 137.000° | 137 radial spokes appear after ~137 seeds | None — straight lines |
| 137.508° (golden) | Uniform disc; no permanent spokes | Consecutive Fibonacci pairs |
| 138.000° | Spokes at 138° intervals | None — straight lines |
| 137.300° | Near-miss; pseudo-spokes at larger n | Weak spiral hints |
| 180.000° | All seeds on two opposite radii | None |
| 120.000° (1/3) | Three-fold symmetry, three radial arms | 3-fold |
| Property | Value |
|---|---|
| Golden ratio φ | (1+√5)/2 ≈ 1.6180339887… |
| Golden angle α | 360°/φ² ≈ 137.5077640500° |
| Seed n position | θ = n·α, r = √n |
| Fibonacci approx to 1/φ | Fₙ/Fₙ₊₁ → 1/φ as n→∞ |
| Spiral arm counts | Always consecutive Fibonacci numbers |
| Packing optimality | Most uniform 2D packing for any angle |
| Biological examples | Sunflower, pinecone, daisy, pineapple, cacti |
| Mathematical basis | Three-distance theorem (Steinhaus 1958) |