Phyllotaxis · the golden angle

Why do sunflowers, pinecones, and daisies place their seeds at the golden angle (≈137.5°) — what is special about it?

▶ Run the simulationSee the measured result

Units: degrees (the golden angle 360°/φ² = 360°·(2−φ), φ = (1+√5)/2)

How the lab tests it

Lay seeds out by Vogel's spiral (seed n at angle n·α, radius √n — uniform areal density for any α), sweep the divergence angle α, and measure the packing uniformity (the minimum nearest-neighbour gap) at each angle.

What it checks

a sharp GLOBAL packing optimum at the golden angle α* = 360°/φ² ≈ 137.508° — the largest minimum gap (most uniform packing); the measured argmax lands at 137.5°, matching it to within the sweep step, and collapses into radial spokes with wasted wedges a degree away. The golden angle wins because φ is the 'most irrational' number, hardest to approximate by a rational p/q that would line every q-th seed into a spoke (other 'noble' angles are weaker local optima)

Golden angle, continued-fraction & rational-spoke calculator

Why 137.5°, and not 137.4° or 90°? This page answers it the way the question is actually shaped — as number theory — and assembles every figure it prints from √5 and integer arithmetic. φ is built as (1+√5)/2 and the angle as 360·(2−φ), the same spelling the shipped module uses for its own GOLDEN static, which the oracle pins bit-for-bit against the reference: 137.50776405003785°. Nothing on the page is stored. Type a divergence angle and it is expanded by the division algorithm into its continued fraction, whose convergent denominators are the candidate spiral counts — for the golden angle those are 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and Fibonacci turns out to be the denominator sequence of one particular number rather than a fact about flowers. Three things here are second routes to numbers the oracle could only reach by building a spiral and searching it. Its rival, the 2/5-turn 144° head, has its minimum gap MEASURED at 0.0792 florets across at N = 1000 and 0.0280 at N = 8000; two square roots give 0.079156 and 0.027955, because along a ray floret k sits at radius √(5k) and the tightest pair is always the last two, q/(√N + √(N−q)). That same expression prints the world's N^(−1/2) decay law without ever calling a logarithm: gap·√N is 2.5033 and 2.5004 at the two head sizes, closing on q/2, and a quantity whose product with √N is constant IS a power law of exponent −½. And the screen's headline, 137.50667°, is rebuilt as arithmetic: the module only looks at α on a grid of 0.7/60·4 = 0.046667° per sample, node 118 from the window start is 137.5066667°, and the 5.0e-12 left over is the drift of 472 float additions, bounded here by an ulp found through halving. The registry's nearest neighbour is ?world=stdmap, which also cares about the golden mean — but there it is the winding number of the last torus to break under a kick, and the shared word is not a shared claim. Four things this page will NOT do. It will not correct a number the lab recovered. It will not invent a closed form for the golden angle's own peak gap, 1.5462 √n-units, which the oracle measured and which has none here. It never enters the plane: floret coordinates, the visible parastichy pair and the packing efficiency all need a sine, and a sine is not among the operations IEEE-754 requires to be correctly rounded, so this page stays on the circle and keeps its digits. And it cannot prove φ optimal — Ridley did that and the oracle measured it; what this page shows is the number theory that makes it unsurprising.