Math
Sunflower
Vogel's sunflower grown seed by seed, r = √n and θ = n·α: sweep the angle by dragging across the disc and the spirals unwind, collapse onto rays, and lock in at the golden angle 137.508°, where the closest pair is 1.602 at any size and the families are Fibonacci numbers - the packing measured live, a task on the angle, and the chart of packing against angle.
What happens
Put each new seed one fixed angle round from the last and a little further out - r = √n, so every seed gets the same area - and watch what the angle does. At a rational turn, a seventh say, the seeds line up on seven rays and the disc is nearly empty between them; the closest pair shrinks toward zero as the disc grows. At 137.3° the spirals unwind into a single family that leaves gaps. At 137.508°, the golden angle 360/φ², the disc fills evenly: the closest pair is 1.602 with 300 seeds and 1.602 with a thousand, and the two spiral families you see are always consecutive Fibonacci numbers - 21 and 34, then 34 and 55, then 55 and 89. Drag across the disc and the angle sweeps by hundredths of a degree; the chart records the closest pair at every angle you visit, a needle at the golden angle, another at the Lucas angle 99.5°, and valleys at every rational turn. The seeds are coloured by the family they belong to, so the spirals are the colours. A sunflower head, a pine cone and a pineapple all do this, because a plant that adds each bud a golden angle from the last never has to plan its packing.
Equations used in the module
Equations Vogel (1979): seed n at r = √n, θ = n·α - equal area per seed, so the mean spacing stays near 1.7 units of √n as the disc grows golden angle: α = 360°/φ² = 137.50776°, φ = (1 + √5)/2; the Lucas angle 360°/(φ + 2) = 99.5016°; 360° − α is the mirror image closest pair: the minimum nearest-neighbour distance over the disc; golden 1.602 at 300 seeds and 1.602 at 1000; 137.3° → 1.10; 137.6° → 0.97; a seventh of a turn → 0.20 at 300, 0.11 at 1000 (seven rays); Lucas 1.574 families (parastichies): the two most common |i − j| among nearest-neighbour pairs; golden: 34 and 21 at 300 seeds, 55 and 34 at 1000, 144 at 4000 - consecutive Fibonacci numbers; Lucas: 47 and 76; 137.3°: one family of 21 with 973 of 1000 seeds task window: 300 seeds, closest pair ≥ 1.5 for α in 137.438 … 137.528 (0.09°), 137.408 reads 1.43; the Lucas angle passes too nearest neighbours: bucket grid of cell 2, first found wins - the same walk in the Python twin and the C# model
