Spiral

A sunflower head, a pinecone, the scales on a pineapple: all of them pack their florets by turning through the same angle, over and over, roughly 137.5° each time. Helmut Vogel gave that pattern an exact formula in 1979 — the nth floret sits at radius c·√n and angle n·137.5077…° from the center, one multiply-and-place per seed, no simulation needed. Below is that formula, drawn live. Drag the angle even a third of a degree away from 137.5077…° and the spiral pattern breaks into obvious straight arms with visible gaps — the interlocking double spiral you'd recognize as a sunflower only exists in a narrow band around one specific number.

Why that number and not, say, a clean 137.5 or 138? Because 137.5077…° is 360°×(1 − 1/φ), where φ is the golden ratio — and the golden ratio is, in a precise sense, the number that rational fractions approximate worst. Any other angle is close to some simple fraction of a full turn (138° is nearly 23/60), and florets spaced by a fraction like that line up into obvious straight rows with gaps between them, since every 60th seed lands almost exactly on top of an earlier one. The golden angle never lines up — no run of seeds ever nearly repeats a previous direction — so new florets keep landing in the widest remaining gaps instead of on top of old rows. That's the packing advantage, and it's the reason a real flower head needs no ruler: it isn't measuring 137.5°, it's just never finding a shortcut around it.

Honest gap: this page runs Vogel's formula, which describes the finished pattern — it does not grow one. A real meristem doesn't know any angle in advance. Each new primordium forms wherever auxin, the plant hormone that existing primordia deplete nearby, is most concentrated — mechanistically confirmed by blocking auxin transport and watching normal phyllotaxis collapse (Reinhardt et al., Nature, 2003). Run that local, iterative rule for long enough and the golden angle falls out on its own, the same way this page's one-line formula draws it directly. Two different processes, the same destination — this room draws the destination, not the walk.

Sources: Vogel, A Better Way to Construct the Sunflower Head, Mathematical Biosciences 44 (1979); Reinhardt, Pesce, Stieger et al., Regulation of Phyllotaxis by Polar Auxin Transport, Nature 426 (2003); the golden angle's own irrationality is the reason Bravais & Bravais first tied phyllotaxis ratios to Fibonacci numbers in 1837.