One angle, a thousand seeds
Grow a seed, turn by a fixed angle, grow the next a little farther out, and repeat. That single rule — turn, step out, repeat — is how a sunflower, a pinecone, and a cactus all arrange themselves.
Almost every angle fails: the seeds collapse into a few radiating spokes, wasting space. One angle near 137.5° packs them perfectly. Drag the dial and hunt for it.
The most irrational number
Why 137.5°? It's the circle divided by the golden ratio φ squared — 360°/φ² = 137.507…°. The golden ratio is the number hardest to approximate by any simple fraction, so this angle never lines up with itself.
Any rational turn — a clean fraction of the circle — makes the seeds repeat, and repetition means spokes. Step through the fractions closing in on φ and watch the spokes multiply, then vanish.
Spirals come in Fibonacci
Look at a finished head and your eye catches spirals — two families, one winding each way. Count them and you always get two consecutive Fibonacci numbers: 21 and 34, or 34 and 55.
It isn't mysticism. Each seed's nearest neighbours sit a Fibonacci number of steps away, so those steps trace the arms you see. Add seeds and the counts climb the sequence. Highlight the arms.
How tight can circles pack?
Phyllotaxis is one answer to an old question: how do you cover the most area with equal circles? Lay them on a square grid and they fill 78.5% of the plane. Shift every other row into the gaps — hexagonal packing — and you reach 90.7%.
That hexagonal figure, π⁄(2√3), was proven optimal: no arrangement of equal circles does better. Toggle between the two and read the density.
The Apollonian gasket
Drop a circle into every gap between three touching circles, then do it again in the smaller gaps it makes, forever. The curvatures obey a 2000-year-old rule of Apollonius, and in this gasket they come out as whole numbers — 2, 3, 6, 11, 18, 27, 38… A packing with no smallest part. Push the depth, recolour, let it turn.