The golden angle

Drop a seed, turn by a fixed fraction of a turn, drop the next one a little further out, and keep going. Most turns pile the seeds into arms with gaps between them. Turn by 0.618034…, the golden ratio's fractional part, and the seeds pack evenly at every size, like a sunflower head. Drag the turn and watch the arms come and go.

A decimal, a fraction like 2/5, or pi, e, phi or √2. Whole turns don't matter.

Turn
0.618034 of a turn (222.5°)
Arms at the edge
55 and 89
Closest seeds
88% of a honeycomb gap
Fractions it's close to
1/2, 2/3, 3/5, 5/8, …

The golden turn. No fraction is close enough to win, so the seeds fill the head evenly at every size, with 55 and 89 spirals at the edge.

Why the golden turn wins

A turn of 2/5 brings every 5th seed back to the same direction, so the seeds stack up in 5 straight arms with empty wedges between them. Any fraction p/q in lowest terms does the same with q arms, once there are enough seeds to go round.

A number close to a fraction nearly does it too. π turns act like 0.14159 of a turn, close to 1/7, so 300 seeds curl into 7 arms. Push it to 3,000 seeds and they straighten into 113, because 16/113 is closer still. Even 0.618, which looks golden, isn't: it's exactly 309/500, and that puts it closer to 55/89 than the golden turn is. At 500 seeds its closest two sit 84% of a honeycomb gap apart; by 3,000 its 89 arms have pulled apart and that's down to 56%.

So the best turn is the number that's hardest to approximate with fractions. Continued fractions measure that, since a big term means a close fraction, and the golden ratio's terms are all 1s, the smallest they can be: φ = 1 + 1/(1 + 1/(1 + …)). In 1891 Adolf Hurwitz proved that the golden ratio and its close relatives are the hardest numbers of all to approximate. The golden turn's closest fractions are ratios of Fibonacci numbers, 1/2, 2/3, 3/5, 5/8, 8/13 and on, which is why the spirals you can count come in Fibonacci numbers: 55 and 89 at the edge of 1,000 seeds.

However many seeds you add, the golden turn keeps the closest two at 88% of a perfect honeycomb gap. Try to beat it: no setting of the turn slider packs the seeds as well, whatever the seed count. The seed head here is H. Vogel's 1979 model of a sunflower: seed n sits √n from the centre, so every seed gets the same room. Turning 0.618 of the way round one way is the same as 0.382 the other, which is 137.5°, the golden angle.