The sunflower turn
The odd turning angle that lets plants pack seeds with no gaps.
What you are looking at
Dots are placed one at a time, each a little further out and turned a certain amount from the one before. Curved spirals show up on their own.
The idea
Imagine placing seeds on a flower head. Put each new seed a bit further from the center, and turn by a fixed amount before you place it. What turn should you pick?
If you turn exactly a quarter of the way around each time, every fourth seed lands on the same line. You get four straight spokes with big gaps between them. Any turn that is a neat fraction does something like this, because eventually the seeds repeat.
So the best turn is one that never repeats. The most "never repeating" turn there is happens to be about 137.5 degrees, or a bit under three eighths of a full circle. With that turn, no two seeds ever line up, so the seeds fill the disc evenly with no gaps. This is what many plants do.
Where else you see it
- Sunflower and daisy centers.
- Pinecones and pineapples, where you can count spirals in two directions.
- The way leaves are arranged around a stem, so each leaf gets sunlight.
Go deeper: the proper names and the math
The angle is 360° × (1 − 1/φ) ≈ 137.5078°, where φ = (1+√5)/2 is the golden ratio. The standard model (Vogel, 1979) places seed number n at radius r = c√n and angle θ = n × 137.5°. The √n makes all seeds take up the same area.
The golden ratio is the number that is hardest to approximate with fractions (its continued fraction is all 1s), which is the exact reason the pattern never lines up. The spiral counts you see in real flowers are neighboring Fibonacci numbers like 21 and 34, because fractions made from consecutive Fibonacci numbers are the best approximations of the angle.