Sunflower Golden Angle
Change the angle between seeds and see why sunflowers use 137.5°.
How to play
- Drag across the flower head to change the angle between one seed and the next. Near 137.5° it settles onto the golden angle.
- Tap a seed to light up the spiral it belongs to, and see how many spirals of that kind there are.
- Open Lab for the exact golden angle and other famous angles.
- Watch the readouts: the spiral counts and how evenly the seeds are packed.
Challenges
- Tap seeds out from the centre until you light up one of the 55 spirals
- Drag the head to an angle that lines the seeds up in straight spokes
- Drag it back to the golden angle, 137.5°, where the seeds pack with no gaps
Try this
- Press 137.3° and then 137.6° in the Lab. A change of a fifth of a degree opens gaps and lines the seeds up in curved spokes.
- At the golden angle, tap a seed near the middle and then one near the rim. Their spirals belong to families of 55 and 89: neighbouring Fibonacci numbers.
- Try 135°, three-eighths of a turn. The seeds line up in 8 straight spokes with wasted space between them.
Fun facts
- The golden angle is about 137.508°. It divides a full turn into two parts in the golden ratio, and it is the smaller part.
- Helmut Vogel proposed the simple model r = c√n, θ = n × 137.508° for the seed head of a sunflower in 1979.
- The numbers of spirals in a sunflower head are usually neighbouring Fibonacci numbers, such as 34 and 55 or 55 and 89.
- A sunflower head is made of hundreds or thousands of tiny individual flowers called florets, each of which can become a seed.
Why the golden angle packs seeds best
Imagine placing seeds one at a time around the centre of a flower head, turning by the same angle before each new seed. Which angle fills the space most evenly? If the angle is a simple fraction of a turn, the seeds pile up in lines. Half a turn (180°) gives two straight rows; three-eighths of a turn (135°) gives eight spokes with empty wedges between them. Any fraction p/q eventually lines the seeds up in q spokes.
The most irrational number
To avoid spokes, the angle must not be a fraction of a turn at all, and more than that, it must stay far from every simple fraction. The number that does this best is the golden ratio, φ ≈ 1.618. In a precise sense it is the hardest number to approximate with fractions, because its continued fraction is made entirely of ones. Turning by the matching angle, 360° × (1 − 1/φ) ≈ 137.508°, places each new seed into one of the biggest gaps left by the earlier ones, so the head fills evenly with no wasted space. The packing readout shows how large round seeds could grow without touching: it peaks at the golden angle and falls quickly as you move away.
Where the spirals come from
Nobody draws spirals on a sunflower, yet our eyes find them. Each seed has close neighbours a fixed number of seeds before and after it, and following those neighbours traces spiral arms called parastichies. At the golden angle those neighbour counts are Fibonacci numbers, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…, because the ratios of neighbouring Fibonacci numbers are the best fractional approximations of the golden ratio. Near the rim of this head you will find 55 spirals one way and 89 the other.
Vogel’s model
In 1979 Helmut Vogel suggested a simple rule: seed number n sits at angle θ = n × 137.508° and at distance r = c√n from the centre. The square root keeps the area per seed constant, so the seeds stay the same size from centre to rim. Real sunflowers are not perfect, but this model reproduces their pattern remarkably well.
For the curious
The golden angle is 360°/φ² ≈ 137.5077°. In this simulation each seed’s two nearest neighbours near the rim give the spiral counts, and the packing figure is the fraction of the head that equal round seeds could cover, using the spacing between nearest neighbours.
Questions people ask
What is the golden angle?
It is about 137.5°: the smaller of the two angles you get by dividing a full circle in the golden ratio. Turning by it again and again never lines points up, which is why it packs seeds so evenly.
Why do sunflowers have Fibonacci spirals?
If each new seed is placed at the golden angle from the last, the nearest neighbours of each seed lie a Fibonacci number of seeds away. Following those neighbours traces spirals whose counts are neighbouring Fibonacci numbers, such as 34 and 55.
Do all sunflowers follow the Fibonacci pattern?
Most do, but not all. Studies of real sunflower heads find that some show other spiral counts or irregular patterns, so the golden-angle model describes the usual case rather than a rule without exceptions.
