Phyllotaxis

Phyllotaxis is the arrangement of leaves, florets or seeds on a plant. In a sunflower head the seeds form two families of spirals, and the two counts are often neighbouring Fibonacci numbers. Here you can count the spirals anywhere on a disc of seeds, see what a small change of angle does, and watch a rule with no angle in it arrive near the golden angle. Under Fixed angle, seed $k$ of $N$, counting from 0, sits at

Each seed turns α further than the last, and the square root gives every new seed the same area of ring. How evenly the seeds spread round the rings depends on the angle. This is Helmut Vogel’s model of a sunflower head (1979).

The blue circle is where you count. On it the page finds the spirals the eye sees, draws them in yellow, solid for the nearest family and dashed for the other, and numbers them where they cross the circle.

No seed is joined to another, and the spirals are the eye’s work. Seed $k + m$ has turned $m\alpha$ further than seed $k$. When $m\alpha$ is close to a whole number of turns, the two point almost the same way, and if their radii are also close, they are neighbours. Joining every seed to the one $m$ later then traces a family of $m$ spirals. The golden angle is

and on the default circle

so seeds 34 apart and 55 apart are the nearest, and the eye sees 34 spirals and 55. A family short of a whole turn winds one way going out, and a family past it winds the other.

Why Fibonacci numbers? The $m$ for which $m\alpha$ comes closest to whole turns are read off the continued fraction of α as a fraction of a turn. For the golden angle every term after the 2 is 1,

so its best fractions have Fibonacci denominators. They fall alternately above and below α/360°, which is why at the golden angle the two families wind opposite ways. Which pair the circle shows depends on where it is. Two seeds $m$ apart differ a little in radius and a little in direction, and going out the difference in radius counts for less and the difference in direction for more. So further out, a family whose direction comes back closer takes over, even though its seeds are more steps apart, and the counts step up one Fibonacci number at a time.

Angles near the golden angle can look nothing like it, and the continued fraction says why. A large term signals an unusually close return in direction, which can make spiral arms stand out. The golden angle, whose terms after the 2 are all 1, keeps the directions even. The inset shows the first seeds’ directions round a circle: however many there are, the longest gap is at most $\varphi^2 \approx 2.618$ times the shortest.

Plants do not measure angles, and Most room gives none. A growing tip makes one primordium after another, each the bud of a leaf or floret. In 1868 Wilhelm Hofmeister observed that a new primordium appears in the largest space between the older ones, which then move away from the tip. Most room runs a simplified model of this rule. From the default start, while the older ones are pushed far out between births, each new primordium appears opposite the last. As the push shrinks, the angle splits away from 180°, swings round 137.5° and settles near 137.47°, and the spirals step up through Fibonacci numbers. From other starts the same rule can settle elsewhere, for example near 99.7°.