Plate 06 · Life

The Arithmetic of a Sunflower

Count the spirals its seeds make, and you will arrive at a famous sequence of numbers. All it takes to produce them is a single angle.

Try it

Chapters · 7

01Count Them
02A Famous Sequence
03Turn by 137.5 Degrees
04Even Slightly Off Won’t Do
05Nobody Is Counting
06657 Sunflowers
07Your Turn

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01

Count Them

A sunflower’s head is packed with seeds. Look along them and they fall into spirals, one set winding left, one set winding right.

On this model sunflower, near the rim, there are 55 spirals one way and 89 the other.

Model sunflower · 1,200 seeds

02

A Famous Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 … each number is the sum of the two before it. The sequence is named for the Italian mathematician Fibonacci.

The spirals counted on a sunflower are very often two neighbors in this sequence: 21 and 34, 34 and 55, 55 and 89.

03

Turn by 137.5 Degrees

Place the seeds one after another, each turned about 137.5 degrees further than the last and set a little farther from the center. That angle divides a full circle in the golden ratio; it is called the golden angle.

Keep placing seeds at that angle and Fibonacci spirals appear on their own. No one needs to know the sequence.

04

Even Slightly Off Won’t Do

Try another angle. Turn by 120 degrees and the seeds line up in three straight arms with gaps between; turn by 137 and the spirals survive, but cracks open between them.

Of all these angles, the golden angle packs the seeds most evenly and most tightly.

05

Nobody Is Counting

Plants do not measure angles. Where the head is growing, each new floret — the future seed — appears where there is the most room, as far as possible from those already there.

In 1992 two French physicists dripped a magnetic fluid, drop by drop, into the center of a dish of oil inside a magnetic field. The drops repelled one another and drifted outward, each heading away from the ones before it. At the right drip rate, the angle between one drop and the next settled near 137.5 degrees on its own.

06

657 Sunflowers

In 2012, a citizen-science experiment in Manchester, England, asked people to grow sunflowers and count their spirals, in honor of the mathematician Alan Turing, who studied the question in the last years of his life. The researchers analyzed 657 flowers.

In the most reliable part of the data, about three in four counts were Fibonacci numbers and nearly one in ten belonged to related sequences; roughly one in five was neither. Nature is not that tidy, and recording the untidy part honestly is what science is.

07

Your Turn

Drag the angle and watch spirals appear and dissolve. Add or remove seeds and watch the spirals you can count step up to the next pair of Fibonacci numbers.

Try thisChange the turn per seed from 137.51° to 135°: how evenly packed are the seeds now?

What the model leaves out

The model follows Vogel (1979): seed n is turned through n equal angles and set at a distance from the center proportional to the square root of n. The model’s head is a regular, shallow dome; a real one bulges, dips or buckles, and its seeds change in size and shape from center to rim; the order in which the seeds actually form is not modeled here.

A common misreading

Not every sunflower has Fibonacci spirals, and not every plant follows the golden angle. Note, too, that the spirals are something we see: every seed lies on several families of spirals at once, and which two your eye picks out depends on which ring of the head you count along.