Why Do Sunflowers Count in Fibonacci Numbers?

Two heads, four numbers

If you counted carefully (or let the widget do it), you found these:

Head	Seeds	Clockwise spirals	Anticlockwise spirals
Small	500	55	34
Big	1,200	55	89

The counts aren’t the same either way, and the bigger head keeps 55 and swaps 34 for 89. Real sunflowers do the same: big heads show 55 and 89 spirals near their edge, and some giant ones 89 and 144.

Now look at the numbers themselves: 34, 55, 89, 144. Is there a pattern? Try adding two of them.

34 + 55 = 89, and 55 + 89 = 144. Each number is the sum of the two before it. Run the rule backwards and see where it came from: 55 − 34 = 21, then 34 − 21 = 13, then 8, 5, 3, 2, 1 and 1. So the counts belong to one list that starts 1, 1 and grows by adding:

$$1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\ 89,\ 144,\ 233,\ \dots$$

Smaller spirals in other plants

Sunflowers have the biggest counts, because their heads hold the most seeds. Plants with fewer, bigger parts show smaller numbers from the same list. A pine cone’s scales also sit in spirals both ways round: a survey of pine cones from many kinds of pine tree in California found 5, 8 or 13 spirals on every one, though a later study found the odd cone that doesn’t fit. Pineapples, artichokes and the leaves of the spiral aloe show the same kind of pattern.

The Fibonacci numbers

The list 1, 1, 2, 3, 5, 8, 13, … is called the Fibonacci numbers, after the Italian Leonardo of Pisa, nicknamed Fibonacci, who wrote about it in 1202. Write F₁ for the first, F₂ for the second, and so on. Then the whole list comes from two starting numbers and one rule, which turns each position n into one number $F_n$, so the list is a function of n:

$$F_1 = 1, \quad F_2 = 1, \qquad F_{n+1} = F_n + F_{n-1}$$

A rule like this, which makes each number from the ones before it, is called a recursive rule: to find $F_{12}$ you need $F_{11}$ and $F_{10}$, and they need the ones before them, down to the two 1s.

n	1	2	3	4	5	6	7	8	9	10	11	12
Fₙ	1	1	2	3	5	8	13	21	34	55	89	144

The same rule with different starting numbers makes a Fibonacci-type sequence. Start with 2 and 5 and you get 2, 5, 7, 12, 19, 31, 50, 81… Whatever two positive numbers you start with, divide each number by the one before it and the answers settle down to the same number, about 1.618, the golden ratio (for Fibonacci’s own list, 89 ÷ 55 = 1.6182 already). Where Did the Missing Square Go? shows where that ratio hides in a famous puzzle.

So why would a flower, which can’t count, put its seeds in spirals whose numbers follow an adding rule?

One seed at a time

A sunflower head doesn’t make its seeds all at once. Each seed starts as a tiny bump, one after another, and every new bump appears at a fixed turn from the one before, like the hand of a clock that jumps by the same angle each time. (On a sunflower the first bumps form near the rim and later ones nearer the middle, so the oldest seeds are at the edge; on a growing shoot, new bumps appear at the tip.) Botanists have measured that turn on many plants. It is very close to 137.5∘°.

Before asking why 137.5∘°, see what other turns would do. Choose a turn, and the widget grows 400 seeds with it, from the middle outwards.

A quarter turn (90∘°) is a disaster for a plant: seed 5 points the same way as seed 1, seed 6 the same way as seed 2, and the seeds line up in 4 straight spokes with wide, empty gaps between them. The same happens with any fraction of a turn. If the turn is $\frac{2}{5}$ of a turn (144∘°), then after 5 seeds the turns add up to exactly 2 whole turns, so every fifth seed points the same way: 5 spokes. A turn of $\frac{3}{8}$ makes 8 spokes. In lowest terms, the bottom of the fraction is the number of spokes.

Now look at the fractions in the widget that get closer and closer to the real 137.5∘°:

Fraction of a turn	1/2	1/3	2/5	3/8	5/13	8/21	13/34	21/55
Turn	180∘°	120∘°	144∘°	135∘°	138.46∘°	137.14∘°	137.65∘°	137.45∘°
Spokes	2	3	5	8	13	21	34	55

Fibonacci numbers on top, Fibonacci numbers on the bottom! And each fraction comes from the two before it by adding the tops and adding the bottoms: from $\frac{1}{3}$ and $\frac{2}{5}$ you get $\frac{1+2}{3+5} = \frac{3}{8}$. The turns zigzag above and below 137.5∘°, each one closer than the last, and never reach it: the real turn is not a fraction at all. It is the golden angle, 137.5077…∘°, the share 0.381966… of a full turn that the golden ratio picks out. Cut a full turn into 137.5∘° and 222.5∘°, and the bigger part divided by the smaller is 1.618, the same as the whole turn divided by the bigger part.

Spirals are near-misses

So the golden turn makes no spokes. Where do the spirals come from? Look closely at one seed and the seeds touching it.

In Figure 20.2, the seeds touching seed 300 are seeds 266 and 334 (34 away), 245 and 355 (55 away), and, a little farther off, 279 and 321 (21 away). None is 1, 10 or 40 away. Why Fibonacci numbers again?

Seed 300 + m has turned m × 0.381966… turns further than seed 300. If that is nearly a whole number of turns, the new seed points nearly the same way as seed 300, just a little farther out: so it is a close neighbor. Here is how near a whole number each step gets:

Seeds later, m	1	2	3	5	8	13	21	34	55
Turns, m × 0.381966	0.382	0.764	1.146	1.910	3.056	4.966	8.021	12.987	21.008
Miss from a whole turn	+0.382	−0.236	+0.146	−0.090	+0.056	−0.034	+0.021	−0.013	+0.008

Every step left out of the table misses by more than a smaller step already did. Try m = 4 (1.528 turns, a miss of nearly half a turn), 10 (3.820, a miss of 0.180) or 18 (6.875, a miss of 0.125). The record near-misses, each closer than any before it, happen only at the Fibonacci numbers.

And look at how the misses go: plus, minus, plus, minus, each smaller than the last. Add two neighbors and you get the next one: the miss for 13 is −0.034 and the miss for 21 is +0.021, and −0.034 + 0.021 = −0.013, the miss for 34. Of course: 34 seeds later is 13 seeds later and then 21 more, so its turns are the two added, and so are its misses. The near-misses obey the Fibonacci rule because the steps do.

Now the spirals. Seed 300 is close to seed 334, which is close to seed 368, which is close to 402, and so on outwards, each a small sideways step from the last: that chain is one spiral. Start from seed 301 instead and you get another. There are 34 starting points, seeds 0 to 33, before the chains repeat, so there are 34 spirals of that kind. The steps of 55 make 55 spirals the other way round, because their miss is the other sign. That is what you counted in the first widget.

Why does the bigger head swap 34 for 89? Farther out, the seeds go round a bigger circle, and the same sideways miss in degrees is a longer distance there, so a neighbor must miss by less to stay close. Out at the edge of the big head, the step of 34 misses by too much, and the next record, 89 with its miss of 0.005 of a turn, takes over. (The rings of seeds are also thinner out there, so a seed 89 steps later is hardly any farther out.) Real heads do this too: the Manchester team below notes that a head’s counts nearer the middle are close to the difference of its two counts at the rim, like 89 − 55 = 34, so they counted at the rim, where the numbers are biggest.

Why would a plant turn by the golden angle?

The flower doesn’t measure 137.5∘°. In 1868 the botanist Wilhelm Hofmeister suggested a simpler rule: each new bud forms in the least crowded place round the growing tip, as far as it can get from the buds already there. In 1992 two physicists, Stéphane Douady and Yves Couder, tested whether a rule like that is enough, with no plant at all. They let drops of a magnetic liquid appear one at a time at the middle of a dish of oil, in a magnetic field that made the drops push each other away: each new drop pushed the older ones outwards and was itself pushed into the emptiest direction. The drops arranged themselves in spirals in Fibonacci numbers, all by themselves. So the golden angle isn’t a number the plant knows. It is what you get when each new thing squeezes into the biggest gap left by the ones before.

That is also why it suits a plant. Your widget showed it: any fraction of a turn eventually lines seeds up in spokes and leaves gaps, while the golden turn, which no fraction matches (mathematicians call the golden ratio the “most irrational” number), packs seeds evenly all the way out, with room for as many as possible. Biologists have found that a plant hormone, auxin, gathers where a new bud will form and drains away from the area round it, and scientists still debate which model best explains the rest.

Real flowers are not perfect models, and the best way to find out how often the rule really holds is to count a lot of them. In 2012, for the hundredth anniversary of Alan Turing’s birth, the Museum of Science and Industry in Manchester asked the public to grow a sunflower and count its spirals, or bring the head in to be counted. Turing, a mathematician, had studied sunflower spirals in the 1950s.

The project collected 657 sunflowers. In the clearest data the counters checked 768 spiral counts:

What the count was	Counts	Share
A Fibonacci number	565	74%
A Lucas number (2, 1, 3, 4, 7, 11, 18, 29, 47, 76…)	41	5%
Twice a Fibonacci number (42, 68…)	25	3%
Another Fibonacci-type number	1	0.1%
None of these	136	18%

So about three counts in four were Fibonacci numbers, and about one in five wasn’t in any of these lists. Among those, a number one less than a Fibonacci number (such as 54 or 88) turned up 49 times, nearly three times as often as one more (17 times).

The Lucas numbers are a Fibonacci-type sequence: the same adding rule, started from 2 and 1 instead of 1 and 1. A head with 29 and 47 spirals has followed the adding rule, but from a different start, and the ratio 47 ÷ 29 = 1.62 is still close to the golden ratio.

There is a warning in that table, too. Small numbers prove very little. Of the whole numbers from 1 to 17, every one except 12, 15 and 17 is in one of these Fibonacci-type lists (Fibonacci, Lucas, twice Fibonacci, or 1, 4, 5, 9, 14…). So a flower with 4, 7 or 11 petals can be “explained” by some list, and that explains nothing. Sunflowers matter because their counts are big: 55 and 89 are hard to hit by chance.

• Where Did the Missing Square Go? The same numbers in a puzzle where a square seems to vanish, and why the ratio of neighbors settles at 1.618.
• Is the Golden Ratio Really Everywhere? (Grades 9–12) The golden ratio in sunflowers is real; in shells, faces and famous buildings, mostly not.
• How Many Ways Can You Climb the Stairs? (Grades 9–12) Question f in full: the Fibonacci numbers as a way of counting.
• How Does the Mind-Reading Card Trick Work? (Computing, Going Further) A card trick built on binary, and a version built on Fibonacci numbers to program.
• Grade 11 Computing: Recursion Counting the ways up a stairway with a program that calls itself.