Sacred geometry · In nature
Is the Fibonacci sequence really in nature?
Yes, in one place above all: the spirals of plants. Count the spirals on a sunflower head, a pine cone or a pineapple and you usually get two neighbouring Fibonacci numbers, such as 34 and 55. Surveys of 650 plant species found Fibonacci spirals in about 92% of plants with spiral patterns. Usually is not always, though: in 2016 a count of 768 sunflower spirals found 565 Fibonacci numbers and nearly 20% with no Fibonacci pattern at all. And the most famous example, the nautilus shell, measures about 1.33, not the golden ratio’s 1.618. Light up the spirals below and count them yourself.
Sunflower
Light a spiral family
The mechanism
Why plants grow Fibonacci spirals
Picture a sunflower placing its seeds one at a time at the centre of the head, each new one turned about 137.5° from the last, while the older ones are pushed outward. That turn is the golden angle: a full circle of 360° divided by φ², where φ is the golden ratio, 1.618... The drawing above is built exactly that way: seed number n sits n × 137.5° round the circle, at a distance from the middle that grows with the square root of n.
The spirals are a side effect. Nobody draws them; your eye joins each seed to its nearest neighbours, and those neighbours sit 34 or 55 seeds apart in the growing order. MathWorld: the florets in the head of a sunflower “form two oppositely directed spirals : 55 of them clockwise and 34 counterclockwise.” Van Iterson worked out why as early as 1907. MathWorld again: he “showed in 1907 that points separated by [the golden angle] on a tightly bound spiral tends to produce interlocked spirals winding in opposite directions, and that the number of spirals in these two families tend to be consecutive Fibonacci numbers.”
Why that angle? 137.5° is 0.381966... of a full turn, and that number can never be written exactly as a fraction. Turn by a simple fraction instead, say 3/8 of a turn, and every eighth seed lands in the same direction: the seeds pile up on 8 straight spokes with empty wedges between them. The golden ratio is, in MathWorld’s word, the “worst” number to approximate with fractions, so seeds placed at the golden angle never line up and fill the head evenly. The Fibonacci fractions 1/2, 1/3, 2/5, 3/8, 5/13, 8/21... creep toward 0.381966 from both sides, which is why the spiral counts come out as Fibonacci numbers. Drag the angle yourself in the Golden Angle Garden and watch the spokes appear the moment you leave 137.5°.
You can get the same pattern without a plant. In 1992 the physicists S. Douady and Y. Couder let drops of magnetic liquid fall, one at a time, into the centre of a dish of silicone oil sitting in a magnetic field. Each drop became a tiny magnet, pushed away from the drops before it and drifted outward. Their abstract: “A specific crystalline order, involving the Fibonacci series, had until now only been observed in plants (phyllotaxis). Here, these patterns are obtained both in a physics laboratory experiment and in a numerical simulation.” The reason they give is the same one: the system’s “trend to avoid rational (periodic) organization, thus leading to a convergence towards the golden mean.”
Alan Turing chased this too. In the 1950s he named it “the hypothesis of geometrical phyllotaxis” but, as a 2016 paper puts it, “could not prove” it, and his notes were not published until after his death. The debate is not closed: a 2022 paper by Yin and Tsukaya carries the title “Fibonacci spirals may not need the Golden Angle.”
The biggest count ever
657 sunflowers, grown by the public
For Alan Turing’s 100th birthday in 2012, the Museum of Science and Industry in Manchester asked the public to grow sunflowers and send in their seed heads. The team “collected data on 657 sunflowers” and, in the most reliable set, counted 768 clockwise or anticlockwise spiral numbers. The results, published in 2016:
- Fibonacci565 · 74% · e.g. 89 and 55
- Double Fibonacci25 · 3% · e.g. 68 and 42
- Lucas numbers41 · 5% · e.g. 76 and 47
- Other Fibonacci type1 · 0.1%
- No Fibonacci structure136 · 18% · 49 were one less than a Fibonacci number, 17 one more, 70 other
So most sunflowers are Fibonacci, and a real minority are not. Double Fibonacci numbers (2, 4, 6, 10, 16, 26, 42, 68...) and Lucas numbers (1, 3, 4, 7, 11, 18, 29, 47, 76...) follow the same add-the-last-two rule from a different start, so the paper counts them as Fibonacci structure: 632 of 768, or 82%. The other 136 had none, “nearly 20% of parastichy numbers” in the paper’s words. The authors say this study “systematically reports for the first time, to the best of our knowledge, seedheads without Fibonacci structure.” Older counts agree on the big picture: “Schoute found numbers from the main Fibonacci sequence 82% of the time and Weise 95%.”
Across all plants the share is higher. A compilation by R. V. Jean (1994) of surveys covering 650 species and 12,500 specimens found that “among plants showing spiral or multijugate phyllotaxis, about 92% showed Fibonacci phyllotaxis.” Phyllotaxis is just the botanist’s word for how leaves, florets and scales are arranged.
Pine cones and pineapples
8, 13 and 21 rows
Pineapples. The eyes of a pineapple sit in three sets of sloping rows. In 1970 P. B. Onderdonk reported in The Fibonacci Quarterly that “The vast majority of pineapples checked had 8 - 13 - 21 rows of fruitlets (eyes)”, and quoted a 1933 report: “no specimen has yet been found in which the basic pattern was other than 8 and 13 spirals.” Pick Pineapple above to light all three sets.
Pine cones. Brother Alfred Brousseau examined cones from a large variety of pine trees in California for The Fibonacci Quarterly in 1968; as Ron Knott summarises it, they “all exhibited 5,8 or 13 spirals.” Turn a cone upside down, pick one scale on the rim and follow its row toward the stalk, marking where you started. The picture above shows a cone with 8 rows one way and 13 the other.
The same works on daisies, cauliflowers and more. MathWorld: “A similar phenomenon occurs for daisies , pineapples, pinecones, cauliflowers, and so on.” The numbers to expect are the neighbouring pairs of the Fibonacci sequence: 5 and 8, 8 and 13, 21 and 34, 34 and 55, 55 and 89.
The famous one, measured
Is the nautilus shell a golden spiral?
No. When it was measured, the nautilus came out near 1.33, not 1.618. In 1999 Clement Falbo measured the nautilus shells in the collection of the California Academy of Sciences in San Francisco. As Science News reported, the spirals “could be inscribed within rectangles with sides in the ratio of about 1.33”, and “The measured ratios ranged from 1.24 to 1.43.” None reached the golden ratio.
The growth rate shows the gap plainly. A golden spiral gets φ times wider every quarter turn, so φ⁴ = 6.854 times wider every full turn. The nautilus, per the same report, “triples in radius with each full turn”. Falbo’s own verdict: “It seems highly unlikely that there exists any nautilus shell that is within 2 percent of the golden ratio, and even if one were to be found, I think it would be rare rather than typical.”
The shell is still a beautiful spiral. It is a logarithmic spiral, the same family as the golden spiral, that keeps its shape as it grows. It just grows at its own rate, about 3 per turn instead of 6.85.
What is proven
Fact and tradition, sorted
| Claim | Status |
|---|---|
| Sunflower seed heads show 34 spirals one way and 55 the other. | Real · MathWorld |
| About 92% of plants with spiral patterns are Fibonacci. | Real · Jean 1994, 650 species |
| Every sunflower is Fibonacci. | No · 18% of 768 counts were not |
| Pineapples show 8, 13 and 21 rows. | Real · Onderdonk 1970 |
| Pine cones show 5, 8 or 13 spirals. | Real · Brousseau 1968 |
| The golden angle packs seeds with no lined-up gaps. | Proven · φ is not a fraction; van Iterson 1907 |
| Non-living drops make the same spirals. | Real · Douady and Couder 1992 |
| Plants need the exact golden angle to make Fibonacci spirals. | Debated · Yin and Tsukaya 2022 |
| The nautilus shell is a golden spiral. | No · measured about 1.33 |
Questions
Fibonacci in nature
Is the Fibonacci sequence really found in nature?
Yes, in plant spirals. Sunflower heads usually show 34 and 55 spirals, pineapples 8, 13 and 21 rows, pine cones 5, 8 or 13. Surveys of 650 species found Fibonacci patterns in about 92% of spiral plants. It is common, not universal: about 18% of sunflower spiral counts in a 2016 study had no Fibonacci structure.
Why does the Fibonacci sequence appear in plants?
Because each new seed or leaf grows about 137.5° round from the last, the golden angle. That angle never lines seeds up, so they pack evenly, and the spirals the eye sees come out in neighbouring Fibonacci numbers. Douady and Couder got the same spirals in 1992 from drops of magnetic fluid, with no plant involved.
Is pineapple a Fibonacci sequence?
Its rows are. The eyes of a pineapple sit in three sets of sloping rows, and Onderdonk reported in 1970 that the vast majority of pineapples checked had 8, 13 and 21 rows, three neighbouring Fibonacci numbers.
Is the nautilus shell a golden spiral?
No. Clement Falbo measured nautilus shells at the California Academy of Sciences and found a ratio of about 1.33, ranging from 1.24 to 1.43, not 1.618. A nautilus roughly triples in size each full turn; a golden spiral grows about 6.85 times.
Does the Fibonacci sequence apply to humans?
Not in a way measurements support. Mathematician George Markowsky measured his family’s height to navel ratios at 1.59, 1.63, 1.65 and 1.66, and called the navel “a scar of no great importance.” A 2010 face study found the most attractive proportions match an average face (36% and 46%), not the golden ratio.
Why is 1.618 so special?
1.618... is the golden ratio φ, the number the ratios of neighbouring Fibonacci numbers approach (55 ÷ 34 = 1.6176). It is the hardest number to approximate with fractions, so the angle built from it, 360° ÷ φ² = 137.5°, never lines seeds up. That is why it shows up in plant spirals.
Sources
Where this comes from
- Eric W. Weisstein, “Phyllotaxis,” MathWorld: mathworld.wolfram.com.
- Eric W. Weisstein, “Golden Angle,” MathWorld, for van Iterson 1907 and 137.507°: mathworld.wolfram.com.
- Eric W. Weisstein, “Golden Ratio,” MathWorld, for φ and the “worst” approximable number: mathworld.wolfram.com.
- J. Swinton, E. Ochu and the MSI Turing’s Sunflower Consortium, “Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment,” Royal Society Open Science 3: 160091 (2016): royalsocietypublishing.org.
- Smith College Phyllotaxis site, “Classification,” citing R. V. Jean (1994): smith.edu.
- P. B. Onderdonk, “Pineapples and Fibonacci Numbers,” The Fibonacci Quarterly 8 (1970): 507-508: fq.math.ca.
- Ron Knott, “Fibonacci Numbers and Nature,” University of Surrey, summarising Brousseau, The Fibonacci Quarterly 6 (1968): surrey.ac.uk.
- S. Douady and Y. Couder, “Phyllotaxis as a physical self-organized growth process,” Physical Review Letters 68(13) (1992): 2098-2101: doi.org.
- Fan and Bursill, University of Melbourne report UM-P-95/61, describing the Douady and Couder experiment: osti.gov.
- X. Yin and H. Tsukaya, “Fibonacci spirals may not need the Golden Angle,” Quantitative Plant Biology 3:e13 (2022): PMC.
- Ivars Peterson, “Sea Shell Spirals,” Science News (2005), reporting Clement Falbo’s nautilus measurements: sciencenews.org.
- George Markowsky, “Misconceptions about the Golden Ratio,” The College Mathematics Journal 23(1) (1992): 2-19: maine.edu.
- P. M. Pallett, S. Link and K. Lee, “New “golden” ratios for facial beauty,” Vision Research 50(2) (2010): 149-154: PubMed.
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