Why Are Sunflower Seeds Arranged in Spirals?
– The Beauty of the "Most Awkward Angle"
When a sunflower finishes flowering, its centre shows a swirling pattern of tightly packed seeds. It's easy to overlook, but if you count the swirls, a particular, curious sequence of numbers appears. Behind it is thought to be a special angle the plant uses as it adds seeds one by one.
Have you ever looked closely at the centre of a big sunflower, packed full of seeds after the bloom is over? The seeds aren't just lined up in rays. They form two kinds of spiral, one running clockwise and one anticlockwise.
Try counting the clockwise spirals and the anticlockwise spirals separately. In many cases you get a fixed pair of numbers, such as 34 and 55, or 55 and 89.
Is that a coincidence? Or is some mathematical reason hidden in the way the plant grows?
Numbers like 34, 55 and 89 are part of a famous sequence, the Fibonacci sequence, where each number is the sum of the previous two.
Each time a plant makes a new seed near the centre, it is thought to place it turned by exactly the same angle from the previous seed.
Let's look at what that "fixed angle" really is, step by step.
Count the swirls, and curious numbers appear
Let's actually count the clockwise and anticlockwise spirals in a sunflower's seed pattern. In many cases the counts are two neighbouring numbers, such as 34 and 55, 55 and 89, or even larger pairs. This has been reported since long ago.
These numbers belong to a famous string of numbers called the Fibonacci sequence. It is built from a simple rule: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… where adding the previous two numbers gives the next. The sequence is thought to be widespread in other plants too, such as the arrangement of scales on a pine cone and the pattern on a pineapple's surface.
Seeds are added one at a time, each turned by a fixed angle
A sunflower's seeds are made one after another from a part at the centre of the flower called the growing point. Each new seed forms turned by a fixed angle from the seed made just before it, and as the plant grows, the older seeds are gradually pushed outward.
Simply repeating this rule of "turn, then add one" is thought to produce a complex-looking, swirling pattern all by itself. The key point is that the turning angle is a special value, right around 137.5 degrees.
It just adds seeds one at a time, turning by the same angle each time.
Why must that angle be the "golden angle"?
This special angle is called the golden angle, and it is linked to the golden ratio, a special value known since ancient times. Suppose the turning angle divided 360 degrees evenly, such as 90 or 120 degrees. Then the seeds would form straight lines radiating out, with big gaps between the lines.
The golden angle, by contrast, has a property of being very hard to divide evenly: no multiple of it makes 360 degrees come out as a neat fraction. Thanks to this, a new seed almost never lands in nearly the same direction as an earlier one, so the seeds fill the space with no gaps, in the most efficient way, it is thought.
The "most awkward number" leaves the fewest gaps
Mathematically, the golden ratio is known to have a special property: it is the "irrational number that is hardest to approximate with fractions". Because of this, even if you keep turning by the golden angle, seeds do not line up in similar directions for an extremely long time. As a result, seeds can be packed most densely into a limited area, with no waste. In the next fold-out, we check the actual angle with numbers.
Spiral arrangements are seen in many plants, such as pine cone scales, pineapple skin patterns, and Romanesco (a swirling relative of cauliflower). A shared growth mechanism, "add each new part one at a time, turned by the golden angle", is thought to produce these patterns.
Something you can check yourself
- Mark one dot in the centre of a sheet of paper
- Using a protractor, mark the next dot a little way from the centre, in a direction about 137.5 degrees around from that dot
- Turn another 137.5 degrees and mark the next dot a little further out. Repeat 20 to 30 times
- Look at the pattern of dots and check that a spiral pattern appears on its own
You can feel for yourself how a complex, sunflower-like pattern arises from the simple rule of "turn by the same angle".
Summary
The spiral patterns in sunflower seeds arise because the plant adds new seeds one at a time, each turned by a fixed angle, the golden angle of about 137.5 degrees. The golden angle is mathematically the angle that is "hardest to divide evenly", so the seeds pack together with the fewest gaps and most efficiently. The appearance of Fibonacci numbers in the spiral counts is also thought to be deeply tied to the properties of this golden angle.
The sunflower's beautiful pattern has no blueprint.
There is only a simple rule: faithfully keep to a single angle.
For those who want more – terms, numbers and links to textbooksFrom middle-school science to active research, each part is labelled by level
- Middle schoolCovered in middle-school maths
- High schoolCovered in high-school "Mathematics B"
- High school+High-school maths, or extension and sidebar material in textbooks
- UniversityNot taught in high school; university specialist content (plant developmental biology)
- ResearchNot taught even at university as settled fact; questions researchers are studying now
Middle schoolTerms: words around sunflower spirals
- Fibonacci sequence: a run of numbers where adding the previous two gives the next (1, 1, 2, 3, 5, 8, 13…).
- Golden ratio: a special ratio long thought beautiful. About 1.618.
- Golden angle: a special angle of about 137.5 degrees, set by the golden ratio.
High schoolChecking with a formula: how is the golden angle calculated?
Let's work out the golden angle from the value of the golden ratio.
Golden angle = 360° × (1 − 1 ÷ golden ratio)
| Golden ratio | The well-known value is about 1.618 |
| Golden angle | Measured in degrees (°) |
| 1 ÷ golden ratio | 1 ÷ 1.618 ≒ 0.618 |
| Subtract from 1 | 1 − 0.618 ＝ 0.382 |
| Multiply by 360 | 360 × 0.382 ≒ 137.5 |
| Golden angle | About 137.5° |
The result is a value near 137.5°, the same as the turning angle observed in sunflower seeds.
Let's also check that the ratio of neighbouring Fibonacci numbers approaches the golden ratio.
| Ratio of 55 to 34 | 55 ÷ 34 ≒ 1.618 |
| Ratio of 55 to 34 | About 1.618 (almost equal to the golden ratio) |
The larger the Fibonacci numbers get, the closer the ratio of neighbouring numbers comes to the golden ratio. The Fibonacci numbers show up in sunflower spiral counts because the golden angle itself is inseparably tied to this sequence, it is thought.
* The golden ratio value (1.618) is a commonly used approximation. The true value is a never-ending decimal (an irrational number).
High school+The golden ratio is the number "hardest to turn into a fraction"
Using the method of approximating numbers by fractions (continued fractions), one can show that the golden ratio is "the hardest-to-approximate irrational number, with the simplest continued fraction". This property leads directly to the fact that even if you keep turning by the golden angle, you hardly ever come back to the original direction.
UniversityHow does a plant produce the "angle"?
In plant developmental biology, a plant hormone (auxin) is thought to play an important role in deciding where new seeds and leaves start at the growing point. A mathematical model has been proposed in which auxin is distributed so that it is repelled from existing seed and leaf beginnings, so that as a result, the next seed or leaf beginning tends to appear near the golden angle.
ResearchWhat is still unclear
- How a plant senses the distribution of auxin at the cell level and achieves an angle so close to the golden angle, the detailed molecular mechanism, is still being studied in plant developmental biology.
- Some plants are reported to show patterns that depart from the golden angle and don't fit the Fibonacci sequence, and why such "exceptions" arise is also being studied.
- Golden-angle layouts are a way to place objects efficiently in a limited area, and applied research in engineering, such as the layout of solar panels, is also under way. It is an attempt to use plants' wisdom in artificial designs.
Behind the familiar sunflower pattern lies a rich topic where mathematics and biology meet, and research continues.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Middle school | Maths: patterns and sequences | Basic terms: Fibonacci sequence, golden ratio |
| High school | Mathematics B: sequences | Finding the golden angle from the golden ratio; convergence of Fibonacci ratios |
| High school+ | Maths: continued fractions (extension) | The golden ratio's "hardest to approximate" property |
| University | Plant developmental biology | Auxin-based model of seed and leaf placement |
| Research | Plant developmental biology, mathematical engineering (ongoing research) | Working out the molecular mechanism; engineering uses of golden-angle layouts |
- Explanations of the Fibonacci sequence and the golden ratio in mathematics textbooks.
- Explanations of leaf arrangement (phyllotaxis) and the golden angle in botany materials.
- Research reviews on auxin-based mathematical models of primordium placement in plant developmental biology.
- Explanations of the continued-fraction form of the golden ratio and its resistance to approximation, in mathematical-history literature.
- Research reviews on solar-panel layouts that apply golden-angle arrangements, in engineering.
* The golden ratio and golden angle values are commonly used approximations. The number of spirals is also said to vary with the individual plant and growing conditions.
* This article is a general-audience science explainer. When observing plants, please be considerate, for example by getting permission from the grower or owner.