Nature is disordered, messy, chaotic. But upon closer look, you might start noticing patterns, sequences, and symmetry on all sorts of varying scales, from the grandest spiral galaxies to the tiniest snail shell.

One popular pattern spotted in many places is the Fibonacci sequence. You’ve likely encountered it before, perhaps as a spiral graphic often superimposed over images of human ears, hurricanes, or nautilus shells. How many of them are actually examples of the Fibonacci sequence arising in nature?

What Is the Fibonacci Sequence?

Simply put, the Fibonacci sequence describes a series of numbers, wherein each successive value is the sum of the two before it. It goes something like 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.

It sprang from the mind of the 13th century Italian mathematician Leonardo Pisano, or Fibonacci. The sequence itself is derived from a math problem: If you put a pair of newly born rabbits – one male and the other female – together, let’s say it takes them one month before they reach the age to reproduce.

Assuming the population never dies, the female rabbit produces another duo of baby male and female rabbits. Once those females grow up, they’ll do the same. If this pattern keeps going, with each female rabbit taking one month to produce another pair, how many pairs of rabbits will there be in one year?

The formula resulting from this problem is Fₙ = Fₙ₋₁ + Fₙ₋₂, Fₙ being the nth Fibonacci number in the series above.

But the Fibonacci sequence has plenty of applications outside of numbers and biologically impossible situations. You might have heard of the golden spiral: a logarithmic spiral deemed one of the most perfect and beautiful patterns you could obtain.

When drawing a line down the middle of the spiral, you’ll see it increases by a factor of 1.618 per iteration. This number is the golden ratio, the irrational number you get when dividing one Fibonacci number by its predecessor. Artists, mathematicians, and architects alike have aspired to this perfect ratio for thousands of years.

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Examples of the Fibonacci Sequence in Nature

Given what we know about the Fibonacci sequence, how common is it really in nature? Some scientists caution against ascribing the golden ratio or spiral too ubiquitously, out of fear of perpetuating false myths with poorly sampled studies or pseudoscience. At the same time, that doesn’t mean that occurrences of the Fibonacci sequence don’t exist at all.

1. Pine Cones

When you pick up a pinecone from the ground, you’ll see two sets of spiraling bracts, rotating out from its base where it was once connected to a tree. Those spirals curl around one another in opposing directions, settling against one another with specific angles of rotation.

When counting the bracts spiraling in opposite directions, you’ll find a hand-sized example of Fibonacci numbers popping up in nature, with each spiral typically taking on adjacent Fibonacci numbers.

2. Sunflowers

Sunflowers are another famous example of Fibonacci at work in nature. Particularly, the arrangement of seedheads on sunflowers often takes on Fibonacci numbers. For example, if 34 seed rows curve clockwise, then either 21 or 55 seed rows will spiral the other way.

But because this is nature and random things happen, not all sunflowers ascribe to this pattern. While Fibonacci numbers are common structures for sunflower seeds, a recent study evaluating data from over 600 citizen-grown sunflowers found some of them were disordered enough that they didn’t follow the Fibonacci sequence.

However, scientists believe that following Fibonacci numbers in the first place allows sunflowers to fit the greatest amount of seeds possible on their heads.

3. Tree Branches

If you look up at a cover of trees, chances are the points where branches split off from their original trunk are also related to the golden section. Within poplar trees, for example, researchers modeling the golden section in plants found that the angle between a poplar branch shooting off a main stem was 34.4 degrees – which, when corresponding to a 90 degree angle, satisfied the golden ratio.

4. Plant Leaves

The arrangement of leaves around a plant stem, otherwise known as phyllotaxy, is another area of biology where the Fibonacci sequence shines. Similarly to how sunflowers can stuff the most amount of seeds into their heads thanks to Fibonacci numbers, researchers believe the arrangement of leaves according to the golden ratio is also a way for plants to maximize their light-gathering potential, thus longevity.

5. Flower Petals

Plenty of other types of flowers, from roses to buttercups to lilies, follow Fibonacci numbers in the petals. Wild roses, for example, typically have five petals, which is one of the early numbers in a Fibonacci series. The amount of petals that flowers have also, similarly to leaves, allows them to maximize the amount of light they can absorb while growing.

6. Human Body

Of course, Fibonacci numbers aren’t just found in the world all around us, but also within our own bodies, too. The role of the golden ratio in the ideal human body has long been studied, since the days of artists like Leonardo da Vinci.

Even today, scientists are still finding new ways in which those numbers manifest. For example, a recent review in 2022 has found that the ratio of the distance from our belly buttons to our feet, to the distance from our belly buttons to our heads, formed the golden ratio.

7. Whales

Recent news at the beginning of 2024 generated buzz about whales’ abilities to create Fibonacci spirals during their hunting process. Specifically at the center of the media spotlight was a pair of great humpback whales who blew a spiral of bubbles toward the ocean surface, casting the bubble net to capture their prey.

The net appeared on the sea surface as a tightly winding spiral, certainly a beautiful sight to onlookers. However, it’s not necessarily a Fibonacci spiral, which relies on specific proportions and ratios. Still, it was a stunning spiral nonetheless and another opportunity for scientists to gain a better understanding about whales’ hunting habits.

8. Galaxies

When we zoom out from Earth for a second to view our existence on the grandest of cosmic scales, spiral galaxies are another example of natural golden spirals that come to mind. However, as researchers in a 2022 article published in the journal Symmetry have noted, real galaxies are imperfect and disordered, and won’t always perfectly align with golden spirals and ratios.

But that doesn’t mean that Fibonacci patterns don’t exist in our cosmos at all: The authors of the article added that the average distances of the planets in our solar system from the sun “approximately relate to each other according to the Golden ratio.”

9. Pineapples

Pineapples are often named as an example of the Fibonacci sequence in fruits, thanks to the spiraling patterns that each bulb within a pineapple fruit creates. One study in 1978 has found that while pineapple spirals consist of Fibonacci numbers, those spirals don’t always follow the same directions across all pineapple species.

10. Nautilus Shells

Nautilus shells have come a long way since the days they were hailed as manifestations of the Fibonacci sequence in seashells. While they may resemble a golden spiral, scientists have since debunked the idea that these shells follow the golden ratio perfectly. The ratio is actually closer to 1.310, for the Nautilus genus as a whole, rather than 1.618.

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Why Does Fibonacci Appear in Nature?

As for why nature seems to like symmetry and patterns, researchers have yet to stumble across a single theory that applies to all cases. It’s possible that biological systems appear symmetrical due to physical forces greater than themselves that impose restraints on how they can operate.

Or, symmetry within the tiniest particles and molecules that make up everything could also play a role. Like how plant leaves utilize the golden ratio to maximize their efficiency at photosynthesis, it could also be a matter of pure survival and evolutionary advantage.

Some researchers believe we might never stumble across the reason. “A general unifying theory of symmetry in biology does not exist and, perhaps, has no chance to be developed,” concluded the authors in a 2022 article published in the journal Biophysica.

That doesn’t discount the importance or prevalence of patterns in a world of disorder. But even in the seemingly most organized things, from sunflowers to galaxies, you will probably find something out of place.

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Article Sources

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