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Close-up shot of the spiky branches of a monkey puzzle tree.
The spiky branches of a monkey puzzle tree. Joshua Bruce Allen/Shutterstock

How a 400 million year old fossil changes our understanding of mathematical patterns in nature

If your eyes have ever been drawn to the arrangement of leaves on a plant stem, the texture of a pineapple or the scales of a pinecone, then you have unknowingly witnessed brilliant examples of mathematical patterns in nature.

What ties all of these botanical features together is their shared characteristic of being arranged in spirals that adhere to a numerical sequence called the Fibonacci sequence. These spirals, referred to as Fibonacci spirals for simplicity, are extremely widespread in plants and have fascinated scientists from Leonardo da Vinci to Charles Darwin.

Such is the prevalence of Fibonacci spirals in plants today that they are believed to represent an ancient and highly conserved feature, dating back to the earliest stages of plant evolution and persisting in their present forms.


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However, our new study challenges this viewpoint. We examined the spirals in the leaves and reproductive structures of a fossilised plant dating back 407 million years. Surprisingly, we discovered that all of the spirals observed in this particular species did not follow this same rule. Today, only a very few plants don’t follow a Fibonacci pattern.

The first author of the study creating digital 3D models of Asteroxylon mackiei.
Holly-Anne Turner, first author of the study, creating digital 3D models of Asteroxylon mackiei at the University of Edinburgh. Luisa-Marie Dickenmann/University of Edinburgh, CC BY-NC-ND

What are Fibonacci spirals?

Spirals occur frequently in nature and can be seen in plant leaves, animal shells and even in the double helix of our DNA. In most cases, these spirals relate to the Fibonacci sequence – a set of numbers where each is the sum of the two numbers that precede it (1, 1, 2, 3, 5, 8, 13, 21 and so on).

These patterns are particularly widespread in plants and can even be recognised with the naked eye. If you pick up a pinecone and look at the base, you can see the woody scales form spirals that converge towards the point of attachment with the branch.

At first, you may only spot spirals in one direction. But look closely and you can see both clockwise and anticlockwise spirals. Now count the number of clockwise and anticlockwise spirals, and in almost every case the number of spirals will be integers in the Fibonacci sequence.

The same pine cone colour coded to show 8 clockwise and 13 anticlockwise spirals.
The same pinecone colour coded to show 8 clockwise and 13 anticlockwise spirals. 8 and 13 are consecutive numbers in the Fibonacci series. Sandy Hetherington, Author provided

This particular instance is not an exceptional case. In a study that analysed 6,000 pinecones, Fibonacci spirals were found in 97% of the examined cones.

Fibonacci spirals are not just found in pine cones. They are common in other plant organs such as leaves and flowers.

If you look at the tip of a leafy shoot, such as that of a monkey puzzle tree, you can see the leaves are arranged in spirals that start at the tip and gradually wind their way round the stem. A study of 12,000 spirals from over 650 plant species found that Fibonacci spirals occur in over 90% of cases.

Due to their frequency in living plant species, it has long been thought that Fibonacci spirals were ancient and highly conserved in all plants. We set out to test this hypothesis with an investigation of early plant fossils.

Three examples of living plants with Fibonacci spirals.
Examples of living plants with Fibonacci spirals. From left to right: spirals in leaves of a monkey puzzle trees, a pine cone and in the flower of a seaside daisy. Sandy Hetherington, Author provided

Non-Fibonacci spirals in early plants

We examined the arrangement of leaves and reproductive structures in the first group of plants known to have developed leaves, called clubmosses.

Specifically, we studied plant fossils of the extinct clubmoss species Asteroxylon mackiei. The fossils we studied are now housed in museum collections in the UK and Germany but were originally collected from the Rhynie chert – a fossil site in northern Scotland.

We took images of thin slices of fossils and then used digital reconstruction techniques to visualise the arrangement of Asteroxylon mackiei’s leaves in 3D and quantify the spirals.

Based on this analysis, we discovered that leaf arrangement was highly variable in Asteroxylon mackiei. In fact, non-Fibonacci spirals were the most common arrangement. The discovery of non-Fibonacci spirals in such an early fossil is surprising as they are very rare in living plant species today.

A digital reconstruction of the fossil Asteroxylon mackiei.
Life reconstruction of fossil Asteroxylon mackiei. Matt Humpage/Northern Rogue Studios, CC BY-NC-ND

Distinct evolutionary history

These findings change our understanding of Fibonacci spirals in land plants. They suggest that non-Fibonacci spirals were ancient in clubmosses, overturning the view that all leafy plants started out growing leaves that followed the Fibonacci pattern.

Furthermore, it suggests that leaf evolution and Fibonacci spirals in clubmosses had an evolutionary history distinct from other groups of living plants today, such as ferns, conifers and flowering plants. It suggests that Fibonacci spirals emerged separately multiple times throughout plant evolution.

The work also adds another piece to the puzzle of a major evolutionary question – why are Fibonacci spirals so common in plants today?

This question continues to generate debate among scientists. Various hypotheses have been proposed, including to maximise the amount of light that each leaf receives or to pack seeds efficiently. But our findings highlight how insights from fossils and plants like clubmosses may provide vital clues in finding an answer.

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