9 Amazing Fractals Found in Nature

Succulent plant with fractal pattern

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Fractals are known as geometric shapes that display similarity through the full range of scale—that is, they look the same no matter how big or how small they are. And they are, in fact, ubiquitous in nature.

There are many examples of fractals that we encounter in everyday life. Pineapples grow according to fractal laws, and ice crystals form in similar fractal shapes. Fractals allow plants to maximize their exposure to sunlight. They allow cardiovascular systems to efficiently transport oxygen to all parts of the body.

Here, we explain 14 fascinating and beautiful examples of fractals in nature.

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Romanesco Broccoli

Close-up of complex patterns of Romanesco broccoli bunch

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Romanesco broccoli (which, despite its name, is a hybrid between broccoli and cauliflower) grows in a golden spiral, a pattern based on the golden ratio. The spiral gets wider by a factor of φ for every quarter turn it makes. This vegetable's tendency to produce buds at an accelerating rate causes this spiral pattern and resulting conical shape. The tip gets taller and taller as the broccoli grows.

Other golden spirals in nature include spiral galaxies and nautilus shells.

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Pine Cones

Pine cone hanging on lush green pine tree

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Pine cones, also called strobili, are woody, scaly fruits that dangle off evergreens and serve as the trees' seed-bearing organs. You have no doubt noticed the satisfying spiral of their scales, which reflect the seeds they protect. They close tightly when it's damp or cold, then open when it's optimal for seeds to spread in the wind.

Again, the fractal design is caused by accelerated growth. It's a natural example of the logarithmic or equiangular spiral.

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Close-up of the pattern of spiral aloe leaves

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In plant biology, fractal designs are called spiral phyllotaxy, the word "phyllotaxy" simply referring to the arrangement of leaves on a plant. There are good reasons for the upturned, coiling leaves of the spiral aloe (Aloe polyphylla) and some Echeveria varieties: They help funnel rainwater to the plant's core and prevent top leaves from shading out bottom leaves.

In the mid-00s, a mathematician hypothesized that the spiral pattern of both plants and fingerprints occurred for the same reason, which was to relieve stress. Forces acting in opposing directions causes skin and plant tissue to buckle inward as it grows, he said.

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Ice and Snow

Macro shot of snow flake on dark blue surface

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No two snowflake designs are alike, but many represent fractals in that the branches of a snowflake spawn their own side-branches, and so on. The snowflake could continue on like this for an eternity, growing the size of Earth itself, if it weren't to stop accumulating moisture and, eventually, melt away.

The most famous fractal snowflake pattern is known as the Koch snowflake, stemming from one equilateral triangle forming another and another and another. This was one of the earliest fractals described.

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Tree Branches

View looking up into lush green branches of large tree

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Trees are one of the most quintessential fractals in nature. As they grow, branches develop from the trunks, and each of these branches is like a smaller tree in itself, developing its own branches and their own branches. If you look at a complex tree, you will notice the repetition of the Y shape throughout.

This kind of fractal design, like the spiraling of succulents, helps trees optimize their exposure to sunshine and prevents top branches from shading out lower ones.

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Copper Crystals

Macro shot of copper crystal formation

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Fractal geometry is also common in chemistry. The phenomenon is expertly demonstrated by copper crystals, which branch out in all directions like tree limbs. Each "twig" is a new growth point—as it branches out, it develops into solid metallic copper. Because of their arborescent nature and unique reddish-brown color, copper crystals are often grown for art.

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Drone image of complex marsh network at low tide

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Rivers branch out into streams because water flows from the main artery downhill, but why always in meandering S shapes? While streams can sometimes be established in a straight line, they quickly become bendy as they adapt to disturbances like wildlife dens. Just one disturbance can throw off the flow of a river and cause it to curve throughout.

How wide these streams are is also extremely formulaic. The curves, experts have found, are always six times the width of the channel. This kind of self-similarity is characteristic of fractals and the reason why rivers look alike all around the world.

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Leaf Veins

Back of leaf with water drop magnifying veins

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Trees are, in fact, fractal from their seeds to their roots to their leaves to their canopy. If you look closely at the veins of the leaves, you'll notice just how self-similar they are. The tiniest ones look like the main midrib (the midline vein), and the midrib looks like the tree trunk with its branches.

This is true only of reticulate venation (weblike, not parallel, veins).

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Macro shot of air bubbles on blue surface

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The foam on your morning latte is fractal. In nature, bubbles that occur when ocean waves break or where raindrops have fallen create a self-similar pattern with thin films of liquid separating various-sized gas pockets. Large bubbles are interspersed with small bubbles and those small bubbles interspersed with even smaller bubbles, and so forth.

Outside of nature, you'll see these fractal designs in a soapy bath or while doing dishes in the sink.

View Article Sources
  1. "UA Mathematicians Predict Patterns in Fingerprints, Cacti." University of Arizona. 2004.

  2. Peng, Sheng-Lung, Rong-Xia Hao, and Souvik Pal. "Proceedings of First International Conference on Mathematical Modeling and Computational Science." Springer Nature. 2021.