“Study the science of art. Study the art of science. Develop your senses – especially learn how to see. Realize that everything connects to everything else.”

― Leonardo da Vinci

Ever since I was a kid, I’ve been fascinated by the similar patterns and shapes that nature repeats all around us. The structure of so much organic life follows self-similar, fractal patterns, which can be observed in flowers, trees, plants, and even mountains and coastlines.

The term fractal was coined by the mathematician Benoit Mandelbrot in 1975. In his seminal work The Fractal Geometry of Nature, he defines a fractal as “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.”

Mandelbrot is best known for his discovery in mathematics of the Mandelbrot Set, which can be programmed into basic lines of code that create an infinite stream of changing, self-similar patterns.

Here’s an excellent example of a Mandelbrot Set:

A fractal is a pattern that the laws of nature repeat at different scales. Examples are everywhere in the forest. Trees are natural fractals, patterns that repeat smaller and smaller copies of themselves to create the biodiversity of a forest.

Each tree branch, from the trunk to the tips, is a copy of the one that came before it. This is a basic principle that we see over and over again in the fractal structure of organic life forms throughout the natural world.

Where To Observe Fractals In Nature:

Walking through a forest, you will find fractal patterns in the network-like branching patterns everywhere among the ferns, trees, roots, leaves, and the fungal mycelium in the soil.

You will also find them throughout the natural world in the patterns of streams, rivers, coastlines, mountains, waves, waterfalls and water droplets.

Here are some examples of fractal patterns in nature:

1. Trees

Trees are perfect examples of fractals in nature. You will find fractals at every level of the forest ecosystem from seeds and pinecones, to branches and leaves, and to the self-similar replication of trees, ferns, and plants throughout the ecosystem.

2. River Deltas

This aerial footage from NASA of the Ayeyarwady River Delta (also referred to as Irrawaddy) in Myanmar is a great example of the fractal branching patterns of river delta ecosystems.

Ayeyarwady River Delta

3. Growth Spirals

You will also find fractal patterns in growth spirals, which follow a Fibonacci Sequence (also referred to as the Golden Spiral) and can be seen as a special case of self-similarity.

Plant Growth Spiral

4. Flowers

Observe the self-replicating patterns of how flowers bloom to attract bees. Gardens are amazing places to explore the fractal nature of growth.Fractal Flower Bloom Patterns

5. Romanesco Broccoli

You won’t find it in the forest, but this edible flower bud of the species Brassica oleracea (broccoli) from Italy is a wholesome and delicious example of fractal geometry.Romanesco Fractal Broccoli

These arrangements have explanations at different levels – mathematics, physics, chemistry, biology. Here’s what Wikipedia has to say about what the sciences have observed about these patterns in nature:

“From the point of view of physics, spirals are lowest-energy configurations which emerge spontaneously through self-organizing processes in dynamic systems. From the point of view of chemistry, a spiral can be generated by a reaction-diffusion process, involving both activation and inhibition. Phyllotaxis is controlled by proteins that manipulate the concentration of the plant hormone auxin, which activates meristem growth, alongside other mechanisms to control the relative angle of buds around the stem. From a biological perspective, arranging leaves as far apart as possible in any given space is favored by natural selection as it maximizes access to resources, especially sunlight for photosynthesis.”

Fractals are hyper-efficient in their construction and this allows plants to maximize their exposure to sunlight and also efficiently transport nutritious throughout their cellular structure. These fractal patterns of growth have a mathematical, as well as physical, beauty.

Fractals, Ecology, and Biomimicry:

So, why are fractals important to ecological awareness? In the ecology book Finding Our Way Home author Myke Johnson notes that our ability to measure fractal patterns in the natural world has also given us:

“The ability to create digital worlds that remind us of our own. Fractal formulas are used to generate computer graphics that look realistically like mountain ranges, and rivers, and forests, and clouds.

Fractals have been used to design antennas in greatly reduced sizes, which enabled the creation of the next generation of cell phones and other electronic communicators. Fractal geometry is enlarging our ability to create new devices that work better because they follow patterns that resonate with the natural patterns around us.”

Isn’t that amazing? Biomimicry in action.

Fractals also inspire awe and wonder, especially when you bring your full attention to exploring and mindfully studying them in natural environments like forests. To expand your understanding of fractals, I highly recommend watching the documentary Fractals: Hunting The Hidden Dimension.

Watching it will help you further develop your pattern recognition skills so you can recognize and understand the fractal patterns all around you.

If you want to experiment with creating your own fractal patterns, try playing around with Xaos, a free tool from the Fractal Foundation for those who want to get creative with fractals.