Fractals appearing in nature

Recently we’ve looked at a number of art forms which create patterns in different ways - paper art, drawing machines, kinetic wind sculptures, illusionary sculptures and the digitial display "Quantum memories." (All links are provided below^).

But what is a pattern, actually?

By definition, a pattern is something that is repeated, and is usually aesthetically pleasing. But there are different sorts of patterns, many occuring naturally in the world around us .......

.....and others created by humans using a series of mathematical equations which can be incredibly complex.

A particular type of these, known as Fractals are, in simple terms, never ending patterns where each smaller element making up the pattern is itself the same pattern as the whole! You’ll see what I mean from the following example:

Shape 1a


Fractals also occur in nature and include things like clouds, ice crystals, mountains, river networks, lightning strikes, flower petals, and even our blood vessel system!

Amazing lightning patterns, courtesy of Unsplash

It may astound you to know that these patterns are actually the visual representation of certain mathematical equations. It is amazing to realise that Mother Nature has known and used these equations for hundreds of thousands of years, and it has been up to humans to discover them, and develop further uses for them!

Fractals are now used in science, maths, finance, geography, art and even music! They inform our understanding of how things work, and lead to further inventions and discoveries.

A very important phenomenon of fractals is that they manifest self-similarity at all levels. Benoit Mandelbrot, one of the fathers of fractal geometry (and the man who coined the term fractal), loosely defined fractals as "shapes that are equally complex in their details as in their overall form. That is, if a piece of a fractal is suitably magnified to become of the same size as the whole, it should look like the whole, either exactly, or perhaps only slightly deformed." 1

He declared: "Clouds are not spheres, mountains are not cones, coastlines are not circles," referring to the chaos and irregularity of the world as roughness. He saw it as something to be celebrated, saying it would be a shame if clouds really were spheres. You can see below that a small cloud is similar to the overall formation and structure:

And if mountains are not cones, they are certainly classic example of the chaos and irregularity that Mandelbrot describes as "roughness".......

You will see some more incredible examples from nature if you click here.

All of these shapes have something in common. They are all complicated and irregular which makes them difficult to tame mathematically but a joy to an artist, particularly photographers and digital creators. Look closely at a fractal, and you will find that the complexity is still present at a smaller scale. You can see the self similarity attributes of the tributaries of a river system, when viewed from above the earth:

We all know of the magnificence of the Grand Canyon, even if only appreciated through photography. The repetition of these kaleidoscopic patterns have a mathematical formula at their centre.

Lastly, a man named Leonardo de Pisa discovered in 1202 that some patterns in nature have exactly the same mathematical basis, which has become known as the Fibonacci Sequence. This is a further subset of fractals and is still used in many aspects of life today. It's a very interesting subject all on its own, which we will cover in a separate post. It explains that the way leaves are arranged on a stem, the number of petals on a flower, the shape of a pineapple or the shape of a nautilus shell is no accident!

Tomorrow we will learn more about the work of Benoit Mandelbrot who discovered in 1979 what is now called the Mandelbrot Set and through this, learned how to imitate the mathematics of nature using a computer.


^Recent blogs on other art forms creating patterns:

  • the visual patterns of paper art click here,

  • the patterns created by James Gandy’s drawing machines click here,

  • kinetic wind sculptures click here, and

  • the illusionary sculptures of Michael Murphy click here.

  • "Quantum memories", digital display at The National Gallery of Victoria click here


  1. With thanks to