How to spot fractal patterns in nature

You don't have to be a mathematician to see maths in nature. All around, there are common patterns and geometries to identify.

Us humans have a natural ability to spot patterns in the world around us, whether it be in face recognition or in predicting weather. Making these connections has been a key tool for us to survive in the wild, when making sense of the environment around.

Though, just as we can match faces, some are able to make connections between mathematical rules, and inanimate objects, whether they be looking at the sky above, or the leaves on the ground below.

One of the most commonly recognisable patterns is the golden ratio, named 'golden' by the ancient Greeks due to it appearing 'divine'.

The Golden Ratio

The famous golden ratio has fascinated not only mathematicians, but scientists and artists throughout history. It is not a recent scientific discovery, but can be found historically through famous artwork, and ancient architecture.

In maths, the golden ratio occurs when the ratio of the small quantity to the larger quantity, is the same as the ratio of the large quantity to the sum of both quantities. This ratio comes to the number 1.618.

It is a simple rule that can be defined mathematically, and is based on the rules of the Fibonacci sequence (named after Leonardo of Pisa, also called 'Fibonacci'). This string of numbers is based on a simple rule - the next number in the sequence must be the sum of the former two numbers.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

The ratio between any pair of numbers here are approximately the same as the golden ratio (1.618), and so have a direct relationship to it.

Keeping an eye out

Curiously, this ratio has an aesthetic appeal to the eye, having been used by artists and architects in the layouts and proportions they use - to create that feeling of awe. This beauty is also observed in nature and its forms, such as the shell of the Nautilus (above).

Can you spot the golden ratio in these images?

What makes a fractal?

When something is fractal it means that it is self-similar. One pattern will grow from a starting point, repeating itself continuously, on different scales.

Computational programmers and mathematicians have taken this principle into their own interpretations, experiments and presentations of fractals, generating some stunning images as a result.

One such discovery is the beautiful Harriss spiral, described in an article published by the Guardian.

Beyond the golden ratio, other varieties of the Fibonacci sequence can be found throughout nature, particularly noticeable in the botanical world; such as the the petals of a flower, or leaves of a shrub.

To see beautiful renditions of the Fibonacci sequence in nature, click play below.

Computer programmers and digital artists have taken advantage of the effects of fractal formulae, to create detailed visuals -

In addition to these imaginative spiral patterns, even the branching of dendritic forms reflects fibonacci based fractal patterns, and can be identified in many different places, such as..

the branches of a tree,

..lightning in the sky..

..frost (on a microscopic


..and even river deltas.

#maths #nature #geometry #fractals #Aesthetics