The amazing fractal patterns of nature

west bay
A photo point for monitoring the changing Dorset coastline at East Beach in West Bay

Why is it so difficult to measure the Dorset coastline?

Science feature by Professor Glenn Patrick

FOR those of us living in Lyme Regis, the ever-present sea provides a mesmerising backdrop as we observe its fluctuating rhythms, shifting light and changing moods.

I am sure we have all stood on the shoreline thinking about distant lands and marvelling at the sheer beauty and power of the ocean.

Just over 70 per cent of the planet is covered by water and the average depth of the oceans is about 4,000 metres. Although we talk of the deep blue sea, this is only 0.06 per cent of the radius of the Earth.

On this grand scale, the seas are comparatively shallow, but any changes in sea level can have profound and complex consequences on the shoreline.

For example, a recent study led by the University of Exeter reconstructed past sea-level rises and revealed that less than a thousand years ago the Isles of Scilly emerged from a single island to form the current arrangement of 140 islands, islets and rocks.

Each winter, we are used to stormy weather constantly rearranging the Dorset coast through cliff falls, eroded beaches and landslips.

It is important that we monitor the shifting features of our coastline so that we can better understand the processes at play, especially with accelerating climate change.

You can play a personal role in this monitoring whenever you visit West Bay. Here, you will find that the Plymouth Coastal Observatory has installed photo points next to the East and West Beaches where you can place your smartphone and upload images of the coast.

This is part of a programme called ‘Coastsnap SouthWest’, which is researching the dynamic nature of the south west coast.

In Lyme Regis, we are lucky that hard engineering solutions have been deployed to build sea walls, rock armour and shingle beaches to hold a firm line against the sea.

Other places are not so lucky in the constant battle fought along the entire British coastline to protect the land from the erosive forces of the sea.

west bay cliffs
Zoom in on the cliffs at West Bay and the fractal structure is revealed – the source of the ‘Coastline Paradox’

Strangely, there is no agreed measurement of the total length of Britain’s coastline let alone the Dorset coast. Using Google, you will find various figures quoted between 12,000 and 31,000 km, but which one is correct?

The answer is none of them – there is no definitive number!

There are, of course, obvious problems, like whether you take the high water or low water mark and how far you go up estuaries for your measurement.

There is though something more profound called the ‘Coastline Paradox’ or the ‘Richardson Effect’, named after the English mathematician Lewis Fry Richardson who first observed it.

Simply put, this paradox states that the coastline of a landmass does not have a well-defined length. Imagine measuring the coastline with a ruler; you would intuitively think by using a shorter and shorter ruler to go round all the bays, inlets, rocky crags, etc. you would eventually converge on the true length of the coastline.

In fact, the length just keeps on increasing as the unit of measurement is made smaller and smaller.

This problem was investigated by the famous mathematician Benoit Mandelbrot in a 1967 paper called ‘How Long is the Coast of Britain?’. He concluded that the coastline just gets wigglier and wigglier as you zoom in and your ruler gets shorter and shorter.

This is an example of what Mandelbrot eventually called a ‘fractal’ – an object that appears similar and looks roughly the same at different sizes or scales. This means that the length of a coast only makes sense if you also quote the length of your ruler.

Flowers often exhibit fractal patterns

You do not need to be a mathematician to appreciate the beauty of fractals. Nature is full of fractal patterns wherever you have a simple process that repeats itself at different scales.

For example, a tree grows from a sapling by repetitively branching from the trunk with the larger branches then splitting into smaller and smaller branches until we arrive at twigs.

The ammonites found along our shores build their spiral shells by adding similar pieces that grow and twist at a constant rate.

A fern is a classic example of a fractal – if you look at the branches coming from the main stem you will see that they look very similar to the entire frond.

Study your garden plants and notice how many of them, like sunflowers, are made up of repeating patterns formed by growing new pieces that look the same shape, but smaller.

In 1820, The Japanese artist Katsushika Hokusai created the ‘The Great Wave Off Kanagawa’. This is widely considered the most famous image in Japanese art and is one of most reproduced artworks in the world.

It depicts a huge wave threatening three boats beneath it with Mount Fuji in the distant background. Look closer though and you realise that the smaller fingers and dripping droplets of water also have the same basic shape as the big wave.

Hokusai clearly appreciated the fractal structure of breaking waves long before the mathematics had even been invented. Watch waves breaking on the Cobb or on the beach and you will see the same effect.

The concept of fractals has spread through many fields including cosmology, medicine, meteorology, financial markets, and image analysis. Even our cell phones contain antennae that are made from fractals.

Fractal patterns often appear in arty magazines and music videos when colourful psychedelic graphics are required or deployed to make CGI animations in movies.

Perhaps most intriguing of all, our own bodies contain fractal patterns. Our lungs are full of tree-like structures, which keep on branching from our bronchial airways down to millions of small sacs called alveoli.

Our amazing brains are basically fractal networks comprising billions of neurons connected by trillions of dendrites and axons in a complex tree-like structure.

Next time you take your daily exercise by the coast, observe how nature is not smooth, but made up from patterns of irregular recurring shapes of different sizes– for example in billowing clouds, rocky cliffs, stormy waves and even layers of tree bark.

Remember, if the walk seemed longer than the last time, it probably was – due to fractals!

Glenn Patrick is a particle physicist and science communicator who explores the quantum world of sub-atomic particles (including at the Large Hadron Collider) and now lives in Lyme Regis.

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