Geometric patterns in nature


A drying polygonal shaped mudflat in a lagoon in the Camargue, France.

IMAGES of the Earth from satellite and aerial photography are readily available to us today. We may often ogle at them wondering how stone circles, settlement patterns and other man-made and natural landscape features have developed and why they take a particular shape. Seldom, however, do we look at the very patterns before our eyes at ground level. This is, no doubt, because we are so busy in our lives and fail to find time to stop, stare and wonder.

I often recall Welsh poet William Henry Davies’ sonnet entitled ‘Leisure’, written in 1911 and referring to the beauty of ‘Mother Nature’. I give you but the first two and last lines of this poem:

What is this life if, full of care,

We have no time to stand and stare …

… A poor life this if, full of care,

We have no time to stand and stare.

As a geomorphologist, it is inherent in me to stand and stare and marvel at the shapes of landscapes. I well remember my first long bike ride with my father when I was five years old. We visited an ancient Bronze Age dwelling at Chysauster, in West Cornwall, England.

He pointed out the shape and patterns of this early settlement tucked into a slight fold in the desolate granitic moorland. It was raining hard, yet we marvelled at the field patterns below in such a bleak environment. Little, did we know then, that 4,000 years earlier, the climate was much warmer than in 1951.

Forty-nine years later, as dawn broke on Low’s Peak at 4,101 metres on Mount Kinabalu, I shared with my 26 A-level students, the patterns in the landscape of the Sabahan countryside and coastline looking down below. As the sun gradually rose all became clearer with ever-changing colours. The granitic rock, over which we had crawled with head torches in earlier darkness, truly revealed its patterns. White veins in that rock were now more visible. We stood and stared.

Hexagonal columns of basalt at the World Heritage site of the Giant’s Causeway in County Antrim, Northern Ireland.

What are nature’s patterns?

These are the visible regularities of forms which repeatedly appear and sometimes can be mathematically modelled. There are cracks, meanders, spirals, symmetries, tessellations, and waves, not to mention spots and stripes. Both Plato and Pythagoras used these patterns to try to explain order in nature.


For many a year, I have observed contraction cracks in granite areas worldwide. The molten magma eventually cooled as it was intruded into overlying country rocks, often below overlying sedimentary rocks, when the land was uplifted.

Granitic rocks exhibit vertical cooling contraction joints and false horizontal bedding planes, whereas sedimentary rocks display vertical joints caused also by contraction as they dry out, upon uplift, from their original deposition on the seabed.

They exhibit true horizontal bedding planes, which demarcate different forms of sedimentary deposits when they were first laid down on the ocean floor.

In igneous rocks, the most spectacular 120 degree vertical cracks are seen in the hexagonal columns of the cooled basalt at the World Heritage site of the Giant’s Causeway, County Antrim, in Northern Ireland.

Mud deposits, cracks of a similar fashion, when in drought, show again 120 degree cracks, taking polygonal shapes. I stood and stared at such a wonder in a shallow, evaporating lagoon in the French Camargue bordering the Mediterranean Sea this summer. Palm trees experience branching vertical cracks and horizontal leaf scars and even the silver birch trees in my English garden have their own patterns of splitting bark as the trees grow upwards.


As a fluvial geomorphologist, these have always fascinated me. To arrive in Kuching on a day flight, I am mesmerised by the sinuosity of the Sungai Sarawak and its many tributaries. Water naturally tends to flow in meandering patterns. Just witness the path of rain drops running down a window pane.

A river, too, flows in a helical pattern, eroding material from its outside bank, where its velocity is strongest, and sweeping the sediment across stream in a corkscrew fashion to deposit it as at mud, gravel and sand on the inside bank of a bend where its velocity is decreased.

Simply, the river is working, within its system, to lose energy as it flows.

Huge meanders of the atmosphere’s jet streams, way above our heads, guide storms, hurricanes and typhoons on their paths of destruction.

Coral reefs exhibit meandering patterns in the form of brain coral (Diploria strigosa), while on land our snakes slither most efficiently in a sinuous path. Why? It is because less energy is needed in moving or developing this pattern.

Spiral growth patterns on a fossilised giant ammonite in Carboniferous limestone rock.


At home, I daily look at my fossil giant ammonite in my fireplace.

This Carboniferous aged creature, in limestone, has each chamber in its shell as a near copy of the preceding one, gradually getting larger towards the mouth of the nautilus shell.

These chambers are so arranged in a logarithmic spiral. The gastropod shellfish, which we so lovingly suck out of their black and white shells in Kuching eateries, also exhibit this spiral shell. This, too, illustrates how its opening moves around, outwards and downwards as the animal grows. The same occurs in rams’ horns and wildebeest horns. The seed heads in sunflowers also grow in a similar spiral form.


Many animals and the leaves of plants and the flowers of orchids have mirror symmetry. A tiger, when photographed face on, shows bilateral symmetry in its stipe patterns, while starfish have a five-fold symmetry much the same as many flowers and fruits. Rock crystals, including gemstones, show equal symmetry in rhombic shapes.

It is, perhaps, snowflakes, with their six-fold symmetry that most appeal to me for, on each of their six limbs, there is an almost identical ice crystal structure growing laterally outwards. Each ice crystal is unique and depends upon its localised atmospheric condition. Similarly, a raindrop splashing in a roadside pool displays radial symmetry, with a crown shape much like that of a sea anemone’s growth pattern.


A hexagonal shape is the most efficient shape to include in a pattern especially, if we are choosing new floor tiles. Actually, this word is derived from the Latin ‘tessella’ meaning a tile. How, so often, do we see this hexagonal shape?

Beehive wax-celled honeycombs demonstrate this to perfection, as do the paper cell patterns of Asian hornets’ nests. Snakes, crocodiles, pangolins are all protected by overlapping scales which may all be of different sizes but are created in almost identical shaped units.



These exhibit energy as they move, either through the air or across the sea, oscillating as they flow. Ripples are seen on sandy sea-beds at both full and ebb tides and in sand deserts (ergs), where they are again obvious in certain sand dune formations. The crescent-shaped sand dune, the barchan, has two ‘horns’ in the crescent and a lee face pointing downwind.

This pattern is formed by unidirectional winds blowing the sand from the windward slope over the dune’s crest to the leeside: slowly the sand dune advances as sand collapses downwards. Ocean waves breaking against coastlines exhibit similar characteristics.

If we can just pause to stand and stare in our daily lives and focus our eyes on the many natural patterns, on the ground we tread, which can be explained mathematically, and seen in the myriad of animal, fish, and vegetation forms, let alone in the skies above our heads, we will fulfil WH Davies’ dream and, moreover, enlighten ourselves to the wonders of nature.