Fractals in Nature: Why the Same Patterns Repeat From Coastlines to Galaxies

Fractals are patterns that repeat their own shape at smaller and smaller scales, and nature uses them everywhere: in coastlines, ferns, rivers, trees, lightning and the branching airways of our lungs. They appear so often because the same simple rules, repeated again and again, can build very complex forms. If you have ever stood on the shore in Malibu and watched the coastline fold into coves and the coves into rocks and the rocks into pebbles, you have already seen a fractal.

What is a fractal, in simple words?

A fractal is a shape that looks similar to itself when you zoom in. Break off a branch of a fern and it looks like a small fern. Break off a floret of Romanesco broccoli and it looks like a small Romanesco. The pattern holds across many scales, at least within the limits of the real world.

The mathematician Benoit Mandelbrot coined the word fractal in 1975, from the Latin fractus, meaning broken. He used it to describe rough, irregular shapes that classical geometry, with its smooth circles and straight lines, could not describe well.

Why can't anyone measure a coastline exactly?

This is one of the most surprising ideas in science, known as the coastline paradox. In 1967 Mandelbrot published a paper in Science asking "How Long Is the Coast of Britain?" The answer depends on the length of your ruler. Measure with a long ruler and you skip over the small coves. Use a shorter ruler and you follow more bends, so the total length grows. Keep shrinking the ruler and the coastline keeps getting longer.

That is why a coastline like Malibu's can be described by a number called a fractal dimension, which measures how rough and folded a line is, rather than by a single fixed length.

Why does nature love branching?

Branching patterns appear in rivers, tree crowns, blood vessels, lightning and the airways of the lungs. The physics and biology behind each one are different, but the logic is similar: a branching network is an efficient way to connect one point to a very large area or volume. Repeating the same split again and again lets a tree reach light in every direction and lets the lungs pack an enormous surface for gas exchange into the chest.

Spirals and Fibonacci: patterns that repeat across scale

Spirals are not always fractals, but they are another pattern nature repeats from small to huge. You find them in snail shells, in the seed heads of sunflowers and in the arms of spiral galaxies, each formed by completely different processes.

• Sunflowers: the seeds are arranged in two sets of spirals, and the number of spirals in each direction is very often a pair of consecutive Fibonacci numbers, such as 34 and 55.
• Shells: many shells grow in a shape close to a logarithmic spiral, getting larger while keeping the same proportions.
• Galaxies: spiral arms are regions where stars and gas gather, shaped by gravity on a scale of many thousands of light years.

Waves: the simplest repeating pattern of all

A sine wave repeats perfectly, crest after crest. Real ocean waves are messier, a sum of many waves of different sizes riding on each other, but each part can still be described with the same simple curves. We wrote more about this in the physics of waves.

From cells to galaxies: how scale inspires jewelry design

Margaret Krinetz has always worked with art and science side by side, with a background in physics and biology alongside a lifelong art practice. One idea runs through her work: the way the smallest structures can mirror the largest ones. That is where the collections come from.

The Wavy collection turns the sine and cosine curves she loves into recycled gold and natural diamonds, like the Wavy Diamond Hoop Earrings, a wave of diamonds curving around the ear in 18K recycled gold. The RAW collection follows the irregular, organic surfaces of nature. The Raw Diamond Ring is an organic gold shape with a raw texture, unpolished on purpose, set with an approx. 0.31 ct natural diamond. Like a coastline, its beauty is in the folds.

To see how the biggest scale and the smallest connect, read stars and diamonds are made of the same carbon.

Frequently asked questions

Are fractals real or only mathematical?
Perfect fractals exist only in mathematics. In nature, the pattern repeats over a limited range of scales, which is why ferns and coastlines are called fractal like.

Is every spiral in nature a Fibonacci spiral?
No. Many natural spirals are close to logarithmic spirals, and Fibonacci numbers appear mainly in how some plants arrange seeds and leaves.

How long does delivery to Malibu take?
Margaret Krinetz Fine Jewelry, handmade in Tel Aviv and shipped to Malibu, is made to order in 4 to 6 weeks, then DHL Express delivers in 3 to 5 business days.

Which pattern feels like you?

The wave or the raw surface, the perfect curve or the beautiful irregular fold? Tell me which one you are drawn to, and I will help you find your piece. Write through the contact page.

Sources: Benoit Mandelbrot, "How Long Is the Coast of Britain?", Science (1967); Mandelbrot, The Fractal Geometry of Nature; standard references on fractal dimension and phyllotaxis.