Fractals in Nature

Infinitely repeating patterns can be found anywhere you look

Table of Contents

Learn about the surprising mix of math and nature

A fractal is nothing but an infinitely repeating pattern. Imagine a triangle. Each triangle is part of a larger triangle which is part of a larger triangle… and so on. A fractal pattern is made up of these smaller patterns, each of which is identical or at least similar to the others! This crucial property is known as self-similarity.

A triangle version of the Menger fractal
A triangle version of the Menger fractal, Credit: Wikimedia/RobertdWc

Notice how each larger triangle is made up of smaller triangles. Although the triangle itself is simple, the patterns it makes just by being repeated can be very complex. Fractals can be quite beautiful.

What are examples of fractals in nature?

On the surface, math and nature seem like opposites. Nature is wild and unpredictable, creating surprising shapes and forms. On the other hand, one math equation always makes a predictable answer.

 

But most things in nature are made up of identical, smaller parts. A honeybee hive is actually a regular pattern of hexagons, made by almost identical worker bees!

The regular, repeating pattern of hexagons
The regular, repeating pattern of hexagons, Credit: publicdomainpictures.net

Branching is another common way that natural fractals are made. Neurons in the human brain branch, each one connecting to another neuron, which has its own branches, and so on. Trees branch in a similar way. Plant parts which are made of similar units, like the disc of a sunflower, also possess fractals. Snowflakes are spectacular fractals, and every single one is unique!

What are the four types of fractals?

The most common kind of fractal found in nature is known as “complexity from simplicity. This is the quintessential fractal. With a simple set of rules and guidelines, you can generate a complex fractal yourself. One example is the triangle we looked at earlier. In the Koch snowflake below, the only guideline is to overlay one triangle with another.

A Koch snowflake
A Koch snowflake, Credit: Wikimedia/Phreneticc

The second kind of fractal is the most dizzying. It is known as the infinite intricacy fractal. No matter how much you zoom in, it never gets simple. This is why the intricacy is described as infinite! It’s difficult to imagine an equation or pattern that is so complex. Consequently, the first example of this fractal was only found in 1872, described by mathematician Karl Weierstrass. Weierstrass’ fractal was a zigzag where each zag was made up of corners. No matter how you twisted, turned, zoomed in or zoomed out, the zag just could not be broken down to simpler components.

The Weierstrass function
The Weierstrass function, Credit: Wikimedia/Eeyore22

A zoom symmetry fractal is almost the opposite of an infinite intricacy fractal. Each tiny bit of the fractal is a reflection of the whole! Another key property of this kind of fractal is its ability to be transformed—or, more accurately, its inability to be transformed. Imagine a square. If you tip the square on its side and rotate it 90 degrees, you end up with something that looks exactly the same. Each component of this fractal, when transformed, creates the same fractal. These fractals were discovered relatively late, in the 1970s, by Polish mathematician Mandelbröt.

The last feature of a fractal is its odd dimensional nature. You’re probably reading these words on some kind of screen. The words themselves are one dimensional. The device is likely three dimensional, with a length, breadth, and width. Fractals occupy a weird dimensional space. They typically lie somewhere between the second and third dimension.

Are fractals important in nature?

Fractals are very important in nature! A fractal pattern allows things in nature to pack far more than they should. For example, the branching of vessels in the lungs follows a fractal pattern. Though they fill only the space in your chest, the unraveled surface area would be more than 70 square meters (700 square feet)!

 

Using fractal logic also helps us understand the natural world. If a whole can be broken down into its parts, each of which is described by the same mathematical equation, we can simplify and comprehend otherwise complex systems.

What are the five patterns in nature?

The key underlying mechanism of most patterns in nature is self-organization. Each unit knows what to do, so a pattern emerges overall!

 

There is a bit of back-and-forth about exactly how many patterns are observed in nature, but the most common five seem to be spiral, meander, explosion, packing, and branching. Branching is exactly what nerves and trees do! Packing is similar to the beehive example, where nature tries to fit as much as possible into as little space as possible.

 

Spirals seem to be very common for organisms that grow throughout their lives, like certain molluscs and plants. Physicists think this is because spirals are one of the lowest-energy arrangements.

The spiral dictated by the famous Fibonacci sequence
The spiral dictated by the famous Fibonacci sequence, Credit: Wikipedia/User:Dicklyon
NautilusCutawayLogarithmicSpiral
A Nautilus mollusc shell, Credit: Wikipedia/Chris 73

Meanders describes the movement of things like rivers and coastlines. As two opposing forces like water and rock interact, they push and pull each other in different directions.

The meandering of the River Negro in Argentina, demonstrated in a satellite image
The meandering of the River Negro in Argentina, demonstrated in a satellite image, Credit: Wikimedia/NASA Astronaut photograph ISS022-E-19513

Explosions are pretty dramatic! Typically, they have a high energy or high density center, from which things originate.

The head of a dandelion
The head of a dandelion, Credit: Wikimedia/Greg Hume

Patterns unfold around us in nature. The reasons for them are varied, from the most energy-efficient layout to the ability to self organize. What patterns do you see around you?

Glossary

Complexity from simplicity: A complex arrangement arising from simple components 

 

Infinite intricacy: A fractal that is very complex regardless of scale

 

Self-organization: The property that each unit behaves independently

 

Self-similarity: Each component is made of smaller, identical components

Flesch Kincaid Grade Level: 7.9

 

Flesch Kincaid Reading Ease: 56.7

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  • Yamini Srikanth

    Yamini's (he/they) interests lie in environmental education, science communication and trying to build a better world. When not languishing in front of his laptop, they can be found outside, poking at any insect, bird or plant. They love making science accessible, especially to those who aren't encouraged to pursue it. Yamini hopes that the young women who read Smore love learning from their articles and get just a little bit more excited about science!