When you think of fractals, you might think of Grateful Dead posters and T-shirts, all pulsating with rainbow colors and swirling similarity. Fractals, first named by mathematician Benoit Mandelbrot in 1975, are special mathematical sets of numbers that display similarity through the full range of scale — i.e., they look the same no matter how big or how small they are. Another characteristic of fractals is that they exhibit great complexity driven by simplicity — some of the most complicated and beautiful fractals can be created with an equation populated with just a handful of terms. Scroll down towards the bottom to see how this works.

The Mandelbrot Set

The Mandelbrot Set

One of the things that attracted me to fractals is their ubiquity in nature. The laws that govern the creation of fractals seem to be found throughout the natural world. Pineapples grow according to fractal laws and ice crystals form in fractal shapes, the same ones that show up in river deltas and the veins of your body. It's often been said that Mother Nature is a hell of a good designer, and fractals can be thought of as the design principles she follows when putting things together. Fractals are hyper-efficient and allow plants to maximize their exposure to sunlight and cardiovascular systems to most efficiently transport oxygen to all parts of the body. Fractals are beautiful wherever they pop up, so there's plenty of examples to share.

Here are 14 amazing fractals found in nature:

Broccoli

Photo: Rum Bucolic Ape/Flickr

Try not to fall into this closeup photo of Romanesco broccoli. Each of the smaller buds is made up of even smaller buds. Here's another.

Romanesco broccoli closeup

Photo: Manuel Noah Angeja/Flickr

You can see some of the same fractality in the spirals of pinecone seeds.

Pinecone

Photo: Aidan M. Grey/Flickr

And in how this plant's leaves grow around each other.

Aloe

Photo: Genista/Flickr

This block of plexiglass was exposed to a strong current of electricity that burned a fractal branching pattern within. This can be best thought of as bottled-lightning.

Fractal caught in plastic

Photo: Wikimedia Commons

That same pattern shows up all over the place. Here are ice crystals forming.

Fractal ice

Photo: Wikimedia Commons

And a 20 times magnification of dendritic copper crystals forming.

dendritic copper crystals forming

Photo: Wikimedia Commons

The pattern below was created by running electricity between two nails sunk in a piece of wet pine.

Fractal woodburn

Photo: Wikimedia Commons

It's in trees.

tree

Photo: Abe Bingham/Flickr

tree

Photo: Burroblando/Flickr

And rivers.

river path as seen from above

Photo: Fabio Mascarenhas/Flickr

And leaves.

leaf veins and fractals

Photo: i5a/Flickr

We see fractals in water drops.

water droplets

Photo: NatJLN/Flickr

And air bubbles.

air bubbles and fractals

Photo: Woodley Wonderworks/Flickr

They're everywhere!

A great example of how fractals can be constructed with just a few terms is my favorite fractal, the Mandelbrot Set. Named for its discoverer, the previously mentioned mathematician Benoit Mandelbrot, the Mandelbrot Set describes a fantastical shape that displays amazing self-similarity no matter what scale it is looked at and can be rendered with this simple equation:
zn+1 = zn2 + c
I won't get into the technicalities of the equation here (you can read this infographic I made about how to render the Mandelbrot Set if you want to dive into more specifics), but basically it means that you take a complex number, square it, and then add itself to the product, over and over again. Do it enough times, translate those numbers to colors and locations on a plane, and baby, you've got yourself a beautiful fractal!

Here's what I mean by fractals looking the same throughout the scale. This shows a zoom into a smaller region on the larger Mandelbrot Set. Notice anything similar between where you start and where you end?

The Mandelbrot Set zooming in

Illustration: Shea Gunther

For an extreme example of how this works, check out this video showing a super deep zoom into the Mandelbrot Set.

Besides the Mandelbrot Set, there are scores of other types of fractals. Here are a few of the more well-known fractals.

The Koch snowflake

The Koch snowflake. (Wikimedia Commons)

The Sierpinski Triangle.

The Sierpinski Triangle. (Wikimedia Commons)

The Dragon Curve.

The Dragon Curve. (Wikimedia Commons<)

Pythagoras tree

Pythagoras tree. (Wikimedia Commons)

The fractal tree

The fractal tree. (Manuel Noah Angeja/Flickr)

What about you? Do you have any favorite natural fractals? Share some links in comments.

Are you on TwitterFollow me (@sheagunther) there, I give good tweets. And if you really like my writing, you can join my Facebook page and visit my homepage.

More cool stuff on MNN: 11 transparent creatures

The opinions expressed by MNN Bloggers and those providing comments are theirs alone, and do not reflect the opinions of MNN.com. While we have reviewed their content to make sure it complies with our Terms and Conditions, MNN is not responsible for the accuracy of any of their information.