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. (More on that later.)

The Mandelbrot Set (Photo: Wikimedia Commons)

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: Bert Hickman/Wikimedia Commons)

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

Fractal ice(Photo: Schnobby/Wikimedia Commons)

And a 20 times magnification of dendritic copper crystals forming.

dendritic copper crystals forming(Photo: Paul/Wikimedia Commons)

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

Fractal woodburn (Photo: Peter Terren/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. (Photo:Wikimedia Commons)

The Sierpinski Triangle. The Sierpinski Triangle. (Photo: Wikimedia Commons)

The Dragon Curve. The Dragon Curve. (Photo: Wikimedia Commons)

Pythagoras treePythagoras tree. (Photo: Wikimedia Commons)

The fractal tree The fractal tree. (Photo: Manuel Noah Angeja/flickr)

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

Shea Gunther ( @sheagunther ) is a podcaster, writer, and entrepreneur living in Portland, Maine. He hosts the popular podcast "Marijuana Today Daily" and was a founder of Renewable Choice Energy, the country's leading provider of wind credits and Green Options. He plays a lot of ultimate frisbee and loves bad jokes.