From sea shells and spiral galaxies to the structure of human lungs, the patterns of chaos are all around us. Fractals are patterns formed from chaotic equations and contain self-similar patterns of complexity increasing with magnification. If you divide a fractal pattern into parts you get a nearly identical reduced-size copy of the whole. The mathematical beauty of fractals is that infinite complexity is formed with relatively simple equations. By iterating or repeating fractal-generating equations many times, random outputs create beautiful patterns that are unique, yet recognizable. We have pulled together some of the most stunning natural examples we could find of fractals on our planet. Above: Romanesco Broccoli This variant form of cauliflower is the ultimate fractal vegetable. Its pattern is a natural representation of the Fibonacci or golden spiral, a logarithmic spiral where every quarter turn is farther from the origin by a factor of phi, the golden ratio. Image: Flickr/Tin.G.

Salt Flats The San Francisco Bay salt flats pictured above have been used for commercial salt production for over a century. Below, an image from the largest salt flat in the world, Salar de Uyuni, located in southern Bolivia. The encrusted salt shows a remarkably consistent but random pattern, a characteristic of fractals. Images: 1) Flickr/Tolka Rover. 2) Flickr/Mike D. Green.

Ammonite Sutures Extinct for 65 million years, ammonites were marine cephalopods that built chambered spiral shells. The walls between these chambers, called sutures, were complex fractal curves. Stephen Jay Gould used the complexity of ammonite sutures over time to argue that there is no evolutionary drive toward greater complexity and that we are a "glorious accident," alone in the universe. The shells of ammonites also grow as a logarithmic spiral, a pattern that appears often in nature, as with romanesco broccoli. Ammonites also inspired this cathedral staircase in Barcelona, Spain. Images: 1) Flickr/cobalt123. 2) Flickr/Didier.bier. 3) Flickr/Edgley Cesar

Mountains Mountains are the result of tectonic forces pushing the crust upward and erosion tearing some of that crust down. The resulting pattern is a fractal. Above is an image of the Himalayan Mountains, home to many of the tallest peaks on Earth. The Himalayas are still being uplifted by the collision of India with the Eurasian plate, which began about 70 million years ago. Image: NASA/GSFC/JPL, MISR Team.

Ferns Ferns are a common example of a self-similar set, meaning that their pattern can be mathematically generated and reproduced at any magnification or reduction. The mathematical formula that describes ferns, named after Michael Barnsley, was one of the first to show that chaos is inherently unpredictable yet generally follows deterministic rules based on nonlinear iterative equations. In other words, random numbers generated over and over using Barnsley's Fern formula ultimately produce a unique fern-shaped object. Images: 1) Flickr/cobalt123. 2) Flickr/CatDancing.

Clouds The marine stratus clouds above were photographed by the Aqua satellite over the South Atlantic Ocean, off the west coast of Africa. A fractal cloud pattern is interrupted by a series of diagonal grooves. According the NASA Earth Observatory, it is highly unusual to see such a sharp boundary in a continual cloud formation such as this, and scientists have yet to explain how it could form. Below, the 200-mile long series of cloud vortexes is perhaps the longest of these formations ever photographed, according to NASA. These cloud vortexes are also called von Kármán cloud streets, for the late fluid dynamicist Theodore von Kármán. Cloud streets form when low-lying clouds are interrupted by an object such as an island. Similar patterns occur in the wind flow downstream of airplane wings. Images: 1) Jeff Schmaltz/MODIS Land Rapid Response Team/NASA. 2) NASA.

Leaves Many plants follow simple recursive formulas in generating their branching shapes and leaf patterns. Ironically, deforestation patterns that occur along primary, secondary and tertiary roadways form a similar pattern, like that pictured below from the state of Rondonia in western Brazil, one of the most deforested regions of the Amazon. Images: 1) Flickr/ CatDancing. 2) NASA.

Canyons This false-color image above, of the Grand Canyon, displays the fractal pattern created by the Colorado River over millions of years. Red indicates areas of vegetation in the image. Below, a false-color image of Canyon de Chelly National Monument in Arizona. The National Park Service calls it one of the longest continually inhabited landscapes in North America. Images: 1) GeoEye/Space Imaging. 2) NASA.

Lightning The path lightning takes is formed step by step as it moves towards the ground, turning air into plasma. Images: 1) Flickr/thefost. 2) Flickr/Brujo+.

Peacock Feathers Peacocks attract mates with the repeating patterns in their plumage. Images: 1) Flickr/Digimist. 2) Flickr/Maia C.

Snowflakes Crystallizing water forms repeating patterns in snowflakes and on frosty surfaces. The patterns have inspired claims about the power of consciousness to affect matter, as well as one of the first described fractal curves, the Koch snowflake. Images: 1) Flickr/mommamia. 2) Flickr/monteregina.

Waterfall Similar to canyons, a combination of rock irregularity and gravity produces repeating patterns as water flows over the side of a sharp ledge. Image: Flickr/catdancing