Fractal patterns in nature

http://www.wired.com/2010/09/fractal-patterns-in-nature/

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.

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.

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.