Romanesco broccoli is likely one of the most classic representations of fractals in nature as its buds spiral in a strikingly elegant self-similar pattern.

Credit: Jon Sullivan via Wikipedia

An NSF mathematician created this fractal to better understand how wind would move oceanographic sensors in an eddy. This is just a snapshot in time, but wind would push the sensors back and forth, making it uncertain where they might go. Because of the stochastic nature -- a kind of organized randomness -- this fractal helps visualize the scenario and come up with an optimal, control so sensors use as little battery as possible and minimize displacement.

Credit: Lora Billings

The Koch snowflake is among the earliest fractal geometry work. Not surprisingly, nature's snowflakes seem to share that self similarity the Swedish mathematician Helge von Koch described.

Credit: Alexey Kljatov via Flickr

Mathematicians use "stochastic" fractals to discern a sort of order in what seems to be random chaos. In other words, they can use fractals to whittle down probabilities in things like the stock market. A likely more beautiful version of a "stochastic" fractal is a sea fan like this one.

Credit: Rob via Flickr

On a gecko's foot, there are two or three iterations of a repeating pattern, and when you look at it through a microscope, it continues repeating. Some scientists believe those very fine structures create so much electrostatic tension.

Credit: Bjørn Christian Tørrissen

Sagrada Familia in Barcelona, designed by Antoni Gaudi. He studied how plants grow and noticed patterns and fractals. He considered all this when he built this cathedral in a way to maximize light. Compared to this modernist Gothic cathedral, 15th century Gothic cathedrals had nowhere as much light come through. He maximized by using geometry and included fractal structures -- a stem that splits up and splits again -- one of the most beautiful man-made fractals.

Credit: Dave Telford