If you take a picture of a stretch of coastline from a satellite, you will see a set of regular convolutions that randomly fall along the edge of the coastline. If you take a closer picture of a portion of the same coastline, you will see similar patterns at a smaller scale. If you look at a section of this picture with a magnifying glass, you might again see this convoluted pattern. These similar patterns occurring at various scales are naturally occurring quasi-self-similar fractals, and can continue to the microscopic level. Fractal geometry refers to this phenomenon as the coastline paradox.
Leaves have a similar branching structure to the trees of which they are a part. Sections of a leaf display this branching at ever-smaller scales. You can also aerially observe this geological branching in watersheds where smaller and smaller tributaries feed watercourses. You can find also find these quasi-self-similar fractals in a head of broccoli as you first observe the whole head, then the next iteration of florets that looks like a miniature version of the whole head, and the single floret that is also similar in morphology to the whole.
Two common geometric fractals that you can draw with a pencil, protractor and ruler are the Sierpinski triangle fractal and the Van Koch line fractal. The Sierpinski fractal is a series of ever-smaller equilateral triangles contained within one another. The Van Koch fractal is a simple line that you sequentially break into smaller equal segments constructed of three right angles. Teachers often use these simple geometric fractals in the classroom.
Adam Lerer, in his page "World of Fractals," lists 20 different kinds of computer-generated geometric fractals. With the incorporation of color, these often-complex fractal designs have become objects of art. The most well known of these computer-generated geometric fractal designs is the Mandelbrot fractal, designed by Benoit Mandelbrot, a world-renowned mathematician who died in October 2010 at the age of 85.