Many of us have heard the term fractal used at one time or another, but what really is a fractal? It was not until 1975, when Benoit Mandelbrot published his book Fractals: Form, Chance and Dimension that he first coined the term fractal. His investigations centered around the fact that there are so many forms in nature that are so irregular or fragmented that the geometric world of Euclid cannot be used to describe them in their entirety. It should be noted that when Mandelbrot coined the term ‘fractal’ , which is from the Latin root fractus meaning and irregular surface like that of a broken bone. However, in his monumental work Mandelbrot refuses to give a precise definition or indication for its use. Despite this, he admits in that this has led to both advances in the field that he dreamed were never possible, as well as a great deal of confusion among physicists and raised suspicions among mathematicians. Perhaps a quote by Mandelbrot himself puts a unique perspective on the vast topic when he said, “Fractals provide a workable new middle ground between excessive geometric order of Euclid and the geometric chaos of roughness and fragmentation”.

Despite this there are many ways, in fact, to identify fractals. One of the key ideas in fractal geometry is the idea of self-similarity. Simply put, an object is self similar if it can be decomposed into smaller copies of itself. Formally in calculus, if you take successive highly magnified views of some curve, if it is continuous and differentiable, it can be viewed as a polygonal arc with infinitely small sides. For a basic curve, the fundamental building block is a collection of infinitely small lines. Most regular geometric objects and be constructed or measured with similar basic building blocks such as lines, squares and cubes. However, fractals usually encompass a different sort of situation. If you highly magnify natural objects such as coast lines, vegetation patches, or a graph of a fluctuating population, you will find that upon any power of magnification they are still as irregular as if viewed with no magnification. Therefore, fractal curves do not have the same idea of length as we commonly know it, they have instead a fractal; dimension, which is described under another section. The basic building block is not an infinitesimally small line, but instead an irregular form, usually an infinitesimally small copy of the whole object.
 
It is interesting to note here the conjugate approaches of fractal and Euclidean geometry. In one sense they are opposites of each other. Fractal geometry employs complex building blocks that use simple recursive or iterated patterns to create a complex object such as a fern. On the other hand, Euclidean geometry could also describe the structure of a fern, but it will use simpler building blocks such as lines, but require enormously complex processes to create the object.