While the classical Euclidean geometry works with objects which exist in integer dimensions, fractal geometry deals with objects in non-integer dimensions. Euclidean geometry is a description lines, ellipses, circles, etc. Fractal geometry, however, is described in algorithims — a set of instructions on how to create a fractal.
The world as we know it is made up of objects which exist in integer dimensions, single dimensional points, one dimensional lines and curves, two dimension plane figures like circles and squares, and three dimensional solid objects such as spheres and cubes. However, many things in nature are described better with dimension being part of the way between two whole numbers. While a straight line has a dimension of exactly one, a fractal curve will have a dimension between one and two, depending on how much space it takes up as it curves and twists.
The more a fractal fills up a plane, the closer it approaches two dimensions. In the same manner of thinking, a wavy fractal scene will cover a dimension somewhere between two and three. Hence, a fractal landscape which consists of a hill covered with tiny bumps would be closer to two dimensions, while a landscape composed of a rough surface with many average sized hills would be much closer to the third dimension.
For the most part, when the word fractal is mentioned, you immediately think of the stunning pictures you have seen that were called fractals. But just what exactly is a fractal? Basically, it is a rough geometric figure that has two properties: First, most magnified images of fractals are essentially indistinguishable from the unmagnified version. This property of invariance under a change of scale if called self-similiarity. Second, fractals have fractal dimensions, as were described above. The word fractal was invented by Benoit Mandelbrot, “I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means to break to create irregular fragments. It is therefore sensible and how appropriate for our needs! – that, in addition to fragmented, fractus should also mean irregular, both meanings being preserved in fragment.”