Fractal dimension

The term "fractal," introduced in 1975 by Benoit Mandelbrot, is an abbreviation for "fractional dimension." We all learned in high school that in classical geometry a line is an one dimensional object and a plane is two dimensional. Strangely, if we put enough kinks in a line, the resulting fractal curve will have a dimension somewhere between one and two, so that it is neither a line nor a plane but something in between. Similarly, an extremely convoluted surface will have a dimension beween two and three. Such a figure is called a fractal. For objects of classical geomery, the classical dimension of the object and its fractal dimension are the same. A fractal, on the other hand, is an object that has a fractal dimension that is strictly greater than its classical dimension. Although they are continuous, fractal curves are so rough that they are nowhere differentiable. The concept of fractal dimension provides a way to measure how rough fractal curves are. The more jagged and irregular a curve is, the higher its fractal dimension, a value betwen one and two. Fractional dimension is related to self-similarity in that the easiest way to create a figure that has fractional dimension is through self-similarity.