We have already seen some pictures of highly irregular geometric objects (for instance, the dragon curve, the Koch curve, the branched fractal, and the bush.) that are commonly called fractals. The term was invented by Mandelbrot in recognition of his fundamental insight that this kind of geometry is a truer reflection of the geometry of nature than Euclid's geometries. Mountains, clouds, river systems, trees, etc., all have fractal shapes. There is a measurement of the complexity of a geometric object which we will commonly call the ``fractal dimension,'' although there are several varying concepts and terms, the big brother of which is the Hausdorff-Besicovitch dimension. The fractal dimension is always a nonnegative real number. There is a more familiar notion of dimension called the topological dimension which is always a nonnegative integer. Continuous non-self-intersecting curves always have topological dimension 1, and smooth surfaces always have dimension 2, for instance. The fractal dimension is always greater than or equal to the topological dimension, and Mandelbrot defines a fractal set as one in which the fractal dimension is strictly greater than the topological dimension. In this part of the course, we shall try to give at least an intuitive idea of what topological and fractal dimension mean.
The fractal dimension alone does not give an idea of what ``fractals'' are really about. Mandelbrot founded his insights in the idea of self-similarity, requiring that a true fractal ``fracture'' or break apart into smaller pieces that resemble the whole. This is perhaps a special case of the idea that there should be a dynamical system underlying the geometry of the set. This is partly why the idea and language of fractals has become so popular throughout science; it is a fundamental goal of science to seek to understand the underlying dynamics of any natural phenomenon. It has now become apparent that relatively simple dynamics can produce the fantastically intricate shapes and behavior that occur throughout nature. Therefore, before moving on to dimension, we shall first give a whirlwind description of the basic terminology of dynamical systems, leading up to the notion of an attractor.