There are several motivations to study fractals. Before entering into any sort of discussion on the underlying motivations though, it is best to define (roughly!) what sort of thing you could call a fractal in the first place. Many texts, people - schools of thought one might say - disagree subtly as to what exactly a fractal is. This discussion will not be included in the book, but I have included several possible definitions here to give you an idea of a fractal and its behavior:
- A set which can be viewed at infinite resolution without loss of detail.
- You can never zoom in far enough to stop seeing the pattern.
- A set which is self-similar, or exhibits self-similarity.
- You can see little copies of a large part of the pattern in smaller sectors.
- A set which can be constructed geometrically by an infinite number of iterations following a simple rule.
- There's a process that can be iterated on a simple set [for brevity, read shape] an infinite number of times to create a fractal set.
As an aside, most interesting fractals have one or more of the above properties, and other closely related properties too.
The motivations as described earlier are a hundredfold or more in number, but basically come down to two broad themes depending on your approach to the mathematics world; they are studied for their applicability and their elegance. The pure mathematicians endeavor to seek ever more beautiful mathematical structures and use analytical techniques to describe their "nice" behaviour and appearance whereas those of a more applied bent (not just mathematicians - also statisticians, and physicists; more recently chemists too) appreciate the way that fractals adequately describe some particularly hard-to-model processes quite nicely. It's quite ironic the way fractals have broadly demonstrated the main two schools of mathematics by their very existence as a subject area - see pure and applied mathematics in Wikipedia.
Another (less rigorous) theme is "Fractals as Artistic Objects." The act of exploring the infinite visual worlds encompassed by fractals can be a very pleasurable activity. Searching for interesting fractals, framing pleasing features within them, and coloring and/or shading them -- as well as sharing the results of this search -- is quite rewarding for many people.
The aim of this text is to build the above definitions into an informal, light introduction to the world of fractal geometry and to inspire further research into the subject, whether your interest is of a pure, applied or even recreational nature. Technical issues will be treated in limited detail, but the mechanics and more interesting behaviours of the subject matter will be examined much more deeply. The intent with such issues is to treat them as separate to the main text in a series of appendices, so that the information is available for those who wish to use it but so that the main text doesn't befuddle the more casual reader.