Fractal geometry is a mathematical theory devoted to the study of complex shapes called fractals. Although an exact definition of fractals has not been established, fractals commonly exhibit the property of self-similarity: the reiteration of irregular details or patterns at progressively smaller scales so that each part, when magnified, looks basically like the object as a whole. The Koch curve is a significant fractal in mathematics and examining it provides some insight into fractal geometry. To generate the Koch curve, one begins with a straight line. The middle third of the line is removed and replaced with two line segments, each as long as the removed piece, which are positioned so as to meet and form the top of a triangle. At this stage, the curve consists of four connected segments of equal length that form a pointed protrusion in the middle. This process is repeated on the four segments so that all the protrusions are on the same side of the curve, and then the process is repeated indefinitely on the segments at each stage of the construction.

Self-similarity is built into the construction process by treating segments at each stage the same way as the original segment was treated. Since the rules for getting from one stage to another are fully explicit and always the same, images of successive stages of the process can be generated by computer. Theoretically, the Koch curve is the result of infinitely many steps in the construction process, but the finest image approximating the Koch curve will be limited by the fact that eventually the segments will get too short to be drawn or displayed. However, using computer graphics to produce images of successive stages of the construction process dramatically illustrates a major attraction of fractal geometry: simple processes can be responsible for incredibly complex patterns.

A worldwide public has become captivated by fractal geometry after viewing astonishing computer-generated images of fractals; enthusiastic practitioners in the field of fractal geometry consider it a new language for describing complex natural and mathematical forms. They anticipate that fractal geometry's significance will rival that of calculus and expect that proficiency in fractal geometry will allow mathematicians to describe the form of a cloud as easily and precisely as an architect can describe a house using the language of traditional geometry. Other mathematicians have reservations about the fractal geometers' preoccupation with computer-generated graphic images and their lack of interest in theory. These mathematicians point out that traditional mathematics consists of proving theorems, and while many theorems about fractals have already been proven using the notions of pre-fractal mathematics, fractal geometers have proven only a handful of theorems that could not have been proven with pre-fractal mathematics. According to these mathematicians, fractal geometry can attain a lasting role in mathematics only if it becomes a precise language supporting a system of theorems and proofs.