Value of Fractals

Like chaos, fractals generate controversy within the mathematics community (Pool 1990a). Some mathematicians charge that fractals lack real mathematical content, theorems, and proofs and that fractals emphasize pretty pictures or computer-generated designs. In addition, fractals, like chaos, don't provide physical explanations. Defenders rejoin that fractals are a new and efficient way of mathematically describing natural objects (e.g. Burrough 1984). That is, fractals are a mathematical way of describing the variability of irregular features over a range of scales.
Fractals have geometric features that make them useful as idealized models for our natural world. Examples include the human circulatory system (Barcellos 1984), soil pollution, cave systems, the seeping of oil through porous rock (Peterson 1988), drainage networks, and coral reefs. The general principle in such modeling usually is to add the same structure at progressively smaller scales.
Scientists see fractals partly as a vehicle for opening up a whole new realm of questions about the physical world. Why is an object fractal? What physical processes produce fractals? Why is the same scaling valid over a wide range of scales? Mathematicians can use fractals to develop an intuition for certain mathematical problems. Such intuition in turn brings new conjectures and new approaches to solving important theorems. Society in general (especially the television and motion picture industries) has taken a keen interest in fractals. Fractals, used with computer graphics, provide beautiful and realistic forgeries of many natural shapes, such as rocks, mountains, landscapes, stars, planets, and the like. The Star Trek television programs and motion pictures are a notable example of applying fractals in that way. Furthermore, fractals are easy and fun to generate, at least for someone who likes personal computers. Commercially available computer software programs show fractals in fascinating color and detail.
These and other important and practical applications came about only as unexpected by-products, after the concept of fractals had crystallized and been developed. That sort of thing happens quite often in the world of research. The applications show that basic research—research that may not promise any obvious benefit when first undertaken—can lead to practical and useful ends.