What is a fractal?

Objects with large surfaces

It is taken for granted that the surface or volume of a traditional geometrical object, for example a sphere or a cube, is well defined. Indeed, filling the object with smaller and smaller cubes leads to better and better approximations, and the total volume of the cubes converges to that of the object in question. It is well known that the surface, S, is proportional to the second, while the volume V is proportional to the third, power of the linear size, L, of the object. Consequently, the surface-to-volume ratio, S/V, is proportional to V−⅓. (For plane figures, the ratio of the perimeter, P, to the area, A, is proportional to A−½.) The surface-to-volume ratio is therefore finite and becomes smaller as the size becomes larger. This is why surface phenomena are of little importance compared with volume phenomena for macroscopic systems of traditional geometry.

On the other hand, it is known that there exist macroscopic objects with large surface area. These are always porous, with ramified or pitted surfaces. Effective chemical catalysts, for example, must have a large surface. The need for rapid gas exchange accounts for the large surface-to-volume ratio of the respiratory organs. The surface area of the human lungs (measured at microscopic resolution), for example, is the same as that of a tennis court (approximately 100 m2), while the volume is only a few litres (10−3 m3).