International Encyclopedia of Systems and Cybernetics

G. MAYER-KRESS writes:" A very important distinction between the mathematical and the physical concepts of dimension is that mathematically the dimension is invariant under smooth transformations like deformations and changes of the coordinate system. If we take a ball of dough and roll it out to form a one micrometer thick pancake, then in a mathematical sense these are both three-dimensional objects. From a physicist's point of view, however, it might be important that the pancake can be treated as a two dimensional object. This apparent change of the dominant dimension plays a significant role in dynamical systems and fractal chaos.

"Dimensional arguments in physics have let to profound discoveries in, for example, turbulence, in phase transition theory and elementary particle physics, as well as in developing renormalization theory" (1988, p.357).

Of course, physical fractalization ad infinitum is impossible, because there are changes in the nature of the elements. Parts of a complex system may be fractal, but each subpart is generally itself another complex system, which in turn can be only partly fractal. At each level, the parts are of different kinds, for example cells, molecules, atoms, etc.