International Encyclopedia of Systems and Cybernetics

The concept of fractalization is now unifying and making clearer a number of geometrical paradoxes which appeared since the end of the 19th Century. All had a common feature: the emergence of fractionary dimensions.

1. The CANTOR set: The removal of the central third of a segment, and the reiteration of the same operation on the remaining parts produced a set of infinitely many points, but their total length at infinity is zero.

2. The von KOCH curve: In this case, starting with a equilateral triangle, constructing on the central third of each side a new equilateral triangle and iterating this process, an apparently infinitely long perimeter surrounds a finite area.

3. The SIERPINSKI carpet: "is constructed by cutting the center one-ninth of a square; then cutting out the centers of the eight smaller squares, and so on" (J. GLEICK, 1987, p.101). The limit, which would imply the complete disappearance of the carpet can obviously never be reached.

4. The MENGER sponge: "is the three dimensional analogue (of the SIERPINSKI carpet)… a solid lattice that has an infinite surface area, and yet zero volume" (p.101).

While these constructs are abstract, they are nonetheless extremely useful to understand natural structures as diverse as a rocky or sandy coast, the branching of trees and the construction of leaves, the enormous surface of the human lungs, the Brownian motion of molecules and, possibly, the distribution of galaxies in the cosmos. From an epistemological viewpoint, one should however be aware of the paradoxes of infinity… which are actually as old of Zenon and his arrow.