Rulers and Boxes

Our measuring or scaling tool in all the above examples was one-dimensional—a ruler or yardstick. Any other object of known size also can be a suitable standard. Common examples are squares, circles, boxes, discs, or spheres. Each has a characteristic diameter or radius. Thus, we can measure the length of a coastline by using squares or boxes instead of a ruler to cover the coastline. A related popular method is to form a grid of squares covering the entire coastline and to count the number of squares that contain any part of the coastline (e.g. Feder 1988). Longley & Batty (1989) describe techniques that are more sophisticated.
Of the measuring devices just mentioned, early chaologists liked the box (which they imaginarily extended to more than three dimensions if necessary). You'll often see the similarity dimension as determined with a grid or lattice of boxes called the box dimension or box-counting dimension.

The similarity or box-counting dimension is somewhat impractical for most strange attractors. The reason is that we need an unrealistically large dataset to make sure that some phase space regions are really empty rather than just visited rarely.