The box-counting dimension [1–4] can be defined as
where () is an initial box (or grid) size, is a natural number representing the box-counting step, is the box size for the scaling step, and is the number of boxes with the same size . It can be applied to any fractal, including wild fractals such as the Brownian motion. This Demonstration visualizes one-dimensional (1D) box-counting steps as a stack diagram.
The test map used in this Demonstration is the well-known logistic map [3–7] , where is an iteration number, is the iterate starting from an initial condition , and is a control parameter value; has been fixed at 5.00001, and for imitating attractors, 4000 iterates are selected from to .
Box counting is a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box" shaped, and analyzing the pieces at each smaller scale .