The boxcounting dimension [1–4] can be defined as , where ( ) is an initial box (or grid) size, is a natural number representing the boxcounting 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 onedimensional (1D) boxcounting steps as a stack diagram. The test map used in this Demonstration is the wellknown 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 [1]. [2] B. Mandelbrot, The Fractal Geometry of Nature, San Francisco: W. H. Freeman, 1982. [3] H.O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science, 2nd ed., New York: Springer, 2004. [4] S. Wolfram, A New Kind of Science, Champaign, IL: Wolfram Media, 2002. [5] R. May, "Simple Mathematical Models with Very Complicated Dynamics," Nature, 261(5560), 1976 pp. 459–467. [6] M. J. Feigenbaum, "Quantitative Universality for a Class of NonLinear Transformations," Journal of Statistical Physics, 19, 1978 pp. 25–52. [7] M. J. Feigenbaum, "The Universal Metric Properties of Nonlinear Transformations," Journal of Statistical Physics, 21, 1979 pp. 669–706.
