Stack Diagram for 1D Box-Counting Steps

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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 .


Contributed by: Ki-Jung Moon (December 2013)
Open content licensed under CC BY-NC-SA



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].


[1] Wikipedia. "Box Counting." (Feb 1, 2013)

[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 Non-Linear 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.

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