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## WHAT IS IT?

This model investigates the box-counting dimension for a variety of fractals.

## HOW TO USE IT

The interface consists of three blocks: Fractal Examples, Box-Counting Controls and Plots/Results. Start by selecting one of the fractal examples, e.g., "Koch Curve". Then click "Iterate". Notice at the first iteration the Hausdorff dimension (1.262 for the Koch Curve) appears in a monitor on the right. Click "Iterate" a few more times to get a developed fractal. Now you are ready to apply box-counting to this fractal.

Under Box-Counting Controls, set "initial box length" (i.e., length of side of each box) and "increment" (i.e., amount that box length increases per iteration of box counting). Press "Box Counting Setup" and then "Box Counting Go" and you should see small red boxes covering the fractal. After the fractal has been completely covered, the log of number of boxes will be plotted versus the log of 1/box-length on the plot on the right. These iterations will continue until you press "Box Counting Go" again to stop the process. At that point you can press "linear-regression" on the right side (under the plot) to fit a line to the plotted points. The slope of this line is the box-counting dimension measured by your run; it appears in the monitor on the right, and can be compared with the Hausdorff dimension.

## THINGS TO NOTICE

The box-counting dimension becomes a more accurate approximation of Hausdorff dimension as the number of iterations increase. The accuracy also can depend on the initial settings for box-length and increment (you should experiment with all of these parameters). However, you have probably noticed the iterations taking longer and longer to generate for each successive iteration. This is because the number of line segments is increasing exponentially. The Koch curve, for instance, begins with four line segments but has over a million line segments after ten iterations.

## CREDITS AND REFERENCES

This model is part of the Fractals series of the Complexity Explorer project.

Main Author: John Driscoll

Contributions from: Vicki Niu, Melanie Mitchell

Netlogo: Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

## HOW TO CITE

If you use this model, please cite it as: "Box Counting Dimension" model, Complexity Explorer project, http://complexityexplorer.org

## COPYRIGHT AND LICENSE

Copyright 2016 Santa Fe Institute.

This model is licensed by the Creative Commons Attribution-NonCommercial-ShareAlike International (http://creativecommons.org/licenses/). This states that you may copy, distribute, and transmit the work under the condition that you give attribution to ComplexityExplorer.org, and your use is for non-commercial purposes.

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