10176

# Box-Counting the Dimension of Coastlines

This Demonstration uses the box-counting method [1] to estimate the fractal dimension of three coastlines. The box-counting dimension of a coastline can be computed using a mesh of evenly spaced squares laid over a map of the coastline.
Let be the number of squares that are required to cover the whole coastline and be the sidelength of one square of the mesh. Then the box-counting dimension is .
can be interpreted as the slope of a straight line in a plot of versus . To approximate this, take five points determined by five values of and compute the slope of the linear fit between those points. Smaller values for and higher map resolutions lead to a more accurate estimate, as can be verified by selecting a finer mesh from the menu. It has been computed that the box-counting dimension of the coastline of Great Britain is 1.25 by using a very fine mesh on a very detailed map. This Demonstration finds values up to 1.18 using mesh sizes and a map resolution compatible with a reasonable computing time.
A detailed description of the box-counting method used in this Demonstration can be found in [2].

### DETAILS

References
[1] V. Kaurov. "Measuring Fractal Dimension of Natural Object from Digital Images." Mathematica Stack Exchange. (Oct 16, 2012) mathematica.stackexchange.com/questions/13125/measuring-fractal-dimension-of-natural-objects-from-digital-images.
[2] M. Mitchell et al. "Introduction to Complexity: 3.6 Box-Counting Dimension." Complexity Explorer. (2013) www.complexityexplorer.org/online-courses/3.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.