Box Counting Method

TL;DR

• Divide an object into progressively smaller boxes and count how many boxes contain part of the object.
• Use the relationship between box count and box size to estimate the fractal dimension with a logarithmic formula.
• A measured value of 1 indicates a simple, straight form; a value greater than 1 indicates increased complexity.

Definition

Box counting is a mathematical method used to calculate the fractal dimension of a geometric object. It involves dividing the object into smaller and smaller boxes and counting the number of boxes that contain at least a part of the object. The fractal dimension is then calculated using a mathematical formula that relates the number of boxes to the size of the boxes.

Explanation

The procedure starts with boxes of a relatively large size and repeatedly decreases the box size. For each box size s, count the number N of boxes that contain at least part of the object. Repeat this process for a range of box sizes to obtain paired values of s and N. The fractal dimension is computed from the relationship between N and s using a logarithmic formula.

The fractal dimension quantifies the object’s complexity: a value of 1 indicates a simple, straight form, while values greater than 1 indicate more complex structure.

Examples

Coastline

To calculate the fractal dimension of a coastline, divide the coastline into smaller and smaller boxes, starting with boxes of a large size and gradually decreasing the size of the boxes. Count the number of boxes that contain at least a part of the coastline for each box size. Repeat this process multiple times to obtain a range of box sizes and corresponding counts.

The fractal dimension of the coastline can then be calculated using the following formula:

$$\text{Fractal Dimension} = \frac{\log(N)}{\log(1/s)}$$

A value of 1 indicates a simple, straight coastline, while a value greater than 1 indicates a more complex, rugged coastline.

Tree branch

To calculate the fractal dimension of a tree branch, divide the tree branch into smaller and smaller boxes, starting with boxes of a large size and gradually decreasing the size of the boxes. Count the number of boxes that contain at least a part of the tree branch for each box size. Repeat this process multiple times to obtain a range of box sizes and corresponding counts.

The fractal dimension of the tree branch can then be calculated using the same formula as above:

$$\text{Fractal Dimension} = \frac{\log(N)}{\log(1/s)}$$

A value of 1 indicates a simple, straight branch, while a value greater than 1 indicates a more complex, branching branch.

Use cases

• Can be applied to a wide range of objects, including coastlines, tree branches, and other natural and man-made structures.

Notes or pitfalls

• Interpreting results: for a coastline, a value of 1 indicates a simple, straight coastline and a value greater than 1 indicates a more complex, rugged coastline; for a tree branch, a value of 1 indicates a simple, straight branch and a value greater than 1 indicates a more complex, branching branch.

Related terms

• Fractal dimension