Dimensão Fractal (box-count)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
About this calculator
The fractal dimension (box-count) calculator estimates the complexity of fractal patterns using the logarithmic formula log(N)/log(1/s). Here, a figure is covered with boxes of size s, where N represents the number of boxes intersecting the structure. This metric reflects how detail density changes with scale, commonly applied to natural patterns, digital images, or chaotic phenomena.
To apply the method, divide the image into grids with decreasing sizes and count how many boxes contain parts of the object in each division. The formula calculates the dimension by comparing how N increases proportionally as s decreases. Results near 1 suggest lines; values above 2 indicate complex 3D structures.
This measure is used in materials science, texture analysis, and ecological fractal studies. Caution: small variations in s selection can alter outcomes. Use logarithmic scales for consistency and validate with complementary methods like Hausdorff dimension for accuracy.
Frequently asked questions
Why test multiple scales?
Fractal dimension depends on N's variation with scale. Testing multiple scales ensures the average reflects the structure across different detail levels.
How accurate is this method?
Accuracy varies with iteration count and data shape. Results can be strengthened by averaging or comparing with other fractal methods.
When used in biology studies?
To analyze patterns like blood vessels, leaves, or corals in images, quantifying morphological complexity.
What does a dimension above 2 mean?
Suggests the structure has details across multiple dimensions, such as rough surfaces or complex porous 3D objects.
How to choose the initial box size?
Start with the largest possible s to capture broad contexts, gradually reducing until N changes minimally.