Box-counting dimension

The box-counting dimension is defined for a bounded set S in a metric space, typically Euclidean space. For a given box size ε, let N(ε) be the minimum number of boxes of side length ε needed to cover S. The dimension is the limit, if it exists, of log N(ε) / log(1/ε) as ε approaches zero.¹ In practice, one computes N(ε) for a sequence of decreasing ε values and estimates the slope of the log-log plot of N(ε) versus 1/ε. This method is robust and easy to implement, making it popular in image analysis and signal processing.²

The box-counting dimension is always at least as large as the Hausdorff dimension, and for many self-similar fractals the two coincide.³ However, there are sets where the box-counting dimension is strictly larger, such as the set of rational numbers in the unit interval, which has box-counting dimension 1 but Hausdorff dimension 0. The box-counting dimension is not countably stable, meaning that the dimension of a countable union of sets can exceed the supremum of the individual dimensions. It is also not invariant under bi-Lipschitz transformations, unlike the Hausdorff dimension, which limits its geometric invariance.⁴

Box-counting dimension is used to characterize the complexity of natural and man-made objects. In medicine, it quantifies the branching of blood vessels and the roughness of tissue surfaces, aiding in diagnosis.⁵ In materials science, it measures the fractal nature of fracture surfaces and porous media. In signal analysis, it helps distinguish noise from deterministic signals. The dimension of a coastline, famously estimated by Mandelbrot, is a classic example: the west coast of Britain has a box-counting dimension of about 1.25, reflecting its intricate irregularity.⁶

One subtlety is that the box-counting dimension can depend on the choice of box shape (e.g., squares vs. disks), though for most sets the limit is the same. Another is that the dimension can be defined using the upper and lower limits, leading to upper and lower box dimensions, which may differ. A little-known fact is that the box-counting dimension of the graph of a continuous function is related to its Hölder exponent: if a function is Hölder continuous with exponent α, then the box dimension of its graph is at most 2 - α. Also, the box-counting dimension of the set of points where a Brownian motion in 2D visits is 2, but the dimension of the path itself is 1.5, a distinction that highlights the difference between the set and its trace.⁷

Hausdorff dimension

A measure-theoretic fractal dimension that is often smaller than or equal to the box-counting dimension.

Self-similar fractal

A set that is exactly or approximately similar to a part of itself, such as the Koch snowflake.

Hölder exponent

A measure of the regularity of a function; a function with exponent α satisfies |f(x)-f(y)| ≤ C|x-y|^α.