Minkowski–Bouligand dimension

, imagine this fractal lying on an evenly spaced grid and count how many boxes are required to cover the set.

The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a box-counting algorithm.

Roughly speaking, this means that the dimension is the exponent

is a smooth space (a manifold) of integer dimension

Only in very special applications is it important to distinguish between the three (see below).

is the minimal number of open balls of radius

, which is defined the same way but with the additional requirement that the centers of the open balls lie in the set S. The packing number

is the maximal number of disjoint open balls of radius

are not exactly identical, they are closely related to each other and give rise to identical definitions of the upper and lower box dimensions.

This is easy to show once the following inequalities are proven:

These, in turn, follow either by definition or with little effort from the triangle inequality.

The advantage of using balls rather than squares is that this definition generalizes to any metric space.

In other words, the box definition is extrinsic – one assumes the fractal space S is contained in a Euclidean space, and defines boxes according to the external geometry of the containing space.

One defines an internal ball as all points of S within a certain distance of a chosen center, and one counts such balls to get the dimension.

(More precisely, the Ncovering definition is extrinsic, but the other two are intrinsic.)

The advantage of using boxes is that in many cases N(ε) may be easily calculated explicitly, and that for boxes the covering and packing numbers (defined in an equivalent way) are equal.

The logarithm of the packing and covering numbers are sometimes referred to as entropy numbers and are somewhat analogous to the concepts of thermodynamic entropy and information-theoretic entropy, in that they measure the amount of "disorder" in the metric space or fractal at scale ε and also measure how many bits or digits one would need to specify a point of the space to accuracy ε.

Another equivalent (extrinsic) definition for the box-counting dimension is given by the formula

is defined to be the r-neighborhood of S, i.e. the set of all points in

is the union of all the open balls of radius r which have a center that is a member of S).

However, it is not countably stable, i.e. this equality does not hold for an infinite sequence of sets.

The Hausdorff dimension by comparison, is countably stable.

The lower box dimension, on the other hand, is not even finitely stable.

If A and B are two sets in a Euclidean space, then A + B is formed by taking all the pairs of points a, b where a is from A and b is from B and adding a + b.

For many well behaved fractals all these dimensions are equal; in particular, these dimensions coincide whenever the fractal satisfies the open set condition (OSC).

The upper box dimension may be bigger than the lower box dimension if the fractal has different behaviour in different scales.

For example, examine the set of numbers in the interval [0, 1] satisfying the condition The digits in the "odd place-intervals", i.e. between digits 22n+1 and 22n+2 − 1 are not restricted and may take any value.

This fractal has upper box dimension 2/3 and lower box dimension 1/3, a fact which may be easily verified by calculating N(ε) for

and noting that their values behave differently for n even and odd.

These examples show that adding a countable set can change box dimension, demonstrating a kind of instability of this dimension.