Hausdorff Dimension - Intuition

Intuition

The intuitive dimension of a geometric object is the number of independent parameters you need to pick out a unique point inside. But you can easily take a single real number, one parameter, and split its digits to make two real numbers, and encode any two real numbers in this way (up to some technicalities). The example of a space-filling curve shows that you can even take one real number into two both surjectively (so all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher dimensional object.

Every space filling curve hits some points multiple times, and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension explains why. The Lebesgue covering dimension is defined as the minimum number of overlaps that small open balls need to have in order to completely cover the object. When you try to cover a line by dropping open intervals on it, you always end up covering some points twice. Covering a plane with disks, you end up covering some points three times, etc. The topological dimension tells you how many different little balls connect a given point to other points in the space, generically. It tells you how difficult it is to break a geometric object apart into pieces by removing slices.

But the topological dimension doesn't tell you anything about volumes. A curve which is almost space filling can still have topological dimension one, even if it fills up most of the area of a region. A fractal has an integer topological dimension, but in terms of the amount of space it takes up, it behaves as a higher dimensional space. The Hausdorff dimension defines the size notion of dimension, which requires a notion of radius, or metric.

Consider the number N(r) of balls of radius at most r required to cover X completely. When r is small, N(r) is large. For a "well-behaved" set X, the Hausdorff dimension is the unique number d such that N(r) grows as 1/rd as r approaches zero. The precise definition requires that the dimension "d" so defined is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer. But Benoît Mandelbrot observed that fractals, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:

clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

The Hausdorff dimension is a successor to the less sophisticated but in practice very similar box-counting dimension or Minkowski–Bouligand dimension. This counts the squares of graph paper in which a point of X can be found as the size of the squares is made smaller and smaller. For fractals that occur in nature, the two notions coincide. The packing dimension is yet another similar notion. These notions (packing dimension, Hausdorff dimension, Minkowski–Bouligand dimension) all give the same value for many shapes, but there are well documented exceptions.