Hausdorff dimension

[2] For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3.

However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions.

More specifically, the Hausdorff dimension is a dimensional number associated with a metric space, i.e. a set where the distances between all members are defined.

, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.

For instance, the Koch snowflake shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new equilateral triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4.

[1] Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=SD.

The intuitive concept of dimension of a geometric object X is the number of independent parameters one needs to pick out a unique point inside.

However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information).

Every space-filling curve hits some points multiple times and does not have a continuous inverse.

A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region.

The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric.

For a sufficiently well-behaved X, the Hausdorff dimension is the unique number d such that N(r) grows as 1/rd as r approaches zero.

More precisely, this defines the box-counting dimension, which equals the Hausdorff dimension when the value d is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

But Benoit Mandelbrot observed that fractals, sets with noninteger Hausdorff dimensions, are found everywhere in nature.

He observed that the proper idealization of most rough shapes one sees 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.

[5]For fractals that occur in nature, the Hausdorff and box-counting dimension coincide.

[examples needed] The formal definition of the Hausdorff dimension is arrived at by defining first the d-dimensional Hausdorff measure, a fractional-dimension analogue of the Lebesgue measure.

has the construction of the Hausdorff measure where the covering sets are allowed to have arbitrarily large sizes (Here, we use the standard convention that

In other words, X and Y have the same underlying set of points and the metric dY of Y is topologically equivalent to dX.

These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.

If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies[12] This inequality can be strict.

Many sets defined by a self-similarity condition have dimensions which can be determined explicitly.

Then there is a unique non-empty compact set A such that The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of Rn with the Hausdorff distance.

[14] To determine the dimension of the self-similar set A (in certain cases), we need a technical condition called the open set condition (OSC) on the sequence of contractions ψi.

Suppose the open set condition holds and each ψi is a similitude, that is a composition of an isometry and a dilation around some point.

In general, a set E which is carried onto itself by a mapping is self-similar if and only if the intersections satisfy the following condition: where s is the Hausdorff dimension of E and Hs denotes s-dimensional Hausdorff measure.

This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally: Theorem.

Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.