box-counting dimension

In chaos theory the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points. For example, if we have a set of random points on the real number line between 0 and 1, the correlation dimension will be ν=1, while if they are distributed on say, a triangle embedded 3-space (or N-space, for that matter), the correlation dimension will be ν=2. This is what we would intuitively expect from a measure of dimension. The real utility of the correlation dimension is in determining ...

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Intuitively, the dimension of a set (for example, a subset of Euclidean space) is the number of independent parameters needed to describe a point in the set. One mathematical concept which closely models this naïve idea is that of topological dimension of a set. For example a point in the plane is described by two independent parameters (the cartesian coordinates of the point), so in this sense, the plane is two-dimensional. As one woul ...

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Hausdorff dimension, Hausdorff dimension - Informal discussion, Hausdorff dimension - Formal definition, Hausdorff dimension - Results, Hausdorff dimension - Examples, Hausdorff dimension - Hausdorff dimension and topological dimension, Hausdorff dimension - Self-similar sets, Hausdorff dimension - Historical references

Read more here: » Hausdorff dimension: Encyclopedia II - Hausdorff dimension - Informal discussion

Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set E is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(E) = E, although the exact definition is given below. The following is Theorem 8.3 of the Falconer reference below: Theorem. Suppose are contractive mappings on Rn with contraction constant rj < 1. Then there is a unique non-empt ...

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Hausdorff dimension, Hausdorff dimension - Informal discussion, Hausdorff dimension - Formal definition, Hausdorff dimension - Results, Hausdorff dimension - Examples, Hausdorff dimension - Hausdorff dimension and topological dimension, Hausdorff dimension - Self-similar sets, Hausdorff dimension - Historical references

Read more here: » Hausdorff dimension: Encyclopedia II - Hausdorff dimension - Self-similar sets

The Hausdorff outer measure Hs is defined for all subsets of X. However, we can in general assert additivity properties, that is for disjoint A, B, only when A and B are both Borel sets. From the perspective of assigning measure and dimension to sets with unusual metric properties such as fractals, however, this is not a restriction. Theorem. Hs is a metric outer measure. Thus all Borel subsets of X are measurable and Hs is a countably ad ...

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Hausdorff dimension, Hausdorff dimension - Informal discussion, Hausdorff dimension - Formal definition, Hausdorff dimension - Results, Hausdorff dimension - Examples, Hausdorff dimension - Hausdorff dimension and topological dimension, Hausdorff dimension - Self-similar sets, Hausdorff dimension - Historical references

Read more here: » Hausdorff dimension: Encyclopedia II - Hausdorff dimension - Results

The Hausdorff dimension gives an accurate way to measure the dimension of an arbitrary metric space; this includes complicated sets such as fractals. Suppose (X,d) is a metric space. As mentioned in the introduction, we are interested in counting the number of balls of some radius necessary to cover a given set. It is possible to try to do this directly for many sets (leading to so-called box counting dimension), but Hausdorff's insight was to approach the problem indirectly using the theory of measure developed earlier ...

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Hausdorff dimension, Hausdorff dimension - Informal discussion, Hausdorff dimension - Formal definition, Hausdorff dimension - Results, Hausdorff dimension - Examples, Hausdorff dimension - Hausdorff dimension and topological dimension, Hausdorff dimension - Self-similar sets, Hausdorff dimension - Historical references

Read more here: » Hausdorff dimension: Encyclopedia II - Hausdorff dimension - Formal definition

Let X be an arbitrary separable metric space. There is a notion of topological dimension for X which is defined recursively. It is always an integer (or +∞) and is denoted dimtop(X). Theorem. Suppose X is non-empty. Then Moreover where Y ranges over metric spaces homeomorphic to X. In other words, X and Y have the same underlying set of points and the metric d< ...

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Hausdorff dimension, Hausdorff dimension - Informal discussion, Hausdorff dimension - Formal definition, Hausdorff dimension - Results, Hausdorff dimension - Examples, Hausdorff dimension - Hausdorff dimension and topological dimension, Hausdorff dimension - Self-similar sets, Hausdorff dimension - Historical references

Read more here: » Hausdorff dimension: Encyclopedia II - Hausdorff dimension - Hausdorff dimension and topological dimension