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Hausdorff dimension

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Idea

Hausdorff dimension is a method of measuring the dimension of a metric space. It is always a non-negative real number, but it need not be an integer; one way to define a fractal is a metric space with a fractional (non-integral) Hausdorff dimension. Hence Hausdorff dimension is an example of fractal dimension.

The Hausdorff dimension of the cartesian space n\mathbb{R}^n (or any inhabited open subset thereof) is nn. The Hausdorff dimension of a self-similar fractal which consists of nn copies of itself reduced in size by a factor of mm is log mn\log_m n.

In general, Hausdorff dimension may be defined using Hausdorff measure?.

References

Last revised on May 15, 2017 at 10:48:00. See the history of this page for a list of all contributions to it.