nLab Hausdorff dimension

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

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

References

Textbook accounts:

• John M. Mackay, Jeremy T. Tyson: Hausdorff measure and dimension, §1.4.1 in: Conformal Dimension: Theory and Application, AMS (2010) [pdf, ISBN-13:978-0-8218-5229-3]

See also:

• Wikipedia: Hausdorff dimension

• Encyclopedia of Mathematics: Hausdorff dimension