# Hausdorff dimension

A Hausdorff dimension is a way to accurately measure the dimension of complicated sets such as fractals.
The Hausdorff dimension, named after Felix Hausdorff,
coincides with the more familiar notion of dimension in the case of well-behaved
sets. For example a straight line or an ordinary curve, such as a circle,
has a Hausdorff dimension of 1; any countable set has a Hausdorff dimension
of 0; and an *n*-dimensional Euclidean
space has a Hausdorff dimension of *n*. But a Hausdorff dimension
is not always a natural number. Think
about a line that twists in such a complicated way that it starts to fill
up the plane. Its Hausdorff dimension increases beyond 1 and takes on values
that get closer and closer to 2. The same idea of ascribing a fractional
dimension applies to a plane that contorts more and more in the third dimension:
its Hausdorff dimension gets closer and closer to 3. As a specific example,
the fractal known as the Sierpinski
carpet has a Hausdorff dimension of just over 1.89.