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Hausdorff dimension




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.


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   • FRACTALS AND PATHOLOGICAL CURVES