Hausdorff Dimension - Examples

Examples

• The Euclidean space has Hausdorff dimension n.
• The circle S1 has Hausdorff dimension 1.
• Countable sets have Hausdorff dimension 0.
• Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension. For example, the Cantor set (a zero-dimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff dimension is which is approximately The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of, which is approximately .
• Space-filling curves like the Peano and the Sierpiński curve have the same Hausdorff dimension as the space they fill.
• The trajectory of Brownian motion in dimension 2 and above has Hausdorff dimension 2 almost surely.
• An early paper by Benoit Mandelbrot entitled How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension and subsequent work by other authors have claimed that the Hausdorff dimension of many coastlines can be estimated. Their results have varied from 1.02 for the coastline of South Africa to 1.25 for the west coast of Great Britain. However, 'fractal dimensions' of coastlines and many other natural phenomena are largely heuristic and cannot be regarded rigorously as a Hausdorff dimension. It is based on scaling properties of coastlines at a large range of scales; however, it does not include all arbitrarily small scales, where measurements would depend on atomic and sub-atomic structures, and are not well defined.
• The bond system of an amorphous solid changes its Hausdorff dimension from Euclidian 3 below glass transition temperature T_g (where the amorphous material is solid), to fractal 2.55±0.05 above T_g, where the amorphous material is liquid.