Fractals

"The fractal geometry of nature"-Benoit Mandelbrot

Definition of fractal

A fractal is by definition a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension} \begin{equation*} D>D_{T} \end{equation*}

Effective Dimension

Effective dimension has a subjective basis (dimensions of veil thread ball as studied by physicists). It is a matter of approximation and, therefore of degree of resolution.

Certain ill-defined transitions between zones of well-defined dimension are reinterpreted as being fractal zones within which \( D>D_{T} \)

Spatial Homogeneity, scaling and self similarity

Homogeneous distribution on a line, plane or space has two very desirable properties. It is invariant under displacement, and it is invariant under change of scale.