Fractal

Hausdorff dimension

The new approach of the notion of dimension can be defined rigorously. Given two real numbers and , one can overlap the studied set by sphere of diameter , all lower that (see the figure), then one calculate the limit :

M_{D}(E) \propto \lim \limits_{\varepsilon \rightarrow 0} inf \sum \limits_{k} \varepsilon_{k}^{D}

The limit when tends to 0 of the minorant, corresponds to all spheres of the sum of power of

spherical overlapping of the Von Kock curve [Takayasu, 1990]

One can show there exist one value of such as:

M_{D}(E) = 0 \quad if \quad D > D_{H} \\ M_{D}(E) = \infty \quad if \quad D < D_{H}

has a finite value for .

is the Hausdorff dimension of the set .

PreviousPreviousNextNext
HomepageHomepagePrintPrintCreated with Scenari (new window)