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== '''[[ | == '''[[Hausdorff dimension]]''' == | ||
''by [[User: | ''by [[User:Melchior Grutzmann|Melchior Grutzmann]] (and [[User:Brandon Piercy|Brandon Piercy]] and [[User:Hendra I. Nurdin|Hendra I. Nurdin]]) | ||
---- | ---- | ||
In [[mathematics]], the '''Hausdorff dimension''' is a way of defining a possibly fractional exponent for all figures in a [[metric space]] such that the dimension describes partially the amount to that the set fills the space around it. For example, a [[plane (geometry)|plane]] would have a Hausdorff dimension of 2, because it fills a 2-parameter subset. However, it would not make sense to give the [[Sierpiński triangle]] [[fractal]] a dimension of 2, since it does not fully occupy the 2-dimensional realm. The Hausdorff dimension describes this mathematically by measuring the size of the set. For self-similar sets there is a relationship to the number of self-similar subsets and their scale. | |||
=== Informal definition === | |||
Intuitively, the dimension of a set is the number of independent parameters one has to pick in order to fix a point. This is made rigorously with the notion of ''d''-dimensional (topological) [[manifold]] which are particularly regular sets. The problem with the classical notion is that you can easily break up the digits of a real number to map it bijectively to two (or ''d'') real numbers. The example of space filling curves shows that it is even possible to do this in a continuous (but non-bijective) way. | |||
The | The notion of Hausdorff dimension refines this notion of dimension such that the dimension can be any non-negative number. | ||
Benoît Mandelbrot discovered<ref>B.B. Mandelbrot: ''The fractal geometry of nature'', Freemann '''(1983)''', ISBN 978-0-716-711-865</ref> that many objects in nature are not strictly classical smooth bodies, but best approximated as fractal sets, i.e. subsets of '''R'''<sup>''N''</sup> whose Hausdorff dimension is strictly greater than its topological dimension. | |||
=== | === Hausdorff measure and dimension === | ||
Let ''d'' be a non-negative real number and ''S'' ⊂ ''X'' a subset of a metric space (''X'',''ρ''). The ''d''-dimesional Hausdorff measure of scale ''δ''>0 is | |||
:<math> H^{d*}_\delta(S) := \inf \{\sum_{i=1}^\infty r_i^d : S\subset\bigcup_{i=1}^\infty B_{r_i}(x_i), r_i\le\delta \}</math> | |||
where B<sub>''r''<sub>''i''</sub>(''x''<sub>''i''</sub>) is the open ball around ''x''<sub>''i''</sub> ∈ ''X'' of radius ''r''<sub>''i''</sub>. The ''d''-dimensional Hausdorff measure is now the limit | |||
:<math> H^{d*}(S) := \lim_{\delta\to0+} H^{d*}_\delta(S)</math>. | |||
As in the Carathéodory construction a set ''S'' ⊂ ''X'' is called ''d''-measurable iff | |||
:<math> H^{d*}(T) = H^{d*}(S\cap T)+ H^{d*}(T\cap X\setminus S)</math> for all ''T'' ⊂ ''X''. | |||
A set ''S'' ⊂ ''X'' is called Hausdorff measurable if it is H<sup>''d''</sup>-measurable for all ''d''≥0. | |||
''[[Hausdorff dimension|.... (read more)]]'' | |||
''[[ | |||
{| class="wikitable collapsible collapsed" style="width: 90%; float: center; margin: 0.5em 1em 0.8em 0px;" | {| class="wikitable collapsible collapsed" style="width: 90%; float: center; margin: 0.5em 1em 0.8em 0px;" | ||
Revision as of 07:30, 20 January 2012
Hausdorff dimension
by Melchior Grutzmann (and Brandon Piercy and Hendra I. Nurdin)
In mathematics, the Hausdorff dimension is a way of defining a possibly fractional exponent for all figures in a metric space such that the dimension describes partially the amount to that the set fills the space around it. For example, a plane would have a Hausdorff dimension of 2, because it fills a 2-parameter subset. However, it would not make sense to give the Sierpiński triangle fractal a dimension of 2, since it does not fully occupy the 2-dimensional realm. The Hausdorff dimension describes this mathematically by measuring the size of the set. For self-similar sets there is a relationship to the number of self-similar subsets and their scale.
Informal definition
Intuitively, the dimension of a set is the number of independent parameters one has to pick in order to fix a point. This is made rigorously with the notion of d-dimensional (topological) manifold which are particularly regular sets. The problem with the classical notion is that you can easily break up the digits of a real number to map it bijectively to two (or d) real numbers. The example of space filling curves shows that it is even possible to do this in a continuous (but non-bijective) way.
The notion of Hausdorff dimension refines this notion of dimension such that the dimension can be any non-negative number.
Benoît Mandelbrot discovered[1] that many objects in nature are not strictly classical smooth bodies, but best approximated as fractal sets, i.e. subsets of RN whose Hausdorff dimension is strictly greater than its topological dimension.
Hausdorff measure and dimension
Let d be a non-negative real number and S ⊂ X a subset of a metric space (X,ρ). The d-dimesional Hausdorff measure of scale δ>0 is
where Bri(xi) is the open ball around xi ∈ X of radius ri. The d-dimensional Hausdorff measure is now the limit
- .
As in the Carathéodory construction a set S ⊂ X is called d-measurable iff
- for all T ⊂ X.
A set S ⊂ X is called Hausdorff measurable if it is Hd-measurable for all d≥0.
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