Hausdorff dimension
E608814
The Hausdorff dimension is a mathematical concept in fractal geometry and measure theory that generalizes the notion of dimension to capture the scaling complexity of irregular sets.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
dimension theory concept
ⓘ
mathematical concept ⓘ |
| allows | distinguishing sets with same topological dimension but different scaling complexity ⓘ |
| appliesTo |
Brownian motion paths
ⓘ
Cantor sets NERFINISHED ⓘ Julia sets NERFINISHED ⓘ attractors of dynamical systems ⓘ fractals ⓘ irregular sets ⓘ percolation clusters ⓘ random fractals ⓘ self-similar sets ⓘ |
| basedOn | Hausdorff measure NERFINISHED ⓘ |
| canBe |
fractional
ⓘ
non-integer ⓘ |
| captures |
fractal scaling properties
ⓘ
metric complexity of sets ⓘ scaling behavior of sets ⓘ |
| characterizes | fine structure of sets at small scales ⓘ |
| definedOn |
metric spaces
ⓘ
subsets of Euclidean space ⓘ |
| definitionIdea | critical value where s-dimensional Hausdorff measure jumps from infinity to zero ⓘ |
| equals |
integer dimension for smooth manifolds
ⓘ
topological dimension for many regular sets ⓘ |
| field |
fractal geometry
ⓘ
geometric measure theory ⓘ measure theory ⓘ |
| generalizes |
Lebesgue covering dimension
NERFINISHED
ⓘ
Minkowski dimension ⓘ topological dimension ⓘ |
| introducedBy | Felix Hausdorff NERFINISHED ⓘ |
| introducedIn | early 20th century ⓘ |
| invariantUnder | bi-Lipschitz maps ⓘ |
| monotoneWithRespectTo | set inclusion ⓘ |
| namedAfter | Felix Hausdorff NERFINISHED ⓘ |
| propertyOf | subsets of a metric space ⓘ |
| relatedTo |
Assouad dimension
NERFINISHED
ⓘ
box-counting dimension ⓘ packing dimension ⓘ |
| requires |
coverings by small sets
ⓘ
outer measure construction ⓘ |
| upperBoundedBy | ambient Euclidean dimension ⓘ |
| usedIn |
dynamical systems
ⓘ
fractal geometry classification ⓘ image analysis ⓘ multifractal analysis ⓘ probability theory ⓘ statistical physics ⓘ turbulence modeling ⓘ |
| usesParameter | Hausdorff measure exponent s ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.