Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff....

the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension...

one-dimensional Hausdorff measure of a simple curve in R n {\displaystyle \mathbb {R} ^{n}} is equal to the length of the curve, and the two-dimensional Hausdorff measure...

leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, but there are also other answers to that...

of the concept of a fractal dimension through the work of Hausdorff in the early 1900s who defined a "fractional" dimension that has come to be named after...

In fractal geometry, the parabolic Hausdorff dimension is a restricted version of the genuine Hausdorff dimension. Only parabolic cylinders, i. e. rectangles...

and lower box dimensions are strongly related to the more popular Hausdorff dimension. Only in very special applications is it important to distinguish...

r i d < δ . {\displaystyle \sum _{i\in I}r_{i}^{d}<\delta .} The Hausdorff dimension of X is defined by dim H ⁡ ( X ) := inf { d ≥ 0 : C H d ( X ) = 0...

named after him: Hausdorff completion Hausdorff convergence Hausdorff density Hausdorff dimension Hausdorff distance Hausdorff gap Hausdorff maximal principle...

Hausdorff (1868–1942), the German mathematician after whom Hausdorff spaces are named Hausdorff dimension, a measure theoretic concept of dimension Hausdorff...

for some constant s > 0 and for every ball B(x, r) in X, then the Hausdorff dimension dimHaus(X) ≥ s. A partial converse is provided by the Frostman lemma:...

should concern the Hausdorff dimension of the set of badly approximable numbers. And indeed, V. Jarník proved that the Hausdorff dimension of this set is...

{\displaystyle a_{0}=1,\ a_{1}=6} . The sponge's Hausdorff dimension is ⁠log 20/log 3⁠ ≅ 2.727. The Lebesgue covering dimension of the Menger sponge is one, the same...

to Hausdorff dimension, since packing dimension is constructed by "packing" small open balls inside the given subset, whereas Hausdorff dimension is constructed...

Self-Similarity and Fractional Dimension", published on 5 May 1967, Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These...

subset of Rn has Hausdorff dimension less than n then it is a null set with respect to n-dimensional Lebesgue measure. Here Hausdorff dimension is relative...

space-filling, its Hausdorff dimension is 2 (precisely, its image is the unit square, whose dimension is 2 in any definition of dimension; its graph is a...

often highly irregular, for example a fractal in the sense that its Hausdorff dimension is not an integer. This occurs even for mappings as simple as f (...

Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible...

differentiability is usually given in terms of fractal dimension, with the Hausdorff dimension the most popular choice. This line of research was started...

equivalent statement of the open set condition is to require that the s-dimensional Hausdorff measure of the set is greater than zero. When the open set condition...

needed] A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails. However, a locally compact Hausdorff space is zero-dimensional...

Granlund & Knutsson (1995). Dimension Fractal dimension Hausdorff dimension Topological dimension Intrinsic low-dimensional manifold Amsaleg, Laurent;...

discrete version of the fact that a Wiener process walk is a fractal of Hausdorff dimension 2.[citation needed] In two dimensions, the average number of points...

\log(1/\varepsilon )}}}=1,\qquad {\text{almost surely}}.} The dimension doubling theorems say that the Hausdorff dimension of a set under a Brownian motion doubles almost...

close, the function will not be monotone. The computation of the Hausdorff dimension D {\textstyle D} of the graph of the classical Weierstrass function...

argument involving Montel's theorem. Let dimH denote Hausdorff dimension and H1 denote 1-dimensional Hausdorff measure. Then H1(K) = 0 implies γ(K) = 0 while...

is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there...

evaluated in radians and n is the number of sides of the n-gon. The Hausdorff dimension of a n-flake is log ⁡ m log ⁡ r {\displaystyle \textstyle {\frac...

is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive...