In mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated to any metric space. The Hausdoff dimension generalizes the notion of the dimension of a real vector space. In particular, the Hausdorff dimension of a single point is zero, the Hausdoff dimension of a line is one, the Hausdoff dimension of the plane is two, etc. The Hausdoff dimension was introduced in 1918 by the mathematician Felix Hausdorff. Many of the technical developments used to compute the Hausdorff dimension for highly irregular sets were obtained by Abram Samoilovitch Besicovitch. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Felix Hausdorff Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory and functional analysis. ... Abram Samoilovitch Besicovitch (Besikovitch) (24 January 1891 - 2 November Russian mathematician, who worked mainly in England. ...

Less frequently it is also called the capacity dimension or fractal dimension (the latter is somewhat ambiguous). In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. ...

Informal discussion

Intuitively, the dimension of a set (for example, a subset of Euclidean space) is the number of independent parameters needed to describe a point in the set. One mathematical concept which closely models this naive idea is that of topological dimension of a set. For example a point in the plane is described by two independent parameters (the Cartesian coordinates of the point), so in this sense, the plane is two-dimensional. As one would expect, the topological dimension is always a natural number. Superset redirects here. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement in which no point is included in more than n+1 elements. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In mathematics, the Lebesgue covering dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement with no point included in more than n+1 elements. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

However, topological dimension behaves in quite unexpected ways on certain highly irregular sets such as fractals. For example, the Cantor set has topological dimension zero, but in some sense it behaves as a higher dimensional space. Hausdorff dimension gives another way to define dimension, which takes the metric into account. The boundary of the Mandelbrot set is a famous example of a fractal. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

Sierpinski triangle. A space having fractional dimension ln 3 / ln 2, or log₂ 3, which is approximately 1.58

To define the Hausdorff dimension for X as non-negative real number (that is a number in the half-closed infinite interval [0, ∞)), we first consider the number N(r) of balls of radius at most r required to cover X completely. Clearly, as r gets smaller N(r) gets larger. Very roughly, if N(r) grows in the same way as 1/r^d as r is squeezed down towards zero, then we say X has dimension d. In fact the rigorous definition of Hausdorff dimension is somewhat roundabout, as it allows the covering of X by balls of different sizes. Image File history File links Sierpinski_triangle_(blue). ... Image File history File links Sierpinski_triangle_(blue). ... Sierpinski triangle The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after WacÅ‚aw SierpiÅ„ski who described it in 1916. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. ...

For many shapes that are often considered in mathematics, physics and other disciplines, the Hausdorff dimension is an integer. However, sets with non-integer Hausdorff dimension are important and prevalent. Benoît Mandelbrot, a popularizer of fractals, advocates that most shapes found in nature are fractals with non-integer dimension, explaining that "[c]louds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." ^[1] Benoît B. Mandelbrot, PhD, (born November 20, 1924) is a Franco-American mathematician, best known as the father of fractal geometry. Benoît Mandelbrot was born in Poland, but his family moved to France when he was a child; he is a dual French and American citizen and was... The boundary of the Mandelbrot set is a famous example of a fractal. ...

There are various closely related notions of possibly fractional dimension. For example box-counting dimension, generalizes the idea of counting the squares of graph paper in which a point of X can be found, as the size of the squares is made smaller and smaller. (The box-counting dimension is also called the Minkowski-Bouligand dimension). The packing dimension is yet another notion of dimension admitting fractional values. These notions (packing dimension, Hausdorff dimension, Minkowski-Bouligand dimension) all give the same value for many shapes, but there are well documented exceptions. In fractal geometry, the Minkowski-Bouligand dimension or Minkowski dimension is a way of determining the fractal dimension of a set S in a Euclidean space , or more generally of a metric space (X,d). ... Graph paper or quad-ruled paper is writing paper that is printed with fine lines making up a regular grid. ... In fractal geometry, the Minkowski-Bouligand dimension or Minkowski dimension is a way of determining the fractal dimension of a set S in a Euclidean space , or more generally of a metric space (X,d). ...

Formal definition

Let X be a metric space. If and , the d-dimensonal Hausdorff content of S is defined by

In other words, is the infimum of the set of numbers such that there is some (indexed) collection of balls with r_i > 0 for each which satisfies . (One can assume, with no loss of generality, that the index set I is the natural numbers .) Here, we use the standard convention that inf Ø =∞. The Hausdoff dimension of X is defined by In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ...

Equivalently, may be defined as the infimum of the set of such that the d-dimensional Hausdorff measure of X is zero. This is the same as the supremum of the set of such that the d-dimensional Hausdorff measure of X is infinite (except that when this latter set of numbers d is empty the Hausdoff dimension is zero). In mathematics, the Hausdorff dimension is an extended non-negative real number, that is in the closed infinite interval [0, ∞], associated to any metric space . ...

Examples

• The Euclidean space Rⁿ has Hausdorff dimension n.
• The circle S¹ has Hausdorff dimension 1.
• Countable sets have Hausdorff dimension 0.
• Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension. For example, the Cantor set (a zero-dimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff dimension is ln2 / ln3, which is approximately 0.63 (see natural logarithm). The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln3 / ln2, which is approximately 1.58.
• Space-filling curves like the Peano and the Sierpiński curve have the same Hausdorff dimension as the space they fill.
• The trajectory of Brownian motion in dimension 2 and above has Hausdorff dimension 2 almost surely.
• An early paper by Benoit Mandelbrot entitled How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension and subsequent work by other authors have claimed that the Hausdorff dimension of many coastlines can be estimated. Their results have varied from 1.02 for the coastline of South Africa to 1.25 for the west coast of Great Britain. However, 'fractal dimensions' of coastlines and many other natural phenomena are largely heuristic and cannot be regarded rigorously as a Hausdorff dimension. It is based on scaling properties of coastlines at a large range of scales, but which does not however include all arbitrarily small scales, where measurements would depend on atomic and sub-atomic structures, and are not well defined.

Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, a countable set is a set with the same cardinality (i. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... In mathematics, the Lebesgue covering dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement with no point included in more than n+1 elements. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... Sierpinski triangle The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Wacław Sierpiński who described it in 1916. ... Space-filling curves or Peano curves are curves, first described by Giuseppe Peano, whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube). ... Intuitively, a continuous curve in the 2-dimensional plane or in the 3-dimensional space can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the... Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... In probability theory, an event happens almost surely (a. ... Beno t Mandelbrot was the first to use a computer to plot the Mandelbrot set. ... How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension is a paper by mathematician Benoît Mandelbrot, first published in Science in 1967. ...

Inequalities

Hausdorff dimension and inductive dimension

Let X be an arbitrary separable metric space. There is a topological notion of inductive dimension for X which is defined recursively. It is always an integer (or +∞) and is denoted dim_ind(X). In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ... For other uses, see Topology (disambiguation). ... In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X). ...

Theorem. Suppose X is non-empty. Then

Moreover

where Y ranges over metric spaces homeomorphic to X. In other words, X and Y have the same underlying set of points and the metric d_Y of Y is topologically equivalent to d_X. This word should not be confused with homomorphism. ...

These results were originally established by Edward Szpilrajn (1907-1976). The treatment in Chapter VIII of the Hurewicz and Wallman reference is particularly recommended. Edward Marczewski (15 November 1907 in Warsaw, Poland - 17 October 1976 in Wrocław, Poland) was a Polish mathematician. ...

Hausdorff dimension and Minkowski dimension

The Minkowski dimension is similar to the Hausdorff dimension, except that for the Minkowski dimension one only considers coverings by balls of the same radius. The Minkowski dimension of a metric space is at least as large as the Hausdorff dimension. In many situations, they are equal. However, the set of rational points in [0,1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact metric spaces for which the Minkowski dimension is strictly larger than the Hausdorff dimension. In fractal geometry, the Minkowski-Bouligand dimension or Minkowski dimension is a way of determining the fractal dimension of a set S in a Euclidean space , or more generally of a metric space (X,d). ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...

Self-similar sets

Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set E is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(E) = E, although the exact definition is given below.

Theorem. Suppose

are contractive mappings on Rⁿ with contraction constant r_j < 1. Then there is a unique non-empty compact set A such that In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number such that, for all x and y in M, The smallest such value of k is called the Lipschitz constant...

.

The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of Rⁿ with the Hausdorff distance^[2]. Stefan Banach ( listen) (Ukrainian: Степан Степанович Банах, 1892-1945) was an eminent Polish mathematician who worked in interwar Poland and briefly in Soviet Ukraine. ... The Banach fixed point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points. ... Hausdorff distance measures how far two compact subsets of a metric space are from each other. ...

To determine the dimension of the self-similar set A (in certain cases), we need a technical condition called the open set condition on the sequence of contractions ψ_i which is stated as follows: There is a relatively compact open set V such that

where the sets in union on the left are pairwise disjoint. In mathematics, two sets are said to be disjoint if they have no element in common. ...

Theorem. Suppose the open set condition holds and each ψ_i is a similitude, that is a composition of an isometry and a dilation around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is s where s is the unique solution of In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... Dilation in physiological context may mean: pupil dilation (mydriasis) dilation of blood vessels (vasodilation) cervical dilation (or dilation of the cervix) in childbirth Dilation and curettage (surgical dilation) In mathematics: Dilation This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

Note that the contraction coefficient of a similitude is the magnitude of the dilation.

We can use this theorem to compute the Hausdorff dimension of the Sierpinski triangle (or sometimes called Sierpinski gasket). Consider three non-collinear points a₁, a₂, a₃ in the plane R² and let ψ_i be the dilation of ratio 1/2 around a_i. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket and the dimension s is the unique solution of A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...

Taking natural logarithms of both sides of the above equation, we can solve for s, that is:

The Sierpinski gasket is self-similar. In general a set E which is a fixed point of a mapping

is self-similar if and only if the intersections

where s is the Hausdorff dimension of E and H^s denotes Hausdorff measure. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally: In mathematics, the Hausdorff dimension is an extended non-negative real number, that is in the closed infinite interval [0, ∞], associated to any metric space . ...

Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.

See also

• List of fractals by Hausdorff dimension, some examples of deterministic fractals, random and natural fractals

A fractal is a geometric object whose Hausdorff dimension (δ) strictly exceeds its topological dimension. ...

Historical references

• A. S. Besicovitch, On Linear Sets of Points of Fractional Dimensions, Mathematische Annalen 101 (1929).
• A. S. Besicovitch and H. D. Ursell, Sets of Fractional Dimensions, Journal of the London Mathematical Society, v12 (1937). Several selections from this volume are reprinted in Classics on Fractals,ed. Gerald A. Edgar, Addison-Wesley (1993) ISBN 0-201-58701-7 See chapters 9,10,11.
• F. Hausdorff, Dimension und äußeres Maß, Mathematische Annalen 79(1–2) (March 1919) pp. 157–179.

The Mathematische Annalen is a German mathematical research journal published by Springer-Verlag. ...

References

1. ^ Mandelbrot, Benoît (1982). The Fractal Geometry of Nature, Lecture notes in mathematics 1358. W. H. Freeman. ISBN 0716711869. 
2. ^ K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985 Theorem 8.3

• M. Maurice Dodson and Simon Kristensen, Hausdorff Dimension and Diophantine Approximation (June 12, 2003).
• W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, 1948.
• E. Szpilrajn, La dimension et la mesure, Fundamenta Mathematica 28, 1937, pp 81-89.