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Hausdorff dimension is a concept in mathematics introduced in 1918 by mathematician Felix Hausdorff, and it serves as a measure of the local size of a set of numbers (i.e., a "space"), taking into account the distance between each of its members (i.e., the "points" in the "space"). Applying its mathematical formalisms provides that the Hausdorff dimension of a single point is zero, of a line is 1, and of a square is 2, of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is a ''counting number'' (integer) agreeing with a dimension corresponding to its topology. However, formalisms have also been developed that allow calculation of the dimension of other less simple objects, where, based solely on its properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular sets, this dimension is also commonly referred to as the ''Hausdorff–Besicovitch dimension.'' The Hausdorff dimension is, more specifically, a further dimensional number associated with a given set of numbers, where the distances between all members of that set are defined, and where the dimension is drawn from the real numbers, ℝ, to which two elements have been added, +∞ and −∞ (read as positive and negative infinity, respectively). The set that provides the Hausdorff dimension is called the extended real numbers, , and a set of numbers where distances between all members are defined is termed a metric space,〔The distances, taken together, are called a metric on the set.〕 so that foregoing can be succinctly stated, saying the Hausdorff dimension is a non-negative extended real number ( ≥ 0) associated with any metric space. In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an ''n''-dimensional inner product space equals ''n''. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that irregular sets can have noninteger Hausdorff dimensions. For instance, the Koch curve summarized earlier is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new equilateral triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4.〔Larry Riddle, 2014, "Classic Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online), see (), accessed 5 March 2015.〕 That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original.〔 Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=SD.〔Keith Clayton, 1996, "Fractals and the Fractal Dimension," ''Basic Concepts in Nonlinear Dynamics and Chaos'' (workshop), Society for Chaos Theory in Psychology and the Life Sciences annual meeting, June 28, 1996, Berkeley, California, see (), accessed 5 March 2015.〕 This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects. The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension. ==Intuition== The intuitive concept of dimension of a geometric object ''X'' is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information.) The example of a space-filling curve shows that one can even take one real number into two both surjectively (so all pairs of numbers are covered) and ''continuously'', so that a one-dimensional object completely fills up a higher-dimensional object. Every space filling curve hits some points multiple times, and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called Lebesgue covering dimension, explains why. This dimension is ''n'' if, in every covering of ''X'' by small open balls, there is at least one point where ''n'' + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension ''n'' = 1. But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A fractal has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space. The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number ''N''(''r'') of balls of radius at most ''r'' required to cover ''X'' completely. When ''r'' is very small, ''N''(''r'') grows polynomially with 1/''r''. For a sufficiently well-behaved ''X'', the Hausdorff dimension is the unique number ''d'' such that N(''r'') grows as 1/''rd'' as ''r'' approaches zero. More precisely, this defines the box-counting dimension, which equals the Hausdorff dimension when the value ''d'' is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant. For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer agreeing with the topological dimension. But Benoît Mandelbrot observed that fractals, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes: Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. For fractals that occur in nature, the Hausdorff and box-counting dimension coincide. The packing dimension is yet another similar notion which gives the same value for many shapes, but there are well documented exceptions where all these dimensions differ. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hausdorff dimension」の詳細全文を読む スポンサード リンク
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