List of fractals by Hausdorff dimension
From Wikipedia, the free encyclopedia According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."[1] Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension. Deterministic fractals[edit] Random and natural fractals[edit] See also[edit] Notes and references[edit] Further reading[edit] External links[edit]
List of fractals by Hausdorff dimension
Deterministic fractals[edit] Random and natural fractals[edit] See also[edit] Notes and references[edit] Further reading[edit] Benoît Mandelbrot, The Fractal Geometry of Nature, W. External links[edit]
Hausdorff measure
Fractal measurement In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue-measurable subset of is proportional to the area of the set. Let be a metric space. , let denote its diameter, that is be any subset of and a real number. where the infimum is over all countable covers of by sets satisfying Note that is monotone nonincreasing in since the larger is, the more collections of sets are permitted, making the infimum not larger. exists but may be infinite. It can be seen that -dimensional Hausdorff measure of . are [edit]
Hausdorff-Dimension
Vereinfachte Definition[Bearbeiten] der Kugeln mit dem Radius des Radius . Je kleiner der Radius ist, umso größer ist , mit der gegen Null anwächst, berechnet sich die Hausdorff-Dimension und zwar nach und damit . . Für den Spezialfall eines geometrischen Objekts, welches aus disjunkten Teilobjekten besteht, die im Maßstab verkleinerte Kopien des Gesamtobjekts darstellen, ergibt sich für die Hausdorff-Dimension . Teilobjekte verschiedene Größe, so ist durch definiert, wobei die einzelnen Maßstäbe sind ( ). Es ist jedoch zu beachten, dass diese vereinfachte Definition sich nicht generell mit der exakten Definition (s. u.) deckt. Für eine numerische Bestimmung der Hausdorff-Dimension einer gegebenen Punktmenge lässt sich der so genannte Boxcounting-Algorithmus verwenden. Definition über das Hausdorff-Maß[Bearbeiten] Eine mathematisch exakte Definition der Hausdorff-Dimension einer beschränkten Teilmenge erfolgt über das Hausdorff-Maß , das dieser Menge zu jeder Dimension zugeordnet wird. , für die Für festes
Intrinsic dimension
In signal processing of multidimensional signals, for example in computer vision, the intrinsic dimension of the signal describes how many variables are needed to represent the signal. For a signal of N variables, its intrinsic dimension M satisfies 0 ≤ M ≤ N. Usually the intrinsic dimension of a signal relates to variables defined in a Cartesian coordinate system. Example[edit] Let f(x1, x2) be a two-variable function (or signal) which is of the form f(x1,x2) = g(x1) for some one-variable function g which is not constant. A slightly more complicated example is f(x1,x2) = g(x1 + x2) f is still intrinsic one-dimensional, which can be seen by making a variable transformation x1 + x2 = y1 x1 - x2 = y2 which gives f(y1,y2) = g(y1) Since the variation in f can be described by the single variable y1 its intrinsic dimension is one. For the case that f is constant, its intrinsic dimension is zero since no variable is needed to describe variation. Formal definition[edit] f=f(x) where x=(x1, x2, ..., xN)
H tree
The first ten levels of an H tree The first 18 levels of an H tree as an animation. The H tree (so called because its repeating pattern resembles the letter "H") is a family of fractal sets whose Hausdorff dimension is equal to 2. An alternative process that generates the same fractal set is to begin with a rectangle with sides in the ratio 1:√2, known as a "silver rectangle", and repeatedly bisect it into two smaller silver rectangles, at each stage connecting the two centroids of the two smaller rectangles by a line segment. The Mandelbrot Tree is a very closely related fractal using rectangles instead of line segments, slightly offset from the H-tree positions, in order to produce a more naturalistic appearance. Applications[edit] The planar H tree can be generalized to the three-dimensional structure via adding line segments on the direction perpendicular to the H tree plane.[7] The resultant three-dimensional H tree has Hausdorff dimension equal to 3. Notes[edit] References[edit]
H-Baum
Die ersten zehn Stufen eines H-Baumes Der H-Baum ist eine FASS-Kurve, d. h. er füllt die ganze Ebene aus. Seine fraktale Hausdorff-Dimension ist 2. Mit jeder neuen Iteration multipliziert sich die Gesamtlänge der Kurve um H-Bäume spielen beim Entwurf von synchronen digitalen Schaltungen zur Signalverteilung eine Rolle. Ein anderes Beispiel für die Verwendung von H-Bäumen ist die Abbildung der Kommunikationsstruktur eines Programms auf die Prozessoren in einem Computercluster.
Uncertainty exponent
Suppose we start with a random trajectory and perturb it by a small amount, , in a random direction. If the new trajectory ends up in a different basin from the old one, then it is called epsilon uncertain. , and we expect it to scale exponentially with Thus the uncertainty exponent, , is defined as follows: The uncertainty exponent can be shown to approximate the box-counting dimension as follows: where N is the embedding dimension. C.
