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# Topic: Hausdorff dimension

 Hausdorff dimension - Wikipedia, the free encyclopedia Intuitively, the dimension of a mathematical structure (for example, a subset of Euclidean space) is the number of independent parameters needed to describe a point in the structure. For example box-counting dimension, generalises 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. Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension. en.wikipedia.org /wiki/Hausdorff_dimension   (1887 words)

 Dimension - Open Encyclopedia   (Site not responding. Last check: 2007-11-07) Dimension (from Latin "measured out") is, in essence, the number of degrees of freedom available for movement in a space. In the SI system, the dimension is given by the seven exponents of the fundamental quantities. The Krull dimension of a commutative ring, named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring. open-encyclopedia.com /Dimension   (636 words)

 Encyclopedia topic: Hausdorff dimension   (Site not responding. Last check: 2007-11-07) To define the Hausdorff dimension for X, we first consider the number N(r) of balls (Round object that is hit or thrown or kicked in games) of radius at most r required to cover X completely. For example box-counting dimension (additional info and facts about box-counting dimension), generalises the idea of counting the squares of graph paper (Paper that has lines to permit drawing graphs) in which a point of X can be found, as the size of the squares is made smaller and smaller. The Hausdorff dimension is a well-defined extended real number (Any rational or irrational number) for any set E and we always have 0 ≤ d(E) ≤ ∞. www.absoluteastronomy.com /encyclopedia/h/ha/hausdorff_dimension.htm   (1877 words)

 Talk:Hausdorff dimension - Wikipedia, the free encyclopedia Hausdorff dimension isn't the only fractal dimension...it should have its own entry, and fractal dimension to talk about fractal dimensions collectively, it seems. I am aware that the metric is often referred to as the Hausdorff dimension; however, to dismiss one researcher because "everybody else does" is POV. From the Hausdorff method point of view this is still 2.0 dimensions but it's far from a two dimensional surface we see. en.wikipedia.org /?title=Talk:Hausdorff_dimension   (1764 words)

 [No title] Like in the case of the topological dimension, if a set A is contained in a set B then a theorem says that Hausdorff dimension of A is less or equal to that of B. Thus every set containing a square (or, what is equivalent, a disk) has Hausdorff dimension greater or equal to 2. The Hausdorff dimension of Sierpinski Carpet is not an integer. Dimension theory is developed for sets in even more general spaces, but some statements valid in the euclidean case might fail in general. www.swin.edu.au /chem/complex/vp/txt/vp07txt.htm   (2245 words)

 Hausdorff   (Site not responding. Last check: 2007-11-07) Hausdorff studied at Leipzig University under Heinrich Bruns and Adolph Mayer, graduating in 1891 with a doctorate in applications of mathematics to astronomy. Hausdorff returned to Bonn in 1921, by this time an emminent mathematician, and he worked there until 1935 when he was forced to retire by the Nazi regime. Hausdorff's paper includes a proof that the dimension of the middle-third Cantor set is log 2/log 3. www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Hausdorff.html   (1823 words)

 Hausdorff dimension   (Site not responding. Last check: 2007-11-07) The Hausdorff dimension agrees with the (topological) dimension on "well-behaved sets" but it applicable to many more sets and is always a natural number. The Hausdorff dimension should not be with the (similar) box-counting dimension. The Hausdorff dimension is a well-defined extended real number for any set E and we always have 0 ≤ d (E) ≤ ∞. www.freeglossary.com /Fractal_dimension   (987 words)

 Dimension - Unipedia   (Site not responding. Last check: 2007-11-07) Theories such as string theory predict that the space we live in has in fact many more dimensions (frequently 10, 11 or 26), but that the universe measured along these additional dimensions is subatomic in size. In the physical sciences and in engineering, the dimension of a physical quantity is the expression of the class of physical unit that such a quantity is measured against. Science fiction texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other planes of existence, or concepts that are beyond the reader. www.unipedia.info /2D.html   (918 words)

 Hausdorff dimension The Hausdorff dimension agrees with the ordinary dimension on well-behaved sets, but it is applicable to many more sets and isn't always a natural number. The Hausdorff dimension is a well-defined real number for any metric space M and we always have 0 ≤ d(M) ≤ ∞. 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 ln(3)/ln(2). www.fastload.org /ha/Hausdorff_dimension.html   (303 words)

 Hausdorff dimension The Hausdorff dimension, named after Felix Hausdorff, coincides with the more familiar notion of dimension in the case of well-behaved sets. For example a straight line or an ordinary curve, such as a circle, has a Hausdorff dimension of 1; any countable set has a Hausdorff dimension of 0; and an n-dimensional Euclidean space has a Hausdorff dimension of n. But a Hausdorff dimension is not always a natural number. www.daviddarling.info /encyclopedia/H/Hausdorff_dimension.html   (248 words)

 ipedia.com: Hausdorff dimension Article   (Site not responding. Last check: 2007-11-07) The Hausdorff dimension agrees with the ordinary dimension... The Hausdorff dimension agrees with the ordinary (topological) dimension on "well-behaved sets", but it is applicable to many more sets and is not always a natural number. The Hausdorff dimension should not be confused with the (similar) box-counting dimension. www.ipedia.com /hausdorff_dimension.html   (750 words)

 PlanetMath: Hausdorff dimension   (Site not responding. Last check: 2007-11-07) Hausdorff dimension is easy to calculate for simple objects like the Sierpinski gasket or a Koch curve. In fact, in the case of the Sierpinski gasket, one can take the individual triangles in each approximation as balls in the covering. This is version 9 of Hausdorff dimension, born on 2002-05-31, modified 2004-07-28. planetmath.org /encyclopedia/HausdorffDimension.html   (163 words)

 Fractal Curves and Dimension The topological dimension of a smooth curve is, as one would expect, one and that of a sphere is two which may seem very intuitive. To understand the notion of the similarity dimension, first observe that, if the initial line segment was 1 unit in length, then the second stage curve that consists of four segments each one third of the initial line, is 4/3 units in length. By one of Brouwer's theorems this function preserves the topological dimension of the segment (which is, of course 1). www.cut-the-knot.org /do_you_know/dimension.shtml   (1068 words)

 Edward Taylor's Publications   (Site not responding. Last check: 2007-11-07) We prove that if M is a compact, hyperbolizable 3-manifold, which is not a handlebody, then the Hausdorff dimension of the limit set is continuous in the strong topology on the space of marked hyperbolic 3-manifolds homotopy equivalent to M. Similarly, we define D(M) to be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups, whose quotients are homeomorphic to the interior of M. The distortion function measures the ratio of the intrinsic to extrinsic metrics, and is defined asymptotically as the length of the ray goes to infinity. ectaylor.web.wesleyan.edu /publist.html   (1373 words)

 Fractal dimension   (Site not responding. Last check: 2007-11-07) Roughly, fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero. Since the dimension 1.261 is larger than the dimension 1 of the lines making up the curve, the snowflake curve is a fractal. A set S has topological dimension k if each point in S has arbitrarily small neighborhoods whose boundaries meet S in a set of dimension k-1, and k is the least nonnegative integer for which this holds. fractals.iut.u-bordeaux1.fr /sci-faq/dimension.html   (520 words)

 Petra Bonfert-Taylor's publications   (Site not responding. Last check: 2007-11-07) We provide new bounds on the exponent of convergence of a planar discrete quasiconformal group in terms of the associated dilatation and the Hausdorff dimension of its conical limit set. Convergence groups, Hausdorff dimension, and a Theorem of Sullivan and Tukia, (with J. Anderson and E.C. Taylor), Geometriae Dedicata 103 (2004), pp. Hausdorff dimension and limit sets of quasiconformal groups, (with E.C. Taylor), Mich. Math. pbonfert.web.wesleyan.edu /publist.html   (1051 words)

 Department of Computer Science The two most important notions of fractal dimension are {\it Hausdorff dimension}, developed by Hausdorff (1919), and {\it packing dimension}, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. In 2000, the speaker proved a simple characterization of Hausdorff dimension in terms of {\it gales}, which are betting strategies that generalize martingales. www.cs.uchicago.edu /events/190   (288 words)

 [No title]   (Site not responding. Last check: 2007-11-07) Subject: Re: A countable set with fractal dimension D > 0 Date: Tue, 16 May 2000 13:15:25 -0400 Newsgroups: sci.math Summary: [missing] The Hausdorff dimension of a countable set is zero. And many others.) The Bouligand dimension (box dimension) of a set is the same as for the closure of the set. In order for the similarity dimension to be useful in the study of a set, the set should be the attractor of the IFS, so in particular it should be a nonempty compact set. www.math.niu.edu /~rusin/known-math/00_incoming/dimension   (500 words)

 Felix Hausdorff   (Site not responding. Last check: 2007-11-07) Hausdorff's main work was in topology and set theory. He also introduced the Hausdorff measure and the term "metric space" is due to him. Hausdorff worked at Bonn until 1935 when he was forced to retire by the Nazi regime. www.stetson.edu /~efriedma/periodictable/html/Hf.html   (256 words)

 Minkowski-Bouligand dimension - Art History Online Reference and Guide For example, the box dimension of a single point is 0, but the box dimension of the collection of rational numbers in the interval [0, 1] has dimension 1. An interesting property of the upper box dimension not shared with either the lower box dimension or the Hausdorff dimension is the connection to set addition. For example, the Hausdorff dimension, lower box dimension, and upper box dimension of the Cantor set are all equal to log(2)/log(3). www.arthistoryclub.com /art_history/Box-counting_dimension   (918 words)

 The Hausdorff Dimension Of The (ResearchIndex)   (Site not responding. Last check: 2007-11-07) 0.3: The Hausdorff Dimension of the Boundary of a Self-Similar Tile - Duvall, Keesling, Vince 4 dimension of the boundary of a self-similar tile (context) - Duvall, Keesling et al. The Hausdorff Dimension of the Boundary of a Self-Similar Tile - Duvall, Keesling, Vince citeseer.ist.psu.edu /617597.html   (349 words)

 Delta Hausdorff Dimension, Mu-Ency at MROB The box-counting dimension for a given neighborhood of a point, which is considered to be a term in a series, the limit of which would be the Hausdorff dimension. for a neighborhood of P is constrained to be in the range containing the Hausdorff dimensions of the Julia Sets for all points within the neighborhood. The number gives the fractal dimension of the image by the box-counting method; it can be anything from 1.00 to 2.00. www.mrob.com /pub/muency/deltahausdorffdimension.html   (457 words)

 3.3 Fractal Dimension This does not agree with the topological dimension of 1 but neither is it 2. Its dimension is not a whole number but a fraction. A study of twelve definitions of dimension appeared in 1981 (see Harrison) and an entire book on Dimension Theory was written in 1941 (Hurewicz and Wallman). hypertextbook.com /chaos/33.shtml   (1028 words)

 The Science House - Chaos & Fractals For example, a straight line has a dimension of 1, a sheet of paper has a dimension of 2, and a solid cube has a dimension of 3. That is, an object with a fractional dimension has a dimension that is not an integer but rather a fraction. To determine the Correlation dimension you apply the same idea as the Hausdorff dimension except use circles instead of squares. www.science-house.org /student/bw/chaos/cfpictures.html   (509 words)

 DC MetaData for: On the Hausdorff Dimension of the Sierpinski Gasket with respect to the Harmonic Metric   (Site not responding. Last check: 2007-11-07) DC MetaData for: On the Hausdorff Dimension of the Sierpinski Gasket with respect to the Harmonic Metric On the Hausdorff Dimension of the Sierpinski Gasket with respect to the Harmonic Metric Abstract: The spectral dimension and the Hausdorff dimension www.esi.ac.at /Preprint-shadows/esi1174.html   (86 words)

 [No title] Subject: Re: Hausdorff distance and fractal dimension Date: Mon, 17 Jan 2000 12:09:41 +0100 Newsgroups: sci.math.research Summary: [missing] I seriously doubt this as there are nowhere differentiable functions arbitrarily close to analytic ones; think of Weierstrass' example. Lower bounds for the Hausdorff dimensions of their graphs were given by Mauldin & Williams, these seem to indicate that d(e)>=1 for all e. Subject: Re: Hausdorff distance and fractal dimension Date: Mon, 17 Jan 2000 14:04:02 +0000 (GMT) Newsgroups: sci.math.research Adam Bryant wrote: : Let A and B be compact subsets of Euclidean space. www.math.niu.edu /~rusin/known-math/00_incoming/haus_dim   (747 words)

 Fractals: Hausdorff-Besicovitch Dimension The Hausdorff-Besicovitch dimension is the mathematical expression of the dimension of objects. This formula relates the fractal dimension with the size of the small boxes that made the overall grid. Plotting log(N) on the y-axis against log(r) on the x-axis should give a straigth line, whose slope is equal to the fractal dimension. users.swing.be /TGMSoft/hausdorffdim.htm   (454 words)

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