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Topic: Fractal dimension

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 Fractal - Wikipedia, the free encyclopedia
In mathematics a fractal is a geometric object that satisfies a specific technical condition, namely having a Hausdorff-Besicovitch dimension greater than its topological dimension.
(Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar.
It should be noted that not all self-similar objects are fractals — e.g., the real line (a straight Euclidean line) is exactly self-similar, but since its Hausdorff dimension and topological dimension are both equal to one, it is not a fractal.
en.wikipedia.org /wiki/Fractal   (1793 words)

 Fractal Dimension
Fractal dimension is a measure of how space filling a pattern is. Any pattern in the two-dimensional space of a plane must have a fractal dimension of between 0 and 2.
The higher the fractal dimension of an object up to its limit (two for a limiting plane), the more complex it is. At a fractal dimension of two, a pattern on a plane is totally random.
Lacunarity is a counterpart to the fractal dimension that describes the texture of a fractal.
www.inspiredeconomies.com /intelligibleecosystems/EcoVis_org/fractal_dimension.htm   (1591 words)

 Fractal Dimension
To explain the concept of fractal dimension, it is necessary to understand what we mean by dimension in the first place.
Now we see an alternative way to specify the dimension of a self-similar object: The dimension is simply the exponent of the number of self-similar pieces with magnification factor N into which the figure may be broken.
Fractal dimension is a measure of how "complicated" a self-similar figure is. In a rough sense, it measures "how many points" lie in a given set.
math.bu.edu /DYSYS/chaos-game/node6.html   (694 words)

 2.0 The Fractal Dimension
The dimension D of a fractal 'trace' on the plane, however, is a continuous function with range 1 <= D <= 2.
Fractal dimensions 1 <= D <= 2 quantify the degree to which a trace 'fills' the plane.
The fractal dimension D is most commonly estimated from the regression slope of a log-log power law plot.
www.umanitoba.ca /faculties/science/botany/LABS/ECOLOGY/FRACTALS/dimension.html   (591 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.
This sets are known as the self-similar fractals and, because of that ease, the property of self-similarity is often considered to be germane to fractals in general.
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.
www.cut-the-knot.org /do_you_know/dimension.shtml   (1290 words)

 AllRefer.com - fractal geometry (Mathematics) - Encyclopedia
fractal geometry, branch of mathematics concerned with irregular patterns made of parts that are in some way similar to the whole, e.g., twigs and tree branches, a property called self-similarity or self-symmetry.
Unlike conventional geometry, which is concerned with regular shapes and whole-number dimensions, such as lines (one-dimensional) and cones (three-dimensional), fractal geometry deals with shapes found in nature that have non-integer, or fractal, dimensions : linelike rivers with a fractal dimension of about 1.2 and conelike mountains with a fractal dimension between 2 and 3.
Fractal geometry has been applied to such diverse fields as the stock market, chemical industry, meteorology, and computer graphics.
reference.allrefer.com /encyclopedia/F/fractalge.html   (305 words)

 Fractals & the Fractal Dimension
Mandelbrot began his treatise on fractal geometry by considering the question: "How long is the coast of Britain?" The coastline is irregular, so a measure with a straight ruler, as in the next figure, provides an estimate.
A strange attractor is a fractal, and its fractal dimension is less than the dimensions of its phase space.
An important (defining) property of a fractal is self-similarity, which refers to an infinite nesting of structure on all scales.
www.vanderbilt.edu /AnS/psychology/cogsci/chaos/workshop/Fractals.html   (887 words)

 Fractal Dimension of Broccoli
The purpose of this lab is to determine the fractal dimension of broccoli.
The Fractal dimension of broccoli is the magnitude of the slope of the graph of ln(number) vs. ln(size).
The fractal dimension of a broccoli is 2.66.
hypertextbook.com /facts/2002/broccoli.shtml   (374 words)

 Fractal Geometry
Mandelbrot discovered that the fourth dimension of fractal forms includes an infinite set of fractional dimensions which lie between the zero and first dimension, the first and second dimension and the second and third dimension.
An understanding of how the fourth dimension includes the infinity of intervals between the other dimensions can be gained by visualizing a few of the better known fractal dimensions (sometimes called Hausdorff dimensions by mathematicians).
The dimension is calculated to have a numerical value of.63 and was discovered by mathematician George Cantor in the beginning of the Twentieth Century.
www.fractalwisdom.com /FractalWisdom/fractal.html   (2550 words)

 FDC - Fractal Dimension Calculator   (Site not responding. Last check: 2007-10-08)
FDC estimates the fractal dimension of an object represented as a fl and white image where the object to be analysed is assumed to be made up of the fl pixels.
The estimate of the fractal dimension is the slope of this graph.
Note that the theoretical fractal dimension for the snowflake curve is log(4)/log(3) = 1.262.
astronomy.swin.edu.au /~pbourke/fractals/fracdim   (1600 words)

 Fractal dimension
This is a figure the Euclidean dimension of which is 1 (it is a broken line) and the fractal dimension of which is greater than 1 and, moreover, is not a whole number.
True because for the fractal made of lines which are simple, as is the von Koch curve, the dimensions of homothety and of Hausdorff-Besicovitch are equal.
Moreover there are other approaches of the notion of dimension which are not all equivalent, and the works of Kolmogorov and Tihomirov show a connection between the covering dimension and the notion of entropy (this may bring back some vague memories, for some of you!).
fractals.iut.u-bordeaux1.fr /jpl/dimension_a.html   (1246 words)

 Fractal dimension
Roughly, fractal dimension can be calculated by taking the limit of the quo- tient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero.
A4b: Topological dimension is the "normal" idea of dimension; a point has topological dimension 0, a line has topological dimension 1, a surface has topological dimension 2, etc. For a rigorous definition: A set has topological dimension 0 if every point has arbitrarily small neighborhoods whose boundaries do not intersect the set.
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.
www.faqs.org /faqs/fractal-faq/section-4.html   (572 words)

 Fractal Dimensions
That is, while you are in the 3rd dimension, looking at this on a flat screen which can be considered more or less the 2nd dimension, fractals are in between the dimensions.
If you make a fractal by drawing lines that obey a certain rule, like Koch's Curve, that fractal can't have a dimension higher than the paper it's drawn on, which would be 2 (it can be assumed that paper is as good as we're gonna get to 2 dimensional.
It can be assumed that for any fractal object (of size P, made up of smaller units of size p), the number of units (N) that fits into the larger object is equal to the size ratio (P/p) raised to the power of d, which is called the Hausdorff dimension.
www.jracademy.com /~jtucek/math/dimen.html   (444 words)

 3. Measuring the Fractal Dimension of Natural Objects
A digital image of a fractal object is Euclidean by virtue of its being placed onto a grid, leading to biased estimates of D. To account for this 'rectangularization', the perimeter P is expressed as P/4, where the value 4 is determined as the proportionality constant for a pixel system.
Thus the fractal dimension D can be estimated from the log-log plot of the first moment as a function of L (Milne 1991b).
As expected, the fractal dimension of 'rough' topographic surfaces (negative correlation of height variation) is high, while smooth surfaces (positive correlation of height variation with distance) have a low fractal dimension.
www.umanitoba.ca /faculties/science/botany/LABS/ECOLOGY/FRACTALS/measuring.html   (5187 words)

 Fractal Coastline   (Site not responding. Last check: 2007-10-08)
Fractals can be constructed by copying a basic design over and over again on a larger or smaller scale.
Wherever there is a need to expose as large an exposed surface as possible, but there is a restriction on the total volume of material available, or a penalty to be incurred by increasing weight, then fractals seem to be selected by the evolutionary process.
Fractal shapes may therefore be extremely robust in situations, like those of trees in the wind, panting lungs, or pounding hearts, where it is necessary to withstand a large amount of vibration.
www.btinternet.com /~connectionsinspace/Mapping/Fractal_Coastline/body_fractal_coastline.html   (388 words)

 The Fractal Dimension
These dimensions are known as topological dimensions, and have been used for many years to describe the shape and position of objects.
The amount of space filled by one of these objects is represented by the fractal dimension or index (D), which can be thought of as a "filling factor".
The idea that the fractal index of trabecular bone might be related to bone strength is an appealing one, since the fractal index is simple to calculate from clinical CT images of a given bone.
cms.dt.uh.edu /Faculty/ChenP/fractal/dimension.htm   (518 words)

 Fractal Dimension Calculator manual   (Site not responding. Last check: 2007-10-08)
Fractal Dimension Calculator takes a fl and white image in the form of a Macintosh PICT file or clipboard contents and uses the box counting algorithm to estimate the fractal dimension of the object represented in the image.
The fractal (box) dimension D is given by the slope of the linear portion of a log(N(s)) vs log(1/s) graph.
They have been used as examples because the fractal dimensions are known and range from near 1 to 2.
astronomy.swin.edu.au /~pbourke/fractals/fracdim/fdc_orig   (2013 words)

 3.3 Fractal Dimension
Fractals produced by simple iterative scaling procedures like the Koch coastline are very easy to handle analytically.
There's no obvious fractal structure to the quadratic mapping, no hint that a "monster" curve lurks inside, and no simple way to extract an exact fractal dimension.
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)

 Fractal dimension
This constant is the fractal dimension of the Koch curve.
In applications to natural science, one usually takes the point of view that the fractals that occur in nature are well-behaved with respect to the calculation of their dimension.
This is somewhat ironic since the genesis of the ``fractal geometry of nature'' was the rejection that nature should be described by the smooth objects of classical geometry.
www.math.okstate.edu /mathdept/dynamics/lecnotes/node37.html   (767 words)

 Multifractality: Fractal Dimension   (Site not responding. Last check: 2007-10-08)
In either case, it is useful to quantify the properties of the sets of singularities in the signal by calculating their fractal dimensions (8).
The distribution is fractal because there are holes of all sizes between the dust.
Hence, the fractal dimension of the set of singularities is one, the dimension of a line.
www.physionet.org /tutorials/multifractal/dimension.htm   (596 words)

 Measuring Fractal Dimension   (Site not responding. Last check: 2007-10-08)
Fractal Dimension allows us to measure the degree of complexity by evaluating how fast our measurements increase or decrease as our scale becomes larger or smaller.
The box-counting dimension measure is equal to the slope of that line.
The Box-counting dimension is much more widely used than the self-similarity dimension since the box-counting dimension can measure pictures that are not self-similar (and most real-life applications are not self-similar).
www.cs.wisc.edu /~ergreen/honors_thesis/dimension.html   (522 words)

 Fractals - Fractal Dimension   (Site not responding. Last check: 2007-10-08)
The concept of "fractal dimension" is attributed to a 20th century mathematician, Benoit Mandelbrot.
His fractal theory was developed in order to try to more precisely quantify the immense complexity of nature in relatively simple equations.
Fractal dimension was developed as a way to quantify this contradictory complexity.
www.cs.cornell.edu /Courses/cs312/2000fa/handouts/fractals.html   (818 words)

 Fractal Dimension
The fractal dimension of a plane region is 2.
It can be proved that the fractal dimension is always greater than or equal to the topological dimension.
Therefore, the fractal dimension serves as an interpolation of the topological dimension.
homepages.cwi.nl /~bens/6dim.htm   (358 words)

 Repeatability of Fractal Dimension   (Site not responding. Last check: 2007-10-08)
A graph of fractal dimension against time for each of the repeats is shown in figure 7.18.
Fragmented sections of film tend to have a lower fractal dimension than continuous sections, since they resemble points to the larger ruler sizes.
The reason for this discrepancy is not clear, though it is noted that the vertical variance of the patch is largest for this repeat, and that the graph contains a region of rapid expansion, as can be seen in figure 7.17.
www.ph.ed.ac.uk /~ted/thesis/node96.html   (183 words)

 Fractal Frequently Asked Questions and Answers
A2: A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole.
The aim of this first international journal on fractals is to bring together the most recent developments in the research of fractals so that a fruitful interaction of the various approaches and scientific views on the complex spatial and temporal behavior could take place.
NOTICE from Noel Giffin (noel@erich.triumf.ca): Welcome to the Spanky Fractal Database This is a collection of fractal's and fractal related material for free distribution on the net.
www.faqs.org /faqs/fractal-faq   (9659 words)

 Fractal Dimension   (Site not responding. Last check: 2007-10-08)
The fractal dimension of the grey- white matter boundary in the human brain can be used to differentiate degrees of hydrocephalus.
The fractal dimension of the grey white matter boundary in the human brain can be used to differentiate degrees of hydrocephalus.
Applying this to the grey- white matter boundary a decrease in fractal dimension reflects an increasing smoothness in the boundary which is symptomatic of hydrocephalus.
www.cs.ubc.ca /nest/magic/fractal.html   (177 words)

 Fractals And Fractal Dimension   (Site not responding. Last check: 2007-10-08)
To determine the fractal dimension of the aggregate, first consider N(R), the number of particles, that are closer than some distance R away from the center of mass.
For a solid object in one dimension, the number is proportional to R. In two dimensions, this number is proportional to R2.
To measure the fractal dimension, consider a log-log plot, that is a plot of the logarithm of N(R) versus the logarithm of R. The plot should be linear and the slope of the line is the fractal dimension.
www.physics.uc.edu /~pinskia/dla/dlapres/tsld004.htm   (296 words)

 Fractal Geometry - Dimension   (Site not responding. Last check: 2007-10-08)
the dimension of the outline of a forest fire is related to its speed of spread and the difficulty of extinguishing it.
The notion of fractal dimension as 'fuzzyness', for lack of a better word, is an operational one, by which is meant that it is practical or useful as opposed being a precisely defined concept.
The numerical value is intended to be measure of the degree of roughness of a boundadry region.To explore this notion a little further, we engage in a gedankeneksperiment.
www.saltspring.com /brochmann/math/Fractals/fractal-1/fractal-1.00.html   (435 words)

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