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Topic: Fractal geometry


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In the News (Wed 26 Jun 19)

  
  fractal geometry. The Columbia Encyclopedia, Sixth Edition. 2001-05
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 developed from Benoit Mandelbrot’s study of complexity and chaos (see chaos theory).
Fractal geometry has been applied to such diverse fields as the stock market, chemical industry, meteorology, and computer graphics.
www.bartleby.com /65/fr/fractalge.html   (254 words)

  
  Fractal - Wikipedia, the free encyclopedia
Fractals are said to possess infinite detail, and some of them have a self-similar structure that occurs at different levels of magnification.
Fractals of many kinds were originally studied as mathematical objects.
Fractal geometry was also used for data compression and for modelling complex organic and geological systems, for example the growth of trees or the development of river basins.
en.wikipedia.org /wiki/Fractal   (1915 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)

  
 Fractal -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-05)
The term fractal was coined in 1975 by (French mathematician (born in Poland) noted for inventing fractals (born in 1924)) BenoƮt Mandelbrot, from the Latin fractus or "broken".
Fractal geometry was also used for (Click link for more info and facts about data compression) data compression and for modelling complex organic and geological systems, for example the growth of trees or the development of river basins.
Fractals are generally irregular (not smooth) in shape, and thus are not objects definable by traditional (The pure mathematics of points and lines and curves and surfaces) geometry.
www.absoluteastronomy.com /encyclopedia/f/fr/fractal.htm   (2216 words)

  
 Fractal Geometry - Crystalinks
The term "fractal" was coined by Benoit Mandelbrot about 1975 to describe a complex geometrical object that has a high degree of "self-similarity" and a fractional dimension that exceeds the normal, or "topological", dimension ("D") for that type of object.
Fractals are said to possess infinite detail, and they may actually have a self-similar structure that occurs at different levels of magnification.
Fractal geometry is the branch of mathematics which studies the properties and behavior of fractals.
www.crystalinks.com /fractal.html   (2219 words)

  
 Fractals
Fractals are geometrical shapes that, contrary to those of Euclid, are not regular at all.
To make sense of fractal geometry we have to find ways of expressing the shape and complexity in terms of numbers, just as Euclidean geometry uses the notions of angle, length, area or curvature, and the notions of one, two or three dimensions.
For fractals, the counterparts of the familiar dimensions (0, 1,2,3) are known as fractal dimensions.
www.fortunecity.com /emachines/e11/86/mandel.html   (3741 words)

  
 Fractal Page
Fractals, geometrical forms that appear the same on all scales, developed independently of Chaos and at first appeared to be unrelated but after a closer examination they are realized as being closely related.
Fractals are a new and useful tool for expressing Chaos Theory and provide a means for us to explore the geometry of irregular shapes of nature.
Fractals and Fractal Geometry were discovered and developed by Benoit Mandelbrot (1924 -), a Polish born French mathematical physicist who worked for IBM and is considered the father of fractal theory.
members.aol.com /SpinChaos/PageFract.html   (751 words)

  
 Your Fractal
Fractal Geometry and high-speed computers have teamed up over the last thirty or so years to examine irregular patterns and structures found in nature that could not be examined by classical Euclidian geometry.
In fractal geometry parlance, this stop is known as "Bailout." All seeds are programmed to Bailout including the human seed - the MONAD - the seat and root of consciousness.
Fractal Artists use number formulas to create visual images that are tweaked into aesthetically pleasing compositions and further rendered through the light spectrum as the fractalian details unfold.
www.yourfractal.com   (697 words)

  
 Welcome to Fractal Landscapes
If the fractal object is a surface, and it is displaced in a third dimension (i.e.: like a fractal landscape), then the fractal dimension is between 2.0 and 3.0.
If we know that a particular type of object is approximately fractal in nature, and we wish to model it, we need to generate some parameterised model, incorporating features present in the real object (in particular, some sort of controlled randomness), without losing the controllability of the fractal model.
A 'high-quality' source fractal process is one which is stationary (all locations have the same statistical properties) and isometric (has the same properties in all directions).
www.fractal-landscapes.co.uk /maths.html   (2147 words)

  
 Read about Fractal at WorldVillage Encyclopedia. Research Fractal and learn about Fractal here!   (Site not responding. Last check: 2007-11-05)
Once computer visualization was applied to fractal geometry, it presented a powerful visual argument for fractal geometry connecting far larger domains of mathematics and science than had previously been considered, particularly in the realm of
Fractals are generally irregular (not smooth) in shape, and thus are not objects definable by traditional
fractal image compression, as well as a variety of scientific disciplines.
encyclopedia.worldvillage.com /s/b/Fractal_geometry   (1667 words)

  
 Fract-Ed Introduction   (Site not responding. Last check: 2007-11-05)
This is an informal introduction to fractal geometry and is intended to provide a foundation for further experimentation.
Fractal geometry, on the other hand, is the 'geometry of nature', and with it we can attempt to describe and mimic nature in a way that was never before possible.
Fractal geometry was founded upon the work of many great mathematicians of the last two centuries.
www.ealnet.com /ealsoft/intro.htm   (500 words)

  
 Fractal Curves and Dimension
However, the definition of fractals is far from being trivial and depends on a formal definition of dimension.
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.
It's interesting to observe that if a fractal curve serves as a boundary of a plane region the region itself will not be fractal.
www.cut-the-knot.org /do_you_know/dimension.shtml   (1068 words)

  
 The Fractal Microscope   (Site not responding. Last check: 2007-11-05)
With fractal geometry we can visually model much of what we witness in nature, the most recognized being coastlines and mountains.
Fractals are used to model soil erosion and to analyze seismic patterns as well.
There are definitely uses for fractals within the classroom, such as introducing similarity (although the Mandelbrot set is only quasi self-similar), density, infinity, vector addition, division and reduction of fractions, scale and magnification, and pattern discovery.
archive.ncsa.uiuc.edu /Edu/Fractal/Fractal_Home.html   (645 words)

  
 [No title]
Fractals often exhibit self-similarity, which means that various copies of an object can be found in the original object at smaller size scales.
Fractal geometry is an important tool in the analysis of phenomena, ranging from rhythms in music melodies to the human heartbeat and DNA sequences.
Fractal geometry is not only fun to play around with, but it is also amazing to look at.
www.facstaff.bucknell.edu /udaepp/090/w3/toddw.htm   (1394 words)

  
 Fractals   (Site not responding. Last check: 2007-11-05)
Mathematically, fractals are pictures that result from iterations of nonlinear equations, usually in a feedback loop.
By studying fractals, mathematicians have a whole new geometry for describing the universe, beyond the boundaries of Euclidean geometry.
Fractals mirror these irregular shapes, thereby allowing us to study and understand nature by understanding fractals.
home.inreach.com /kfarrell/fractals.html   (480 words)

  
 Fractal Evolution
Fractal geometry, as its name implies, is a geometry focusing on the description of geometrical structures, and structuring, in fract[ion]al space.
Another important implication of fractal biology and evolution is that at long last, we have solid support for the Gaia hypothesis that the planet is one organism.
The human being is a fractal of the single cell, the planet is a fractal of the human being.
www.fractal.org /Bewustzijns-Besturings-Model/Fractal-Evolution.htm   (3752 words)

  
 Fractals and Fractal Architecture - City Planning   (Site not responding. Last check: 2007-11-05)
As shown in the previous chapters, fractal geometry is able to describe complex forms, finding out their underlying order and regularity - self-similarity, simple algorithms -, by reproducing the real world and not by an abstraction into pure mathematics - "clouds are not spheres".
Therefore fractal geometry offers a good field for application on cities, moreover, even most of the "planned" cities, using the geometry of Euclid and showing simplicity of form, have been adapted to their context in more natural ways and therefore also contain some "organic" growth and irregularity[01].
Thus the fractal dimension may be between 1 and 2, the final value depending upon what is measured - which is the final definition of the course of the boundary - and which method of measuring is used - e.g.
www.iemar.tuwien.ac.at /modul23/Fractals/subpages/62Cityplanning.html   (4682 words)

  
 Fractal Geometry
Hence, a fractal landscape which consists of a hill covered with tiny bumps would be closer to two dimensions, while a landscape composed of a rough surface with many average sized hills would be much closer to the third dimension.
Fractal dimensions are not limited to being between zero and one.
Hence, a fractal image is a graphical representation of the points which diverge, or go out of control, and the points which converge, or stay inside the set.
library.thinkquest.org /3493/noframes/fractal.html   (3472 words)

  
 ipedia.com: Fractal Article   (Site not responding. Last check: 2007-11-05)
Fractals were originally studied as mathematical objects, and the term "fractal" has been given various precise definitions by mathematicians.
Fractal geometry, the branch of mathematics which studies fractals, has recently found numerous applications to science, technology, and computer-generated art.
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 few would argue that a definition of "fractal" should include the real line.
www.ipedia.com /fractal.html   (1778 words)

  
 The Fractal Microscope: Fractal Geometry   (Site not responding. Last check: 2007-11-05)
So a fractal landscape made up of a large hill covered with tiny bumps would be close to the second dimension, while a rough surface composed of many medium-sized hills would be close to the third dimension (Peterson, 1984).
We can demonstrate this by first defining a fractal set according to Nn = C/rnD where Nn is the number of fragments with a linear dimension rn, C is a constant, and D is the fractal dimension (Turcotte, 1992).
Just as it was possible to find a fractal dimension between zero and one, we can apply the same methods to a square and find dimensions between zero and two.
archive.ncsa.uiuc.edu /Edu/Fractal/Fgeom.html   (1089 words)

  
 Fractal Geometry at Polytechnic School
Fractal geometry is the geometry of chaos theory in the sense that fractal geometry may be used to visually illustrate the behavior of chaotic dynamical systems.
Its "father" is indisputably Benoit Mandelbrot who coined the word "fractal" in 1975, the same year in which James Yorke gave chaos theory its name.
fractal dimension, a way of characterizing the roughness of fractal shapes.
home.earthlink.net /~srrobin/fractal.html   (523 words)

  
 Fractal Geometry
But Mandelbrot conceived and developed a new fractal geometry of nature based on the fourth dimension and Complex numbers which is capable of describing mathematically the most amorphous and chaotic forms of the real world.
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.
For a fractal as a geometric figure not only has irregular shapes - the zig zag world of nature - but there is lurking in the disorder a hidden order in these irregular shapes.
www.fractalwisdom.com /FractalWisdom/fractal.html   (2550 words)

  
 sci.fractals FAQ   (Site not responding. Last check: 2007-11-05)
Fractals also describe many real-world objects, such as clouds, mountains, turbulence, coastlines, roots, branches of trees, blood vesels, and lungs of animals, that do not correspond to simple geometric shapes.
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.
Topics include: challenges of using fractals in the classroom, new ways of generating art and music, the use of fractals in clothing fashions of the future, fractal holograms, fractals in medicine, fractals in boardrooms of the future, fractals in chess.
www.faqs.org /faqs/sci/fractals-faq   (12472 words)

  
 Encyclopedia: Fractal geometry   (Site not responding. Last check: 2007-11-05)
What is truly surprising is that the boundary of the Mandelbrot set also has a Hausdorff dimension of 2.
Altivec Fractal Carbon (http://www.daugerresearch.com/fractaldemos/altivecfractalcarbon.html) Mac-based benchmarking utility, using fractals to determine performance.
Babya Fractal Studio (http://babyasystem.portal.dk3.com) free Windows-based fractal generator, using fractals to create bitmap images.
www.nationmaster.com /encyclopedia/Fractal-geometry   (1741 words)

  
 22fall~1.htm Fractal Geometry
The purpose of this unit is to provide an initial experience with fractals that is rich in creativity and inquiry for geometry students and to provide an introduction into fractal geometry for teachers.
Historically, one of the early questions leading to the geometry of fractals concerned the length of the coastline of Britain.
After calculating values of D for various fractals, students can be guided to compare them, and to observe that larger values of D belong to "rougher" fractals.
www.woodrow.org /teachers/mi/1993/22fall.html   (2886 words)

  
 FRACTAL CHAOS: the Philosophy of Freedom and Self Determination
It is possible to apply this new knowledge to better understand your life, to live autonomously, based on freedom, and your own contact with the Source of the Universe, the Infinite.
These fractals are in space, as shown by the graphics on these webs, and in time, as shown in all life, including your own.
The Story of Benoit B. Mandelbrot and the Geometry of Chaos: an overview of the life story and key discoveries of the mathematician who started the Science of Chaos.
www.fractalwisdom.com /FractalWisdom/index.html   (2073 words)

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