61. The Math Forum - Math Library - Fractals World of fractals Adam Lerer A general site about fractals, including image galleries,animations, descriptions of different types of animations, explanations http://mathforum.org/library/browse/static/topic/fractals.html

Browse and Search the Library Home Math Topics Dynamical Systems : Fractals Library Home Search Full Table of Contents Suggest a Link ... Library Help Selected Sites (see also All Sites in this category The Fractal Microscope - Education Group, National Center for Supercomputing Applications (NCSA) An interactive tool for exploring the Mandelbrot set and other fractal patterns. With the Fractal Microscope students can enjoy the art as they master the science of mathematics, and can address a variety of topics in the K-12 curriculum including scientific notation, coordinate systems and graphing, number systems, convergence, divergence, and self-similarity. The program is designed to run in conjunction with NCSA imaging tools such as DataScope and Collage. more>> Fractals - Cynthia Lanius This lesson plan for exploring fractals is designed so 4th through 8th grade students can work independently and be assessed innovatively. It conforms to the 1989 NCTM standards, and provides links to other fractal sites. Contents: Why study fractals? Making fractals: Sierpinski Triangle, Sierpinski Meets Pascal, Jurassic Park Fractal, Koch Snowflake. Fractal Properties: Self-similarity, Fractional dimension, Formation by iteration. Teacher-to-Teacher notes; Fractals on the Web. more>> Fractals (Mathematics Archives) - University of Tennessee, Knoxville (UTK)

62. The Math Forum - Math Library - Fractals for K12 general mathematics education. ClarisWorks and Acrobat (PDF) files to download;videotapes. more . Genuine fractals Software - Altamira Group, Inc. http://mathforum.org/library/topics/fractals/?start_at=201

63. The Science House - Chaos & Fractals Spring 2001 Chaos and fractals Resources general Fractal and ChaosResources. Dynamical Systems and Technology Project this is http://www.science-house.org/student/bw/chaos/resources.html

Spring 2001 Chaos and Fractals Resources General Week 1 Week 2 Week 3 ... Programs General Fractal and Chaos Resources Dynamical Systems and Technology Project - this is where the books we use came from Center for Polymer Studies at BU - an excellent site that shows real live fractals and chaos. Good hands on activities too. Fractal Frequently Asked Questions and Answers Fractals Unleashed - excellent descriptions and examples in the tutorial section The Chaos Experience - good real world examples of chaos Fantastic Fractals The Chaos Metalink - links to every conceivable chaos related site The Fractal World of Astrology - why not?

64. Tutorials On Self-Organisation, Complexity And Artificial Life a fivepart online course for everyone - by Matthew A. Trump. Exploring Chaos fractals - a web Book by Informit. Complexity and general Systems Theory. http://www.calresco.org/tutorial.htm

FAQs, Introductions and Tutorials (For more general introductions on these subjects see our Themes page) Artificial Intelligence Artificial Life Autopoiesis Cellular Automata ... Systems Thinking Frequently Asked Questions (FAQs) Artificial Intelligence - comp.ai newsgroup (7 part Text FAQ) Artificial Life - comp.ai.alife newsgroup (Original FAQ) Cellular Automata - comp.theory.cell-automata newsgroup (Original FAQ) Evolutionary Computation - comp.ai.genetic newsgroup Fractals - sci.fractals newsgroup Fuzzy Systems - comp.ai.fuzzy newsgroup Genetic Programming - comp.ai.genetic newsgroup Neural Nets - comp.ai.neural-nets newsgroup Non-Linear Systems - sci.nonlinear newsgroup Robotics - comp.robotics newsgroup Self-Organizing Systems - comp.theory.self-org-sys newsgroup (listed in order of difficulty or detail per subject category) For more general introductions see Themes and for more specialised treatments see Online Papers Artificial Intelligence Artificial Intelligence for the Beginner - incl NNs and GAs by Mark Lambourne An Introduction to AI - essays on most areas by Generation 5 An Introduction to the Science of Artificial Intelligence - by Thinkquest Introduction to Artificial Intelligence - course notes (some Postscript) by Michael Gasser The Pattern Recognition Basic of AI - introductory Book by Donald Tveter Artificial Life Introduction to CNS/Ph175: Artificial Life - by Chris Adami Artificial Life - introduction by Anders Kaplan Artificial Life an Interactive Essay - by Stewart Dean An Introduction to Artificial Life - paper by Moshe Sipper

65. Fractals In general, fractals arising in a chaotic dynamical system have a far more complexscaling relation, usually involving a range of scales that can depend on http://www.drchaos.net/drchaos/Book/node9.html

Next: References and Notes Up: Some Terminology: MapsFlows, Previous: Binary Arithmetic Fractals Nature abounds with intricate fragmented shapes and structures, including coastlines, clouds, lightning bolts, and snowflakes. In 1975 Benoit Mandelbrot coined the term fractal to describe such irregular shapes. The essential feature of a fractal is the existence of a similar structure at all length scales. That is, a fractal object has the property that a small part resembles a larger part, which in turn resembles the whole object. Technically, this property is called self-similarity and is theoretically described in terms of a scaling relation. Chaotic dynamical systems almost inevitably give rise to fractals. And fractal analysis is often useful in describing the geometric structure of a chaotic dynamical system. In particular, fractal objects can be assigned one or more fractal dimensions, which are often fractional ; that is, they are not integer dimensions To see how this works, consider a Cantor set , which is defined recursively as follows (Fig.

66. Science Watch - Mathematics Links List of Great Math Programs computer algebra, geometry, fractals, AI, games an expressionevaluator, unit converter, 2D/3D graphing, general ledger, financial http://cgd.best.vwh.net/home/sci/mathl.htm

Mathematics Links Table of Contents General Indexes Organizations College and University Mathematics Departments General Resources ... TeX Information General Indexes Yahoo Mathematics links Excite's Mathematics links InfoSeek Guide: Mathematics ZIA Mathematics Resources - well done Galaxy Mathematics links WWW Virtual Library: Mathematics - Based at Florida State University Mathematics Information Servers - A large worldwide list, including university math departments, mathematical societies, research institutes, commercial pages, journals, preprint servers, and subject area pages. Based at PSU. Mathematical Resources on the Web - Extensive list, based at the University of Florida. Includes academic sites that had particularly good indexes. Math Sites on the Internet - Part of The Math Forum Useful Internet Math Resources - At Oklahoma State U, includes extensive list of links by David Wright Math-Net Links to the Mathematical World - Mostly lists of indexes and other higher-level lists University of Pennsylvania Electronic Sources for Mathematics Caltech Guide to Mathematical Resources CSC Mathematical Topics - Traditional list of links CSC TopMath - Allows anyone to add their own site. However, it is rather unusual as an index, since the interface doesn't allow direct access to the URLs.

67. Past Top Ten Mathematics Links of A+ Math. ; Go to A+ Math. DiscoverySchool WebMath Description Top. fractals, Chaos. The Beauty of Chaos...... general. A+ Math http://www.learn.motion.com/lim/links/matlinks/matlinks.htm

Learning in Motion's Top Ten List Mathematics General Arithmetic Number, Data, Chance Geometry, Proofs ... Current Top Ten List General A+ Math Description of "A+ Math." Go to A+ Math Discovery School WebMath Description of "Discovery School WebMath." Go to Discovery School WebMath Eric Weisstein's World of Mathematics Description of "Eric Weisstein's World of Mathematics." Go to Eric Weisstein's World of Mathematics Explore Math Description of "Explore Math." Go to Explore Math Game Theory.net Description of "Game Theory.net." Go to Game Theory.net Interactive Mathematics Online Description of "Interactive Mathematics Online." Go to Interactive Mathematics Online The Math Forum Description of "The Math Forum." Go to The Math Forum Mathematics Lessons that are fun fun fun! Description of "Mathematics Lessons that are fun fun fun!" Go to Mathematics Lessons that are fun fun fun! Platonic Realms Description of "Platonic Realms." Go to Platonic Realms Purple Math Description of "Purple Math."

68. Modelling/Simulation Of Randomly And Partially Ordered Cond.matter: General Department of XRay Structural Research general aspects of the simulation and analysisof random structures. The figure shows two examples for random fractals. http://www.ifw-dresden.de/ifs/32/mod_sim/mod_sim_hhma_e.htm

Department of X-Ray Structural Research General aspects of the simulation and analysis of random structures IFS (Institute of Solid State Analysis and Structural Research) IFW Dresden Deutsch Models for random and partially ordered structures We consider both analytical models and computer simulations. The analytical models are essentially random point fields, random mosaics, random germ-grain models, and others whereas computer models are especially applied for the simulation of partially ordered systems. Our tunable random surface fractal gives a good example for the variability of the analytical methods. The fractal is constructed by the set-theoretical union of a series of random germ-grain models which are self-similar in a statistical sense. The model can be generated in two or three dimensions. It is variable with respect to the maximum size of the grains, the shape (spheres, random polyhedra etc) of the (convex) grains, the volume fractions of the two phases separated by the fractal interface (or border line in two dimensions), the parameters of the self-similarity transformation and the fractal dimension. Analytical expressions for several structure parameters are given, e.g., for the fractal dimension and the correlation function. The figure shows two examples for random fractals. This figure shows two examples for random fractals. These images were genereated in the two-dimensional space. Disks were chosen as grains. The area fraction of the black regions is 0.3 and 0.95, and the fractal dimension is 1.3 and 1.7 for the image on the left and the right hand side, respectively.

69. World Of Fractals - Understanding Fractals If you'd like a more indepth look into fractals, check out Types of fractals andThey're Equations or go to general Interest fractals or The Geometry Junkyard http://www.angelfire.com/art2/fractals/learning.htm

World of fractals By Adam Lerer Home Image Galleries Animations Types of Fractals ... Links Understanding Fractals These three lessons have been created to give people with no knowledge of the mathematics of fractals a basic understanding. If you'd like a more in-depth look into fractals, check out Types of Fractals and They're Equations or go to General Interest Fractals or The Geometry Junkyard Creating Fractals: The Basics The concepts used to create fractals - no mathematical knowledge required Creating Fractals: The Mathematics The mathematics required to create fractals - some algebra knowledge required Fractals and Chaos Theory in the Real World How fractals relate to the world around us and what mathematicians use them for Enjoy! Home Image Galleries Animations Types of Fractals ... Links

70. Order In Chaos: Fractals And Iterated Function Systems fractals and Iterated Function Systems In general, an IFS is a set of such functions,called affine transformations, over a space like R 2 . The notation for a http://www.geocities.com/chelikuzhiyil/fractal.html

Fractals and Iterated Function Systems A fractal is an object that is made up of an infinite number of smaller copies of itself. An example of a well known fractal is the Sierpinski Triangle: The Sierpinski Triangle contains an infinite number of black and white triangles. It contains an infinite number of smaller copies of itself. Thus, it is a fractal. Now that you have seen a picture of a fractal, look at the sequence of pictures below. These pictures show how a fractal is made. Theoretically, we would need to make an infinite number of pictures to make a fractal, since a fractal is supposed to have infinite complexity, but for all practical purposes, we only need a few repetitions. After a few pictures, our precision is limited by the size of the pixel on the computer screen. The process of generating one picture from the previous one in the sequence of images above is called an iteration . So, doing several iterations will yield an image that looks like a fractal. So a fractal can be described by its iteration process. That's kind of like saying that a spring is described by its coiling process. But it's true. Given an iteration process, we can draw the fractal. So how do we describe an iteration? For example, how would we describe the iteration process for the Sierpinski Triangle above? Well, each picture is obtained from the last by taking the last picture, shrinking it down to a quarter of its size (shortening each side by a half of its original length), making three copies of this shrunken image, and then placing one copy of the image in the top left corner of the new image, another copy in the bottom left, and the last copy in the bottom right. Now we have formed the new image from the previous image.

71. Citation This general introduction to fractals and chaos is similar in style toan article in Scientific American. Its main goal is to generate http://portal.acm.org/citation.cfm?id=98104&coll=Portal&dl=GUIDE&CFID=11111111&C

72. Lecture Notes On Physics And Fractals Talk into account quantum mechanics and general relativity, and describes quantum numbersgeometrically. Goal of this talk to show that fractals have something to http://www.arches.uga.edu/~mathclub/fractalnotes.html

Lecture Notes on Fractals and Physics Alan Dion October 2, 2001 Introduction and Motivation Two successful theories: General Relativity Based on physical principles "The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion." A very geometrical theory, based in Riemannian space-time. But whence T mn A purely local theory - the metric is differential What's the global topology of the universe? Quantum Mechanics Based on strange mathematical postulates Probabilistic DxDp ³ Ñ/2 l = Ñ/p t = Ñ/E The observed properties of the quantum world cannot be reproduced by Riemannian geometry. Quantum field-particles approach has not successfully described gravitation. We need to try and extend these theories Observations In quantum mechanics, changing the resolution of measurement dramatically affects the results. Indeed, if your ruler measure centimeters, what sense does an angstrom make? General relativity accounts for all differentiable transformations of coordinate systems. Maybe we could extend to continuous transformations. And, noting the above remark on resolutions, relativity should account for transformation of resolution. Scale dependence in quantum mechanics is only implicitly included through Born's statistical interpretation.

73. Learning About Fractals A1 _Chaos_ is a good book to get a general overview and history. _fractals Everywhere_is a textbook on fractals that describes what fractals are and how to http://www.faqs.org/faqs/fractal-faq/section-1.html

Single Page Top Document: Fractal Frequently Asked Questions and Answers Previous Document: News Headers Next Document: What is a fractal? Learning about fractals http://millbrook.lib.rmit.edu.au/exploring.html Exploring Chaos and Fractals http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html Fractal Microscope http://is.dal.ca:3400/~adiggins/fractal/ Dalhousie University Fractal Gallery http://acat.anu.edu.au/contours.html "Contours of the Mind" http://www.maths.tcd.ie/pub/images/images.html Computer Graphics Gallery http://wwfs.aist-na.ac.jp/shika/library/fractal/ SHiKA Fractal Image Library http://www.awa.com/sfff/sfff.html The San Francisco Fractal Factory. http://spanky.triumf.ca/www/spanky.html Spanky (Noel Giffin) http://www.cnam.fr/fractals.html Fractal Gallery (Frank Rousell) http://www.cnam.fr/fractals/anim.html Fractal Animations Gallery (Frank Rousell) Top Document: Fractal Frequently Asked Questions and Answers Previous Document: News Headers Next Document: What is a fractal? Single Page By Archive-name By Author ... Help Send corrections/additions to the FAQ Maintainer: stepp@marshall.edu

74. Zeal.com - United States - New - Computing - Multimedia - Graphics - Fractals - A great resource for United States New - Computing - Multimedia- Graphics - fractals - Art Images - general Galleries. Find http://www.zeal.com/category/preview.jhtml?cid=10024815

75. Books Part I is concerned with the general theory of fractals and their geometry, coveringdimensions and their methods of calculation, plus the local form of http://geometricarts.freeyellow.com/books.htm

Welcome to my little bookshop.I have tried to compile some useful books to help you to enrich your understanding of geometry and to get full enjoyment out of this wonderful arena: Geometric Arts. Good hunting! Index Op art (optical art) Escher Geometry Fractals ... Geometrical Patterns Fractals Fractals : The Patterns of Chaos : A New... by John Briggs Scientists have discovered that systems in transitional states between order and chaos possess certain patterns with unique, predictable qualities. These patterns are called "fractals." In essence, they are visual images or pictures of chaos at work. Now comes a breathtaking visual tour of this exciting new scientific frontier. 207 photographs, 178 in full color; 49 line drawings Chaos and Fractals : New Frontiers of... by Heinz-Otto Peitgen, Dietmar Saupe Fascinating and authoritative, Chaos and Fractals: New Frontiers of Science is a truly remarkable book that documents recent discoveries in chaos theory with plenty of mathematical detail, but without alienating the general reader. In all, this text offers an extremely rich and engaging tour of this quite revolutionary branch of mathematical research. The Computational Beauty of Nature :...

76. Sierpiñski Fractals (S) In one way they produce images that are more truly fractal than general CAT fractals,because if you magnify a Sierpinski fractal, the structure of the whole http://www.alunw.freeuk.com/sierpinskiroom.html

Sierpiñski fractals (S) Sierpiñski was the mathematician who discovered the first fractal, the Sierpiñski triangle. You can find out more about it on the Spirofractal tour The process used to construct the triangle can be generalised slightly by picking a number of fixed points and picking one of them at random as a a starting point. Then we pick one of the other fixed points at random, and move a fixed proportion (less than one) of the distance between the two points, either towards the point or away from it. We repeat this indefinitely, coloring points according to how often they are visited. In fact Sierpiñski fractals are really just a special kind of C.A.T. fractal. However they are usually very recognisable. In one way they produce images that are more truly fractal than general C.A.T. fractals, because if you magnify a Sierpinski fractal, the structure of the whole is repeated in each part, without the distortions that occur in general C.A.T. fractals. This is a randomly generated Sierpiñski pentagon. Notice that whereas the Sierpiñski triangle can be decomposed into its component parts, which are all identical (except half size) copies of the whole image, the same cannot be done for the Pentagon. Different scaled down copies of the fractal overlap. Fractals that do not overlapping parts are called

77. Historical Notes: History Of Fractals somewhat mixed success, leading to the introduction of multifractals with more parameters,but Mandelbrot’s general idea of the importance of fractals is now http://www.wolframscience.com/reference/notes/934a

From: Stephen Wolfram, A New Kind of Science Notes for Chapter 5: Two Dimensions and Beyond Section: Substitution Systems and Fractals Page 934 History of fractals. The idea of using nested 2D shapes in art probably goes back to antiquity; some examples were shown on page 43. In mathematics, nested shapes began to be used at the end of the 1800s, mainly as counterexamples to ideas about continuity that had grown out of work on calculus. The first examples were graphs of functions: the curve on page 920 was discussed by Bernhard Riemann in 1861 and by Karl Weierstrass in 1872. Later came geometrical figures: example (c) on page 191 was introduced by Helge von Koch in 1906, the example on page 187 by Waclaw Sierpinski in 1916, examples (a) and (c) on page 188 by Karl Menger in 1926 and the example on page 190 by Paul Lévy in 1937. Similar figures were also produced independently in the 1960s in the course of early experiments with computer graphics, primarily at MIT. From the point of view of mathematics, however, nested shapes tended to be viewed as rare and pathological examples, of no general significance. But the crucial idea that was developed by Benoit Mandelbrot in the late 1960s and early 1970s was that in fact nested shapes can be identified in a great many natural systems and in several branches of mathematics. Using early raster-based computer display technology, Mandelbrot was able to produce striking pictures of what he called fractals. And following the publication of Mandelbrot’s 1975 book, interest in fractals increased rapidly. Quantitative comparisons of pure power laws implied by the simplest fractals with observations of natural systems have had somewhat mixed success, leading to the introduction of multifractals with more parameters, but Mandelbrot’s general idea of the importance of fractals is now well established in both science and mathematics.

78. Fractals-TMR Network Topics In this sense, it is still missing a general and precise definition of the circumstancesleading to fractals and SOC and the identification of common features http://pil.phys.uniroma1.it/eec3.html

Fractals-TMR Network- cn:FMRXCT980183 TMR NETWORK: FRACTAL STRUCTURES AND SELF-ORGANIZATION Research Topic Project objectives Scientific Originality Research Method ... Work Plan RESEARCH TOPIC In the last years there has been a growing interest in the understanding a vast variety of scale invariant and critical phenomena occurring in nature. Experiments and observations indeed suggest that many physical systems develop spontaneously correlations with power law behaviour both in space and time. Pattern formation, aggregation phenomena, biological systems, geological systems, disordered materials, clustering of matter in the universe are just some of the fields in which scale invariance has been observed as a common and basic feature. However, the fact that certain structures exhibit fractal and complex properties does not tell us why this happens. A crucial point to understand is therefore the origin of the general scale-invariance of natural phenomena. This would correspond to the understanding of the origin of fractal structures and of the properties of Self- Organized Criticality (SOC) from the knowledge of the microscopic physical processes at the basis of these phenomena.

79. Final Topics Photorealistic Rendering. 3D Modeling fractals (general descriptions)Uses; Methods Recursive Subdivision (with random displacement); Iterative http://www.cs.rit.edu/~ncs/Courses/570/Topics_Final.shtml

20011 Graphics Final Topics Tuesday, 2/25/03, 6:00-8:00 P.M. Room: 12-3215 The OpenGL Graphics Pipeline What processes take place in the OpenGL graphics pipeline and in what order? 3-D graphics World to camera/eye coordinate transformation - why? Why are two coordinate systems generally used for 3-D graphics? How many pieces of info are needed to establish location of the camera/eye coordinate system, what are they? What are the 4 steps in world to camera/eye coordinate transformation and what is accomplished by each? Viewing Parameters: eye, look-at, up What is the function of each in the camera/eye coordinate system? How are u v , and n specified for the camera/eye coordinate system? What is the affect on image when each is altered? (homework 2) Which coordinate system are they specified in? 3D Transformations: scale, rotate, translate What are the equations of transformation? the matrices? 3D Clipping When done and why? How are 2-D and 3-D clipping the same? How different? (compare and contrast)? How is the view volume specified: components/coordinates, i.e., what sets its boundaries? Which coordinate system specified in? How are clipping and hidden surface/line removal the same? How different? (compare and contrast)

80. Wiley Canada :: Fractals, Random Shapes And Point Fields: Methods Of Geometrical Wiley Canada Mathematics Statistics Probability Mathematical Statistics general Probability Mathematical Statistics fractals, Random Shapes and http://www.wileycanada.com/cda/product/0,,0471937576,00.html

Shopping Cart My Account Help Contact Us By Keyword By Title By Author By ISBN By ISSN Wiley Canada Fractals, Random Shapes and Point Fields: Methods of Geometrical Statistics Related Subjects Bayesian Analysis Multivariate Analysis Related Titles By These Authors Comparison Methods for Stochastic Models and Risks (Hardcover) Stochastic Geometry and Its Applications, 2nd Edition (Hardcover) Aspects of Statistical Inference (Hardcover) A. H. Welsh Design and Inference in Finite Population Sampling (Hardcover) A. S. Hedayat, Bikas K. Sinha Modern Mathematical Statistics (Hardcover) Edward J. Dudewicz, Satya N. Mishra Modern Probability Theory and Its Applications (Paperback) Emanuel Parzen Limit Theorems in Change-Point Analysis (Hardcover) Fractals, Random Shapes and Point Fields: Methods of Geometrical Statistics Dietrich Stoyan, Helga Stoyan ISBN: 0-471-93757-6 Hardcover 406 Pages September 1994 US $160.00

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