81. Fractals Julia Set code. mandelbrot Set. The mandelbrot set is the most complex object in mathematics, its admirers like to say. mandelbrot Set code. Related Information. http://www.cecm.sfu.ca/~kghare/numeric/fractal.html

82. ThinkQuest : Library : Yarra Valley Fractals Arguably the most famous of all the computergenerated images of fractals, the mandelbrot set was discovered by Benoit B. mandelbrot, the god-father of fractals http://library.thinkquest.org/12631/info/datasets.htm

Index Math Fractals Yarra Valley Fractals Welcome to our web page on Fractals! Visit Site 1997 ThinkQuest Internet Challenge Languages English Students Steven Yarra Valley Anglican School, Ringwood, Australia Myles Yarra Valley Anglican School, Ringwood, Australia Jason Yarra Valley Anglica School, Ringwood, Australia Coaches Ian Yarra Valley Anglican School, Ringwood, Australia Ian Yarra Valley Anglican School, Ringwood, Australia Ian Yarra Valley Anglican School, Ringwood, Australia Want to build a ThinkQuest site? The ThinkQuest site above is one of thousands of educational web sites built by students from around the world. Click here to learn how you can build a ThinkQuest site. Privacy Policy

83. Second Story. Wheels The nature of fractals is reflected in the word itself, coined by mathematician Benoit B. mandelbrot from the Latin verb frangere, to break, and the related http://library.thinkquest.org/18222/root/story3/page1/page1.htm?tqskip=1&tqskip1

84. Stomp*3 - Fractals Technical SubSection. This description of fractals will be limited to the two most common, popular, and simplest fractals the mandelbrot and Julia set. http://stompstompstomp.com/fractals/

Weblogs Pictures Technical Other ... Links Fractals I've always been curious about fractals, and recently I've spent the time necessary to be familiar with the basics of fractals. You don't need to be a geek, a mathematics God, or a computer programmer to appeciate the beauty of fractal images. This section of my personal website is split into two distinct subsections: a gallery of beautiful and interesting fractal images for your viewing pleasure, and a technical section describing the theroy of fractals and linking to some simple software I've created to generate fractal images. Gallery Sub-Section Technical Sub-Section This description of fractals will be limited to the two most common, popular, and simplest fractals: the Mandelbrot and Julia set. The theroy behind these fractals is applicable to other fractal sets, though. A fractal is a set of complex numbers. Which numbers are part of the set is determined by an iterative function which must be evaluated for an infinite number of iterations at every complex number. Gee, it sounds like that would take a long time. The Mandelbrot and Julia sets are defined by the equation z n+1 = z n + c z and c are both complex numbers.

85. Mandelbrot And Julia Fractals > MagentaStudios | CafePress , Flower I, », Flower II, », Poppies, », mandelbrot and Julia fractals, », Fractal Snowflake, », Fractal Heart, », Fractal Heart II, », Your Brain on Drugs, http://www.cafeshops.com/magentastudios/137793

Help Order Status Shop Home Politics ... Sacred Symbols Mandelbrot and Julia Fractals Fractal Snowflake Fractal Heart Fractal Heart II Your Brain on Drugs ... Black and White Mandelbrot Set This store is powered by CafePress.com

86. Fractals, Chaos, And Cosmic Autopoiesis mandelbrot s fractals are the brainchild of mathematics and computergenerated technology. They help to illustrate iterations, bifurcations http://www.bizcharts.com/stoa_del_sol/plenum/plenum_5.html

Home The Logos Continuum The Cosmic Plenum The Imaginal Within The Cosmos ... The Cosmic Plenum : Fractals, Chaos, and Cosmic Autopoiesis In his theory of the Implicate Order, the late quantum physicist David Bohm refers to fractals in his study of the holomovement, the plenum that powers the inner universe almost in the sense of a feedback loop of unfolding-enfolding between the implicate and explicate orders of the cosmos. Fractal geometry shows that *shapes have self-similarity at descending scales.* Fractals can be generated by iteration; they are characterized by "infinite detail, infinite length, no slope or derivative, fractional dimension and self-similarity." Basically, the "system point folds and refolds in the phase space with infinite complexity." [John Briggs and F. David Peat, TURBULENT MIRROR, Harper & Row, 1989. p. 95] Benoit Mandelbrot, one of the world's mathematical giants on fractals, said that "fractal shapes of great complexity can be obtained merely by repeating a simple geometric transformation, and small changes in parameters of that transformation provokes global changes." In essencethrough a predictable, orderly process the "simple iteration appears to liberate the complexity hidden within it, thus giving access to creative potential." [Ibid, p. 104] Thus, in that misnomer called chaos theory, mathematicians and physicists have discovered an *underlying order,* a kind of memory operating in non-linear, evolving systems. Fractal geometry illustrates that shapes have self-similarity at descending scales. In other words, the form, the *information,* is enfoldedalready present in the depths of the cosmos. So this is reminiscent of the Implicate Order. Iteration liberates the complexity hidden within it. It is not dissimilar to Bohm's law of holonomy: a "movement in which new wholes are emerging." [David Bohm, WHOLENESS AND THE IMPLICATE ORDER, Ark Paperbacks, 1983, pp. 156-157.]

87. Mandelbrot On Fractals, Academia, And Industry mandelbrot on fractals, Academia, and Industry. By Akshay Patil staff writer. The Tech had an opportunity to talk to math and physics http://www-tech.mit.edu/V121/N63/Mandelbrot.63f.html

Mandelbrot on Fractals, Academia, and Industry By Akshay Patil staff writer The Tech had an opportunity to talk to math and physics legend Benoit B. Mandelbrot during his short visit to MIT. One of the fathers of fractal science, Mandelbrot discovered a mathematical set of numbers whose graphical representation is so stunning that it is often considered the face of fractals and chaos today. The Tech: Do you have any personal heroes and inspirations that have driven you over the years? Benoit Mandelbrot: For a long time my hero was John von Neumann, who was, among other things, one of the pioneers of computers. I was a post-doc with von Neumann when Dr. von Neumann died and he was my hero because he succeeded during his life in doing work in mathematics and application based technologies; all without compromising his perfectly rigorous manner of doing things. In time, more heroes appeared. One that is not so widely known I think, a pity, is a Spaniard who lived a hundred years ago, his name was Santiago Ramon y Cajal. Do you know his name? Ramon y Cajal was a doctor in Spain who described the structures of the nervous system, which is made of molecules, if you wish, which are the neurons, and atoms, which are parts of neurons, and how they interact. He then drew pictures of all these neurons. It was so perfect, so early, that in the early 1950s when neuron anatomy awoke again, because of new progress here at MIT, my friends at MIT were using as the reference for the nervous system, a book, first published in Spanish 60 years before. They were using the French translation from 1903 .

88. Sprott's Fractal Gallery fern (16,353 bytes) Fractal fern; henon (10,764 bytes) - Henon map with basin manbrot (21,408 bytes) - mandelbrot set; predator (36,669 bytes) - Predator-prey http://sprott.physics.wisc.edu/fractals.htm

Sprott's Fractal Gallery Awards Received MIDI Fractal background music courtesy of Forrest Fang Fractal of the Day Every day at a few minutes past midnight (local Wisconsin time), a new fractal is automatically generated by a variation of the program included with the book Strange Attractors: Creating Patterns in Chaos by Julien C. Sprott . The figure above is today's fractal displayed in low (320 x 200) resolution. Click on it or on any of the cases below to see them at higher (640 x 480) resolution with a code that identifies them according to a scheme described in the book. Older Fractals of the Day are saved in an archive , which you can access. If your browser supports Java, you might enjoy the applet that creates a new fractal image every five seconds or so. If you would like to place the Fractal of the Day on your Web page, you may do so provided you mention that it is from Sprott's Fractal Gallery and you provide a link back to this page. fracday0.gif Today's fractal of the day fracday1.gif

89. Citations Fractals And Scaling In Finance - Mandelbrot BB mandelbrot, fractals and Scaling in Finance (SpringerVerlag, New York, 1997). 11 BB mandelbrot, fractals and Scaling in Finance. http://citeseer.ist.psu.edu/context/520431/0

90. 2003_Mon_File NewYork),1982. 2. BB.mandelbrot, fractals and Scaling in Finance Discontinuity, Concentration, Risk, New York Springer, 1997. http://www.japanprize.jp/e_2003_mon_file.htm

Dr. Benoit B. Mandelbrot Academic Degrees: Ing nieur dipl m , Ecole Polytechnique, Paris California Institute of Technology, Master of Science California Institute of Technology, Professional Engineer in Aeronautics, Facult des Sciences de Paris, Docteur d'Etat s Sciences Math matiques Professional Career: Staff member (Attach , then Charg , then Ma tre de Recherches), Centre National de la Recherche Scientifique, Paris, France Ma tre de Conf rences de Mathematiques Appliqu es, Universit , Lille, France Ma tre de Conf rences d'Analyse Math matique, Ecole Polytechnique, Paris, France Research Staff Member, IBM Thomas J. Watson Research Center, Yorktown Heights NY. IBM Fellow, IBM Thomas J. Watson Research Center Abraham Robinson Adjunct Professor of Mathematical Sciences, Yale University, New Haven, CT. 1993-present IBM Fellow Emeritus, IBM Thomas J. Watson Research Center 1999-present Sterling Professor of Mathematical Sciences, Mathematics Department: Yale University Major Books and Papers: B.B.Mandelbrot

91. Fractals, Chaos And The Mandelbrot Set Workshop Pt. 5 . Here is a short description of the......fractals, Chaos and the mandelbrot Set Workshop Pt. 5. InfoPak Computer Programs http://www.grenvillecc.ca/faculty/jchilds/cmprgdes.htm

Fractals, Chaos and the Mandelbrot Set Workshop - Pt. 5 InfoPak Computer Programs Description Here is a short description of the programs that are available on the disk that accompanies the InfoPak. The first four were written by me, and can be considered freeware. Most of the programmes have detailed explanation screens describing the processes that they illustrate. Try running them with different values and explore! They will all run, menu driven, directly off the floppy. Nothing is zipped, and nothing has to be installed. When you get the disk, type "menu" or "go" at the A:>_ DOS prompt and you're off! It will run faster if you copy the whole thing to your hard drive, but you can do that later. :-) I would be glad to hear from you, and will consider minor custom programing or sharing the source code. Chaos Game : Explains the basic idea of combining randomness and iteration to produce a complex pattern. It lets you play three, four or six corners, fractional three corners, and has good explanation screens. It is the perfect tutorial for introducing the study of chaos at any level. E-mail me and I'll send it to you directly. Population Simulation : This program is a population growth model. It illustrates the ideas discussed on pages 69-80 in Chaos, The Making of a New Science, or chapter 3 in Turbulent Mirror. See fantastic compelxity arise out of an extremely simple math formula repetition. It's what bug poplulations really do!

92. Fractals sci/fractalsfaq/; Canadian National Lab for Particle and Nuclear Physics - spanky.triumf.ca - maintained by Noel Giffin. Encyclopedia of the mandelbrot Set http://www.patlyons.com/research/Fractals.htm

Fractal info Click here for Pat Lyons Home Page Email to Pat Lyons Top Level: [ Teaching Research Community Links ... Feedback Research Level: Neural Nets Fractals My Research Links Book ... Articles Links Level: References Student Business Health ... Miscellaneous Fractals Information about Above Graphic Original Images - click here to see the original sequence of eight images that I created using the Fractal Microscope. I then used Macromedia Flash to create the above graphic showing the source of each image. Introduction to Fractal Microscope - www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html - an interactive tool designed by the Education Group at the National Center for Supercomputing Applications (NCSA) for exploring the Mandelbrot set and other fractal patterns. Direct Link to Fractal Microscope - compute2.shodor.org/cgi-bin/mandy/cnew.pl Madelbrot Set Definition Mathematical Definition - The Mandelbrot set is defined as the set of points c in the complex plane for which the iteratively defined sequence z n z n c with z = does not tend to infinity.

93. J.P. Louvet : Fractals - A History tend when the step tends towards 0 ? Richardson s merit was to have found empirically that the function is (in Les objets fractals mandelbrot write but this is http://fractals.iut.u-bordeaux1.fr/jpl/history.html

The discovery of fractals for this page Home page The history of fractals begins with Benoît Mandelbrot. Nevertheless, as far as a certain number of things were known before his works, these have to be pointed out so that we can better see what his contribution was. This is no more than telling a story, in an incomplete way, at second... or third hand, and intended to provide a few landmarks for those who just discover this domain. In some cases I do not have the precise dates of the work mentioned and therefore, I would be grateful to anyone who could help me to improve this paper What follows is not the history of fractals but, in a more limited way, the history of their discovery. What was known before Mandelbrot The knowledge can be grouped into three fields : natural objects, geometrical figures, and mathematical theories . Of course this distinction is quite artificial, especially for the last two groups, but it will simplify the account. Geometrical figures The first fractals described are dated at the end of the XIX th century.

94. File Library At Channel 1: Fractals EGA required. 341, mandec75.zip, 45262, 0719-97, mandelbrot Eye Candy v.75 Free mandelbrot viewer. fractals. DOS-based. 342, mandel.zip, http://www.filelibrary.com/Contents/DOS/94/9.html

First Shareware Hall of Fame Award Winner June 1997! New Files Top 20 File Search Message Board F ractals: files Search for concept filenames only Jump to files beginning with: A B C D ... Z Jump to page number NEW this page Filename (click to download) Size Date Description lax8.gif GENESIS:Fractal gif 360x480 VGA lfn2.gif NATURE'S ELOQUENCE: A 360x480VGA fractal. life050.zip Bela Lubkin's supurb Life w/rule edit - FAST! lifec.exe Game of life in 4 colors limebal.gif 3 - D Fractal Crossing the lossus and lamda rendered in a radar.par. Very different. [640x400x256] ll_land.zip Lord Logics - 3D Fractal Landscape generator with palette and source and other features longmand.zip Picture of the complete Mandelbrot set, using 64k iterations on each pixel, 12 hours of computing on a Sequent Balance were used. lorentz.zip The Lorentz' strange attractor the first discovery of a chaotic system. Need Color VGA. lorenzww.zip Lorenzian water wheel micheal monagle Interactive chaos demonstrator lortdp.zip CHAOS Time Delay Plots from Lorenz Attractor Way Cool VGA Graphics. man-jul.zip

95. When A Butterfly Flaps Its Wings 1 Image (left) Fig. 4. mandelbrot set The Icon of fractals. Now let me make some compensation by hybriding Laotzu s Taiji with mandelbrot s fractals. http://sunsite.nus.edu.sg/mw/iss06/fract1.html

Iteration Two: Fractals When wandering at the vegetable department of a supermarket, did you ever pay attention to a fresh and clean cauliflower and get intrigued by it? If not, take a look at Fig. 5 now, or simply buy a fresh and clean cauliflower, then zoom in and out using your eyes at its surface structure. Despite the elegant spiral arrangement of the small buds, what more can you see? The whole cauliflower consists of smaller cauliflowers, and the smaller cauliflowers in turn consist of even smaller cauliflowers, so on and on ...! If, as you look closer, your size shrinks according to the size of the cauliflower buds you focus on, can you tell whether you are looking at the whole cauliflower? A small bud of it? A smaller bud on a small bud? ... Most certainly you cannot, because you are looking at a self-similar structure, a scaling-invariant object, ... a fractal. For definition, a fractal object is self-similar in that subsections of the object are similar in some sense to the whole object. No matter how small a subdivision is taken, the subsequent subsection contains no less detail than the whole. Image (right): Fig. 5. Cauliflower a living fractal.

96. ? MagentaStudios ? SYMBOLS ? Fractals ? Mandelbrot Set MagentaStudios SYMBOLS fractals mandelbrot Set mandelbrot Set Long Sleeve TShirt, symbols fractals, mandelbrot set, mandelbrot set long sleeve t-shirt. http://www.magentastudios.com/page.asp?p=6949823

97. Mandelbrot And Julia Set Explorer Julia and mandelbrot Set Explorer For background on Julia and mandelbrot sets, see the introduction. There Also, check out the Applet to explore the mandelbrot set. mandelbrot Set http://aleph0.clarku.edu/~djoyce/julia/explorer.html

Julia and Mandelbrot Set Explorer David E. Joyce For background on Julia and Mandelbrot sets, see the introduction. There is detailed help available for using this form. For more information on complex numbers, see Dave's Short Course on Complex Numbers . Also, check out the Applet to explore the Mandelbrot set Mandelbrot Set x in [-1.0,2.0]; y in [-1.5,1.5]. Parameters Clicks on the Mandelbrot set image will get a Julia set magnify the Mandelbrot set by a factor of Alternate Mandelbrot parameter plane mu lambda (mu = lambda^2/4-lambda/2) 1/mu 1/(mu+.25) 1/lambda 1/(lambda-1) 1/(mu-1.40115) 1/(mu-2) Width= by Height= in pixels. Maximum number of iterations= Note that it may take a while if you set your parameters to ask for a big image or a lot of iterations. Image rendition Image colors Color images Grayscale images Wrap through the colors times. Escape shape: circular (standard) square half plane Level pattern: plain feathered binary Index of recently created images . Many interesting images are found by others. Click on html files for the full form; click on gif files for just the image. Table of contents Introduction Julia and Mandelbrot set Explorer Mandelbrot applet Generation form Alternate parameter planes Bibliography Web references Images produced by this Julia and Mandelbrot Explorer are in the public domain and may be used for any purpose whatsoever.

98. A Fractals Unit For Elementary And Middle School Students A fractals unit for elementary and middle school students This web site, for students and teachers in grades 4 to 8, provides a unit on fractals. The unit is designed to introduce students to http://rdre1.inktomi.com/click?u=http://math.rice.edu/~lanius/frac/&y=020A62

99. Julia And Mandelbrot Sets Julia and mandelbrot Sets. David E. Joyce August, 1994. mandelbrot Sets. Consider a whole family of functions parameterized by a variable. http://aleph0.clarku.edu/~djoyce/julia/julia.html

Julia and Mandelbrot Sets David E. Joyce August, 1994. Last updated May, 2003. Function Iteration and Julia Sets Gaston Julia studied the iteration of polynomials and rational functions in the early twentieth century. If f x ) is a function, various behaviors can arise when f is iterated. Let's take, for example, the function f x x We will iterate this function when initially applied to an initial value of x , say x a . Let a denote the first iterate f a ), let a denote the second iterate f a ), which equals f f a )), and so forth. Then we'll consider the infinite sequence of iterates a a f a a f a a f a It may happen that these values stay small or perhaps they don't, depending on the initial value a . For instance, if we iterate our sample function f x x a = 1.0, we'll get the following sequence of iterates (easily computed with a handheld calculator) a a f a f a f a f a f a It helps to see what's going on graphically. In the diagram above, the graph y x y x is drawn in green. Then the values a a a a , and a are shown grapically, starting with our first value of a , namely, 1.0. To find

100. Java Applets This applet allows you to string together a collection of fractal images generated Applets associated with the book The mandelbrot and Julia Sets include the http://math.bu.edu/DYSYS/applets/

JAVA Applets As part of the Dynamical Systems and Technology Project, we have developed several JAVA Applets for use in exploring the topics of chaos and fractals. These applets are designed to accompany the four booklets in the series A Toolkit of Dynamics Activities , published by Key Curriculum Press Applets associated with the book Fractals include: The chaos game . Yes, this is a game. Try to beat the computer by hitting specific targets via the moves of an iterated function system. This game allows students to understand the construction of the Sierpinski triangle via the chaos game. Fractalina. This applet allows you to set up the vertices, compression ratios, and rotations associated to an iterated function system and then compute and view the resulting fractal. If you have black and white monitors, be sure that you choose the appropriate colors for the points from the Color Selection button inside the applet so that you will be able to see the picture! Fractanimate. This applet allows you to string together a collection of fractal images generated by Fractalina into a movie. We encourage you to become quite familiar with Fractalina before trying to use this applet.

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