61. MANDELBROT, BENOIT B. - CIRS BBMandelbrot, fractals and Scaling in Finance Discontinuity, Concentration, ML Frame BBMandelbrot, fractals, Graphics and Mathematical Education, http://www.cirs-tm.org/researchers/researchers.php?id=644

62. Angell's Heaven Presents Fractals There are two main types of fractals, mandelbrot and Julia- fractals, The mandelbrot fractal always looks the same, but the Julia fractal can be http://www.angells.com/fun/games/tetris/index3.htm

Games Downloads Jigsaw Puzzles ... Search This page has not been converted to the new site format. Please be patient with us during this time of change. Home Smokey the Bear Wallpapers Subscribe to List ... Angell's Heaven Julia and Mandelbrot Fractals Press button to start. Drag on picture to zoom. The fractal viewer is not a game. It just looks good, and it came free with Tetris. Press the button to start the interactive fractal program. There are two main types of fractals, Mandelbrot- and Julia- fractals, use the radio-buttons to switch between the two. The Mandelbrot fractal always looks the same, but the Julia fractal can be altered by changing the two constants and pressing enter You can zoom in into the fractals by dragging with the mouse. Zoom back out by pressing the radio-button once again. Look at Fractals here with a java enabled browser. It takes a few seconds to load the applet. Please be patient.

63. Chaffey's Fractals - The Mandelbrot Chaffey s Fractal mandelbrot mandelbrot Questions and Answers When did the mandelbrot surface? Who found the thing? How does the mandelbrot come to form http://www.chaffey.org/fractals/mandelbrot/

Chaffey High School's FRACTALS on the Web http://www.chaffey.org/fractals/ Chaffey High School's Mandelbrot Home Page Last Updated February 28, 2003 MANDELBROT Questions and Answers When did the mandelbrot surface? Who found the thing? How does the Mandelbrot come to form? What is ITERATION? Does the mandelbrot come from an equation? Where can I find out more about these mandelbrots? The mandelbrot was first discovered about 1980, so it is fairly new to the mathematical world. Here are some of the first sets that were seen using computer generated software. The first image is from Asking who found the Mandelbrot? Well, Mr. Mandelbrot himself (depicted to the here on the right). To be more exact, Benoit B. Mandelbrot. He is a mathematician, born in 1924 in Warsaw. He studied at the Ecole Polytechnique, Paris, and at the California Institute of Technology What is know as the mandelbrot set is.. Iteration is.. The Equation is Z=z*z+c CHS Home Fractals Home About Fractals Awards ... Theory

64. Java Fractal Generator And Introduction To Fractal Mathematics mandelbrot Fractal Generator Java Applet by Nick Lilavois It generates the most widely recognized fractal (the mandelbrot set), and a few other fractal http://www.lilavois.com/nick/fractals/

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65. Fractal - Wikipedia, The Free Encyclopedia mandelbrot defined fractal as a set for which the HausdorffBesicovitch dimension strictly Some common examples of fractals include the mandelbrot set, http://en.wikipedia.org/wiki/Fractal

Fractal From Wikipedia, the free encyclopedia. The Mandelbrot set , named after its discoverer, is a famous example of a fractal. The Mandelbrot set , named after its discoverer, is a famous example of a fractal. A fractal is a geometric object which is rough or irregular on all scales of length, and so which appears to be 'broken up' in a radical way. Some of the best examples can be divided into parts, each of which is similar to the original object. Fractals are said to possess infinite detail, and they may actually have a self-similar structure that occurs at different levels of magnification. In many cases, a fractal can be generated by a repeating pattern, in a typically recursive or iterative process. The term fractal was coined in by Beno®t Mandelbrot , from the Latin fractus or "broken". Before Mandelbrot coined his term, the common name for such structures (the Koch snowflake , for example) was monster curve Fractals of many kinds were originally studied as mathematical objects. Fractal geometry is the branch of mathematics which studies the properties and behaviour of fractals. It describes many situations which cannot be explained easily by classical geometry, and has often been applied in science technology , and computer-generated art . The conceptual roots of fractals can be traced to attempts to measure the size of objects for which traditional definitions based on Euclidean geometry or calculus fail.

66. Benoît Mandelbrot - Wikipedia, The Free Encyclopedia 1 Early years; 2 Later years; 3 mandelbrot and fractals; 4 Pronunciation Far from being unnatural, mandelbrot held the view that fractals were, http://en.wikipedia.org/wiki/Benoît_Mandelbrot

Beno®t Mandelbrot From Wikipedia, the free encyclopedia. Beno®t Mandelbrot was the first to use a computer to plot the Mandelbrot set Beno®t B. Mandelbrot (born November 20 ) is a Polish -born French mathematician and leading proponent of fractal geometry . He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University and IBM Fellow Emeritus at the Thomas J. Watson Research Center Contents Early years Later years Mandelbrot and fractals Pronunciation ... edit Early years Born in Warsaw , Mandelbrot lived in France from the age of 12 to the end of his college studies. He was born into a family with a strong academic tradition - his mother was a medical doctor and his uncle, Szolem Mandelbrojt , was a famous Parisian mathematician. His father, however, made his living trading clothing. His family left Poland for Paris in . There, Beno®t was introduced to mathematics by his two uncles. Mandelbrot attended the Lyc©e Rolin in Paris until the start of World War II , when his family moved to Tulle . In he returned to Paris to attend the ‰cole Polytechnique , where he studied under Gaston Julia and Paul L©vy . He graduated from the ‰cole Polytechnique in , and spent two years at the California Institute of Technology where he studied aeronautics. Back in France, he studied for a Ph.D. in Mathematical Sciences at the

67. Fractals Many people hope that the mandelbrot set and other fractal pictures, now appearing on Tshirts and posters, will help to give the young a feeling for the http://www.fortunecity.com/emachines/e11/86/mandel.html

web hosting domain names photo sharing Fractals-a geometry of nature Fractal geometry plays two roles. It is the geometry of deterministic chaos and it can also describe the geometry of mountains, clouds and galaxies Benoit Mandelbrot Science and geometry have always progressed hand in hand. In the 17th century, Johannes Kepler found that he could represent the orbits of the planets around the Sun by ellipses. This stimulated Isaac Newton to explain these elliptical orbits as following from the law of gravity. Similarly, the back-and-forth motion of a perfect pendulum is represented by a sine wave. Simple dynamics used to be associated with simple geometrical shapes. This kind of mathematical picture implies a smooth relationship between an object's form and the forces acting on it. In the examples of the planets and the pendulum, it also implies that the physics is deterministic, meaning that you can predict the future of these systems from their past. Two recent developments have deeply affected the relationship between geometry and physics, however. The first comes from the recognition that nature is full of something called deterministic chaos. There are many apparently simple physical systems in the Universe that obey deterministic laws but nevertheless behave unpredictably . A pendulum acting under two forces, for example.

68. Mandelbrot Set -- From MathWorld Branner, B. The mandelbrot Set. In Chaos and fractals The Mathematics Behind the Computer Graphics, Proc. Sympos. Appl. Math., Vol. http://mathworld.wolfram.com/MandelbrotSet.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Applied Mathematics Complex Systems Fractals ... Dickau Mandelbrot Set The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected and not computable "The" Mandelbrot set is the set obtained from the quadratic recurrence equation with , where points in the complex plane for which the orbit of does not tend to infinity are in the set . Setting equal to any point in the set that is not a periodic point gives the same result. The Mandelbrot set was originally called a molecule by Mandelbrot. J. Hubbard and A. Douady proved that the Mandelbrot set is connected . Shishikura (1994) proved that the boundary of the Mandelbrot set is a fractal with Hausdorff dimension 2. However, it is not yet known if the Mandelbrot set is pathwise-connected. If it is pathwise-connected, then Hubbard and Douady's proof implies that the Mandelbrot set is the image of a

69. Fractal -- From MathWorld mandelbrot, BB fractals Form, Chance, Dimension. mandelbrot, BB The Fractal Geometry of Nature. New York WH Freeman, 1983. http://mathworld.wolfram.com/Fractal.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Applied Mathematics Complex Systems Fractals Fractal A fractal is an object or quantity that displays self-similarity , in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the fractal dimension . The prototypical example for a fractal is the length of a coastline measured with different length rulers . The shorter the ruler , the longer the length measured, a paradox known as the coastline paradox Illustrated above are the fractals known as the Gosper island Koch snowflake box fractal Sierpinski sieve ... Barnsley's fern , and Mandelbrot set SEE ALSO: Attractor Backtracking Barnsley's Fern Box Fractal ... [Pages Linking Here] REFERENCES: Barnsley, M. F. and Rising, H.

70. Mandelbrot Set And Fractals My Fractal Program will let you explore not only the mandelbrot Set (z z2) but also (z- z3) and (z- z4) as well as a number of more esoteric fractals. http://ourworld.compuserve.com/homepages/pagrosse/mandelb.htm

Index Mah Jong Mandelbrot Water Rockets SIRDS Weird ... E-Mail Fractals Fractals have always been interesting to people who like to explore but getting your hands on a program that will let you do that has always been a problem. The link at the bottom of this page allows you to download a copy of my mandelbrot program which is postcardware (you send me a post card to license it) Technical Mumbo Jumbo:- ) as well as a number of more esoteric fractals. Exploration is not limited to the x : i plane but extends to the y : z plane as well giving you Julia sets and Fatou Dusts reflecting the earlier work of the mathematicians Gaston Julia and Pierre Fatou whilst extending it into higher order mappings. Background Mumbo Jumbo:- E xisting between the -1 and +1 in four dimensions, the ginger bread man fractal figures of the Mandelbrot set (based on two of the planes, the x plane and the imaginary plane) have become familiar to us all. M andelbrot's work was a result of trying to unify the work of Gaston Julia and Pierre Fatou during the First World War. The mathematics is based upon repetitive mapping of points in the imaginary plane. T he imaginary plane was invented to explain away problems in expressing time in relativity as imaginary numbers were the only solution to the four dimensional Pythagorean solution.

71. Mandelbrot Explorer More fractals can be found at the mandelbrot Exhibition, part of the Virtual Museum of Computing Panagiotis Christias christia@softlab.ntua.gr / Last http://www.softlab.ece.ntua.gr/miscellaneous/mandel/mandel.html

Selected images created by *you* using Mandelbrot Explorer are available at the Mandelbrot Explorer Gallery Page. Zoom Factor : ZoomIn x16 ZoomIn x8 ZoomIn x4 ZoomIn x2 None ZoomOut x2 ZoomOut x4 ZoomOut x8 ZoomOut x16 Set the Zoom Factor as desired and then click at the point you like to zoom in (or out) in the image area above. Drawing Area : X Min : X Max : Y Min : Y Max : Commands : Alternatively, you can set the desired Drawing Area and press the ``Draw New Area'' button to see it. More Fractals can be found at the Mandelbrot Exhibition , part of the Virtual Museum of Computing Panagiotis Christias NTUA/SoftLab Home Page

72. Fractals And The Mandelbrot Set This Microworld is an interactive introduction to fractals and the mandelbrot set. It steps through the construction of that set, developing the notion of http://www.mathwright.com/book_pgs/book660.html

Complimentary Microworld: Fractals and the Mandelbrot Set Click the Hyperlink above to visit the Microworld in your Browser. Author Jim Swift This 12 page Microworld is an interactive introduction to Fractals and the Mandelbrot set. It steps through the construction of that set, developing the notion of complex iterated maps, and provides many exercises that can illustrate the basic ideas. The book is accompanied by a number of dazzling pictures that support exploration of well-known properties of the Mandelbrot set, and that lead the reader to investigate some mysterious connections with the Fibonacci sequence. There are three kinds of pages in this Microworld. Descriptive material giving background information and/or instruction about the interaction on a following page. This includes Windows Help that you may pop up on each page. Interactive pages that give you the opportunity to explore hypotheses about the Mandelbrot iteration, and observe how these iterations go, step-by-step. Along thw way, you will learn a little about complex numbers and Fractals. Exercise pages where you can practice what you have learned on the interactive pages.

73. Fractals, PHOTOVAULT Graphics: Mandelbrot And Julia Fractals fractals mandelbrot and Julia Sets by Wernher Krutein fractals were discovered by Benoit B. mandelbrot who decided to name this bold new area of http://www.photovault.com/Link/WordsGraphics/FractalsMandelbrot.html

PHOTOVALET (tm) Enter search term Fractals: Mandelbrot and Julia Sets by Wernher Krutein S even years ago my friend Mathemetecian/Computer Wiz/Space-Cadet-at-large, Mark Burstein, told me about this bold new branch of mathematics called Fractals. From that moment on I went completly crazy on the visual exploration of the wondrous multi-dimensional world of these Fractals. The Mandelbrot and Julia subsets have some of the most rewarding moments of discovery of the many landscapes and environments that these mathematical functions can create. I believe that we have only just begun to understand and apply fractals into our everday living. What is a Fractal? A fractal is a kind of repeating structure displaying properties of self similarity. This means taht you can explore a Fractal shape itno virtualy any depth of magnification and you will come across many similar shapes of the original. Fractals were discovered by Benoit B. Mandelbrot who decided to name this bold new area of mathematics "Fractals". He chose that word from the Latin word fractus, which means "to break". Some of the most typical fractals are: flowers water , clouds, trees, waves , smoke, flames lightning leaves galaxies , etc.

74. Java Fractals written by James Henstridge. These include an IFS fractal generator, mandelbrot Sets and Julia Sets using a variety of different formulas. http://www.jamesh.id.au/fractals/

Home Software Talks Articles Fractals IFS fractals Mandelbrot Orbit Fractals Photos ... Diary Java Fractals Update - As well as the new look, I have removed the old mandel0 applet, since it has been superceded by the Classic Mandelbrot/Julia Set applet. Also the code has been moved to the end of this contents page to make rest of the site look more user friendly. You will also notice that I have added a section on orbit fractals that contains six new applets. IFS Fractals What is an IFS fractal My Java IFS Applet Some IFS fractals Sierpinski Triangle Fern Spirals Dragon ... IFS Formula Complex Number Fractals Update - These fractals now work with Windows. Also, if they worked for you before, they will prbably run faster with more colours. (If you are interested in what changed, I have switched over to using an ImageProducer interface) If you want a better resolution, select a different pixel size from the list box (1 is best), and press the redraw button. What Does This Applet Do How the Formula is Used Different Sets: Classic Mandelbrot/Julia Set Cubic Mandelbrot/Julia Set Quartic Mandelbrot/Julia Set Phoenix Set ... Newton-Raphson type Fractal Orbit Fractals Orbit Fractals Introduction Fractal Popcorn Fractal Gingerbreadman Hopalong Orbit Fractal ... Inverse Julia Set Fractal Code Code for all my fractal applets IFS Applet Code Mandel0 Applet Code - My first attempt at a Mandelbrot Set applet.

75. Mandelbrot Benoit mandelbrot was largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html

Benoit Mandelbrot Born: 20 Nov 1924 in Warsaw, Poland Click the picture above to see five larger pictures Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Version for printing Benoit Mandelbrot was largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature. Mandelbrot was born in Poland in 1924 into a family with a very academic tradition. His father, however, made his living buying and selling clothes while his mother was a doctor. As a young boy, Mandelbrot was introduced to mathematics by his two uncles. Hadamard in this post, took responsibility for his education. In fact the influence of Szolem Mandelbrojt was both positive and negative since he was a great admirer of Hardy and Hardy 's philosophy of mathematics. This brought a reaction from Mandelbrot against pure mathematics, although as Mandelbrot himself says, he now understands how Hardy 's deep felt pacifism made him fear that applied mathematics, in the wrong hands, might be used for evil in time of war.

76. Background Information About Mandelbrot's Fractals Andreas Meile s homepage, fractal gallery; Background information, how these pictures arise at all. http://www.hofen.ch/~andreas/Englisch/Fraktalgalerie/Hintergrundinformation.html

How are these nice pictures arising altogether? The whole calculation is based on the behaviour of recursive formulas with complex numbers (numbers in the form a + bj where j=sqrt(-1) i.e. defined square root of -1). Similar the number straight line of real numbers (our normal, ordinary numbers like 1, 2, 3, 4.766, 3.1415926... etc.), all complex numbers form a plane, the Gauss's number plane called from the famous mathematician Carl-Friederich Gauss (1777-1855). Each one of these picture are resulted from the simple formula where z(0) (start value) is the picture constant, c the point of the plane. The number of iterations will counted until the condition depth of computation xmax [e.g. 1000] to avoid an endless loop during the computing process). This procedure will be taken on each point (screen pixel on the computer), so they form these pictures at the finish of computation. This chaotic behaviour was already discovered from the French mathematician Gaston Julia but his knowledge's found only a further interest since the age of graphic computers. Go back to the first picture

77. Java Applets This applet allows you to string together a collection of fractal images 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.

78. High Quality Mandelbrot Images High Quality mandelbrot Images. All images on this page have been computed by a simple C++ program. The emphasis lies on quality, not on computation speed. http://www.cosy.sbg.ac.at/~gwesp/fractals/

High Quality Mandelbrot Images Contrary to other fractal sites, we put all our emphasis on image quality. All computations are done to the highest possible precision. An uncompromising degree of oversampling (a.k.a. anti-aliasing) is used. The images are stored in the PNG format to avoid lossy compression. Only the previews are JPEG-compressed. Highly experienced photographic experts are in charge of print quality. Print order information We offer prints of our fractals on high quality photographic paper. For ordering, please send email with the following information: Image name(s), number of copies, price per print (PPP; see below). Your full name. The shipping address. The billing address, if different from the shipping address. Any comments or suggestions you may have. For more information about placing orders, please see the Orders FAQ The YYY(TM) (Yellow Yin Yang) High quality rendering PPP: EUR 98. The rotating spiral ( MPEG-1 video; 6 MB) Blue Spear High quality rendering PPP: EUR 128. Candle.

79. Fractals Perhaps the king of all the fractals is the mandelbrot Set (named after its mandelbrot set The classic fractal; a mathematical equation named after its http://www.kheper.net/cosmos/fractals/fractals.htm

Kheper Home Topics Index Esotericism Kheper Forum ... Guestbook Fractals The Fractal nature of Reality Fractal Cosmologies Fractal Links The Fractal nature of Reality The universe around us is not linear but fractal in nature. That is, we see the same pattern appearing time and again, no matter what the scale it is examined on. Look at a river with its tributories. Each tributory is itself a river, with smaller tributories, which in turn are themselves rivers with smaller tributories, and so on down to creeks. Or a fern frond; the main frond of which consists of a two rows of sub-fronds, each of which consists of two rows of smaller fronds or leaves. Or the pattern of a coastline: bays and peninsulas contain smaller bays and peninsulars, right down to grains of sand. Fractals were discovered and described by a mathematician by the name of Benoit Mandelbrot Some famous fractals simple fractals The Koch Snowflake A good example of a simple fractal. Take an equilateral triangle. Add a smaller but identical triangle in the middle of each of the three sides. Repeat ad infinitum. The Sierpinski Triangle The Mandelbrot Set The classic fractal; a mathematical equation named after its discover, Benoit Mandelbrot. What makes this equation interesting is that it is recursive; that is, it continually refers back to itself, so that each successive step employs as one of its parameters the outcome of the preceding step. When the Mandelbrot equation is fed into a computer, and the computer is instructed to paint the various points that arise on a screen or a print-out, the most astonishing patterns emerge.

80. Fractals 101, Part 1 || Kuro5hin.org The mandelbrot Set—the bestknown fractal of all—is based on the chaotic system Z = Z^2 + C, where Z and C are complex numbers, and Z is the value of Z http://www.kuro5hin.org/story/2005/7/19/22140/9950

create account help/FAQ contact links ... MLP We need your support: buy an ad premium membership k5 store Fractals 101, Part 1 ... Science By jd Wed Jul 20th, 2005 at 11:36:19 AM EST Fractals are well-known in rave art, popular culture and even the occasional movie. But what are they, where do they come from and what are we going to do with them now they are here? In order to cover fractals in any meaningful way, I am going to first examine what lies behind them. The first, and most important, of these elements is "Chaos Theory" , a branch of mathematics that is rather better known than it is understood. Chaos Theory covers a class of mathematical systems that don't follow the usual rules. (Anarchistic equations!) Normally, mathematical functions are relatively well-behaved. You can find the gradient at a given point, for example. If you change the values you start with (your initial conditions) by just a little, you will change the values you end up with by just a little and in a predictable way. Chaotic systems, on the other hand, throw the rules out the window, without opening the window first. For small enough initial conditions, they behave themselves and generally go to some steady state value. As you increase the values, though, something happens. The system will start to oscillate with period 2. Keep increasing the value, and it stays at that frequency, until you cross a specific threshold. Then the frequency doubles. It keeps on doing this, with the intervals getting shorter each time. Translated into English, this means that if you take some function and then use the results as the new inputs to the same function, it will first bounce between two values. Then, as you increase the initial conditions, it will bounce between four possible values. Then eight, sixteen, thirty-two and so on. It will always be a power of two and it will always be in that order.

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