61. Fractal -- From MathWorld mandelbrot, B. B. fractals Form, Chance, Dimension. San Francisco, CA W. H. Freeman, 1977. mandelbrot, B. B. The Fractal Geometry of Nature. http://mathworld.wolfram.com/Fractal.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Applied Mathematics Complex Systems Fractals Fractal An object or quantity which 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 Attractor Backtracking Barnsley's Fern ... search Barnsley, M. F. and Rising, H.

62. Mandelbrot And Julia Set Explorer clicked. The mandelbrot set is kindof an index to the Julia sets. Centre, Recentres the fractal on the point where you clicked. Zoom http://pjt33.f2g.net/fractals.html

Home Quotes Papers Fractals ... Site Map Mandelbrot Set Filled Julia Set Button selected Action performed by mouse click Julia Causes the Julia applet to draw the Julia set corresponding to the point where you clicked. The Mandelbrot set is kind-of an index to the Julia sets. Centre Recentres the fractal on the point where you clicked Zoom in Zooms in by a factor of 2, recentring at the point where you clicked Zoom out Zooms out by a factor of 2 without recentring Further info on Mandelbrot and Julia sets This applet started life as a university exercise drawing the Mandelbrot set, but I expanded it to Julia sets (modern computing power is wonderful - I used to draw 3 black-and-white Julia sets overnight using my Amiga 500, and then set the parameters for three more to draw while I was at school). I then came across the 5k competition, and spent a while adding features and optimising for size - the final entry didn't do as well as hoped, but it was an interesting exercise. If you want to play with the code , feel free. (Note: that code might not be the same as the applet here - I didn't version my code back then). If you make any improvements to it, I'd appreciate a copy. Last updated 23rd Mar, 2002. Comments etc. to

63. [ Wu :: Fractals | Mandelbrot Set ] fractals The mandelbrot Set http://www.ocf.berkeley.edu/~wwu/fractals/mandelbrot.html

Fractals: The Mandelbrot Set Home Introduction to Fractals Mandelbrot Sierpinski ... Gallery Construction: The Mandelbrot set is the set of all complex numbers c such that iterating z <= z^2 + c does not ascend to infinity, starting with z=0. The terms z and c are complex numbers (see Note 1 for an overview of complex numbers, if necessary). "Ascend to infinity" means that z will continue to grow with each iteration; in calculus terms, it means that z diverges; more on this, as well as the initial condition z=0 , later. I will proceed with explaining simply how to graph the set, and place explanations for the mathematical intricacies in footnotes for those curious. We could probably find a few elements in the Mandelbrot set by pencil and paper, but in order to crank out enough iterations to produce even a semi-decent graph before dying of either old age or a mental segmentation fault, we will want to determine the elements of the Mandelbrot set using a simple computer program. I will use Java-like pseudocode. Let us first declare a complex number class complexNumber // A class for complex numbers.

64. Mandelbrot De mandelbrotset is de oerMoeder van alle fractals. Nagenoeg Zoals boven al gezegd is de mandelbrot-set de Moeder van alle fractals. Door http://proto.thinkquest.nl/~klb045/fractalmap/mandel.html

MANDELBROT EN JULIA Gaston Julia (1893 - 1978) is de echte grondlegger van de fractaltheorie. Hij is vooral bekend geworden door de naar hem genoemde Julia-set. (een verzameling formules die de vorm van een fractal bepalen) Benoit Mandelbrot is echter verantwoordelijk voor het ontstaan en de grote bloei van de fractalmeetkunde. Met behulp van computers toonde hij aan dat Julia's werk een grote bron is voor het maken van fractals. De specifieke vorm van de Julia-set De specifieke vorm van de Mandelbrot-set De Mandelbrot-set is de oerMoeder van alle fractals. Nagenoeg alle fractals die je ooit hebt gezien of nog zult zien hebben als basis een Mandelbrot, al of niet met een Julia invloed. Vroeger, zo'n twintig jaar geleden, waren computers nog niet zulke enorme rekenbeesten als vandaag de dag. We waren al blij als we met behulp van een basic-programmaatje een plaatje op het scherm konden toveren. (zie boven) Door heel voorzichtig aan de code te sleutelen ontstond een nieuwe afbeelding. Dat moest echt heel voorzichtig, want de kans dat de computer op hol sloeg was groot. (zie onder) Julia Mandelbrot Door bovenstaande afbeeldingen aan te klikken zie je de basic-code die bij deze fractals hoort.

65. Efg's Fractals And Chaos -- Fractals Show 2 Lab Report , Number of Parameter Sets. Beauty of fractals mandelbrot, Images from the book The Beauty of fractals Beautyfractals.jpg (5171 bytes), 14....... Group, http://www.efg2.com/Lab/FractalsAndChaos/FractalsShow2.htm

Fractals and Chaos Fractals Show 2 Lab Report NOTE: This program requires additional validation tests, which will be conducted in the next few months. Other items to be addressed include: 1. Re-validation of complex number routines (rewritten from old TP 7 code). 2. Fix data entry for floats (esp. negative values) 3. Fix localization to treat DecimalSeparator correctly outside the U.S. 4. Fix aspect ratio problems in certain "Quick Settings" 5. Allow printing of fractal images 6. Save/Restore iteration map for coloring independently of the calculations "Intermingle" Images from Fractals Show 2 Program Purpose The purpose of this project is to create Mandelbrot and Julia sets for a number of complex math functions. Mathematical Background The famous Mandelbrot Set is formed by the iteration z z + c over an area of the complex plane. A Julia Set is formed using a similar iteration. But instead of iterating only with the function z , many other complex math function can be explored. Here's a summary of how to create a Mandelbrot or Julia Set:

66. Mandelbrot Benoît Translate this page et suivantes , Ed. du Seuil, Paris, 1982. Les objets fractals par Benoît mandelbrot Ed. Flammarion, Paris - 1980. Zoom sur mandelbrot http://www.sciences-en-ligne.com/momo/chronomath/chrono2/Mandelbrot.html

MANDELBROT Benoît, français, 1924- Neveu de Szolem Mandelbrot . Polytechnicien, il s'installe aux Etats-Unis pour fuir le "terrorisme Bourbaki " et professe à l'université de Yale (Yale University, située à New Haven, Connecticut). Il est célèbre pour ses travaux concernant la géométrie fractale (on lui doit ce qualificatif et substantif en 1975, de l'adjectif latin fractus provenant de frangere briser ), initiée au début du siècle par Sierpinski et Julia , qu'il étudia tout particulièrement chez le constructeur IBM (International Business Machines), fabricant de calculateurs depuis 1911. La notion d'objet fractal Les irrégularités de la nature, d'apparence chaotique, sont en fait l'expression d'une géométrie très complexe de l'infiniment petit où la notion de dimension fractionnaire se substitue à celle de dimension euclidienne usuelle (étude des irrégularités des côtes maritimes, de la forme des nuages, d'un arbre, d'une feuille de fougère, etc.). Une courbe fractale est telle que toute portion est identique au tout ! Leur étude confirme les doutes des mathématiciens du début du 20è siècle, comme Lebesgue et Hausdorff , sur le concept de dimension.

67. Fractals: Mandelbrot Curve Construction, As almost all fractals curves, the construction of the mandelbrot curve is based on a recursive procedure. The fisrt http://users.swing.be/TGMSoft/curvemandelbrot.htm

DisplayHeader( "Geometric Fractals", "The Mandelbrot Curve", 0, "main_fractals.htm", "Back to Fractals Main Page"); Content Introduction Construction Properties Variations Author Biography All pictures from WinCrv Introduction The major contribution of Benoit Mandelbrot was to open the fascinating field of the fractal geometry using facts known long before he wrote his first book about factals: Peano Curve, Von Koch Curve, Sierpinsky Objects, Hausdorff-Besicovitch dimension, ... However, as many others, his name is also attached to an intersting factal curve based on a simple iterator. Construction As almost all fractals curves, the construction of the Mandelbrot curve is based on a recursive procedure. The fisrt iteration is obtained by dividing a straight line into three equal segments and then applying the Mandelbrot iterator. The first iteration gives the following picture: The procedure is then repeated with the eight segments generated by the previous iteration. In the third iteration, it's already hard to find out the way through.

68. Fractals does lightning travel in straight lines. mandelbrot. fractals and Computer Graphics Interface Magazine Article Colouring Attractors. http://astronomy.swin.edu.au/~pbourke/fractals/

F r a c t a l s , C h a o s Index Geometry Formats Curves Surfaces Polyhedra Fractals, Chaos Projection Stereographics Rendering Radiance PovRay OpenGL Modelling Terrain Colour Textures Other Data Formats Analysis Fun Puzzles Old Stuff Papers POVRay Fractal Raytracing Contest Fractal Dimension Calculator - FDC Box counting software for the Macintosh and Linux. Ruler or Compass Dimension Multifractal spectrum Lacunarity Recurrence plots ... Rendering Wada-type basins of attraction Here's a party trick for over Christmas. Get 4 large shiny Christmas balls, some coloured wrapping paper, some lights from the Christmas tree, and you have you own fractal laboratory to amuse yourself with over the festive season. Self similarity Examples of self similarity in fractals with examples from mathematics and photos of the physical world. I wonder whether fractal images are not touching the very structure of our brains. Is there a clue in the infinitely regressing character of such images that illuminates our perception of art? Could it be that a fractal image is of such extraordinary richness, that it is bound to resonate with our neuronal circuits and stimulate the pleasure I infer we all feel? P. W. Atkins

69. Article: Fractals 1983. BB mandelbrot, fractals form, chance, and dimension, Translation of Les objets fractals, WH Freeman, San Francisco, 1977. http://www1.physik.tu-muenchen.de/~gammel/matpack/html/Mathematics/Fractals.html

1 Fractals 1.1 Monofractals Introduction A short walk through the Mandelbrot Set References 1.1.1 Introduction 1.1.3 A short walk through the Mandelbrot Set All picture in this section have been created with Matpack's fractal explorer Mandel . The documentation and the source code for this program are available. -plane Quadratic polynomials can be parameterized in different ways which lead to different shapes for the Mandelbrot sets. The typical parameterization is in terms of a complex parameter , and the function being iterated is f(z) = z . If the set 0, f(0), f(f(0)), ... is bounded, then lies in the Mandelbrot set. With this parameterization, the most notable feature of the set is a cardioid studded with circles. To reproduce the image call: mandel -B -r -2 1 -1.5 1.5 -n 500 -c cool-256 mapout log revcmapout size 150 150 export gif mu0.gif -plane If , then the function being iterated is f(z)=z . With respect to this parameterization, a point belongs to the Mandelbrot set in the -plane if its inverse belongs to the Mandelbrot set in the -plane. The inverse of the caridiod is the exterior of a teardrop shape: The circles on the outside of the cardioid are inverted to circles on the inside of the teardrop. The cusp of the cardioid becomes the cusp of the teardrop. To reproduce the image call:

70. Fractals-a Geometry Of Nature Benoit Mandelbrot :How-to.tk: Physics :Free Downlo My bestknown contribution to this area of nonlinear fractals is called the mandelbrot set (see Box 1). The set results from iterating a relatively simple http://www.groovyweb.uklinux.net/index.php?page_name=Fractals-a geometry of natu

71. Mandelbrot - Chaos And Fractals hewgill.com Chaos and fractals mandelbrot Search. mandelbrot Chaos and fractals. Oh no! Your browser does not appear to support Java applets. http://www.hewgill.com/chaos-and-fractals/c14_mandelbrot.html

hewgill.com Chaos and Fractals mandelbrot Search mandelbrot - Chaos and Fractals Oh no! Your browser does not appear to support Java applets. You will not be able to view the samples unless your browser supports Java. (view source) iteration sierpinski koch ... mandelbrot

72. | Mandelbrot And Julia Set Fractals This shockwave generates both mandelbrot and Julia set fractals. Unlike other fractal generators in Director/Shockwave this uses http://venuemedia.com/mediaband/collins/bothfractals.html

This shockwave generates both Mandelbrot and Julia set fractals. Unlike other fractal generators in Director/Shockwave this uses imaging lingo to create fractals up to five times faster. Use the picons on the left and right of the control panel to select the fractal. The real and imaginary constants only effect the Julia set fractals. Use the color gradient to change the color table used to draw the image. To activate any changes to the constants or the color table click the 'Set' button. The progress bar at the bottom of the control panel turns green when the image is complete. Click anywhere on the fractal to zoom in at that location, shift click to zoom out and click the fractal picon to restart that fractal.

73. Fractals & The Fractal Dimension mandelbrot began his treatise on fractal geometry by considering the question How long is the coast of Britain? The coastline is irregular, so a measure http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html

Fractals and the Fractal Dimension Mandelbrot and Nature "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."(Mandelbrot, 1983). The Concept of Dimension So far we have used "dimension" in two senses: The three dimensions of Euclidean space (D=1,2,3) The number of variables in a dynamic system Fractals, which are irregular geometric objects, require a third meaning: The Hausdorff Dimension If we take an object residing in Euclidean dimension D and reduce its linear size by 1/r in each spatial direction, its measure (length, area, or volume) would increase to N=r D times the original. This is pictured in the next figure. We consider N=r D , take the log of both sides, and get log(N) = D log(r). If we solve for D. D = log(N)/log(r) The point: examined this way, D need not be an integer, as it is in Euclidean geometry. It could be a fraction, as it is in fractal geometry. This generalized treatment of dimension is named after the German mathematician, Felix Hausdorff. It has proved useful for describing natural objects and for evaluating trajectories of dynamic systems. The length of a coastline Mandelbrot began his treatise on fractal geometry by considering the question: "How long is the coast of Britain?" The coastline is irregular, so a measure with a straight ruler, as in the next figure, provides an estimate. The estimated length, L, equals the length of the ruler, s, multiplied by the N, the number of such rulers needed to cover the measured object. In the next figure we measure a part of the coastline twice, the ruler on the

74. Fractal Links - Amazing Seattle Fractals! mandelbrot Set Explore details of the mandelbrot set. Julia and mandelbrot Set Explorer Generate online fractals. Coolmath.com Great site for kids adults. http://www.fractalarts.com/ASF/Fractal_Links.html

Amazing Seattle Fractals! Home Fractal Art Galleries Fractal Tutorials Fractal Of The Week ... About Fractal Links I've included many resources on this page if you are looking for more information on fractals, including other fractal artists, tutorials or more information about fractals in general. If you are looking for fractal software programs check out my software page. Enjoy! Seattle Fractals Digital Art If you are interested in any of my art prints or downloading any of my screensavers for a free evaluation, you can find them here. High resolution art prints, fractal art galleries, fractal screensavers, custom made to order screensavers and more! Link Spectrum Fractal Tutorials and Related Links Fractal Types Explantions and illustrations of various types of fractals. UF Spiral Tutorial Dr. Joseph Trotsky's excellent tutorial on creating the classic fractal spiral form as well as other helpful UF info. He has also written helpful info on the program Fractal Explorer. Janet Parke Preslar's excellent tutorials on using the Ultra Fractal Program. Prof. John Matthew's

75. Fractals Section 2 - Mandelbrot And Julia Sets For example, the mandelbrot fractal has characteristics nodules, which appear in similar configurations when you zoom in. fractals http://members.tripod.com/Gavin_Winston/FRAFRACT.HTM

var cm_role = "live" var cm_host = "tripod.lycos.com" var cm_taxid = "/memberembedded" 2. Mandelbrot and Julia Sets Introduction Fractals are images, made up of a set of points which are produced by mathematical calculation. Fractals are generated by a process called iteration , explained below, and the results are images with infinite complexity. The more you zoom in to a particular area, the more detail that is revealed, rather like the coastline around a country. The fractals also have regions of self-similarity. For example, the Mandelbrot fractal has characteristics nodules, which appear in similar configurations when you zoom in. Fractals can have finite area, but infinite perimeter, so they do not fall in the normal categories of one, two or three dimensions, but have a non-integral number of dimensions. Iteration Iteration is the repetition of the same fairly simple calculation over and over again, each calculation using the result of the previous one. The terms generated by an iteration are defined as functions of the previous term, and hence the first term must be fixed. For example, the iteration z(n+1) = z(n) + 2 where z0 = 1 generates the terms: z(0) = 1, z(1) = 3, z(2) = 5 and so on (the odd numbers).

76. Dave's Fractals Page - Mandelbrot Set Fractals Created By David J. Grossman Dave s fractals Page. mandelbrot Set and Julia Set fractals Created by David J. Grossman. The mandelbrot Set. There are currently numerous http://www.unpronounceable.com/fractals/mandelbrot-set.html

Dave's Fractals Page Mandelbrot Set and Julia Set Fractals Created by David J. Grossman The Mandelbrot Set There are currently numerous sites that explain the calculations involved in creating the Mandelbrot Set. I plan on eventually writing up my own page with enough information to help even those with little advanced mathmatics understand the calculation of this set. In the meantime, check out this Google search for other Mandelbrot Set sites. Back

77. Dave's Fractals Page - Mandelbrot Set Fractals Created By David J. Grossman Dave s fractals Page. mandelbrot Set and Julia Set fractals Created With Fractint by David J. Grossman. Updated Febrary 21, 2004 · Created http://www.unpronounceable.com/fractals/

Dave's Fractals Page Mandelbrot Set and Julia Set Fractals Created With Fractint by David J. Grossman Contact Links Dave's Quaternion Julia Fractals Page ... Productions Non-commercial private use including printing, wallpaper and derived works is permitted. Please e-mail me if you would like to use any of these images for any public use.

78. Chaos, Fractals And The Mandelbrot Set Chaos, fractals and the mandelbrot Set. Here is a list of books you may be interested in Benoit B. mandelbrot. The Fractal Geometry of Nature. Freeman. http://students.bath.ac.uk/ma2lhh/MandelbrotSet.html

Chaos, Fractals and the Mandelbrot Set Click here to learn about Chaos Theory Click here to find out about Fractals Click here to discover what the Mandelbrot Set actually is Click here to see more intruiging pictures of the Mandelbrot Set Introduction "The Mandelbrot Set is the most complex object in mathematics, its admirers like to say. An eternity would not be enough time to see it all, its disks studded with prickly thorns, its spirals and filaments curling outward and around, bearing bulbous molecules that hang, infinitely variegated, like grapes on God's personal vine." [James Gleik, "Chaos: Making a New Science"] Many people who have an interest in mathematics (and many who don't!) will have heard of the Mandelbrot set. This web page will descibe the basic elements of Chaos Theory and Fractals , and will describe what the Mandelbrot Set actually is. Click on the above links to go to these pages. Below is a picture of the whole Mandelbrot set. Here is a list of books you may be interested in: Benoit B. Mandelbrot.

79. Fractals -- An Introduction It is also important to note that several images, now considered fractals predate the work of mandelbrot. Last Updated Friday, 02Feb-2001 051705 GMT. http://ejad.best.vwh.net/java/fractals/intro.shtml

Fractals An Introduction The purpose of this lesson is to learn very simple methods to construct fractals, to do and practice some math, to appreciate the beauty of fractals, and finally to have fun with math. The lesson includes several Java applets for hands-on construction and interaction with simple fractals. Lets start with definitions. In very simple terms, fractals are geometrical figures that are generated by starting with a very simple pattern that grows through the application of rules. In many cases, the rules to make the figure grow from one stage to the next involve taking the original figure and modifying it or adding to it. This process can be repeated recursively (the same way over and over again) an infinite number of times. The fractals' growth mechanism can be visualized very easily with a simple example. Start with a + sign and grow it by adding a half size + in each of the four line ends. Repeat the exact same process recursively as many times as desired. We'll call this the Plusses fractal: Notice how the + sign grows into a rhombus (popularly known as diamond) in very few simple steps. Further in the lesson we'll count the number of +'s in each of the stages to see how quickly its complexity grows.

80. Mandelbrot Set Fractals The mandelbrot set is that iterating z z 2 + c does not go to infinity (starting with z = 0). from Michael C. Taylor s The Sci.fractals FAQ available http://home.san.rr.com/jayrhill/MSet/MSet.html

Mandelbrot Fractal Zone Here are some images and files. The Mandelbrot set + c does not go to infinity (starting with z = 0)." from Michael C. Taylor' s The Sci.Fractals FAQ available from his website View and download parameters for calculating Midget Mandelbrot Set Images Also, parameter files for calculating Mandelbrot set images centered on Misiurewicz points Read Dr. J's fractal bed time story "The Mysterious Fractal Comet" (about 1 MByte). What are critical points Home

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