41. Fractalized! Java animations of different fractals, including the mandelbrot and Julia sets. http://www-unix.oit.umass.edu/~dtillber/

Fractalized! DLA Builder Windows Java Explorer Windows Small Java Medium Java Large Java ... Very Large Java Java Animations Mandelbrot Zoom Julia Zoom 1 Julia Zoom 2 Home Dan Tillberg These Windows programs do not require installation and can be run directly from this webpage. Download DLA Builder for Windows (100K) If you have trouble running the Windows programs on your system, please contact me at dtillber@student.umass.edu , including your display settings (resolution and color depth) and Windows version. About me (presumably you either know me or are very bored): I'm a Senior Physics major at the University of Massachusetts at Amherst. Some of my interests are in the use of computation to solve difficult problems in Physics and Mathematics. To that end, I have created this site and the programs on it to facilitate public access to some of the beauty of very simple mathematics. Thanks for visiting! reports that I have devout fans.

42. Fracture : A Fractal Screensaver Fracture is a screensaver for Mac OS X that can generate a wide variety of fractal images, including the mandelbrot Set, Julia sets, SelfSquared Dragons, and Attraction Basin fractals. http://www.sticksoftware.com/software/Fracture.html

Fracture 1.2 : A fractal screensaver Download Fracture (276K) VersionTracker's Page on Fracture Price: $10 ( Kagi PayPal Fracture is a screensaver for OS X that creates a wide variety of fractal images. It can render the Mandelbrot Set, Julia Sets, Self-Squared Dragons, and Attraction Basin fractals generated using Newton's Method, Halley's Method, and two other root-finding algorithms. It can make images using advanced fractal imaging techniques like orbit traps, periodicity analysis, and binary decomposition, and it can use several different parameterizations for even greater image variety. Here's one sample image (visit Stick Software's gallery to see more): It will use multiple processors if you have them, to make its calculations faster, and it will display fractals on multiple monitors if you have them. It can generate a wide variety of color schemes under user control. And it can even antialias its images, for the maximum in image quality! It can be controlled by the user while it is running, so you can find a color scheme you like, skip images you don't like, hold images you do like, and save images you particularly like to disk. As you can rely on with Stick Software products, it has a suitably overengineered configuration panel that lets you control many details of its operation. Here's a snapshot of one tab of that panel:

43. The Fractal Microscope blue. But we can appreciate the beauty of the fractals encompassed in the mandelbrot set without the specific mathematics behind it. http://archive.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html

The Fractal Microscope A Distributed Computing Approach to Mathematics in Education The Fractal Microscope is 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. By combining supercomputing and networks with the simple interface of a Macintosh or X-Windows workstation, students and teachers from all grade levels can engage in discovery-based exploration. The program is designed to run in conjunction with NCSA imaging tools such as DataScope and Collage. With this program students can enjoy the art of mathematics as they master the science of mathematics . This focus can help one address a wide variety of topics in the K-12 curriculum including scientific notation, coordinate systems and graphing, number systems, convergence, divergence, and self-similarity. Why Fractals? Many people are immediately drawn to the bizarrely beautiful images known as fractals . Extending beyond the typical perception of mathematics as a body of sterile formulas, fractal geometry mixes art with mathematics to demonstrate that equations are more than just a collection of numbers. With fractal geometry we can visually model much of what we witness in nature, the most recognized being coastlines and mountains. Fractals are used to model soil erosion and to analyze seismic patterns as well. But beyond potential applications for describing complex natural patterns, with their visual beauty fractals can help alter students' beliefs that mathematics is dry and inaccessible and may help to motivate mathematical discovery in the classroom.

44. Don Archer Digital Art Traditional fractals, music, fractals combined with verse. Photo of Benoit mandelbrot taken April 6, 2001. http://www.donarcher.com/

Thursday, 6/10/2004 , 4:52:33 AM ET, Brooklyn and Prattsville, NY Updated June 9, 2004 This site is an exercise in love, vanity and art. It includes my current images, vintage fractals, animations, fractal music, ceramic tiles, digital photographs, digital postcards, and more... There are 3 viewers online now. Enjoy! And thanks for visiting! YEAR 2003 FRACTALS Gallery December 2003 Gallery November 2003 ... January 2003 YEAR 2002 FRACTALS Gallery December 2002 Gallery November 2002 ... January 2002 YEAR 2001 FRACTALS Gallery December 2001 Gallery November 2001 ... Jan-April 2001 NEWEST IMAGES Gallery Stout Series Gallery Trichon Series ... Lake Series COLOR SHIFT ANIMATIONS Don Archer, director CREDITS Online ABS gallery 2001-2003 crowns me "fractal guru" and publishes comprehensive exhibit of my art. Print art included in the collection of Ball State University Art Museum, Muncie, IN. Some 180 fractals included in a CD-ROM, Fractal Frenzy II, Postcards from the Edge of Space, published 1995 by Walnut Creek. Images included in Wirehead's CD-ROM, Virtual Media Gallery by Quantum Access, 1995.

45. The Fractal Microscope The mandelbrot Set going to learn to calculate complex number functions and see how these fumctions lead to the creation of fractals such as the Julia set and the mandelbrot set. http://www.shodor.org/master/fractal/software/mandy/

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46. Benoit Mandelbrot, Fractals And Astronomy (Part 1) Benoit mandelbrot, fractals and Astronomy (Part 1). by Dave Snyder Printed in Reflections November, 1998. fractals are mathematical http://www.umich.edu/~lowbrows/reflections/1998/dsnyder.3.html

Benoit Mandelbrot, Fractals and Astronomy (Part 1) by Dave Snyder Printed in Reflections: November, 1998. Fractals are mathematical objects with strange properties. They have been known for many years, but had been relegated to an obscure corner of mathematics. In the beginning fractals were curiosities, very few people thought they had any real applications (Ludwig Boltzmann and Jean Perrin were among the exceptions). All that changed when Benoit Mandelbrot began his career. Mandelbrot discovered that complex phenomenon in a variety of sciences, including astronomy, could be understood in terms of fractals. Fractal geometry along with several other sciences were motivated by examining human senses. For example, the sense of sight led to the study of electromagnetic radiation and the sense of hearing led to the study of acoustics. However until recently, there had never been any science of roughness. Starting in the late 1800's and into the early 1900's, a number of strange mathematical objects were developed by Georg Cantor, Helge von Koch, David Hilbert, Giuseppe Peano, Carl Ludwig Sierpinski and others. They were called "monster curves" as if they were unruly beasts who needed to be locked up before they did some real damage (the word fractal would come later). Unlike other objects like circles and sine curves which are smooth, these objects are rough and this roughness persists even as the object is magnified. As the object is magnified more and more, the same amount of roughness is present. They are created using a simple process known as aggregate replacement. By repeating this process indefinitely images of these objects form, showing that a complex object can result from a simple procedure.

47. Mathematics Archives - Topics In Mathematics - Fractals KEYWORDS mandelbrot Set, Quaternionic fractals, Iterated Function Systems, Selfsimilar Structures, Lyapunov Exponents, Period Doubling, Reaction-Diffusion; http://archives.math.utk.edu/topics/fractals.html

Topics in Mathematics Fractals Chaos, Fractals, and Arcadia ADD. KEYWORDS: The Chaos Game, The Sierpinski Hexagon, Iterated Function Systems Chaos in the Classroom, Boston University ADD. KEYWORDS: The Chaos Game, Self-similarity, Fractal Dimension, Iterated Function Systems Clifford A. Pickover's Homepage ADD. KEYWORDS: Cellular Automata, Carotid-Kundalini Functions, Books, Puzzles, JAVA Geometry Explorer The Dance of Chance ADD. KEYWORDS: Random Processes, Branching Structures, Patterns in Nature, Student Activities, Laboratory Experiments, Visualization, Software, Fractal Dimension, Fractal Coastlines Diffusion-limited Aggregation Exploring Fractals ADD. KEYWORDS: Lesson Plan, Tutorial, Bibliography Fantastic Fractals ADD. KEYWORDS: Tutorials, Gallery, Fractal Music, Bibliography, Software, Newsgroups Fract-ED ADD. KEYWORDS: Iteration, Attractor, Software, Tutorial, Bifurcation, Self-similarity, Strange attractor Fractal Clouds The Fractal Geometry of the Mandelbrot Set, The Periods of the Bulbs

48. Fractal Gallery: What Is A Fractal? mandelbrot derived the term fractal from the Latin verb frangere, meaning to break or fragment. Basically, a fractal is any pattern http://www.glyphs.com/art/fractals/what_is.html

What Is a Fractal? And who is this guy Mandelbrot? Images and text by Alan Beck The word "fractal" was coined less than twenty years ago by one of history's most creative mathematicians, Benoit Mandelbrot, whose seminal work, The Fractal Geometry of Nature , first introduced and explained concepts underlying this new vision. Although prior mathematical thinkers like Cantor, Hausdorff, Julia, Koch, Peano, Poincare, Richardson, Sierpinski, Weierstrass and others had attained isolated insights of fractal understanding, such ideas were largely ignored until Mandelbrot's genius forged them at a single blow into a gorgeously coherent and fruitful discipline. Lamp (63 k / jpg) Mandelbrot derived the term "fractal" from the Latin verb frangere , meaning to break or fragment. Basically, a fractal is any pattern that reveals greater complexity as it is enlarged. Thus, fractals graphically portray the notion of "worlds within worlds" which has obsessed Western culture from its tenth-century beginnings. Traditional Euclidean patterns appear simpler as they are magnified; as you home in on one area, the shape looks more and more like a straight line. In the language of calculus such curves are differentiable. The trajectory of an artillery shell is a classic example. But fractals, like dendritic branches of lightning or bumps of broccoli, are not differentiable: the closer you come, the more detail you see. Infinity is implicit and invisible in the computations of calculus but explicit and graphically manifest in fractals.

49. Fractals, PHOTOVAULT Graphics: Mandelbrot And Julia Fractals PHOTOVALET (tm) Enter search term. fractals mandelbrot and Julia Sets by Wernher Krutein. Seven years ago my friend Mathemetecian 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.

50. Fractal Explorer: Mandelbrot And Julia Sets (by Fabio Cesari) Keywords fractals, mandelbrot set, fractal, julia sets, quaternion, quaternion julia sets, mandelbrot, Julia Your browser doesn t support frames. http://www.geocities.com/CapeCanaveral/2854/

Keywords: fractals, mandelbrot set, fractal, julia sets, quaternion, quaternion julia sets, Mandelbrot, Julia Your browser doesn't support frames. This site is best viewed with , or an equivalent browser that supports JavaScript and frames You can always access this no-frames version of this site. If you have troubles accessing it, please let me know Many people have probably been fascinated by the infinite complexity and beauty of fractals. I wrote this brief tutorial to explain, in simple terms, how the Mandelbrot set and Julia sets are generated. This document provides an informal introduction to these subjects, and is only intended to be a starting point to learn more about fractals and fractal geometry. You can contribute to the future development of this site by filling out the feedback form Comments and suggestions are very appreciated. Have fun! About complex numbers Mandelbrot set Julia sets Images gallery ... Quaternion Julia sets images gallery" Other pages: About the author Links Feedback form Sign my guestbook ... View my guestbook This page hosted by Get your own Free Home Page

51. The Mandelbrot Set Area Lower Bound counting (Robert Munafo, sci.fractals, 1993, 1996; web page, 1997). See also Kerry Mitchell s A Statistical Investigation of the Area of the mandelbrot Set; http://www.geocities.com/CapeCanaveral/Lab/3825/Period-Area-16.html

Area of Mandelbrot Set Components and Clusters The area lower bound passes 1.5063036. The area of the Mandelbrot Set is one of those useless measures which, since it exists, some feel compelled to determine. If this were too easy, we would quickly move on to other things. In my chapter in Fractal Horizons: The Future Use of Fractals, (Cliff Pickover) , editor (St. Martin's Press, 1996) I reported the lower bound is 1.50585063. This article reports the area evaluation of 430809 components of the Mandelbrot Set yielding an improved lower bound of 1.506303622. In Table 1, the areas of components and clusters are summarized for components with periods 1 through 16. The second column shows the total area of all components with the period shown in column one. The third column shows the total area in clusters whose base 'cardioid' has the period in first column. The last column shows the total number of components used to evaluate the cluster areas Table 1. Area of components and clusters by period Period Area in Components Components in Period Area in Clusters Components in Clusters Total Method of area evaluation The area of the Mandelbrot Set question has been discussed at length on the internet ( sci.fractals

52. Math.com Wonders Of Math Interactive Fractal Sites mandelbrot Set Zoom into a fractal in your browser window. mandelbrot and Julia Set Explorer Zoom into fractals. http://www.math.com/students/wonders/fractals.html

Home Teacher Parents Glossary ... Email this page to a friend More Wonders Fractals Spirograph Conway's Game of Life Roman Numeral Calculator ... Lissajous Lab Resources Cool Tools References Test Preparation Study Tips ... Wonders of Math Search Fall Into Fractals The word FRACTAL was invented by Benoit Mandelbrot Fractals are interesting because as you zoom in closer, the pattern is just as beautiful and complex as when you start. Learn about fractals and create your own beautiful fractal images by following the links below. Interactive Fractal Sites Mandelbrot Set Zoom into a fractal in your browser window. Mandelbrot Explorer Make and post your own images. The Fractory A site built by students for the Thinkquest contest. Build your own fractals and learn about the math behind the images. Mandelbrot and Julia Set Explorer Zoom into fractals. Fractal Galleries Fractalus The fractal from an artist's point of view. Sprott's Fractal Gallery You won't believe the fractal art, animations, and even music! Be sure to visit

53. The Mandelbrot Set Above is the famous mandelbrot set, discovered by Benoit mandelbrot (the discoverer of fractals). But many people created fractals, long before mandelbrot. http://www.jimloy.com/fractals/mandel.htm

Return to my Mathematics pages Go to my home page The Mandelbrot Set Above is the famous Mandelbrot set, discovered by Benoit Mandelbrot (the discoverer of fractals). The set is a fractal. This is an image made by Fractint , an amazing program that is free on the Internet. The Mandelbrot set is the central dark part of the picture. The colored part is drawn to show the great detail of the black part. The different colors are produced by the program, to show how many iterations (repetitions) were needed to decide that this particular point was not in the set. So, just what is the Mandelbrot set? It is a graph, of the simple function z=z +c (or z=z^2+c), where c is the point in question, on the complex plane. The origin is in the right center of the image. z starts at zero. And the formula gives the value of the next z. For most points on the plane, z quickly blows up, goes toward infinity. Near the Mandelbrot set, z blows up more slowly. In the Mandelbrot set, z never blows up. To graph the Mandelbrot set, the program selects a point c and a z=0, and then it solves for z over and over again. If z gets large, the program assumes that the point is not in the set, colors the point, and goes on to the next point. If z stays small for a long time, the program assumes that the point is in the set, colors the point black, and goes on to the next point. The picture above shows approximately where the origin (0,0) is. The above picture uses different colors (to emphasize different features), and is the upper right portion of the set, above and slightly to the left of the origin.

54. History Of Mandelbrot And Julia Fractals history of the mandelbrot and julia fractals. Fractal is a term coined by Benoit mandelbrot (1924) to describe an object which has partial dimension. http://www.icd.com/tsd/fractals/beginner1.htm

history of the mandelbrot and julia fractals "Fractal" is a term coined by Benoit Mandelbrot (1924-) to describe an object which has partial dimension. For example, a point is a zero-dimensional object, a line is a one-dimensional object, and a plane is a two-dimensional object. But what about a line with a kink in it? Or a line that has an infinite number of kinks in it? These are mathematical constructs which don't fit into normal (Euclidean) geometry very well, and for a long time mathematicians considered things like these as "monsters" to be avoided - lines of thought that defied rational explanation in known terms. Within the past few decades, "fractal math" has exploded, and now there are "known terms" for describing objects which heretofore were indescribable or inexplicable. There are an infinite variety of fractals and types; those that I focus on in my gallery are Mandelbrot and Julia fractals. Gaston Julia (1893-1978) was a French mathematician whose work (published in 1918) inspired Mandelbrot in 1977 (the second time Mandelbrot looked at Julia's work). Mandelbrot used computers to explore Julia's work, and discovered (quite by accident) the most famous fractal of all, which now bears his name: the Mandelbrot set. Menu Next Topic

55. The Mandelbrot Set a class (1), (2), or (3) Julia set, you get a different fractal sort of Because the detail in the mandelbrot set (there s only one) is an amalgamation of all http://www.icd.com/tsd/fractals/beginner3.htm

the mandelbrot set As I mentioned, Mandelbrot sets and Julia sets are related in a very special way. There are several types of Julia sets; the broadest distinction, though, is whether there is an "inside" to it or not. Remember, for any given Julia set, all sequences of z run through the iterative equation will fall into one of three classes: 1. the values increase without bound (towards infinity) 2. the values collapse (to zero) 3. the values change, but do not seem be (1) or (2) Julia sets are strictly defined as class (3) points, which just sort of drift around to other class (3) points and don't really go towards zero or infinity. Some Julia sets just consist of a bunch of disconnected points - any z you pick that is part of the set will move around to a different, but totally disconnected point. These are "dust" sets: Other Julia sets are sort of a wiggly line that outlines one or more areas (the "inside") - these areas are all connected. These are called "solid": There are also some which are borderline between the two - the Julia set is a wiggly line, all connected, but it doesn't outline anything. These sets are "dendritic" types: Mandelbrot was looking for some sort of clue as to which c numbers made disconnected sets, and which made connected sets. It turns out that the test is easy. You just start with a

56. Simply Fractals - Fractal Gallery: Mandelbrot Set below this paragraph are images of a mandelbrot set. The image at the bottom is a java applet. With this applet, you can see exactly why this set is a fractal. http://pages.infinit.net/garrick/fractals/gallery4.html

MANDELBROT SET Fig 1: Mandelbrot Set Seen above ( fig 1 )and below this paragraph are images of a Mandelbrot set. The image at the bottom is a java applet. With this applet, you can see exactly why this set is a fractal INSTRUCTIONS: You can zoom into the image by creating a window around the area in which you want to zoom in. (To do this, clicking on the applet and drag.) Mandelbrot Set Applet

57. Benoit Mandelbrot - Wikipedia, The Free Encyclopedia In 1975, mandelbrot published Les objets fractals, forme, hasard et dimension ( The fractal objects, form, randomness and dimension ). http://en.wikipedia.org/wiki/Benoit_Mandelbrot

Benoit Mandelbrot From Wikipedia, the free encyclopedia. Benoît B. Mandelbrot (born November 20 ) is a Polish -born French mathematician , discoverer and leading proponent of fractal geometry Born in Warsaw, Poland , he has lived in France for much of his life. Mandelbrot was born into a family with a strong academic tradition - his mother was a doctor and his uncle, Szolem Mandelbrojt , was a famous Parisian mathematician. His father, however, made his living buying and selling clothes. His family left Poland for Paris in the 1930s. There, Mandelbrot was introduced to mathematics by his two uncles. He studied at École Polytechnique Educated in France , he developed the mathematics of Gaston Julia , and began the (now common) graphing of equations on a computer. Mandelbrot originated what is now known as fractal geometry and the fractal called the Mandelbrot set is named after him. In 1975, Mandelbrot published Les objets fractals, forme, hasard et dimension ("The fractal objects, form, randomness and dimension"). His work on fractals as a mathematician at IBM earned him an Emeritus Fellowship at the T.J. Watson Research Laboratories.

58. Fractal - Wikipedia, The Free Encyclopedia fractals such as the mandelbrot set are more loosely selfsimilar they contain small copies of the entire fractal in distorted and degenerate forms. http://en.wikipedia.org/wiki/Fractal

Fractal From Wikipedia, the free encyclopedia. The term fractal is now used as a scientific concept , as well as a strictly mathematical idea. In the first sense, it means a geometric shape that is self-similar on all scales. In other words, no matter how much you magnify a fractal, it always looks the same (or at least similar). Table of contents 1 History 2 Categories of fractals 3 Definitions 4 Examples ... edit History Objects that we now call fractals were discovered and explored long before we had a word for them. In 1872 Karl Weierstrass found an example of a function with the non-intuitive property that it is everyhere continuous but nowhere differentiable - the graph of this function would now be called a fractal. In 1904 Helge von Koch , dissatisfied with Weierstrass's very abtract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch snowflake . The idea of self-similar curves was taken further by Paul Pierre Lévy who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole , described two fractal curves, the Lévy C curve and the Lévy dragon curve Georg Cantor gave examples of subsets of the real line with unusual properties - these Cantor sets are also now recognised as fractals. In an attempt to understand objects such as Cantor sets, mathematicians such as

59. Java Fractals old mandel0 applet, since it has been superceded by the Classic mandelbrot/Julia Set also notice that I have added a section on orbit fractals that contains http://www.daa.com.au/~james/fractals/

Home Software Talks Articles Fractals IFS fractals Mandelbrot Orbit Fractals 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.

60. Fractals And The Mandelbrot Set This Microworld is an interactive introduction to fractals and the mandelbrot set. It steps Microworld fractals and the mandelbrot Set. This 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.

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