1. Math Forum: Suzanne Alejandre - MandelBrot Activity Studying mandelbrot fractals. Fractals NOTE Use of Internet Explorer 5.0is recommended. What is a fractal? Alan Beck in What Is a Fractal? http://mathforum.org/alejandre/applet.mandlebrot.html

Studying Mandelbrot Fractals Fractals NOTE: Use of Internet Explorer 5.0 is recommended. What is a fractal? Alan Beck in What Is a Fractal? And who is this guy Mandelbrot? writes: "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." 1. Click on the button Col+ or Col- to change the colors of the fractal image. 2. Now that you have the colors set to your liking, it is time to investigate the fractal itself! 3. Using the mouse, draw a small rectangle on the fractal image. Click on Go and watch as the smaller section of the image is redrawn to fill the fractal screen. 4. What do you notice? How do the images compare? Click on the Out button to revisit the first image and the In button to return to the enlarged image. 5. Continue going into the fractal image. What do you observe? 6. It has been stated that fractals have finite areas but infinite perimeters . Do you agree? Why?/Why not?

2. Math Forum: Suzanne Alejandre - MandelBrot Activity Studying mandelbrot fractals NOTE Use of Internet Explorer 5.0 is recommended. What is a fractal? "Basically, a fractal is any pattern that reveals greater complexity as it is enlarged. http://forum.swarthmore.edu/alejandre/applet.mandlebrot.html

Studying Mandelbrot Fractals Fractals NOTE: Use of Internet Explorer 5.0 is recommended. What is a fractal? Alan Beck in What Is a Fractal? And who is this guy Mandelbrot? writes: "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." 1. Click on the button Col+ or Col- to change the colors of the fractal image. 2. Now that you have the colors set to your liking, it is time to investigate the fractal itself! 3. Using the mouse, draw a small rectangle on the fractal image. Click on Go and watch as the smaller section of the image is redrawn to fill the fractal screen. 4. What do you notice? How do the images compare? Click on the Out button to revisit the first image and the In button to return to the enlarged image. 5. Continue going into the fractal image. What do you observe? 6. It has been stated that fractals have finite areas but infinite perimeters . Do you agree? Why?/Why not?

3. About "Studying Mandelbrot Fractals" Studying mandelbrot fractals. Library Home Full Table of Contents Suggest a Link Library Help Visit this site http//mathforum http://mathforum.org/library/view/7747.html

Studying Mandelbrot Fractals Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://mathforum.org/alejandre/applet.mandlebrot.html Author: Suzanne Alejandre Description: What is a fractal? A definition and a Java applet to help in exploring the Mandelbrot set, redrawing small areas to fill the fractal screen and noticing how the images compare. Also links to other sites with fractal information for middle-schoolers. Levels: Elementary Middle School (6-8) High School (9-12) Languages: English Resource Types: Lesson Plans and Activities Internet-Based Projects Link Listings Web Interactive/Java Math Topics: Fractals Suggestion Box Home The Math Library ... Search http://mathforum.org/ webmaster@mathforum.org

4. Kosmoi: Mandelbrot Fractals Fractals, Fractal Gallery, Mathematics. mandelbrot fractals. Life http://www.kosmoi.com/Science/Mathematics/Fractals/Mandelbrot

Fractals Fractal Gallery Mathematics Mandelbrot Fractals Nature Agriculture Animals Biology ... Kosmoi Press the left mousebutton ro "Z" where you want to zoom in, press the right mouse button key or "z" when you want to zoom back out again. You may enter coordinates yourself. Apparently undocumented feature: pan with the arrow keys. Applet by Steffen Thorsen. See also: Fractals Fractal Gallery Mathematics Book and Website Recommendations ... Contact Us

5. Some Art Applets By Daniel Pitts A growing collection of small applets with source code. mandelbrot fractals, Conways Game of Life, and spinning stars are just some of the applets appearing here. http://turquoise.coloraura.com/cgi-bin/artwork

Art applets Here are a few applets I've written. Most of them are just quick "hacks" I've put on here so that I have something on this page. I'll put some more up when I get some free time. Applet name Description CircleInversion This is called a Circle Inversion fractal. Life Conway's game of Life simulation. This is a simple cellular automata. Challenge A dimonstration of what the screen should look like in the 8086 assembly challenge Mandelbrot The popular "Mandelbrot Set" fractal. Triangle A recursive triangle fractal. A different version of the FractalTree. Uses 3d math, but doesn't look very 3d. Spinning "Stars" spinning around the screen. A little fun. MovingCircles Recursive circles that move (similar to the Circles applet) Circles Recursive circles which are static (not moving). FractalTree A Recursive fractal tree that is partially randomized to give it a more organic feel. Squares Recursive squares, pretty boring actually. Return to the main page

6. Kosmoi: Mandelbrot Fractals mandelbrot fractals. Life. Featured Bestselling Books mandelbrot fractals. Clickhere for Amazon's page, Harry Potter and the Order of the Phoenix (Book 5). http://kosmoi.com/Science/Mathematics/Fractals/Mandelbrot/

Fractals Fractal Gallery Mathematics Mandelbrot Fractals Nature Agriculture Animals Biology ... Kosmoi Press the left mousebutton ro "Z" where you want to zoom in, press the right mouse button key or "z" when you want to zoom back out again. You may enter coordinates yourself. Apparently undocumented feature: pan with the arrow keys. Applet by Steffen Thorsen. See also: Fractals Fractal Gallery Mathematics Book and Website Recommendations ... Contact Us

7. Mandelbrot Fractals; A Brief History Of Fractal Geometry mandelbrot fractals; a brief history and illustrated explanation of fractalgeometry. This page also features an interactive fractal zoom. http://www.sunleitz.com/whatarefractals.html

scratching the surface Fractal geometry is a relatively new branch of mathematics whose name was coined by Benoit B. Mandelbrot. Working as a research mathematician in I.B.M.'s Thomas Day Watson laboratory in upstate New York, Mandelbrot was experimenting with the theories of another French mathematician (Gaston Julia) when on March the 1st, 1980 the Mndelbrot set was discovered. Gaston Julia's theories were published in 1917 but could not be put to the test until the advent of modern super computers allowed the millions of necessary calculations to be performed. In lay terms, the Mandelbrot set is a set of coordinates whose representative numbers feed back on themselves when the equation z = z*z + c is applied. After thousands of iterations, a number either goes off in the direction of infinity or back to zero. To visualize the set, the numbers which are unable to escape and are destined to return to zero are represented by the color black, while those that soar off to infinity are assigned various colors whose values are determined by the rate at which they accelerate towards infinity. The colors chosen, can be any. The Mandelbrot set looks like this: Unfortunately, you don't have the Java language, so you can't see it work.

8. Mandelbrot Fractals For Linux/X Issue 10 discovered the Mandelbrot set in 1981. By the mideighties personal computers hadevolved to the point that anyone could experiment with various fractals, and http://www.linuxgazette.com/issue10/xaos.html

XaoS: A New Fractal Program for Linux by Larry Ayers Published in Issue 10 of the Linux Gazette Transforming certain recursive complex-number formulae into images of unlimited depth and complexity was only made possible by the development of the modern computer. Benoit Mandelbrot, a Belgian researcher working for IBM, first discovered the Mandelbrot set in 1981. By the mid-eighties personal computers had evolved to the point that anyone could experiment with various fractals, and programmers soon discovered that the 8-bit 256-color vga palette could be mapped to various parameters, which allowed the creation of stunning animated images. The most comprehensive and feature-filled of all fractal-generation programs is Fractint, a freeware program originally written for DOS. Fractint is maintained by a far-flung group of developers, rather like Linux. It was ported to unix by Ken Shirriff and a Linux version is commonly included in many Linux distributions. Not all features of the DOS version work in Linux, and if you just want to see what fractals are all about Fractint is probably overkill. It has such a multitude of options and features that it can be somewhat overwhelming to a new user. Recently Jan Hubicka (developer of the Koules X-window game) and Thomas Marsh have released a small fractal program for Linux called XaoS. This is an efficient program, with the option to compile both X-Windows and SVGA-console versions. XaoS can't render the dozens of fractal types which Fractint can, but it does the basic Mandelbrot and Julia sets quickly, with several keyboard options.

9. Art And Mandelbrot Fractals ART WITH MANDELBROT PICTURES? I have explained why I consider that the most beautifulMandelbrot pictures are not true art, but, at most, collector's pieces. http://perso.wanadoo.fr/charles.vassallo/en/art/art_3.html

ART WITH MANDELBROT PICTURES? I must say, I was seldom convinced in the field of mere pictures. The trouble with the best Mandelbrot pictures is their richness and their too high symmetry. They are perfect in themselves and there is nothing to be added. Surely, one can always put a pretty girl in front of them; they were put into so many strange sceneries... But undoubtly, questions will arise about the relationship between the girl and the scenery; answers will be provided, again undoubtly, but I am afraid they might be somewhat artificial. Mandelbrot pictures have nothing to do with life and bodies; they cannot be said inhuman, they simply belong somewhere else. Nevertheless, I find a few virtues to the picture on the left, because I read it mainly at a graphical level. First of all, this is the contrast of two movements. The opposition then is reinforced with the contrast of natures, life on the one hand and mathematical abstraction on the other hand. Of course, the gymnast was carefully located in the composition. Notice that the Mandelbrot picture is not among the most complicated and that it is not quite rich enough to completely fill the frame on its own. The picture on the right shows another attempt at using Mandelbrot pieces in an artistic way. It comes from

10. Mandelbrot Fractals And Pretty Girls Mai 1992, p.9), and specially a picture Marylin Julia showing a solarizedface of Marylin above a Julia fractal, ie a close relative of Mandelbrot. http://perso.wanadoo.fr/charles.vassallo/en/art/art_4.html

... Sure, one can always put a pretty girl in front of a beautiful fractal... A simple sally? I used to think so but... please read the following. I am less affirmative now. I was browsing through old papers when I stopped short in front of a series of Marylin portraits that were cooked through various recipes by Ilene Astrahan ( IEEE Computer Graphics and Applications, In fact, though the paper was devoted to the computer art, the occurrence of other Marylin variants rather made them a kind of exercice, all the more as the artistic possibilities of personal computers were then under investigation. I am afraid that this picture -not reproduced here- would not be very attractive by now, but one must advocate that Ilene was working with a mere Amiga 2000, a rather primitive machine by current standards, and also that the paper was poorly printed. So it is tempting to resume the exercice with today's tools -only six years later, but six years of incredible technical development. Let us go! Let us stick a movie face, the same Marylin to begin with, over a

11. Kosmoi: Mandelbrot Fractals mandelbrot fractals. Artzia. Featured Bestselling Books mandelbrot fractals. Clickhere for Amazon's page, Harry Potter and the Order of the Phoenix (Book 5). http://encyclozine.com/Science/Mathematics/Fractals/Mandelbrot

Fractals Fractal Gallery Mathematics Mandelbrot Fractals Artzia Arts Fantasy History ... EncycloZine Press the left mousebutton ro "Z" where you want to zoom in, press the right mouse button key or "z" when you want to zoom back out again. You may enter coordinates yourself. Apparently undocumented feature: pan with the arrow keys. Applet by Steffen Thorsen. See also: Fractals Fractal Gallery Mathematics Book and Website Recommendations ... Contact Us

12. MuSoft Builders: Mandelbrot Fractals Created With A Musical Generator Zooming into the Mandelbrot set. Below you find six successive zoomsof the Mandelbrot set with a Musical Generator 3.0. All pictures http://www.musoft-builders.com/links/mandelbrot.shtml

Logo: Giaco Parkinson Home Products Register ... Contact Welcome visitor Zooming into the Mandelbrot set Below you find six successive zooms of the Mandelbrot set with a Musical Generator 3.0. All pictures were generated with a Musical Generator. If you have a plot of the Mandelbrot or Julia set, just select an area with the mouse and the plotter automatically zooms in. Music related to that fractal automatically changes too, so be aware. The last picture is plotted again in rainbow and 4 colors. Start Zoom 1 Zoom 2 Zoom 3 Zoom 4 Zoom 5 Zoom 5 with rainbow colors Zoom 5 with four colors. Home Products Register Support ... MuSoft Builders

13. Marty Taylor's Mandelbrot Fractals Computer Science 126 Princeton University February 2002 mandelbrot fractals Aquick (and not very presentable) overview of what I've done with them A Mini http://www.princeton.edu/~mstaylor/fractals/

14. Mandelbrot Fractals Toxic Swamp I figured that there was something interesting going on here, so I decidedto give it a go with one of my color schemes A MiniMandelbrot in the http://www.princeton.edu/~mstaylor/fractals/toxic.htm

15. An Explanation Of Julia And Mandelbrot Fractals of the Mandelbrot set, see below. At other times the Julia set is one or more continuousfractal lines, usually enclosing an 'interior' set. Julia Fractals http://www.felicite-parmentier.freeserve.co.uk/page3.htm

A basic explanation of fractals A plane consists of points P which will be either represented by their real Cartesian co-ordinates ( x , y ) or more usually by the complex number Z = x + iy. You do not need to understand complex numbers to enjoy fractals. The points in the plane are subjected to a transformation by means of a formula or procedure which converts any point P into its image point P . This process is then repeated to convert P into P and so on, thus creating the sequence of points P P P P . . . which is called the orbit of the initial point P . Each conversion is called an iteration, so n iterations will get you as far as the point P n Julia sets The orbits themselves are sometimes the interest, but more often they are divided into two classes based on the long term behaviour of the orbit. The points of this boundary make up the Julia set. The whole plane excluding the Julia set is known as the Fatou set. These are named after the mathematicians who first studied them. Sometimes the Julia set is just a disconnected cloud of points, known as 'Fatou dust'. It is this situation which defines the outside of the Mandelbrot set, see below. At other times the Julia set is one or more continuous fractal lines, usually enclosing an 'interior' set.

16. Examples Of Julia And Mandelbrot Fractals Drawn By Genfract. Home page. A basic explanation of Julia and mandelbrot fractals. A short descriptionof what Genfract can do. Critical values and compound Mandelbrots. http://www.felicite-parmentier.freeserve.co.uk/page2.htm

Example outputs from Genfract The full size pictures on this page average 70 kb. This is a re-creation of Beauty Of Fractals map 48. Note that two palette sections have been allocated to the escape fractal. All eleven sections could be used, if desired. This is the full picture for BOF fig 58. (Note that the book is 90 degrees out.) The internal colouring of the Mandelbrots shows that it does not possess L-R symmetry. In the orange body to the L, there is an attractor of period two. But in the green body to the R, there are two attractors of period one. Similarly, the mauve body top L has two attractors of period 3, while the yellow at top R has one attractor of period 6. Green has the attractor Z = 1 whereas plum-red has the attractor infinity. A third attractor lives in the orange region, which includes Z = 0. Here is BOF fig 45, a detail in the Mandelbrot for the Newton formula for The points in the cyan area, which include the critical point, go to an attractor of period 2. The other three colours are parts of the basins of attraction of the three roots of the cubic for the value of C chosen.

17. Fractals Julia and mandelbrot fractals. There are two main types of fractals, Mandelbrotand Julia- fractals, use the radiobuttons to switch between the two. http://hem.passagen.se/mnomn/fractal.html

drawNavibar('fractal.html'); Julia and Mandelbrot Fractals Press button to start. Drag on picture to to zoom. The fractal viewer is not a game. It just looks good. OK! You need Internet e xplorer 4 or N etscape 4.5 or higher to view the fractal. Look at Fractals here with a java enabled browser. Press the button to start the interactiv fractal program. There are two main types of fractals, Mandelbrot- and Julia- fractals, use the radiobuttons 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 draging with the mouse. Zoom back out by pressin the radiobutton once again. The performens is horrible,but Im gonna modify it some or these days. since sept 98:

18. Wierd G-Mandelbrot Fractals -- MN Karthik ~ wierd G mandels ~ I said Innovate, not imitate! . http://www.metlin.org/graphics/gm1/

~ wierd G mandels ~ " ....I said Innovate, not imitate! " Home Graphics Programming Downloads ... About Wierd G-Mandels Here are some cool and wierd mandelbrots created in GIMP in Linux. Mmm... Eye Candy!!! mnkarthik@yahoo.com mnkarthik@yahoo.com

19. Plasmic G-Mandelbrot Fractals -- MN Karthik ~ plasmic G mandels ~ I said Innovate, not imitate! . http://www.metlin.org/graphics/gm2/

~ plasmic G mandels ~ " ....I said Innovate, not imitate! " Home Graphics Programming Downloads ... About Plasmic G-Mandels Mandel mania!!! Attack of the plasma textured Mandelbrots from fractal dimension... Isn't mathematics truly beautiful? Questions/comments? You can e-mail me at mnkarthik@yahoo.com mnkarthik@yahoo.com

20. Mandelbrot And Julia Sets More on Fractals Mandelbrot and Julia Sets. Studying mandelbrot fractalsactivityto create and study the Mandelbrot Set and links to supplemental sites. http://home.inreach.com/kfarrell/mandelbrot.html

More on Fractals: Mandelbrot and Julia Sets Valuable Sites to Visit: Fractal Music Fractional Dimension Mandelbrot Information What is a Fractal? ... Julia Set The First Fractals Benoit Mandelbrot was one of the first to discover fractals. Madelbrot was examining the shapes created by a Gaston Julia, a mathematician in the 1920's who was working without the benefit of computers. Julia could not describe these shapes using Euclidean geometry. His work was obscure and largely forgotten. The image in this text is one of the shapes discovered by Julia. Others looked like pinched circles, some like brambles, some like spots of dust. Mandelbrot tried to classify these shapes. By iterating a simple equation and mapping this equation in the complex plane, Madelbrot discovered the fractal you see at the top of this page. This discovery lead to further information about Julia Sets and fractional dimensions. Find out more by clicking on "Mandelbrot Information" . Compare the Mandelbrot Set to the Julia Sets by clicking on "Julia Sets". Activities: Create a Fractal : using FracInt, a downloadable program, create a fractal.

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