1. Mandelbrot Fractals; A Brief History Of Fractal Geometry mandelbrot fractals; a brief history and illustrated explanation of fractal geometry. This page also features an interactive fractal zoom. fractals fractal geometry? Fractal geometry is a relatively new branch of mathematics whose name was coined by Benoit B. Mandelbrot New York, Mandelbrot was experimenting with 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.

2. Math Forum: Suzanne Alejandre - MandelBrot Activity Studying mandelbrot fractals. Fractals. NOTE What is a fractal? Alan Beck in What Is a Fractal? And who is this guy Mandelbrot? writes 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?

3. Studying Mandelbrot Fractals Studying mandelbrot fractals 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 http://rdre1.inktomi.com/click?u=http://mathforum.org/alejandre/applet.mandlebro

4. ColorAura Networks -- Applets 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/artwork/

Art Applets Name Description Fractal Tree V1.2 This is the second version of the Colorful Fractal Tree Fractal Tree V1.0 The original Fractal Tree Spinning Stars Stars spinning around in 3D Moving Circle Fractal Recursivly drawn circles, with a varying angle. Blazing Fire A pretty fire simulator Moving Plasma A dynamic display of plasma. Wavy Pattern A pattern that waves around. Note: Some applets require a newer version of the java plug in. You can download and install the Java runtime environment if any applet doesn't work. Main page This page has been viewed 1190 times.

5. Mandelbrot Fractals mandelbrot fractals. HOME. About Us. Site Map. Top Pages. Talk to Us. Link to Us. Recommended Books/Sites • Search web or Kosmoi for mandelbrot fractals . http://kosmoi.com/Science/Mathematics/Fractals/Mandelbrot/

EncycloZine Astronomy Biology Chemistry ... Web Design Mandelbrot Fractals About Us A - Z Site Map Top Pages ... Cell Phones See also: Mathematics Fractals Fractals Gallery ... Mandelbrot Click the left mousebutton or "Z" where you want to zoom in, Click the right mouse button key or "z" when you want to zoom back out. You may enter coordinates yourself. Apparently undocumented feature: pan with the arrow keys. Applet by Steffen Thorsen. Fractals In mathematics fractals are a class of complex geometric shapes that commonly exhibit the property of self-similarity, such that a small portion of it can be viewed as a reduced scale replica of the whole. The term fractal is derived from the Latin word fractus ("fragmented," or "broken"). Fractals are distinct from the simple figures of classical, or Euclidean, geometry (the square, the triangle, the circle, the sphere, etc.) They are capable of describing the many irregularly shaped objects or spatially nonuniform phenomena in nature that cannot be accommodated by Euclidean geometry. Recommended Books/Sites web or Kosmoi for 'Mandelbrot Fractals'.

6. An Explanation Of Julia And Mandelbrot Fractals of the Mandelbrot set, see below. At other times the Julia set is one or more continuous fractal 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.

7. Examples Of Julia And Mandelbrot Fractals Drawn By Genfract. Home page. A basic explanation of Julia and mandelbrot fractals. Critical values and compound Mandelbrots. An explanation of Volterra Lotka fractals. 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.

8. Fractals Julia and mandelbrot fractals. There are two main types of fractals, Mandelbrot and 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:

9. Art And Mandelbrot Fractals fractal picture. Indeed, the dark blue shape has a complicated contour, but we are far from the richness in the details of the most beautiful Mandelbrot 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 This page shows three attempts of collage of a girl face above a Mandelbrot fractal. Sure, one can always put a pretty girl in 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. WisFaq! Iteratie en Mandelbrot. Julia fractals begrensd en mandelbrot fractals convergent? Julia spiraal. Mandelbrot fractaal. Mandelbrotfractal. Mandelbrotset. http://www.wisfaq.nl/overzicht.asp?categorie=Fractals

12. Mandelbrot Fractals in Java. To view this page, you must use a Javacompatible browser. Click in the Mandelbrot set to view the corresponding Julia set. http://math.polytechnique.fr/cmat/auroux/java/mand3.html

Fractals in Java To view this page, you must use a Java-compatible browser. Click in the Mandelbrot set to view the corresponding Julia set. Shift-drag mouse to zoom in given box. Control-click to translate to a new center. Shift-Control-click to revert to global view. This applet is slow and I am still experimenting with multithreading, so please be patient ! The source.

13. Complex Fractals The exponent can be changed to produce different mandelbrot fractals and so can the function, as long as it retains the Z=Z +C format. http://www.ga.k12.pa.us/academics/us/math/geometry/stwk98/ABIDEN/complex.html

Complex Fractals I The Mandelbrot set is one of the most commonly seen complex fractals, it applies a simple nonlinear algebraic equation Zn = Zn-1^2 + C which is called the recursion law. All fractals using this equation belong to the Mandelbrot set. The following graph displays the complex plane. Image obtained from Fractal Explorer The Mandelbrot set is built starting with a point C on the complex plane. Calculate the value of the expression: Z^2 + C using zero as the value of Z (the result would be C obviously). Assign the result to Z and repeat the calculation. It would then be C^2 + C. Assign this value to Z and repeat this process many times. This is called the iteration of the function Zn = Zn-1^2 + C.

14. Visual Prolog 6 Examples And Demos: Drawing Mandelbrot Fractals Visual Prolog 6 Examples and Demos. Drawing mandelbrot fractals. Written by Kari Rastas. Over 10 years ago, when Chaos theories and http://www.visual-prolog.com/vip6/Community/userExamples/mandel2.htm

var toRoot=""; Navigation without JavaScript Site Map Visual Prolog 6.1 (Build 6108) ... Examples and Demos Drawing Mandelbrot Fractals Written by Kari Rastas Over 10 years ago, when Chaos theories and fractals were popular, I tested PDC Prolog capabilities in calculation speed and drawing. I used this old code in learning Visual Prolog 6 With this little application you can draw fractals, save and print pictures. I made this program for learning purposes, so it is quite limited. Feel free to modify and enlarge the program. Download the project Visual Prolog 6.1 (Build 6108) Commercial Edition ... Site Map Prolog Development Center A/S - H.J. Holst Vej 3-5C - 2605 Broendby, Denmark - Tel +45 3636 0000 - Fax +45 3636 0001 - sales@visual-prolog.com

15. Fractals - Amazing Seattle Fractals! - Screensavers, Tutorials, Free Software of the Molecular Biology Institute at UCLA, has written an excellent paper complete with illustrations on HIGHER ORDER mandelbrot fractals EXPERIMENTS IN http://www.fractalarts.com/ASF/

Amazing Seattle Fractals! Home Fractal Art Galleries Fractal Tutorials Fractal Of The Week ... Animation Theatre E-Mail: Fractal Software Guest Galleries Awards Will Engdahl Galleries ... About Welcome to Amazing Seattle Fractals! If you came to learn more about fractals, learn how to create your own fractal art, download free fractal software programs or just to view some fractal art galleries, you've come to the right place! These unique digital art images are created using a computer and fractal software generating programs. Fractals are created from mathematical formulas resulting in images of an astounding diversity of form, detail, color and light. I believe you will agree, as you view this digital art, that these images truly are AMAZING! Caution! This is a graphics intensive site! May 2004 News! Due to the amount of SPAM these days many e-mails I receive get eliminated before even opening. If you write please be sure to write something regarding fractals in the subject line or your e-mail may get deleted inadvertantly! Do not just put hi, N/A, hello etc. in the subject line as this is common in many SPAM e-mails and they get deleted before opening. Thanks! New galleries scheduled to be added soon!

16. MuSoft Builders: Mandelbrot Fractals Created With A Musical Generator 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 http://www.musoft-builders.com/links/mandelbrot.shtml

Logo: Giaco Parkinson Home Register Support ... 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 Register Support Contributions ... MuSoft

17. Mandelbrot Fractals PlanetMike.com. Home Fractals mandelbrot fractals. mandelbrot fractals. Page 1 2 3 4 5 6 7 . Fracture 126 (Mandelbrot).tiff, http://www.planetmike.com/fractals/mandelbrot/

Home Fractals Mandelbrot Fractals Page 1 Click on a picture to enlarge it. Generated by Galerie Sunday, December 7, 2003 10:26:54 PM December 12, 2003 http://www.planetmike.com/fractals/mandelbrot/index.shtml

18. Mandelbrot Information (3d Delphi Graphics Mandelbrot Pascal Programming ) code. mandelbrot fractals, Conways Game of Life, and spinning stars are just some of the applets appearing here. Alienorganic An http://www.programming-x.com/programming/mandelbrot.html

programming-x.com mandelbrot Information STEEL's Programming Resource Page A website with information, explanations, and source code about 3D graphics. Klosiewicz, Przemek - SweetSheep Homepage About. Graphics programming, mandelbrot - Julia set explorer, RayMax - Raytracer, OpenGL. Photos. efg's Reference Library Algorithms, dates and times, files, directories, disks, I/O, graphics, math functions, math information and links, printing. efg's Computer Lab A large number of Delphi projects involving Image Processing, Color, Graphics, Mathematics, Fractals and Chaos. colorForth 2.0 programs in Linux GNU Assembly, and colorForth; source and binary versions; downloads; Heapsort, and mandelbrot set. By Mark Slicker. Thompson, Nathanael mandelbrot viewer and chemical engineering Java applets (with source), a forum and chatroom, resume. A collection of stuff that is of interest to Nate. 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. Alienorganic An interactive mandelbrot fractal applet.

19. MATLAB Central File Exchange - 3D Mandelbrot The code generates the mandelbrot fractals and displays the computed sets in 3D as a graphical image onto your screen. File Details,...... http://www.mathworks.nl/matlabcentral/fileexchange/loadFile.do?objectId=3197&obj

20. Mandelbrot Fractals For Linux/X Issue 10 discovered the Mandelbrot set in 1981. By the mideighties personal computers had evolved to the point that anyone could experiment with various fractals, and http://linuxgazette.net/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.

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